United States Office of Research and EPA/600/R-97/053
Environmental Protection Development August 1997
Agency Washington DC 20460
&EPA 3DFATMIC
of
1.0
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3DFATMIC: User's Manual of a Three-Dimensional
Subsurface Flow, Fate and Transport of Microbes and
Chemicals Model
Version 1.0
by
Gour-Tsyh (George) Yeh and Jing-Ru (Ruth) Cheng
Department of Civil and Environmental Engineering
The Pennsylvania State University
University Park, PA 16802
and
Thomas E. Short
U. S. Environmental Protection Agency
Robert S. Kerr Environmental Research Center
Subsurface Protection and Remediation Division
Ada, Oklahoma 74820
Cooperative Agreement CR-818322
Project Officer
Thomas E. Short
U. S. Environmental Protection Agency
Robert S. Kerr Environmental Research Center
Subsurface Protection and Remediation Division
Ada, Oklahoma 74820
National Risk Management Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Cincinnati, OH 45268
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DISCLAIMER
The U. S. Environmental Protection Agency through its Office of Research and Development
partially funded and collaborated in the research described here under assistance agreement number
CR-818322 to The Pennsylvania State University. It has been subjected to the Agency's peer and
administrative review and has been approved for publication as an EPA document. Mention of trade
names or commercial products does not constitute endorsement or recommendation for use.
When available, the software described in this document is supplied on "as-is" basis without
guarantee or warranty of any kind, express or implied. Neither the United States Government (United
States Environmental Protection Agency, Robert S. Kerr Environmental Research Center), The
Pennsylvania State University, nor any of the authors accept any liability resulting from use of this
software.
11
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FOREWORD
The U.S. Environmental Protection Agency is charged by Congress with protecting the Nation's
land, air, and water resources. Under a mandate of national environmental laws, the Agency strives
to formulate and implement actions leading to a compatible balance between human activities and
the ability of natural systems to support and nurture life. To meet these mandates, EPA's research
program is providing data and technical support for solving environmental problems today and
building a science knowledge base necessary to manage our ecological resources wisely, understand
how pollutants affect our health, and prevent or reduce environmental risks in the future.
The National Risk Management Research Laboratory is the Agency's center for investigation of
technological and management approaches for reducing risks from threats to human health and the
environment. The focus of the Laboratory's research program is on methods for the prevention and
control of pollution to air, land, water, and subsurface resources; protection of water quality in public
water systems; remediation of contaminated sites and ground water, and prevention and control of
indoor air pollution. The goal of this research effort is to catalyze development and implementation
of innovative, cost-effective environmental technologies; develop scientific and engineering
information needed by EPA to support regulatory and policy decisions; and provide technical support
and information transfer to ensure effective implementation of environmental regulations and
strategies.
Bioremediation is unique among remediation technologies in that it degrades or transforms
contaminants through the use, possibly with manipulative enhancement, of indigenous
microorganisms. Bioremediation can be used in many ways - degradation on concentrated organic
contaminants near their sources, as a secondary remediation strategy following physical or chemical
treatment methods, for sequestration of metals through microbially mediated transformation
processes, and for remediating large plumes of dilute contaminants that are broadly dispersed in the
environment. Thus, bioremediation has the potential to be one of the most cost-effective
technologies for dealing with environmental remediation problems. Yet, realistically quantitative
predictions and assessments of bioremediation technologies appear lacking. In order to meet the
objectives of having a realistic tool for predicting and assessing if a bioremediation technology can
be successfully implemented, the 3DFATMIC model has been developed. This numerical model
simulates 1) the fate and transport of multiple microbes, electron acceptors, substrates, and nutrients
and density-dependent fluid flow in saturated-unsaturated subsurface media under either steady-state
or transient conditions; 2) multiple distributed and point sources/sinks as well as boundary sources;
and, 3) processes which degrade and transform contaminants, cause the growth and death of
microbes, and control the fluid flow.
Clinton W. Hall, Director
Subsurface Protection and Remediation Division
National Risk Management Research Laboratory
in
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ABSTRACT
This document is the user's manual of 3DFATMIC, a ^-Dimensional Subsurface Flow, FAte
and Transport of Microbes and Chemicals Model using a Lagrangian-Eulerian adapted zooming and
peak capturing (LEZOOMPC) algorithm. This 3-dimensional model can completely eliminate peak
clipping, spurious oscillation, and numerical diffusion; i.e., solve the advective transport problems
exactly, within any prescribed error tolerance, using very large mesh Courant numbers. The size of
mesh Courant number is limited only by the accuracy requirement of the Eulerian step. Since this
model also includes diffusion zooming in solving diffusion elemental matrix, the accuracy is
improved by specifying the number of local subelements in every global element. In other words,
the more subelements zoomed in diffusion step, the more accuracy at Eulerian step. To sum up, a
better solution with respect to advection transport can be obtained with larger time-step sizes; the
time-step sizes are only limited by the accuracy requirement with respect to diffusion/dispersion
transport and chemical reaction terms. However, the limitation of time-step size imposed by
diffusion/dispersion transport is normally not a very severe restriction.
The model, 3DFATMIC, is designed to obtain the density-dependent fluid velocity field, and
to solve the advective-dispersive transport equation coupled with biodegradation and microbial
biomass production. Water flow through saturated-unsaturated media and the fate and transport of
seven components (one substrate, two electron acceptors, one trace element, and three microbial
populations) are modeled. For each specific application, 74 maximal control-integers must be
assigned using PARAMETER statements in the MAIN program. In addition, if a user uses different
analytical forms of boundary conditions, source/sink strength value functions, and soil property
functions from those used in this program, he is instructed to modify subroutines ESSFCT,
WSSFCT, VBVFCT, DBVFCT, NBVFCT, CBVFCT, and SPFUNC, respectively. The input data
to the program include the control indices, properties of the media either in tabular or analytical
form, the geometry in the form of elements and nodes, initial conditions and boundary conditions
for flow and transport, and microbe-chemical interaction constants. Principal output includes the
spatial distribution of pressure head, total head, moisture content, Darcy velocity component,
concentrations, and material fluxes at any desired time step. Fluxes through various types of
boundaries are shown in the mass balance table. In addition, diagnostic variables, such as the
number of non-convergent nodes and residuals, may be printed, if desired, for debugging purposes.
IV
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TABLE OF CONTENTS
Page
ii
Foreword iii
iv
vii
ix
1. 1
2. OF 3
2.1 Mathematical Statement of 3DFATMIC 3
2.2 Numerical Approximation 17
2.3 Description of 3DFATMIC Subroutines 26
3. -. ::> -.:r" -vnoi i .;-F TO 69
3.1 Parameter Specifications 69
3.2 Soil Property Function Specifications 79
3.3 Input and Output Devices 81
4. 83
4.1 Example 1 : One-Dimensional Column Flow Problem 83
4.2 Input and Output for Example 1 87
4.3 Example 2 : Two-Dimensional Drainage Flow Problem 91
4.4 Input and Output for Example 2 96
4.5 Example 3 : Three-Dimensional Pumping Flow Problem 100
4.6 Input and Output for Example 3 105
4.7 Example 4 : One-Dimensional Single Component Transport Problem 118
4.8 Input and Output for Example 4 123
4.9 Example 5 : Two-Dimensional Single Component Transport Problem 126
4.10 Input and Output for Example 5 129
4.11 Example 6 : Two-Dimensional Multicomponent Transport in a Uniform
Flow Field 133
4.12 Input and Output for Example 6 135
4.13 Example 7 : Two-Dimensional Coupled Flow and Multicomponent
Transport Problems 146
4.14 Input and Output for Example 7 149
4.15 Example 8 : Three-Dimensional Multicomponent Transport in a
Uniform Flow Field 162
4.16 Input and Output for Example 8 163
185
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A: Data A-l
B: B-l
C: The for
C-l
D: D-l
VI
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LIST OF FIGURES
Figure Page
2.1 The basic structure for coding transport part of 3DFATMIC 24
2.2 Program Structure of 3DFATMIC (MAIN) 28
2.3 Program Structure of 3DFATMIC (Flow Part) 36
2.4 Program Structure of 3DFATMIC (Transport Part 1 of 3) 44
2.4 Program Structure of 3DFATMIC (Transport Part 2 of 3) 45
2.4 Program Structure of 3DFATMIC (Transport Part 3 of 3) 46
4.1 Problem definition and sketch for Example 1 84
4.2 Finite element discretization for Example 1 85
4.3 Pressure head profiles at various times 88
4.4 Problem definition and sketch for Example 2 92
4.5 Finite element discretization for Example 2 93
4.6 Pressure head distribution for Example 2 95
4.7 The velocity field for Example 2 97
4.8 Problem definition and sketch for Example 3 101
4.9 Finite element discretization for Example 3 102
4.10 Pressure head distribution for Example 3 104
4.11 Velocity distribution for Example 3 105
4.12 Problem definition and sketch for Example 4 119
4.13 Finite element discretization for Example 4 120
4.14 The concentration profiles of Example 4 122
4.15 Problem definition and sketch for Example 5 127
4.16 Finite element discretization for Example 5 127
4.17 Contours of 50% concentration at various times 128
4.18 The x-z cross-section of the region of interest and the associated
physical parameters 134
4.19 The discretization of the region of interest 134
4.20 Dissolved plumes at 100 days : (a) substrate and (b) oxygen 143
4.21 Dissolved plumes at 200 days : (a) substrate and (b) oxygen 144
4.22 Total microbial mass distributions : (a) 100 and (b) 200 days 145
4.23 The x-z cross-section of the region of interest 148
4.24 The discretization of Example 7 149
4.25 The velocity field at time = 2 days and time = 4 days 158
4.26 The concentration profiles of microbes and chemicals at
time = 2 days and time = 4 days 159
4.27 The region of interest for Example 8 162
4.28 Dissolved plumes at 100 days : (a) substrate and (b) oxygen on x-y
cross-section 179
4.29 Dissolved plumes at 200 days : (a) substrate and (b) oxygen on x-y
cross-section 180
vn
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4.30 Total microbial mass distributions : (a) 100 and (b) 200 days on x-y
cross-section 181
4.31 Dissolved plumes at 100 days : (a) substrate and (b) oxygen on x-z
cross-section 182
4.32 Dissolved plumes at 200 days : (a) substrate and (b) oxygen on x-z
cross-section 183
4.33 Total microbial mass distributions : (a) 100 and (b) 200 days on x-z
cross-section 184
vin
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LIST OF TABLES
Table Page
4.1 The list of input parameters for Example 1 89
4.2 Input Data Set for Example 1 90
4.3 The list of input parameters for Example 2 97
4.4 Input Data Set for Example 2 98
4.5 The list of input parameters for Example 3 106
4.6 Input Data Set for Example 3 107
4.7 The list of input parameters for Example 4 123
4.8 Input Data Set for Example 4 124
4.9 The list of input parameters for Example 5 130
4.10 Input Data Set for Example 5 131
4.11 The list of input parameters for Example 6 135
4.12 Input Data Set for Example 6 138
4.13 The list of input parameters for Example 7 149
4.14 Input Data Set for Example 7 152
4.15 The list of input parameters for Example 8 163
4.16 Input Data Set for Example 8 166
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1. INTRODUCTION
3DFATMIC (A ^-Dimensional Subsurface Flow, FAte and Transport of Microbes and Chemicals
Model) can be used to investigate saturated-unsaturated flow alone, contaminant transport alone, combined
flow and transport, or the fate and transport of microbes and chemicals in ground-water environment. For the
flow module, the Galerkin finite element method is used to discretize the Richards' equation and for the
transport module, the hybrid Lagrangian-Eulerian approach with an adapted zooming and peak capturing
algorithm is used to discretize the transport equation. This approach can completely eliminate spurious
oscillation, numerical dispersion, and peak clipping due to advection transport. Large time-step sizes as well
as large spatial-grid sizes can be used and still yield accurate simulations. The only limitation on the size of
time steps is the requirement of accuracy with respect to dispersion transport, which does not pose much severe
restrictions.
The purpose of this manual is to provide guidance to users of the computer code for their specific
applications. Section 2.1 lists the governing equations, initial conditions, and boundary conditions for which
3DFATMIC is designed to solve. Section 2.2 describes the numerical procedure used to simulate the
governing equations. Section 2.3 contains the description of all subroutines in 3DFATMIC. Since occasions
may arise that require the user to modify the code, this section should help the user to trace the code so the user
can make necessary adjustments for individual purposes. Section 3.1 contains the parameter specification.
For each application, the user needs to assign 74 maximal control-integers in the MAIN program. Section 3.2
describes the required modification of the code so that one might use a different analytical form of soil property
function from the ones used in this report. Section 3.3 describes files required for the execution of
3DFATMIC. Appendix A contains the data input guide that is essential for any specific application.
The users may choose whatever consistent set of units. Units of mass (M), length (L), and time (T)
are indicated in the input description.
1
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The special features of 3DFATMIC are its flexibility and versatility in modeling as wide a range of
problems as possible. This model can handle: (1) heterogeneous and anisotropic media consisting of as many
geologic formations as desired; (2) both spatially distributed and point sources/sinks that are spatially and
temporally dependent; (3) the prescribed initial conditions by input or by simulating a steady state version of
the system under consideration; (4) the prescribed transient concentration over Dirichlet nodes; (5) time
dependent fluxes over Neumann nodes; (6) time dependent total fluxes over Cauchy nodes; (7) variable
boundary conditions of evaporation, infiltration, or seepage on the soil-air interface for the flow module and
variable boundary conditions of inflow and outflow for the transport module automatically; (8) two options
of treating the mass matrix - consistent and lumping; (9) three options (exact relaxation, under- and over-
relaxation) for estimating the nonlinear matrix; (10) automatically time step size reset when boundary
conditions or sources/sinks changed abruptly; (11) two options, Galerkin weighting or upstream weighting for
advection term in transport module; (12) two options for the Lagrangian numerical scheme in transport
module, which are enabling and disabling adapted zooming scheme; (13) two options for solving Eulerian step
including the enable and disable of diffusion zooming; (14) the mass balance checking over the entire region
for every time step; and, (15) modification of program if different conditions are used.
-------
P deah
! y
Pw dh at
r -1
xV. j\^ *( vh ~t~ \/Z )
. S " Pw
+
p
2. DESCRIPTION OF 3DFATMIC MODEL
2.1 Mathematical Statement of 3DFATMIC
3DFATMIC is designed to solve the following system of governing equations, along with initial and
boundary conditions, which describe flow and transport through saturated-unsaturated media. The governing
equations for flow (detailed derivation shown in Appendix B.), which describes the flow of variable-density
fluid, are basically the Richards' equation.
Governing Flow Equation
. Hfl ah _L _ ._. n _ .1 n* p .
q) (2.1)
.. , .. Pw
The saturated hydraulic conductivity Ks is given by
(P/Pw)
S SW f I \
(H/Hw)
where h is the referenced pressure head defined as p/pwg in which p is pressure (M/LT2), t is time (T), Ks is
the saturated hydraulic conductivity tensor (L/T), I\ is the relative hydraulic conductivity or relative
permeability, z is the potential head (L), q is the source and/or sink (L3/T), and 6 is the moisture content, p and
(i are the density (M/L3) and dynamic viscosity (M/LT) at microbial concentrations Q, C2, C3, and chemical
concentrations Cs, C0, Cn, and Cp (M/L3); and Ksw, pw and (iw are the referenced saturated hydraulic
conductivity tensor, density, and dynamic viscosity, respectively. The strength of the source/sink is the
discharge or withdraw flow rate q, and p* is the density of the injected fluid. These referenced values are
usually taken as the saturated hydraulic conductivity at zero microbial and chemical concentrations. The
density and dynamic viscosity of fluid are functions of microbial and chemical concentrations and are assumed
to take the following form
-------
= 1+E -- C:; i = 1, 2, 3, s, o, n, p (2.2b)
Pw ' Pw f
-^ = i+ Pscs+Poco + Pncn + Ppcp + p1c1+p2c2 + p3c3 (2.2c)
^w
where Cs and ps are dissolved concentration and intrinsic density of substrate, respectively (M/L3); C0 and p0
are dissolved concentration and intrinsic density of oxygen (M/L3), respectively; Cn and pn are dissolved
concentration and intrinsic density of nitrate (M/L3), respectively; Cp and ppare dissolved concentration and
intrinsic density of nutrient (M/L3), respectively; Q and pj are dissolved concentration and intrinsic density
of microbe #1 (M/L3), respectively; C2 and p2 are dissolved concentration and intrinsic density of microbe #2
(M/L3), respectively; C3 and p3 are dissolved concentration and intrinsic density of microbe #3 (M/L3),
respectively; and PS, PO, Pn, Pp, Pl5 P2, and P3 are viscosity-effecting factor of associated species (L3/M). It is
assumed that microbe #1 utilizes substrate under aerobic conditions, microbe #2 utilizes substrate under
anaerobic conditions, and microbe #3 utilizes substrate under both aerobic and anaerobic conditions.
The Darcy velocity is calculated as follows:
V = -KK-I Vh+Vz| (2.3)
r
Initial Conditions for Flow Equation
h = hj(x,y,z) in R (2.4)
where R is the region of interest and h; is the prescribed initial condition, which can be obtained by either field
measurement or by solving the steady-state version of Eq.(2.1).
Boundary Conditions for Flow Equation
Dirichlet Conditions:
h = hdCXb'Yb'Zb't) on Bd (2.5)
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Neumann Conditions (gradient condition) :
-n-KK-^Vh = qn(xb,yb,zb,t) on Bn (2.6)
Cauchy Conditions (flux condition) :
/ \
-n-KsKr- -^Vh+Vz = qc(xb,yb,zb,t) on Bc (2.7)
Variable Conditions - During Precipitation Period:
h = h (xb,yb,zb,t) iff -n-KsKr- -^Vh +Vz > q on Bv (2.8a)
\ P )
or
n-KsKr- -Vh+Vz = qp(xb,yb,zb,t) iff h0 on By (2.8c)
\ P
or
h = hm(xb,yb,zb,t) iffn-Kfrl -Vh+Vz < qe on Bv (2.8d)
P
or
-n-KK
= qe(xb,yb,zb,t) iff h > hm on By (2.8e)
V P
where (xb, yb, zb) is the spatial coordinate on the boundary; n is an outward unit vector normal to the boundary;
hd, c^, and qc are the prescribed Dirichlet functional value, Neumann flux, and Cauchy flux, respectively; Bd,
Bro and Bc are the Dirichlet, Neumann, and Cauchy boundaries, respectively; Bv is the variable boundary; hp
is the allowed ponding depth and qp is the throughfall of precipitation on the variable boundary; h m is the
allowed minimum pressure, and qe is the allowed maximum evaporation rate on the variable boundary, which
is the potential evaporation.
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Only one of Eqs. (2.8a) through (2.8e) is used at any point on the variable boundary at any time. It
normally occurs at air-soil interface. During precipitation period, it is assumed that only seepage or infiltration
can occur for any point on the air-soil interface. No evapotranspiration is allowed. If seepage happens, the
Dirichlet boundary condition, Eq. (2.8a), must be imposed. On the other hand, if infiltration occurs, either the
Dirichlet boundary condition, Eq. (2.8a), or the Cauchy boundary condition, Eq. (2.8b), may be specified
depending on the soil property and throughfall rate qp in Eq. (2.8b). The problem is which equation, Eq. (2.8a)
or Eq. (2.8b), should be prescribed for a point on the boundary. This problem is settled by iteration. The
procedure adopted is as follows. At each iteration, the solution is examined at each node along the variable
boundary and test whether the existing boundary condition is still consistent. Specifically, if the existing
condition is Eq. (2.8b) (Cauchy boundary condition), the pressure head at the boundary node is computed.
If the head is greater than the allowed ponding depth hp in Eq. (2.8a), too much water has been forced into the
region through the node. In other words, the throughfall rate is greater than that which the media can absorb.
To account for this, the boundary condition is changed to Eq. (2.8a), which in practice should result in
infiltration at a rate less than that qp in Eq. (2.8b) or result in seepage. If the computed head is less than the
ponding depth, the media is capable of absorbing all throughfall and no change of boundary condition is
required. On the other hand, if the existing boundary condition is Eq. (2.8a) (Dirichlet boundary condition),
Darcy's flux at the node is computed. If the computed Darcy's flux is going out of the region (seepage) or into
the region (infiltration) but its magnitude is less than qp in Eq. (2.8b), no change of boundary condition is
needed. However, if the computed Darcy's flux is directed into the region (infiltration) with a rate greater than
the throughfall rate qp, a change of boundary condition to Eq. (2.8b) is required since Eq. (2.8a) would force
more water than available into the region. By changing the boundary condition to Eq. (2.8b), it should in
practice result in a pressure head less than hp. The iteration outlined above is discontinued when no change-
over of boundary condition is encountered along the entire boundary.
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Similarly, during non-precipitation period, it is assumed that only evapotranspiration or seepage can
occur and no infiltration is allowed. If seepage actually occurs at a node, Eq. (2.8c) (Dirichlet boundary
condition) must be specified at the node. On the other hand, if evapotranspiration happens, either Eq. (2.8d)
(Dirichlet boundary condition) or Eq. (2.8e) (Cauchy boundary condition) may be imposed at the node. The
problem is again to determine which of the three equations should be used as boundary conditions. Iteration
procedure is used to solve the problem. If the existing boundary condition is Eq. (2.8c), the Darcy's flux is
calculated. When the computed Darcy's flux is going out of the region, the existing boundary condition is
consistent and no change on boundary condition is necessary. When the Darcy's flux is directed into the region
(remember no infiltration is allowed), the application of Eq. (2.8c) implies the infiltration and prohibits
evapotranspiration. Hence, the boundary condition is changed to Eq. (2.8e), which in practice would generate
evapotranspiration and would result in a pressure head lower than the ponding depth in Eq. (2.8c). If the
existing boundary condition is Eq. (2.8d), the Darcy's flux is computed. Since the minimum pressure is
prescribed on the boundary, it is unlikely that this computed Darcy's flux will be directed into the region.
Thus, when the computed outgoing Darcy's flux is less than qe in Eq. (2.8e), the existing boundary condition
is consistent and no change on boundary condition is needed. When the computed Darcy's flux is greater than
qe in Eq. (2.8e), the application of Eq. (2.8d) implies the imposition of too much suction at the node. Hence,
the boundary condition is changed to Eq. (2.8e), which in practice should result in a pressure greater than ^
in Eq. (2.8d). If the existing boundary condition is Eq. (2.8e), pressure head at the node is calculated. If this
computed pressure head is not lower than t^ in Eq. (2.8d), the boundary condition is consistent and no change
is required. However, if the computed head is lower than ^ in Eq. (2.8d), the application of Eq. (2.8e) implies
too much water is removed through the node yielding a too low pressure head. Hence, the boundary condition
is changed to Eq. (2.8d), which should yield an evapotranspiration rate less than qe in Eq. (2.8e). This iteration
process is completed only when consistent boundary conditions have been applied to all nodes on the variable
boundary.
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The governing equations for transport are derived based on the continuity of mass and flux laws. The
major processes are advection, dispersion/diffusion, adsorption, decay, source/sink, and microbial-chemical
interactions.
Governing Equations for Transport
Transport of the carbonaceous substrate, oxygen, nitrate, and nutrient in the bulk pore fluid is
expressed by advection-dispersion equations coupling sink terms that account for biodegradation. The four
nonlinear transport and fate equations are (derivation shown in Appendix B)
^ + V-VCs = V-6D-VCs - As(6+PbKds)Cs
(i)
(2)
(3)
(4)
(5)
P Pw P
(6)
o so
(7)
Y
(2)
(2)
+cs
(8)
V +CPJ
c
c
Y(3)
(3)
(9)
C.
(10)
(2.9)
-------
(8 + PbKdohr + V'VCo = V'0D'VCo - A0(6+PbKdo)Co + qmCom
dt
(1) (2) (3)
(4) (5)
(6)
c
c
c
(7)
(8)
(2.10)
/3V3)
lo Ho
cs
K(3) + C
"so ^s
C0
K(3) + C
. ฐ ฐ.
y~l
P
K (3) + r
. Pซ ^p
(9)
(10)
-2 + V-VCn = V-0D-VCn - An(0+pbKdn)Cn + qmCmn + pV-V(^) -^qm Cr
at P Pw P
(1) (2) (3)
(4) (5)
(6)
PbKd2)C2]
(2) (2)
YnX
cs
Kฎ+C._
cn
(7)
cp
Kp(? + Cp
(8)
(2.11)
V(3)U(3)
In Hn
cs
Kl3)+Cs
cn
(9)
cp
Kpn) + Cp
43)+cr
(10)
-------
dp/
at
(1) (2)
V-VCp = V-0D-VCp - Ap(0+PbKdp)Cp
(3) (4)
..d)r ^
-------
PbKd2hr + V'VC2 = V'0D'VC2 - A2(0+PbKd2)C2 + qmC2m + ^VV(-H-)_iL
9t ^ P Pw P
(1) (2) (3) (4) (5) (6)
+ (0 + pbKd2)C2
[ r T r '
(2) ซ"s S,
(2) (2)
sn + ^s n + ^n
f CP 1
(2)
. pn + p
- Aฎ
(7) (8).
(2.14)
^d3/
,303
at
V-VC3 = V.0D-VC3 - A3(0+PbKd3)C3 + qmC3m + VV(-^) - C3
(1)
(2)
(3)
(4)
(5)
Pw P
(6)
C
(7)
(8)
(9)
(10)
(2.15)
where 6 is the moisture content, pb is the bulk density of the medium (M/L3), t is time, V is the discharge
(L/T), V is the del operator, D is the dispersion coefficient tensor (L2/T). The As, A0, Ap, A^ A1; A2, A3 and K^,
Kdo, K^, Kdp, Kdl, K^, K^ are transformation rate constants and distribution coefficients of dissolved substrate,
oxygen, nitrate, nutrient, microbe #1, microbe #2, and microbe #3, respectively; q^ is the source rate of water;
and CS1^ C01^ C,^ Cm Clm^ C2l^ and C3l^ are the concentrations of substrate, oxygen, nitrogen, nutrient, microbe
#1, microbe #2 and microbe #3 in the source, respectively.
In each of Eqs. (2.9) through (2.15), term (1) represents the rate of material increase per unit medium
volume, term (2) is the rate of transport by advection, term (3) is the rate of transport by dispersion-diffusion,
term (4) represents the rate of first order transformation, term (5) is due to the rate of artificial injection, and
term (6) is the rate due to the rewriting of the transport equation from conservative form to advective form.
In Eq. (2.9), term (7) through term (10) are the substrate removal rates under aerobic condition of microbe #1,
11
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anaerobic condition of microbe #2, aerobic condition of microbe #3, and anaerobic condition of microbe #3,
respectively. In Eq. (2.10), term (7) through term (10) represent the oxygen utilization rates resulting from the
energy requirement for the growth of microbe #1, the energy maintenance of microbe #1, the energy
requirement of microbe #3, and the energy maintenance of microbe #3, respectively. In Eq. (2.11), term (7)
through term (10) are the nitrate utilization rates resulting from the energy requirement for the growth of
microbe #2, the energy maintenance of microbe #2, the energy requirement of microbe #3, and the energy
maintenance of microbe #3, respectively. In Eq. (2.12), term (7) through term (10) represent the nutrient
removal for the synthesis of microbe # 1 under aerobic condition, microbe #2 under anaerobic condition,
microbe #3 under aerobic condition, and microbe #3 under anaerobic condition, respectively. Term (7) and
term (8) in Eqs. (2.13) through (2.15) are growth rate and decay rate of microbe #1 under aerobic condition,
microbe #2 under anaerobic condition, and microbe #3 under aerobic condition, respectively. Term (9) and
term (10) in Eq. (2.15) represent the growth rate and decay rate of microbe #3 under anaerobic condition,
respectively.
The dispersion coefficient tensor D in Eq. (2.9) to Eq.(2.15) is given by
6D =aT V 6 +(aL-aT)VV/1V +6amT6 (2.16)
where V is the magnitude of V , 6 is the Kronecker delta tensor, % is lateral dispersivity, aL is the
longitudinal dispersivity, a,,, is the molecular diffusion coefficient, and T is the tortuosity.
1
is an inhibition function which is under the assumption that denitrifying enzyme
inhibition is reversible and noncompetitive, where I\, is the inhibition coefficient (M/L3). ^r), (jj-2-1, (jD(3) and
l^3' are the maximum specific oxygen-based growth rates for microbe #1, the maximum specific nitrate-based
growth rate for microbe #2, the maximum specific oxygen-based growth rate for microbe #3, and the maximum
specific nitrate-based growth rate for microbe #3 (1/T), respectively; Y0(1), Yn(2), Y0(3), and Yn(3) are the yield
coefficient for microbe # 1 utilizing oxygen, the yielding coefficient for microbe #2 utilizing nitrate, the
12
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yielding coefficient for microbe #3 utilizing oxygen and nitrate, in mass of microbe per unit mass of substrate
(MM); Kj1', Kso(3), Kjฎ, Ksn(3), Kp0(1), Kp0(3), Kpn(2), Kpn(3) are the retarded substrate saturation constants under
aerobic conditions with respect to microbe #1, microbe #3, the retarded substrate saturation constants under
anaerobic conditions with respect to microbe #2, microbe #3, the retarded nutrient saturation constants under
aerobic conditions with respect to microbe #1, microbe #3, and the retarded nutrient saturation constants under
anaerobic conditions with respect to microbe #2, microbe #3, respectively; K0(1), K0(3), Kn(2), K^3' are the
retarded oxygen saturation constants under aerobic conditions with respect to microbe #1, microbe #3, and the
retarded nitrate saturation constant under anaerobic conditions with respect to microbe #2 and microbe #3
(M/L3), respectively. A0(1), A0(3), An(2), and An(3) are the microbial decay constants of aerobic respiration of
microbe #1 and microbe #3, and the microbial decay constants of anaerobic respiration of microbe #2 and
microbe #3 (1/T), respectively; y0(1), y0(3), yn(2), and yn(3) are the oxygen-use or nitrate-use for syntheses by
microbe #1, microbe #2, or microbe #3, respectively; ce0(1), ce0(3), cen(2), and cen(3) are the oxygen-use or nitrate-use
coefficient for energy by microbe #1, microbe #2, or microbe #3, respectively; F0(1), F0(3), Fn(2), and Fn(3) are the
oxygen or nitrate saturation constants for decay with respect to microbe #1, microbe #2, or microbe #3 (M/L3),
respectively; and e0(1), e0(3), en(2), and en(3) are the nutrient-use coefficients for the production of microbe #1,
microbe #2, or microbe #3 with respect to aerobic or anaerobic respiration, respectively.
13
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Initial Conditions of Transport
Cs = Csi(x,y,z)
C0 = C01(x,y,z)
Cn = Cm(x,y,z)
CP = cpi(x,y,z) in R (2.17)
q = Cjifry.z)
C2 = C2i(x,y,z)
C3 = C3i(x,y,z)
Prescribed Concentration (Dirichlet) Boundary Conditions
Cs = Csd(xb,yb,zb,t)
C0 = Cod(xb,yb,zb,t)
Cn = Cnd(xb,yb,zb,t)
Cp = Cpd(xb,yb,zb,t) on Bd (2.18)
Ci = Cld(xb,yb,zb,t)
C2 = C2d(xb,yb,zb,t)
C3 = C3d(xb,yb,zb,t)
14
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Variable Boundary Conditions
n-(VCs - 6D-VCs) = n-VCsv(xb,yb,zb,t)
n-(VC0 - 6D-VC0) = n-VCov(xb,yb,zb,t)
n-(VCn - 6D-VCn) = n-VCnv(xb,yb,zb,t)
n-(VCp - 6D-VCp) = n-VCpv(xb,yb,zb,t) if n-\
-------
Cauchy Boundary Conditions
n-(VCs-6D-VCs) = qsc(xb,yb,zb,t)
n-(VC0-6D-VC0) = qoc(xb,yb,zb,t)
n-(VCn-6D-VCn) = qnc(xb,yb,zb,t)
n-(VCp-6D-VCp) = qpc(xb,yb,zb,t) on Bc (2.20)
n-CVCj-OD-VCj) = qlc(xb,yb,zb,t)
n-(VC2-6D-VC2) = q2c(xb,yb,zb,t)
n-(VC3-6D-VC3) = q3c(xb,yb,zb,t)
Neumann Boundary Conditions
n-(-6D-VCs) = qsn(xb,yb,zb,t)
n-(-6D-VCo) = qon(xb,yb,zb,t)
n-(-6D-VCn) = qnn(xb,yb,zb,t)
n-(-6D-VCp) = qpn(xb,yb,zb,t) on Bn (2.21)
iK-eD-VC^) = qln(xb,yb,zb,t)
n-(-6D-VC2) = q2n(xb,yb,zb,t)
n-(-6D-VC3) = q3n(xb,yb,zb,t)
where CS1, C01, Cm, Cpl, CH, C2l, and C3l, are the initial concentrations of substrate, oxygen, nitrogen, nutrient,
microbe #1, microbe #2, and microbe #3; and R is the region of interest; (x^y^) is the spatial coordinate on
the boundary; n is an outward unit vector normal to the boundary; Csd, Cod, Cnd, Cpd, Cld, C2d, C3d, and Csv, Cov,
Cnv, Cpv, Clv, C2v, C3v, are the prescribed concentrations of substrate, oxygen, nitrogen, nutrient, microbe #1,
microbe #2, and microbe #3, on the Dirichlet boundary and the specified concentrations of water through the
variable boundary, respectively; Bd, and Bv are the Dirichlet and variable boundaries, respectively; q,c, q^, q^,,
qpc, qic, q2c, q3c and qsn, qon, q m, q pn, q ln, q 2n, q 3n, are the prescribed total flux and gradient flux of substrate,
16
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oxygen, nitrogen, nutrient, microbe #1, microbe #2, and microbe #3 through the Cauchy and Neumann
boundaries Bc and Bn, respectively.
2.2 Numerical Approximation
Flow Equation
The pressure head can be approximated to Eq.(2.22) by the finite element method.
h * ฃhj(t)Nj(x,z)
(2.22)
where N is the total number of nodes in the region and Nj and hj are the basis function and the amplitude of
h, respectively, at nodal point j. Substituting Eq.(2.22) into Eq.(2.1) and choosing Galerkin finite element
method, the governing flow equation can be approximated to the following.
Pw
dL
~dT
/'(VNi)-K(VNj)dR
/Nj-^-(or - -H-)qdR - f(VNi)-K-(-t-Vz)dR + f n-K-(Vh + -H-Vz)N;dB ,
(2.23)
r\v
where i = l,2,...,N. F =
dt
Equation (2.23) written in matrix form is:
dh
dt
(2.24)
where {dh/dt} and {h} are the column vectors containing the values of dh/dt and h, respectively, at all nodes;
[M] is the mass matrix resulting from the storage term; [S] is stiffness matrix resulting from the action of
conductivity; and {G}, {Q} and {B} are the load vectors from the gravity force, internal source/sink, and
boundary conditions, respectively. The matrices, [M] and [S] are given by
17
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ee
E /Nae^FNpedR
S'J \ J T'ฐ/" T'Pr" (2.26)
e J^
where
R, = the region of element e,
Me = the set of elements that have a local side ce-p coinciding with the global side i-j,
Nae = the a-th local basis function of element e.
Similarly, the load vectors {G}, {Q} and {B} are given by
ir._P-
/"(VN^-K'-t^-VzdR
Qi = E fNae-^(or--^) (2 29)
eeMeJ Pw Pw
where
Be = the length of boundary segment e,
Nse = the set of boundary segments that have a local node a coinciding with the global node i.
The reduction of the partial differential equation Eq. (2.1) to the set of ordinary differential equations
Eq. (2.24) simplifies the evaluation of integrals on the right hand side of Eqs. (2.25) through (2.29) for every
element for boundary surface e. The major tasks that remain to be done are the specification of base and
18
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weighting functions and the performance of integration to yield the element matrices. Linear hexahedral
elements are demonstrated in the Appendix C of users' manual of 3DFEMFAT (Yeh et al., 1994).
The following steps demonstrate the incorporation of boundary conditions into a matrix equation by
the finite element method.
For the Cauchy boundary condition given by Eq.(2.7), Eq.(2.7) is simply substituted into Eq.(2.28)
to yield a boundary-element column vector {Bce} for a Cauchy segment:
IBC6} = {qce} (2.30)
where {qce} is the Cauchy boundary flux vector given by
qcซ = -/Nซe qcdB , a = i or 2
(2 31)
The Cauchy boundary flux vector represents the normal fluxes through the two nodal points of the segment
Be on Bc.
For the Neumann boundary condition given by Eq.(2.6), Eq.(2.6) is substituted into Eq.(2.28) to yield
a boundary-element column vector {Bne} for a Neumann segment:
lBne} = {qne} (2.32)
where {qnae} is the Neumann boundary flux vector given by:
e
q
-na
N>K-^Vz-Nacqn|dB ; a = 1 or 2 (2 33)
B Pw
which is independent of pressure head.
The implementation of the variable-type boundary condition is more involved. During the iteration
of boundary conditions on the variable boundary, one of Eqs.(2.8a) through (2.8e) is used at a node. If either
Eq.(2.8b) or (2.8e) is used, it is substituted into Eq.(2.28) to yield a boundary element column vector {Bve}
for a variable boundary segment:
19
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{Bve} = {qve} (2.34)
where {qve} is the variable boundary flux given by:
ae-^qpdB, or qva = -JN(Xe-^qedB ; a = 1 or 2 (2 35)
Assembling over all Neumann, Cauchy, and variable boundary segments, one obtains the global boundary
column vector {B} as:
(Bl = {q} (2.36)
in which
(q) = E {qne) + E (qce} + E {qve} n 3?)
^Nne eeNce eeNve
where Nne, Nce, and Nve are the number of Neumann boundary segments, Cauchy boundary segments, and
variable boundary segments with flux conditions imposed on them, respectively. The boundary flux {B} given
by Eqs.(2.36) and (2.37) should be added to the right hand side of Eq.(2.24).
At nodes where Dirichlet boundary conditions are applied, an identity equation is generated for each
node and included in the matrices of Eq.(2.24). The Dirichlet nodes include the nodes on the Dirichlet
boundary and the nodes on the variable boundary to which either Eq.(2.8a), (2.8c), or (2.8d) is applied.
Transport Equation
To simplify the notation, the subscript s, o, n, p, 1, 2, or 3 in Eqs. (2.9) to (2.15) will be dropped for
the development of numerical algorithm in this section. Since the hybrid Lagrangian-Eulerian approach is used
to simulate Eq. (2.9) to (2.15), it is written in the Lagrangian-Eulerian form as
(6 +pbKd)^ =V-(6D-VC) - A(6C +PbKd) +QCm --^QC + -^ VV(-P-)C
Dt p p p
W (2.38)
- f(Cp C2, C3, Cs, Co, Cn, Cp)C + g(Cp C2, C3, Cs, Co, Cn, Cp)C
20
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<2'39)
where f(Q, C2, C3, Cs, C0, Cn, Cp) is a microbial-chemical interaction function and g(Q, C2, C3, Cs, C0, Cn, Cp)
is amicrobial growth function . Applying the Galerkin finite element method to Eq. (2.9) through Eq. (2.15),
one obtains
[D] + [K]
= {Q} + {B}
(2.40)
where {C} is a vector whose components are the concentrations at all nodes, {DC/Dt} is the time derivative
of {C} with respect to time, [M] is the mass matrix associated with the time derivative term, [A] is the stiffness
matrix associated with the velocity term which is only computed as steady-state is considered, [D] is the
stiffness matrix associated with the dispersion term, [K] is the stiffness matrix associated with the decay term
and microbial-chemical interaction, [B] is the stiffness matrix resulting from boundary conditions, {Q} is the
load vector associated with all source/sink terms, and {B} is the load vector associated with boundary
conditions. These matrices and vectors are given as follows.
E/Nae(6+pbKd)NpedR
E /w>-
VNodR
E
KiJ =
eeM
A(8 + PbKd) + f(C1,C2,C3,Cs,C0,Cn,Cp)
NpedR
(2.41d)
21
-------
BiJ
ฃ fN>-V)NpedB + ฃ fN>-V)NpedB
R * J
(2.41e)
Qi = E fNaeqmCindR+E fNaeg(C1,C2,C3,Cs,Co,Cn,Cp)dR
eeMe J eeMe ฃ
ฃ fN>-V)CvdB + ฃ fNaeqcdB + ฃ fNaeqndB
. + J R eR J R PR -1
(2.41f)
D D +
BeeBv
(2.4 Ig)
where Bv+ is that part of variable boundary for which the flow is directed into the region, C \s the
concentration of the incoming fluid through the variable boundary segment Bv+, and Bc, Bn are the Cauchy and
Neumann boundary segments.
The numerical algorithm for solving this partial differential equation is a modified Lagrangian-
Eulerian method with adapted zooming and peak capturing (LEZOOMPC). Before the LEZOOMPC
algorithm is described, two terms need to be defined, namely smooth elements and rough elements, which shall
be used throughout this document. A smooth element is defined as an element within which any physical
quantity at all points can be interpolated with its node values to within error tolerance. A rough element is
defined as an element within which there exists at least one point for which the physical quantity cannot be
interpolated with its node values to within error tolerance. Basically, LEZOOMPC is a modified method of
the Lagrangian-Euleran decoupling with zoomable (LEZOOM) hidden fine-mesh approach (Yeh, 1990) and
exact peak capturing and oscillation-free scheme (EPCOF) (Yeh et al., 1992) to solve advection-dispersion
transport equations. To compute the concentration Cn+1 at time t^, the Lagrangian concentration, Q*'s, must
be determined first [It is noted that the Lagrangian concentrations Q*'s at all global nodes are exact if
concentrations Cn at time ^ represent the exact solution through the region, not just at the nodes]. The diffu-
sion transport problem is then solved over all the global nodes and activated forward-tracked nodes at time t^
22
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with the Galerkin finite element method, taking the Lagrangian concentrations C;* at all global nodes and
activated forward-tracked nodes as the initial condition, to complete the computation.
Figure 2.1 shows the basic concept structure of solving transport of 3DFATMIC. It contains two main
steps, namely the Lagrangian and Eulerian steps.
First, the concentrations at the last time step f , Cn's, are the known quantities for the computation of
this time step. Second (GNTRAK module), one computes the Lagrangian concentration C;*'s at global nodes
using the backward node tracking as
Q* = E Cj-N/Xi'.Yi'.Zi*), i = l,2,,N (2.42)
j=i
in which
tn+i
Xi* =Xi-JVxdt
tn+i
Yi* = Yr/Vydt (2.43)
where ~Nj(x?,y',z') is the base function associated with node (x^y^Zj) evaluated at (x^y/*,^*); Vx, Vy, and Vz
are the velocities along x-, y- , and z-directions, respectively.
23
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Figure 2.1 The basic structure for coding transport part of 3DFATMIC
Third (FiPTRAK modules), all the activated fine grid nodes and the global nodes are forwardly tracked
to obtain the Lagrangian concentration C/ by the following equations:
(2-44)
in which
24
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(2.45)
It should be noted that C/'s are exact if Cjn's are exact and Eq.(2.45) is integrated exactly.
Fourth (SFDET and FGDET modules), it is determined whether an element is a rough element in the
SFDET module (Yeh et al., 1992) based on prescribed error tolerance. The criteria is shown in the following
formula:
(2.46)
where C/ is the approximate value determined from finite element interpolation, el3 e2 are two error
tolerances and CMf is the maximum concentration of C/'s. FGDET module generates the regular fine-mesh
points in every rough element determined in the SFDET module.
Fifth (ISEFflL module), if the element is a smooth element, all forward-tracked nodes for dispersion
computation at the present time t^ and advection computation at the next time tn+2 are removed. Otherwise,
the number of regular fine grids, which is determined by users, is imbedded into every rough element. The
indices of subelements are stored in the ISE array. In addition to regular fine grids refinement, this module
also captures all the highest and lowest concentrations within each subelement. This demonstrates the idea
called adapted zooming and peak capturing. The above five steps form the Lagrangian computation of the
advective transport.
The next three steps (DFPREP, ASEMBL, and SOLVE modules) are Eulerian steps to solve the
dispersion matrix equation. The module DFPREP prepares all the needed information for assembling the
subelemental matrices, which are zoomed in the Eulerian step. The inclusion of this module gets rid of the
inaccuracy due to the dispersion calculation. In this module, the number of fine grids generated in each global
element is determined by users and may be different from that in the module FGDET. If the element is a
25
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smooth element, no fine grids are imbedded. If the element is a rough element, the element is zoomed and
connected with the surrounding smooth elements. At the end of this module, the nodal connection information
of each point has to be prepared to compose the matrix and solve the matrix equation.
The module ASEMBL is designed to yield the following element matrix equation
(A 6}{C e} = {R e} (2.47)
which is based on Eqs.(2.9) to (2.15), where [Ae] is the element coefficient matrix, {Ce} is the unknown vector
of the concentration, and {Re} is the element load vector. Element e can be a global element or a subelement
generated in a rough region by DFPREP. Then, this module assembles all the element matrix equations to a
global matrix equation with the slave point concept to take care of the interface between rough regions and
smooth regions. The resulting matrix equation will be solved by a diffusion solver.
The module SOLVE solves the assembled global matrix equation by a block iterative solver, pointwise
iteration, or preconditioned conjugate method. If the diffusion zoomed approach is activated in the Eulerian
step, the block iteration method is changed to pointwise iteration solver forcefully.
At the very end of this time step, i.e., at tn+1, the concentrations at all activated fine grid nodes
generated in the Lagrangian step are obtained by finite element interpolations as follows:
N , , , ,
pn+l _ /-i +\^ IPn+1 -f *INbr v 7 I k-17 N +1 (7 4.8*1
k ~~ k / j ' ' i r i\ k'^K> kp ^^ ^->^->---->^^n * \-^-^o)
i = l
2.3 Description of 3DFATMIC Subroutines
3DFATMIC consists of a MAIN program and 120 subroutines. The MAIN is utilized to specify the
sizes of all arrays. The control and coordinate activity are performed by the subroutine HTMICH. Figure 2.2
shows the structure of the program. The functions of these subroutines are described below.
26
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Program MAIN
The MAIN is used to specify the sizes of all arrays. The flow of data input for the model is also
anchored by the MAIN. The subroutine RDATIO is called to read the geometric and material data. MAIN
then calls subroutine PAGEN to generate pointer arrays; SURF to identify the boundary sides and compute
the directional cosine. The source/sink data for flow and transport computations are read in by the subroutines
FSSDAT and TSSDAT, respectively. The boundary conditions for flow and transport calculations are then
read in by the subroutines FBCDAT and TBCDAT, respectively. Control is then passed to subroutine
HTMICH to coordinate and perform flow and/or transport computations.
Subroutine RDATIO
The subroutine RDATIO is called by the program MAIN to read in the soil property functions and
geometric data for the area of interest.
Subroutine FSSDAT
The subroutine FSSDAT is called by the program MAIN to read in the sources/sinks profiles, nodes,
and/or elements for flow simulations. The source/sink type for each node/element is also assigned in this
subroutine according to the data given by the users.
27
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Figure 2.2 Program Structure of 3DFATMIC (MAIN)
28
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Subroutine TSSDAT
The subroutine TSSDAT is called by the program MAIN to read in the sources/sinks profiles, nodes,
and/or elements for transport simulations. The source/sink type for each node/element is also assigned in this
subroutine according to the data given by the users.
Subroutine FBCDAT
The subroutine FBCDAT controls the input of boundary condition, in time and space, assigned to each
boundary node/element for flow simulations. Users need to give the boundary profiles, to specify the global
node/element numbers of the boundary, and to assign boundary profile type to each node/element.
Subroutine TBCDAT
The subroutine TBCDAT controls the input of boundary condition, in time and space, assigned to each
boundary node/element for transport simulations. Users need to give the boundary profiles, to specify the
global node/element numbers of the boundary, and to assign boundary profile type to each node/element.
Subroutine CKBDY
This subroutine checks all the boundary sides and generates the arrays, including NBDYB and IBDY,
for later use in along boundary tracking on both the unspecified and the Neumann boundary sides. NBDYB(I)
represents the accumulated number the unspecified/Neumann boundary sides connecting with the 1-st through
the (I-l)-th global node. IBDY(I) indicates the global boundary side to which the I-th unspecified/Neumann
boundary side relates.
Subroutine HTMICH
The subroutine HTMICH controls the entire sequence of operations, a function generally performed
by the MAIN program. It is, however, preferable to keep a short MAIN and supplement it with several
subroutines with variable storage allocation. This makes it possible to place most of the FORTRAN deck on
a permanent file and to deal with a site-specific problem without making changes in array dimensions
throughout all subroutines.
29
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Depending on the combinations of the parameters KSSf, KSSt, NTI, and IMOD, the subroutine
HTMICH will perform either the steady state flow and/or transport computations only, or the transient state
flow and/or transport computations using the flow and/or transport steady-state solution as the initial
conditions, or the transient flow and/or transport computation using user-supplied initial conditions.
HTMICH calls the subroutines ESSFCT, WSSFCT, CBVFCT, NBVFCT, VBVFCT, and DBVFCT
to obtain sources/sinks and boundary values; subroutine SPROP to obtain the relative hydraulic conductivity,
water capacity, and moisture content from the pressure head; subroutine VELT to compute Darcy's velocity;
subroutine FSFLOW to calculate flux through all types of boundaries and water accumulated in the media;
subroutine FPRINT to print out the results; and subroutine FSTORE to store the flow variables for plotting;
subroutine HMCHYD to perform the flow computations; subroutine FLUX to compute material flux;
subroutine AFABTA to obtain upstream weighting factor based on velocity and dispersivity; subroutine
TSFLOW to calculate material flux through all types of boundaries and water accumulated in the media;
subroutine TPRINT to print out the transport computation results; subroutine TSTORE to store the transport
computation results for plotting; subroutine THNODE to compute the value of moisture content plus bulk
density times distribution coefficient in the case of linear isotherm, or the moisture content in the case of
nonlinear isotherm at all nodes; subroutine DISPC to compute the dispersion coefficients; and subroutine
HMCTRN to perform the transport computations.
Subroutine READR
This subroutine is called by the MAIN as well as subroutines FBCDAT and HTMICH to automatically
generate real numbers if required. Automatic generation of regular patterned data is built into the subroutine
(see Appendix A).
Subroutine READN
This subroutine is also called by the subroutines RDATIO, FBCDAT, and TBCDAT to generate integers if
required (see Appendix A).
30
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Subroutine PAGEN
This subroutine is called by the controlling program MAIN to preprocess pointer arrays that are needed
to store the global matrix in compressed form and to construct the subregional block matrices. The pointer
arrays automatically generated in this subroutine include the global node connectivity (stencil) GNOJCN(J,N),
regional node connectivity LNOJCN(J,I,K), total node number for each subregion NTNPLR(K), bandwidth
indicator for each subregion LMAXDF(K), and partial fill-up for the mapping array between global node
number and local subregion node number GNPLR(I,K) with I = NNPLR(K) + 1 to NTNPLR(K). Here
GNOJCN(J,N) is the global node number of J-th node connected to the global node N; LNOJCN(J,I,K) is the
local node number of the J-th node connected to the local node I in K-th subregion; NTNPLR(K) is the total
number of nodes in the K-th subregion, including the interior nodes, the global boundary nodes, and intra-
boundary nodes; LMAXDF(K) is the maximum difference between any two nodes of any element in K-th
subregion; and GNPLR(I,K) is the global node number of I-th local-region node in the K-th subregion. These
pointer arrays are generated based on the element connectivity IE(M,J), the number of node for each subregion
NNPLR(K), and the mapping between global node and local-region node GNLR(I,K) with 1=1, NNPLR(K).
Here IE(M,J) is the global node number of J-th node of element M; NNPLR(K) is the number of nodes in the
K-th subregion including the interior nodes and the global boundary nodes but not the intraboundary nodes.
If a preconditioned conjugate gradient solver is selected to solve linear matrix equations, this subroutine will
rearrange LRN(J,N) such that LRN(1,N) = N.
Subroutine ELENOD
This subroutine determines the number of nodes, the number effaces, and the elemental shape index
of element M by using the IE(M,5) and IE(M,7) information.
Subroutine LRL3D
This subroutine is called by subroutines PAGEN and ADVW3D. This subroutine generates the node-
element connection pointer arrays NLRL, LRL, NLRLW, and LRLW used in particle tracking.
31
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Subroutine LRN3D
This subroutine is called by subroutine PAGEN. This subroutine generates the node-node connection
pointer arrays NLRN and LRN used in composing the linearized matrix equations.
Subroutine SURF
Subroutine SURF identifies the boundary sides, sequences the boundary nodes, and computes the
directional cosine of the surface sides. The mappings from boundary nodes to global nodes are stored in
NPBB(I) (where NPBB(I) is the global node number of the I-th boundary node). The boundary node numbers
of the four nodes for each boundary side are stored in ISB(I,J) (where ISB(I,J) is the boundary node number
of I-th node of J-th side, I = 1 to 4). There are six, five, or four sides for each hexahedral, triangular prism,
or tetrahedral element, respectively. Which of these sides is the boundary side is determined automatically in
the subroutine SURF and is stored in ISB(5,J). The global element number, to which the J-th boundary side
belongs, is also preprocessed in the subroutine SURF and is stored in ISB(6,J). The directional cosines of the
J-th boundary side are computed and stored in DCOSB(I,J) (where DCOSB(I,J) is the directional cosine of
the J-th surface with I-th coordinate, I = 1 to 3). The information contained in NPBB, ISB, and DCOSB, along
with the number of boundary nodes and the number of boundary sides, is returned to subroutine MAIN
program for other users.
Subroutine IBE3D
The subroutine IBE3D is used to generate the index of boundary element stored in IBE array. If
IBE(M) = 0, it means no boundary element side in the M-th element. If IBE(M)=12, the element side 1 and
2 are the boundary element sides of the M-th element globally.
Subroutine ESSFCT
This subroutine is called by the subroutine HTMICH to compute the elemental source strength. It uses
the linear interpolation of the tabular data or it computes the value with analytical function. If the latter option
is used, the user must supply the function into this subroutine.
32
-------
Subroutine WSSFCT
This subroutine is called by the subroutine HTMICH to compute the well source strength. It uses the
linear interpolation of the tabular data or it computes the value with analytical function. If the latter option is
used, the user must supply the function into this subroutine.
Subroutine VBVFCT
This subroutine is called by the subroutine HTMICH to compute the variable boundary values. It uses
the linear interpolation of the tabular data or it computes the value with analytical function. If the latter option
is used, the user must supply the function into this subroutine.
Subroutine DBVFCT
This subroutine is called by the subroutine HTMICH to compute the Dirichlet boundary values. It
uses the linear interpolation of the tabular data or it computes the value with analytical function. If the latter
option is used, the user must supply the function into this subroutine.
Subroutine CBVFCT
This subroutine is called by the subroutine HTMICH to compute the Cauchy fluxes. It uses the linear
interpolation of the tabular data or it computes the value with analytical function. If the latter option is used,
the user must supply the function into this subroutine.
Subroutine NBVFCT
This subroutine is called by the subroutine HTMICH to compute the Neumann fluxes. It uses the
linear interpolation of the tabular data or it computes the value with analytical function. If the latter option is
used, the user must supply the function into this subroutine.
Subroutine FPRINT
This subroutine is used to line-print the flow variables. These include the fluxes through variable
boundary surfaces, the pressure head, total head, moisture content, and Darcy's velocity components.
33
-------
Subroutine FSTORE
This subroutine is used to store the flow variables on Logical Unit 11. It is intended to for plotting
purposes. The information stored includes region geometry, subregion data, and hydrological variables such
as pressure head, total head, moisture content, and Darcy's velocity components.
Subroutine TPRINT
This subroutine is used to line-print the simulation results of contaminant transport. These include
the material flux components and the concentration at each global node.
Subroutine TSTORE
This subroutine is used to store the simulation results of contaminant transport on Logical Unit 12.
It is intended for plotting purpose. The information stored includes region geometry, concentrations, and
material flux components at all nodes for any desired time step.
Subroutine ADVW3D
This subroutine is called by HTMICH to generate all the working arrays including IBW, IEW,
NLRLW, and LRLW, for 'in-element' tracking in the Lagrangian step computation used in the transient-state
simulation. The more subelements generated for particle tracking, the more accurate result obtained. In this
subroutine, the working arrays are for the following types of elements: (1) tetrahedral elements (if ISHAPE=4
or 0), (2) triangular prism elements (if ISHAPE=6 or 0), and (3) hexahedral elements (if ISHAPE=8 or 0).
Subroutine HMCHYD
HMCHYD calls subroutine SPROP to obtain the relative hydraulic conductivity, water capacity, and
moisture content from the pressure head; subroutine VELT to compute Darcy's velocity; subroutine BCPREP
to determine if a change of boundary conditions is required; subroutine FASEMB to assemble the element
matrices over all elements; subroutine FBC to implement the boundary conditions; subroutine BLKITR, PISS,
PPCG, or ILUCG to solve the matrix equations; subroutine FSFLOW to calculate flux through all types of
boundaries and water accumulated in the media; subroutine FPRINT to print out the results; and subroutine
34
-------
FSTORE to store the flow variables in binary format for plotting. Figure 2.3 shows the flow chart of this
subroutine.
Subroutine SPROP
This subroutine calculates the values of moisture content, relative hydraulic conductivity, and the water
capacity. This subroutine calls subroutine SPFUNC to calculate soil property function by either tabular input
or analytical functions.
Subroutine BCPREP
This subroutine is called by FiMCFiYD to prepare the infiltration-seepage boundary conditions during
a rainfall period or the seepage-evapotranspiration boundary conditions during non-rainfall periods. It decides
the number of nodal points on the variable boundary to be considered as Dirichlet or Cauchy points. It
computes the number of points that change boundary conditions from ponding depth (Dirichlet types) to
infiltration (Cauchy types), or from infiltration to ponding depth, or from minimum pressure (Dirichlet types)
to infiltration during rainfall periods. It also computes the number of points that change boundary conditions
from potential evapotranspiration (Cauchy types) to minimum pressure, or from ponding depth to potential
evapotranspiration, or from minimum pressure to potential evapotranspiration during non-rainfall periods.
Upon completion, this subroutine returns the Darcy flux (DCYFLX), infiltration/potential evapotranspiration
rate (FLX), the ponding depth nodal index (NPCON), the flux-type nodal index (NPFLX), the minimum
pressure nodal index (NPMIN), and the number of nodal points (NCHG) that have changed boundary
conditions.
35
-------
HMCHYD
SPROP
BCPREP
FASEMB
FBC
BLK1TR
PISS
PPCG
ILUCG
VELT
FSFLOW
FPRINT
FSTORE
<
-
<
Q34S
ELENOD
FQ468
Q34S
SOLVE
POLYP
LLTINV
ELENOD
FQ468DV
Q34S
ELENOD
Q468TH
, pi Fwm
!_,!_/ !_,! >l \JLJ
C< T T A TIT1
SHAPE
- SPFUNC
CU \ ID ID
orlArii
CTT A T"ปT7
Mi A Ft
PIT 4 rmn
SHAPE
Figure 2.3 Program Structure of 3DFATMIC (Flow Part)
36
-------
Subroutine SPFUNC
This subroutine calculates the soil property function by either tabular input or analytical functions.
When analytical functions are used, the users must supply the functional form and modify this subroutine.
Subroutine FASEMB
This subroutine calls FQ468 to evaluate the element matrices. It then sums over all element matrices
to form a global matrix equation governing the pressure head at all nodes.
Subroutine FQ468
This subroutine is called by the subroutine FASEMB to compute the element matrix given by
QA(I,J) = N; Nj dR , (2.49a)
J p dh
QB(I,J) = J(VN1e)-KsKr-(VN/)dR , (2.49b)
Subroutine FQ468 also calculates the element load vector given by
RQ(I) = [ - (VN-K--CVz) +N1-q]dR , (2 49c)
R i O I O
where q is the source/sink.
Subroutine SHAPE
This subroutine is called by subroutines SPROP, FQ468, FQ468DV, Q468TH, TQ468DV,TQ468,
and Q468R to evaluate the value of the base and weighting functions and their derivatives at a Gaussian point.
The computation is straightforward.
Subroutine FBC
This subroutine incorporates Dirichlet, Cauchy, Neumann, and variable boundary conditions. For a
Dirichlet boundary condition, an identity algebraic equation is generated for each Dirichlet nodal point. Any
37
-------
other equation having this nodal variable is modified accordingly to simplify the computation. For a Cauchy
surface, the integration of the surface source is obtained by calling the subroutine Q34S, and the result is added
to the load vector. For a Neumann surface, the integrations of both the gradient and gravity fluxes are obtained
by calling the subroutine Q34S. These fluxes are added to the load vector. The subroutine FBC also
implements the variable boundary conditions. First, it checks all infiltration-evapotranspiration-seepage points,
identifying any of them that are Dirichlet points. If there are Dirichlet points, the method of incorporating
Dirichlet boundary conditions mentioned above is used. If a given point is not the Dirichlet point, the point
is bypassed. Second, it checks all rainfall-evaporation-seepage points again to see if any of them is a Cauchy
point. If it is a Cauchy point, then the computed flux by infiltration or potential evapotranspiration is added
to the load vector. If a given point is not a Cauchy point, it is bypassed. Because the infiltration-evaporation-
seepage points are either Dirichlet or Cauchy points, all points are taken care of in this manner.
Subroutine Q34S
This subroutine is called by the subroutines BCPREP, FBC and FSFLOW to compute the surface node flux
of the type
RQ(I) = NqdB , (2.50)
Be ^ฐ
where q is either the Cauchy flux, Neumann flux, or gravity flux.
Subroutine BLKITR
This subroutine is called by the subroutines HMCHYD and HMCTRN to solve the matrix equation
with block iteration methods. For each subregion, a block matrix equation is constructed based on the global
matrix equation and two pointer arrays GNPLR and LNOJCN (see subroutine PAGEN), and the resulting
block matrix equation is solved with the direct band matrix solver by calling subroutine SOLVE. This is done
for all subregions for each iteration until a convergent solution is obtained. This subroutine and the subroutine
SOLVE, to be described in the next paragraph, are needed only when the block iteration option is used.
38
-------
Subroutine SOLVE
This subroutine is called by the subroutine BLKITRto solve for the matrix equation of the type
[C]{x} = {y} (2.51)
where [C] is the coefficient matrix and {x} and {y} are two vectors, {x} is the unknown to be solved, and {y}
is the known load vector. The computer returns the solution {y} and stores it in {y}. The computation is a
standard banded Gaussian direct elimination procedure.
Subroutine PISS
This subroutine is called by subroutine HMCHYD and HMCTRN, if necessary, to solve the linearized
matrix equation with pointwise iteration solution strategies.
Subroutine PPCG
This subroutine is called by the subroutines HMCHYD and HMCTRN, if necessary, to solve the
linearized matrix equation with the preconditioned conjugate gradient method using the polynomial as a
preconditioner. It calls to POLYP to invert the preconditioner.
Subroutine POLYP
This subroutine is called by the subroutine PPCG to solve for a modified residual that will be used in
the preconditioned conjugate gradient algorithm.
Subroutine ILUCG
This subroutine is called by the subroutines HMCHYD and HMCTRN, if necessary, to solve the
linearized matrix equation with the preconditioned conjugate gradient method using the incomplete Cholesky
decomposition as a preconditioner. It calls to LLTINV to invert the preconditioner.
Subroutine LLTINV
This subroutine is called by the subroutine ILUCG to solve for a modified residual that will be used
in the preconditioned conjugate gradient algorithm.
39
-------
Subroutine VELT
This subroutine calls FQ468DV to evaluate the element matrices and the derivatives of the total head.
It then sums over all element matrices to form a matrix equation governing the velocity components at all nodal
points. To save computational time, the matrix is diagonalized by lumping. The velocity components can thus
be solved point by point. The computed velocity field is then returned to HTMICH or HMCHYD through the
argument. This velocity field is also passed to subroutine BCPREP to evaluate the Darcy flux across the
seepage-infiltration-evapotranspiration surfaces.
Subroutine FQ468DV
Subroutine FQ468DV is called by the subroutine VELT to compute the element matrices given by
QB(I,J) = jN^/dR , (2.52)
Be
where N;e and Nje are the basis functions for nodal point i and j of element e, respectively. Subroutine
FQ468DV also evaluates the element load vector:
QRX(I) = -/Nj'rK-(VNje)hjdR - fN^i-K-VzdR (2.53a)
R. R.
QRY(I) = -N^j-K'-CVN/^dR - N^j-K-VzdR (2.53b)
QRZ(I) = - fN:ek-K-(VN:6)h.dR - fN:ek-K-VzdR n 53^
*< V / I l /-> J J I1 \ฃ.J3(*)
J i) J
Re P Re
where
hj = the referenced pressure head at nodal point j,
i = the unit vector along the x-coordinate,
j = the unit vector along the y-coordinate,
k = the unit vector along the z-coordinate,
K = the hydraulic conductivity tensor.
40
-------
Subroutine FSFLOW
This subroutine is used to compute the fluxes through various types of boundaries and the increasing
rate of water content in the region of interest. The function of FRATE(7) is to store the flux through the whole
boundary enclosing the region of interest. It is given by
FRATE(7) = J(Vxnx + Vyny + Vznz)dB , (2.54)
B
where B is the global boundary of the region of interest; Vx, Vy, and Vz are Darcy's velocity components; and
nx, ny, and nz are the directional cosines of the outward unit vector normal to boundary B. FRATE(l) through
FRATE(5) store the flux through Dirichlet boundary BD, Cauchy boundary Bc, Neumann boundary BN, the
seepage/evapotranspiration boundary Bs, and infiltration boundary B,, respectively, and are given by
FRATE(l) = J(Vxnx + VA + Vznz)dB , (2.55a)
FRATE(2) = J(Vxnx + V^y + Vznz)dB , (2.55b)
FRATE(3) = J(Vxnx + VA + Vznz)dB , (2.55c)
FRATE(4) = (V^x + VA + Vznz)dB , (2.55d)
FRATE(5) = J(Vxnx + VA + Vznz)dB , (2.55e)
BR
FRATE(6), which is related to the numerical loss, is given by
41
-------
5
FRATE(6) = FRATE(7) - ฃ FRATE(I) (2.56)
1=1
FRATE(8) and FRATE(9) are used to store the source/sink and increased rate of water within the
media, respectively:
and
FRATE(8) = --qdR, (2.57)
J p
P
rT^/rix f p d9 dh ,_
FRATE(9) = /-ฃ- - dR , (2.58)
If there is no numerical error in the computation, the following equation should be satisfied:
FRATE(9) = -[FRATE(7) + FRATE(8)] (2.59)
and FRATE(6) should be equal to zero. Equation (2.58) simply states that the negative rate of water going
out from the region through the entire boundary and due to a source/sink is equal to the rate of water
accumulated in the region.
Subroutine Q468TH
This subroutine is used to compute the contribution of the increasing rate of the water content from
an element e
/-ปTUU r p d9 dh,
QTHP = -ฃ-dR , (2.60)
J p dh dt
RePฐ
The computation of the above integration is straightforward.
Subroutine HMCTRN
The subroutine HMCTRN controls the entire sequence of transport computations. HMCTRN calls
subroutine AFABTA to obtain upstream weighting factor based on velocity and dispersivity; subroutine
42
-------
DISPC to calculate the dispersion coefficient associated with each Gaussian point in every element; subroutine
THNODE to compute the value of moisture content plus bulk density times distribution coefficient in the case
of linear isotherm, or the moisture content in the case of nonlinear isotherm at all nodes; subroutine GNTRAK
to compute the Lagrangian concentrations at all global nodes, subroutine HPTRAK to perform forward particle
tracking to obtain the Lagrangian concentrations at all activated forward nodes; subroutine ADVBC to
implement boundary conditions in the Lagrangian step; subroutine ADVRX to calculate the Lagrangian
concentrations with microbial-chemical involved; subroutine SFDET to determine sharp front elements;
subroutine FGDET to imbed fine grids into every sharp front element; subroutine ISEFflL to prepare ISE array
which stores the indices of subelements and to determine the activation of the points with the highest or lowest
concentrations in each subelement; subroutine DFPREP to prepare the fine mesh nodes and elements for
diffusion zooming; subroutine TASEMB to assemble the element matrices over all elements; subroutine TBC
to implement the boundary conditions globally; subroutine TBC 1 to apply intra-boundary conditions which
implement the slave point concept to overcome the incompatibility; subroutine BLKITR, PISS, PPCG, or
ILUCG to solve the resulting matrix equations; subroutine FLUX to compute material flux; subroutine
TSFLOW to calculate flux through all types of boundaries and water accumulated in the media; subroutine
TPRINT to print out the results; and subroutine TSTORE to store the results for plotting; Figure 2.4 shows
the flow chart of this subroutine.
Subroutine THNODE
This subroutine is called by HMCTRN to compute the (6 +pbdS/dC).
Subroutine AFABTA
This subroutine calculates the values of upstream weighting factors along 12 , 9, or 6 sides of all
hexahedral, triangular prism, and tetrahedral elements, respectively.
43
-------
Figure 2.4 Program Structure of 3DFATMIC (Transport Part 1 of 3)
44
-------
D
/ \
c V-
v_y
/ ~\
E )
REPLAS
ELENOD
MOVCHK
FCOS
BASE
VALBDL
WRKARY
ELTRK4
CTTDf f*
JiLlKJs.o
ELTRK8
Figure 2.4 Program Structure of 3DFATMIC ( Transport Part 2 of 3)
45
-------
FCOS
REPLAS
MMLOC
1
REPLAS
WRKARY
ONPLAN
TRAK2T
1
PLANEW
CKSIDE
LOCQ3N
REPLAS
CKCNEL
ONPLAN
NEWXE
CKCOIN
ONLINE
TRAK1T
PLANEW
BDYPLN
LOCQ3N
FCOS
DNPLAN
FCOS
REPLAS
1
MMLOC WRKARY
REPLAS
ONPLAN
TRAK2P
PLANEW
CKSIDE
LOCQ3N
CKCNEL
ONPLAN
LOCQ4N
CKCOIN
ONLINE
TRAK1P
i i
PLANEW
LOCQ3N
BDYPLN
LOCQ4N
FCOS
ONPLAN
1
BASE1 MMLOC
1
i
REPLAS
1
RF.PT.AS
T7rr>Q
1
WRKARY
ONPLAN
TRAK2H
PLANEW
CKSIDE
LOCQ4N
REPLAS
CKCNEL
ONPLAN
NEWXE
1
CKCOIN
ONLINE
TRAK1H
1
PLANEW
BDYPLN
LOCQ4N
FCOS
1
ONPLAN
Figure 2.4 Program Structure of 3DFATMIC (Transport Part 3 of 3)
46
-------
Subroutine DISPC
Subroutine DISPC calculates the dispersion coefficient associated with each Gaussian point of an
element.
Subroutine TASEMB
This subroutine calls TQ468 to evaluate the element matrices. It then sums over all element matrices
to form a global matrix equation governing the concentration distribution at all nodes.
Subroutine TQ468
This subroutine is called by the subroutine TASEMB to compute the element matrix given by
QAaJ)=|Nie0NjedR ,
(2.6 la)
(2.61b)
QB(I,J)=J(VN1e)-6D-(VNJe)dR ,
(2.61c)
QV(I,J)=fNieV-(VNje)dR ,
(2.61d)
dS
P P
(2.61e)
where dS/dC should be evaluated at Cw, the dissolved concentration at previous iteration. Subroutine TQ468
also calculates the element load vector given by:
47
-------
-^Cl +qC
(2.61f)
where Cw and Sw are the dissolved and adsorbed concentrations at previous iteration, respectively.
Subroutine TBC
This subroutine incorporates Dirichlet, variable boundary, Cauchy, and Neumann boundary
conditions. For a Dirichlet boundary condition, an identity algebraic equation is generated for each Dirichlet
nodal point. Any other equation having this nodal variable is modified accordingly to simplify the
computation. For a variable surface, the integration of the normal velocity times the incoming concentration
is added to the load vector and the integration of normal velocity is added to the matrix. For the Cauchy
boundaries, the integration of Cauchy flux is added to the load vector and the integration of normal velocity
is added to the matrix. For the Neumann boundary, the integration of gradient flux is added to the load vector.
Subroutine Q34CNV
This subroutine is called by the subroutines TBC to compute the surface node flux of the type
RQO)=|NieqdB , (2.62)
where q is either the Cauchy flux, Neumann flux, or n VCV. It also computes the boundary element matrices
BQ(I,J) = N^VN/dR (2.63)
R
Subroutine TBC1
This subroutine is called whenever the total number of nodes for composing matrix is greater than the
total number of global nodes, i.e., the diffusion zooming scheme is employed. The "slave point" concept takes
care of the incompatibility for the intraboundary points between rough and smooth regions. This subroutine
implements the concept so that the entries for the intraboundary points of the matrix equation can be modified.
If there are diffusion fine grids falling on the global boundaries, the "slave point" concept also resolves the
48
-------
problems of implementation of boundary conditions for these fine grids. Subroutine LOCPLN is called to
obtain the basis functions of the intraboundary point in the intraboundary surface which may be a four point
quadrilateral or three point triangular. Subroutine SLAVPT is called to implement the spirit of "slave point"
concept. For simplicity, two point line segment is used for obtaining basis functions instead of surface after
calling subroutine CKSIDE.
Subroutine SLAVPT
This subroutine implements the "slave point" concept on the intraboundary points between rough and
smooth regions. This subroutine is called by subroutine TBC1 to modify the entries of the matrix equation
related to these points.
Subroutine FLUX
This subroutine calls TQ468DV to evaluate the element matrices and the derivatives of concentrations.
It then sums over all element matrices to form a matrix equation governing the flux components at all nodal
points. To save computational time, the matrix is diagonalized by lumping. The flux components due to
dispersion can thus be solved point by point. The flux due to the velocity is then added to the computed flux
due to dispersion. The computed total flux field is then returned to HMCTRN through the argument.
Subroutine TQ468DV
Subroutine TQ468DV is called by the subroutine FLUX to compute the element matrices given by
QBaJ)=/NieNjedR , (2.64)
Re
where N;e and Nje are the basis functions for nodal point i and j of element e, respectively. Subroutine
TQ468DV also evaluates the element load vector:
|Niei-eD-(VNje)CjdR , (2.65a)
49
-------
|Niej-0D-(VNje)CjdR , (2.65b)
|Niek-0D-(VNje)CjdR , (2.65c)
where Cj is the concentration at nodal point j, i is the unit vector along the x-direction, j is the unit vector along
the y-coordinate, k is the unit vector along the z-coordinate, 6 is the moisture content, and D is the dispersion
coefficient tensor.
Subroutine TSFLOW
This subroutine is used to compute the flux rates through various types of boundaries and the
increasing rate of material in the region of interest. FRATE(7) is to store the flux through the whole boundary
FRATE(7)=J(Fxnx + Fyny + Fznz)dB , (2.66)
where B is the global boundary of the region of interest; Fx, Fy, and Fzare the flux components; and nx, riy, and
nz are the directional cosines of the outward unit vector normal to the boundary B. FRATE(l) stores the flux
rates through Dirichlet boundary Bd. FRATE(2) and FRATE(3) store the flux rate through Cauchy and
Neumann boundaries, respectively. FRATE(4) and FRATE(5) store incoming flux and outgoing flux rates,
respectively, through the variable boundaries Bv~ and Bv+, as given by
(2.67a)
FRATE(2)=J(Fxnx + Fyny + Fznz)dB , (2.67b)
50
-------
FRATE(3)=J(Fxnx + FA + Fznz)dB , (2.67c)
FRATE(4)= / (F n +F n +F n )dB , n 67Hx
^ i ^ x x y y z z ^ \ ' /
B _
FRATE(5)= J (Fxnx + Fyny + Fznz)dB , (2 67e)
B +
V
where By- and By + are that part of variable boundary where the fluxes are directed into the region and
out from the region, respectively. The integration of Eqs. (2.67a) through (2.67e) is carried out by the
subroutine Q34BB.
FRATE(6) stores the flux rate through unspecified boundaries as
5
FRATE(6)=FRATE(7)-^FRATE(I) (2.68)
1=1
FRATE(8) and FRATE(9), which store the accumulate rate in dissolved and adsorbed phases, respectively,
are given by
FRATE(8)=J^pdR , (2.69)
FRATE(9)=f-^dR, (2.70)
J at
R
FRATE(IO) stores the rate loss due to decay and FRATE(11) through FRATE(13) are set to zero as given by
FRATE(10)=jA(6C+PbS)dR
51
-------
FRATE(11)=FRATE(12)=FRATE(13)=0 , (2.72)
FRATE(14) is used to store the source/sink rate as
f
J
1 +sign(Q) ^^ 1 -sign(Q) ^
in - - - +QC - - - dR
If there is no numerical error in the computation, the following equation should be satisfied:
14
ฃFRATE(I)=O (2.74)
1=7
and FRATE(6) should be equal to zero.
Subroutine Q34BB
This subroutine is called by the subroutine TSFLOWto perform surface integration of the following
type
RRQ(0=/NieFdB , (2.75)
where F is the normal flux.
Subroutine 0468R
This subroutine is used to compute the contributions to FRATE(8), FRATE(9), FRATE(IO), and
FRATE(14):
QRM=j6CdR , (2.76a)
QDM=JSdR , (2.76b)
52
-------
+sign(Q))+QC(l -sign(Q))
- - dR , (2.76c)
The computation of the above integration is straightforward.
Subroutine ADVBC
This subroutine is called by HMCTRN to implement the boundary conditions. For Dirichlet boundary,
the Lagrangian concentration is specified. For variable boundaries, if the flow is directed out of the region,
the fictitious particle associated with the boundary node must come from the interior nodes. Hence the
Lagrangian concentration for the boundary node has already computed from subroutine GNTRAK and the
implementation for such a boundary segment is bypassed. For variable boundaries, if the flow is directed into
the region, the concentration of incoming fluid is specified. An intermediate concentration C" is calculated
according to
iI1 , (277a)
B; B;
where C" is the concentration due to the boundary source at the boundary node i, Vn is the normal vertically
integrated Darcy's velocity, and Cm is the concentration of incoming fluid.
Cauchy boundary conditions are normally applied to the boundary where flow is directed into the
region, where the material flux of incoming fluid is specified. The intermediate concentration is thus
calculated according to
Ci"=/NiqcdB/|NiVndB , (2.77b)
where C" is the concentration due to Cauchy fluxes at the boundary node i, Vn is the normal Darcy's velocity,
and qc is the Cauchy flux of the incoming fluid.
53
-------
The Lagrangian concentration is obtained by using the value Q" and Qn (the concentration at previous
time step) as follows
|Ni8NjCi"dB+|NipbKdNjCjndB
B
C; = for linear isotherm (2.78a)
C;* = Cj** for nonlinear isotherm (2.78b)
Subroutine 034ADB
This subroutine is used to perform surface integration of Eqs. (2.77a), (2.77b), and (2.78a) for Cauchy
and variable boundary conditions. Each surface elemental matrix is returned to subroutine ADVBC to
compose a global surface elemental matrix equation so that the Lagrangian concentrations of all the specified
boundary points can be solved.
Subroutine GNTRAK
This subroutine is called by HMCTRN to control the process of backward particle tracking starting
from global nodes. In the subroutine, each particle is tracked one element by one element until either the
tracking time is completely consumed or the particle encounters a specified boundary side. During the particle
tracking, this subroutine calls (1) subroutine ELTRK4 to track a particle in a tetrahedral element, (2)
subroutine ELTRK6 to track a particle in a triangular prism element, and (3) subroutine ELTRK8 to track a
particle in a hexahedral element. When the particle can not be tracked by normally elemental tracking, it calls
subroutine FIXCHK to check if it hits specified or unspecified boundaries. In order to make the particle
tracking complete and remedy the given velocity field error on the unspecified boundaries, subroutine
FIXCHK calls subroutine ALGBDY to continue tracking particles along the unspecified/Neumann boundaries.
At the end of backward particle tracking, the concentrations are obtained by interpolation executed in
subroutine INTERP.
54
-------
Subroutine HPTRAK
This subroutine is called by HMCTRN to compute the locations and concentrations of all forward-
tracked find-mesh nodes. Basically, the algorithm of this subroutine is the same as that of subroutine
GNTRAK.
Subroutine ADVRX
This subroutine solves the following seven nonlinear simultaneous ordinary differential equations
IDCป
t
n sn
pn ~p|
u(3)
"o
Y(3)
0
+ "? '
n
C
s
K(3)+C
. so s.
f Cง 1
Ks(n3)^
C
o
K (3) + r
. ฐ ฐ
f C" 1
K (3) r
Kn +Cn
c
p
K (3) + r
.. pฐ p.
cp |
K (3) . r
Kpn +Cp
fo)
.DC
^do/
Dt
c
c
.(3)
(3) + r
po +CP
c
r(3)
1
(2.79a)
(2.79b)
55
-------
Dt
.(2)
(2)
C
(2) + r
pn +Cp
(2)1(2)
(2.79c)
.(3)
cs
K (3) + r
^sn ^s
cn
Kn(3) + Cn_
f CP 1
K (3) + r
Kpn +^p
c
r(3)+c
^
"dp/
,DC
Dt
p _
-(1)
po
,(2)
2) ^n
Y
x
(2)
(2)
+ r
(2) r
pn +Cp
(2.79d)
+
,,(3)
e(3) ^o
ฐ Y0(3)
O
(3)
(3) "n
" v(3)
1 n
cs
K(3) + c
SO s
c
^s
Ks(n3)+C^
co
K(3) + c
.. ฐ ฐ
c T
n
Kn(3)-Cn^
I CP 1
KD(3)+C
Pฐ P.
C 1
P
KS)+CP.
fo)
^
cs
SO s
co
K0(1)-Co
f CP 1
po +^p
)(!)
Ao
(2.79e)
,(2)
c
(2.79Q
56
-------
PbK
d3/
DC3
Dt
(3)
.(3)
f Cs 1
so + ^s
f cฐ 1
K (3) r
0 o.
f Cp 1
K (3) p
No +Lp.
tf
(2.79g)
pn +p
This subroutine is called right after the Lagrangian concentrations have been obtained.
Subroutine RXRATE
This subroutine is called by subroutine ADVRX, and TASEMB at steady state simulations. Basically,
the subroutine calculates the removal rate of substrate which is represented as the terms within the braces on
the right hand side of Eqs. (2.79a) to (2.79g). The values of each bracket within the braces are returned to the
calling subroutines for each component.
Subroutine SFDET
This subroutine determines if an element is a rough element based on the prescribed error tolerance
criteria shown in Eq.(2.46). If the M-th element is a rough element, the array IE(M,11) is activated to M.
Subroutine FGDET
This subroutine generates regular fine grids prescribed by users within each rough element based on
the information of IE(M,11) resulted from subroutine SFDET. It calls subroutine HPTRAK to obtain the
Lagrangian concentrations of each activated fine grid.
Subroutine ISEHIL
This subroutine removes all the forward-tracked nodes in smooth elements and stores the indices of
subelements into ISE array. In addition to regular fine grids refinement, subroutine ISEHIL also captures all
the highest and lowest concentrations within each subelement. The located subelements of the high-low points
57
-------
are determined by subroutine KGLOC. Once these high-low points are activated, subroutine TRIANG is
called to tetrangulate this subelement and the indices of each tetrahedral are also stored in the ISE array.
Subroutine TRIANG
This subroutine is called by ISEHIL for tetrangulating the subelement including the points with
peak/valley values. The indices of new created tetrahedrals are also stored in the ISE array.
Subroutine DFPREP
This subroutine prepares all the needed information for assembling the fine grid elemental matrices.
It calls subroutines GLBCHK to check those points on the elemental boundary connecting to the outermost
layer of rough region, which is smooth after the determination of subroutine SFDET bout rough in the Eulerian
step; FPLUS1 to imbed diffusion fine grid points prescribed by users and calculate the associated
concentrations; GRISED to prepare element indices for each subelement in the Eulerian step for composing
the matrix equation and to store the arrays for the intra-boundary points between rough and smooth regions
to overcome the incompatibility by implementing the "slave point" concept.
Subroutine BASEXI
This subroutine is called by subroutine DFPREP to calculate the coordinates of imbedded grids
according to the passed global coordinates, the associated local coordinate, and computed base functions in
the element.
Subroutine GLBCHK
This subroutine is called by subroutine DFPREP to check those generated fine grid points located on
the elemental boundary sides of each global element to see if the fine grids coincide with global nodes, locate
on the global boundaries, or intraboundaries between rough and smooth regions. The concentrations are
interpolated by calling subroutine INTERP for all generated fine grids.
Subroutine FPLUS1
58
-------
This subroutine is called by subroutine DFPREP to calculate concentrations of the generated fine grids
which are not located on the elemental boundary sides of the element.
Subroutine GRISED
This subroutine is called by subroutine DFPREP to generate ISED array which stores the indices of
each fine mesh for the Eulerian step. The information associated with the intraboundary points, which includes
the global nodes composing this intraboundary surface and nodal connection data locally, is also prepared in
this subroutine.
Subroutine REPLAS
This subroutine replaces the last six arguments with the first six arguments orderly.
Subroutine WRKARY
This subroutine prepares six working arrays for later usage.
Subroutine WARMSG
The arguments passed to this subroutine are N, MAXN, SUBNAM, VARNAM, and NO. The stop
statement is activated whenever N is greater than MAXN, and a message is written in the output file to indicate
which variable is overflow in subroutine SUBNAM.
Subroutine VALBDL
This subroutine calculates three interpolated values by the passed working arrays and basis functions.
Subroutine MOVCHK
This subroutine determines the concentrations and travel time of a fixed particle.
Subroutine ELTRK4
This subroutine counts the particle tracking in atetrahedral element. In the subroutine, the subelement
in which the starting point locates is dug out first. Starting from that subelement, the particle is tracked one
subelement by one subelement until either the tracking time is completely consumed or the particle encounters
a boundary side of the element being considered. During the particle tracking, this subroutine calls (1)
59
-------
subroutine TRAK1T to track a particle in the considered subelement if that particle is right standing on anode
of the subelement, and (2) subroutine TRAK2T to track a particle if that particle is not on any nodes of the
subelement. In the particle tracking process, the average velocity approach is used if IJUDGE=1; the single
velocity approach is used if IJUDGE=2.
Subroutine ELTRK6
This subroutine counts the particle tracking in a triangular prism element. In the subroutine, the
subelement in which the starting point locates is dug out first. Starting from that subelement, the particle is
tracked one subelement by one subelement until either the tracking time is completely consumed or the particle
encounters a boundary side of the element being considered. During the particle tracking, this subroutine calls
(1) subroutine TRAK1P to track a particle in the considered subelement if that particle is right standing on a
node of the subelement, and (2) subroutine TRAK2P to track a particle if that particle is not on any nodes of
the subelement. In the particle tracking process, the average velocity approach is used if IJUDGE=1; the single
velocity approach is used if IJUDGE=2.
Subroutine ELTRK8
This subroutine counts the particle tracking in a hexahedral element. In the subroutine, the
subelement in which the starting point locates is dug out first. Starting from that subelement, the particle is
tracked one subelement by one subelement until either the tracking time is completely consumed or the particle
encounters a boundary side of the element being considered. During the particle tracking, this subroutine calls
(1) subroutine TRAK1H to track a particle in the considered subelement if that particle is right standing on
a node of the subelement, and (2) subroutine TRAK2H to track a particle if that particle is not on any nodes
of the subelement. In the particle tracking process, the average velocity approach is used if IJUDGE=1; the
single velocity approach is used if IJUDGE=2.
60
-------
Subroutine FIXCHK
This is a control panel to check the ongoing process when a particle hits the boundary of the region
of interest. The backward tracked concentrations are obtained by interpolation if the boundary is specified
including Dirichlet, Cauchy, and variable types. Otherwise, the particle tracking continues along the
unspecified boundary till either the specified boundary is encountered or tracking time is consumed.
Function FCOS
This function computes the inner product of the normal vector of a given plane with a specified vector
whose starting point stands on the plane. The result helps to determine where the endpoint of the specified
vector is located.
Subroutine MMLOC
This subroutine is called by ELTRK4, ELTRK6, and ELTRK8 to locate the particle associated with
a specific subelement for subsequent elemental tracking. If this particle coincides with the nodes of a
subelement, ICODE=0 is returned. In addition, the information of the particle location with respect to each
surface of this element is also registered.
Subroutine BDYPLN
This subroutine locates the four global nodal numbers for returning to GNTRAK when the particle
hits a boundary of the working element. In addition, it calls subroutine ONPLAN to adjust the coordinate so
that these five points are really on the same plane.
Subroutine TRAK1T
This subroutine computes the particle tracking in a specified tetrahedral subelement when the starting
point coincides with a node of the subelement. This subroutine calls subroutine PLANEW to determine (1)
whether the particle would move into the subelement or not, and (2) which side (a triangular side) of the
subelement the particle would head onto if the particle does move into the subelement. After determining
which side the particle is going to move onto, this subroutine calls subroutine LOCQ3N to compute the exact
61
-------
location of the target point on the side. For accuracy, using the average velocity of both the starting point and
the target point to locate the target point is firstly considered in the subroutine. However, if this average
velocity approach is not able to deal with very complex velocity fields, the single velocity of the starting point
is used to determine the location of the target point.
Subroutine TRAK2T
This subroutine computes the particle tracking in a specified tetrahedral subelement when the starting
point does not coincide with a node of the subelement. This subroutine calls subroutine PLANEW to
determine (1) whether the particle would move into the subelement or not, and (2) which side (a triangular
side) of the subelement the particle would head onto if the particle does move into the subelement. After
determining which side the particle is going to move onto, this subroutine calls subroutine LOCQ3N to
compute the exact location of the target point on the side. For accuracy, using the average velocity of both the
starting point and the target point to locate the target point is first considered in the subroutine. However, if
this average velocity approach is not able to deal with very complex velocity fields, the single velocity of the
starting point is used to determine the location of the target point.
Subroutine CKCNEL
This subroutine checks the elements connecting to a specific side plane.
Subroutine CKCOIN
This subroutine checks if a specific point coincides with a global node.
Subroutine ONPLAN
This subroutine adjusts the particle coordinates to be on the same plane with the element side.
Subroutine CKSIDE
This subroutine checks if a specific point is on a side line of a side plane.
Subroutine ONLINE
This subroutine adjusts the particle coordinates to be on the same line with the other two points.
62
-------
Subroutine PLANEW
This subroutine determine which one of the two sides, separated by a specified plane, the particle
would move onto. All the computations are made according to the average velocity approach and the single
velocity approach, as the index parameter IJUDGE is 1 and 2, respectively.
Subroutine LOCQ3N
This subroutine locates the target point of a particle tracking in a specified element, which is either
a tetrahedral or a triangular prism element. All the computations are made according to either the average
velocity approach or the single velocity approach as the index parameter IJUDGE is 1 and 2, respectively. The
Newton-Raphson method is used to solve a set of two simultaneous nonlinear algebraic equations such that
the natural coordinates of the target point on the pre-determined element side (a triangular side) can be
determined. With these natural coordinates, the location of the target point can be easily determined based on
both the velocity of the source point and the geometrical relationship between the source point and the pre-
determined element side. This subroutine also calls subroutine NEWXE to compute the new guess of this pair
of natural coordinates.
Subroutine NEWXE
This subroutine is called by subroutines LOCQ3N and LOCQ4N for taking a new guess of local
coordinates within the iteration loop built with Newton-Ralphson scheme.
Subroutine BASE2D
This subroutine is called by LOCPLN to compute the base function values associated with a specified
point based on the given two-dimensional global coordinates. For the cases of quadrilateral elements, it calls
XSI2D to calculate the local coordinates, and computes base functions with these determined local coordinates.
For the cases of triangular elements, the base functions can be analytically determined based on the given
global coordinates.
63
-------
Subroutine XSI2D
This subroutine is called by BASE2D to compute the local coordinate of a quadrilateral element given
the global coordinate within that element.
Subroutine TRAK1P
This subroutine computes the particle tracking in a specified triangular prism subelement when the
starting point coincides with a node of the subelement. This subroutine calls subroutine PLANEW to
determine (1) whether the particle would move into the subelement or not, and (2) which side (either a
quadrilateral or a triangular side) of the subelement the particle would head onto if the particle does move into
the subelement. After determining which side the particle is going to move onto, this subroutine calls
subroutine LOCQ4N (if the side is a quadrilateral one) or subroutine LOCQ3N (if the side is a triangular one)
to compute the exact location of the target point on the side. For accuracy, using the average velocity of both
the starting point and the target point to locate the target point is first considered in the subroutine. However,
if this average velocity approach is not able to deal with very complex velocity fields, the single velocity of
the starting point is used to determine the location of the target point.
Subroutine TRAK2P
This subroutine computes the particle tracking in a specified triangular prism subelement when the
starting point does not coincide with a node of the subelement. This subroutine calls subroutine PLANEW
to determine (1) whether the particle would move into the subelement or not, and (2) which side (either a
quadrilateral or a triangular side) of the subelement the particle would head onto if the particle does move into
the subelement. After determining which side the particle is going to move onto, this subroutine calls
subroutine LOCQ4N (if the side is a quadrilateral one) or subroutine LOCQ3N (if the side is a triangular one)
to compute the exact location of the target point on the side. For accuracy, using the average velocity of both
the starting point and the target point to locate the target point is first considered in the subroutine. However,
64
-------
if this average velocity approach is not able to deal with very complex velocity fields, the single velocity of
the starting point is used to determine the location of the target point.
Subroutine LOCQ4N
This subroutine locates the target point of a particle tracking in a specified element, which is either
a hexahedral or a triangular prism element. All the computations are made according to the average velocity
approach and the single velocity approach, as the index parameter IJUDGE is 1 and 2, respectively. When
the average velocity approach is considered, the Newton-Ralphson method is used to solve a set of two
simultaneous nonlinear algebraic equations such that the local coordinates of the target point on the pre-
determined element side (a quadrilateral side) can be determined. With these local coordinates, the location
of the target point can be easily determined based on both the velocity of the source point and the geometrical
relationship between the source point and the pre-determined element side. This subroutine also calls
subroutine NEWXE to compute the new guess of this pair of natural coordinates.
Subroutine BASE1
This subroutine is called by ELTRK8 to compute the base functions for hexahedral elements.
Subroutine TRAK1H
This subroutine computes the particle tracking in a specified hexahedral subelement when the starting
point coincides with a node of the subelement. This subroutine calls subroutine PLANEW to determine (1)
whether the particle would move into the subelement or not, and (2) which side (a quadrilateral side) of the
subelement the particle would head onto if the particle does move into the subelement. After determining
which side the particle is going to move onto, this subroutine calls subroutine LOCQ4N to compute the exact
location of the target point on the side. For accuracy, using the average velocity of both the starting point and
the target point to locate the target point is first considered in the subroutine. However, if this average velocity
approach is not able to deal with very complex velocity fields, the single velocity of the starting point is used
to determine the location of the target point.
65
-------
Subroutine TRAK2H
This subroutine computes the particle tracking in a specified hexahedral subelement when the starting
point does not coincide with a node of the subelement. This subroutine calls subroutine PLANEW to
determine (1) whether the particle would move into the subelement or not, and (2) which side (a quadrilateral
side) of the subelement the particle would head onto if the particle does move into the subelement. After
determining which side the particle is going to move onto, this subroutine calls subroutine LOCQ4N to
compute the exact location of the target point on the side. For accuracy, using the average velocity of both the
starting point and the target point to locate the target point is first considered in the subroutine. However, if
this average velocity approach is not able to deal with very complex velocity fields, the single velocity of the
starting point is used to determine the location of the target point.
Subroutine ALGBDY
This subroutine is called by FIXCHK to control the process of backward particle tracking along the
unspecified boundaries. In the subroutine, the particle tracking is executed one boundary side by one boundary
side based on the nodal velocity component along the side being considered. The tracking will not be stopped
until either the tracking time is completely consumed or the particle encounters a specified boundary side. This
subroutine calls BNDRY to track a particle along a predetermined boundary side. For accuracy, using the
average velocity of both the source point and the target point to locate the target point is first considered in the
subroutine. However, if this average velocity approach is not able to deal with very complex velocity fields,
the single velocity of the source point is used to determine the location of the target point.
Subroutine BNDRY
This subroutine is called by ALGBDY to locate the target point of a particle tracking along a specified
boundary side. All the computations are made according to the average velocity approach and the single
velocity approach, when the index parameter IJUDGE is 1 and 2, respectively. For both approaches, the
location of the target point can be determined by calling subroutine LOCQ2N. However, when the velocity
66
-------
field is very complex, there might be no solution with the average approach. Thus, IJUDGE is originally set
to 1 and is changed to 2 if the average approach fails. This control is executed in ALGBDY.
Subroutine LOCQ2N
This subroutine locates the target point of a particle tracking on a line segment in a specified element.
All the computations are made according to the average velocity approach and the single velocity approach,
as the index parameter IJUDGE is 1 and 2, respectively. When the average velocity approach is considered,
the Newton-Raphson method is used to solve nonlinear algebraic equations such that the local coordinates
of the target point on the pre-determined element segment can be determined. With these local coordinates,
the location of the target point can be easily determined based on both the velocity of the source point and the
geometrical relationship between the source point and the pre-determined element side.
Subroutine INTERP
This subroutine computes the contaminant concentrations by interpolation with the basis functions
calculated by subroutine BASE. Prior to preforming the interpolation, this subroutine may call subroutine
KGLOC to locate the subelement on which the point falls if the global element is a rough element. This
subroutine can also get the interpolated concentrations for a multi-component system.
Subroutine KGLOC
This subroutine is called by subroutine INTERP to obtain the subelement on which the point falls if
the global element is a rough element. This subroutine also calls subroutine ONPLAN to guarantee
the point is exactly on the plane if it has been checked to be on the plane within a very small distance.
Subroutine BASE
This subroutine calculates basis functions and the derivatives of basis functions for a specific point.
The element shape can be either hexahedral, triangular prism, or tetrahedral. It also calls subroutine XSI3D
and XSI3DP for transferring the global coordinate to local coordinate in a hexahedral element and triangular
prism element, respectively.
67
-------
Subroutine XSI3DP
This subroutine computes the local coordinate (in the vertical direction) and the natural coordinate (in
the horizontal direction) of a triangular prism element given the global coordinates for both the specified and
element nodes.
Subroutine XSI3D
This subroutine computes the local coordinates of a hexahedral element given the global coordinates
for both the specified point and element nodes.
68
-------
3. ADAPTATION OF 3DFATMIC TO SITE SPECIFIC APPLICATIONS
The following describes how one should apply the 3DFATMIC code for site-specific applications and
how the data file should be prepared.
3.1 Parameters Specifications
For each site-specific problem, the users only need to specify the size of the problem by assigning 74
maximum control-integers with PARAMETER statement in the MAIN program. The list and definitions of
the maximum control-integers required for both flow and transport simulations are given below:
Maximum Control-Integers for the Spatial Domain
MAXNPK = maximum no. of nodes,
MAXELK = maximum no. of elements,
MXBESK = maximum no. of boundary-element surfaces,
MXBNPK = maximum no. of boundary nodal points,
MXJBDK = maximum no. of nodes connected to any node,
MXKBDK = maximum no. of elements connected to any node,
MXTUBK = maximum no. of accumulated unspecified boundary sides which connected to each
global node (used for transport part with the Lagrangian approach),
MXADNK = maximum no. of points used to solve matrix equation for transport part;
Maximum Control-Integers for the Time Domain
MXNTIK = maximum no. of time steps,
MXDTCK = maximum no. of times to reset the time step size;
Maximum Control-Integers for Subregions
LTMXNK = maximum no. of total nodal points in any subregion, including interior nodes, global
boundary nodes, and intraboundary nodes. LTMXNK = 1 if the block iteration is not used.
LMXNPK = maximum no. of nodal points in any subregion, including interior nodes and global
boundary nodes. LMXNPK = 1 if the block iteration is not used.
LMXBWK = maximum no. of the bandwidth in any subregion. LMXBWK = 1 if the block iteration
is not used.
MXRGNK = maximum no. of subregions. MXRGNK = 1 if the block iteration is not used.
69
-------
Maximum Control-Integers for Material and Soil Properties
MXMATK = maximum no. of material types,
MXSPMK = maximum no. of soil parameters per material to describe soil characteristic curves,
MXMPMK = maximum no. of material properties per material;
The maximum control-integers for flow simulations and their definitions are given as the following:
Maximum Control-Integers for Source/sinks, flow
MXSELh = maximum no. of source elements,
MXSPRh = maximum no. of source profiles,
MXSDPh = maximum no. of data points on each element source/sink profile,
MXWNPh = maximum no. of well nodal points,
MXWPRh = maximum no. of well source/sink profiles,
MXWDPh = maximum no. of data points on each well source/sink profile;
Maximum Control-Integers for Cauchy Boundary Conditions, flow
MXCNPh = maximum no. of Cauchy nodal points,
MXCESh = maximum no. of Cauchy element surfaces,
MXCPRh = maximum no. of Cauchy-flux profiles,
MXCDPh = maximum no. of data points on each Cauchy-flux profile;
Maximum Control-Integers for Neumann Boundary Conditions, flow
MXNNPh = maximum no. of Neumann nodal points,
MXNESh = maximum no. of Neumann element surfaces,
MXNPRh = maximum no. of Neumann-flux profiles,
MXNDPh = maximum no. of data points on each Neumann-flux profile;
Maximum Control-Integers for Rainfall-Seepage Boundary Conditions, flow
MXVNPh = maximum no. of variable nodal points,
MXVESh = maximum no. of variable element surfaces,
MXVPRh = maximum no. of rainfall profiles,
MXVDPh = maximum no. of data point on each rainfall profile;
Maximum Control-Integers for Dirichlet Boundary Conditions, flow
MXDNPh = maximum no. of Dirichlet nodal points,
MXDPRh = maximum no. of Dirichlet total head profiles,
70
-------
MXDDPh = maximum no. of data points on each Dirichlet profile;
The maximum control-integers for transport simulations and their definitions are given as the
following:
Maximum Control-Integers for Source/sinks, transport
MXSELc = maximum no. of source elements,
MXSPRc = maximum no. of source profiles,
MXSDPc = maximum no. of data points on each element source/sink profile,
MXWNPc = maximum no. of well nodal points,
MXWPRc = maximum no. of well source/sink profiles,
MXWDPc = maximum no. of data points on each well source/sink profile;
Maximum Control-Integers for Cauchy Boundary Conditions, transport
MXCNPc = maximum no. of Cauchy nodal points,
MXCESc = maximum no. of Cauchy element surfaces,
MXCPRc = maximum no. of Cauchy-flux profiles,
MXCDPc = maximum no. of data points on each Cauchy-flux profile;
Maximum Control-Integers for Neumann Boundary Conditions, transport
MXNNPc = maximum no. of Neumann nodal points,
MXNESc = maximum no. of Neumann element surfaces,
MXNPRc = maximum no. of Neumann-flux profiles,
MXNDPc = maximum no. of data points on each Neumann-flux profile;
Maximum Control-Integers for Flowin-Flowout Boundary Conditions, transport
MXVNPc = maximum no. of variable nodal points,
MXVESc = maximum no. of variable element surfaces,
MXVPRc = maximum no. of rainfall profiles,
MXVDPc = maximum no. of data point on each rainfall profile;
Maximum Control-Integers for Dirichlet Boundary Conditions, transport
MXDNPc = maximum no. of Dirichlet nodal points,
MXDPRc = maximum no. of Dirichlet total head profiles,
MXDDPc = maximum no. of data points on each Dirichlet profile;
71
-------
Control-Integers for Number of Components in the system
MXNCCK = maximum no. of components in this system,
Maximum Control-Integers for Refined System
MXKGLDK = maximum no. of subelements in the Eulerian step;
MXLSVK = maximum no. of subelement sides located on the intra-boundaries between extended
rough and smooth regions;
MXMSVK = maximum no. of global element sides located on the intra-boundaries between extended
rough and smooth regions;
MXNDBK = maximum no. of diffusion fine nodal-points located on the global boundary;
MXNEPK = maximum no. of all forward tracked nodal points in the region of interest when the exact
peak capture and oscillation free (EPCOF) numerical scheme is used. When EPCOF is not
used, set MXNEPK =1;
MXEPWK = maximum no. of forward tracked nodal points in any rough element when the exact
peak capture and oscillation free (EPCOF) numerical scheme is used. When EPCOF is not
used, set MXEPWK =1;
MXNPWK = maximum no. of fine nodal-points in any global element for particle tracking;
MXELWK = maximum no. of subelements in any global element for particle tracking;
MXNPWS = maximum no. of fine nodal-points in any global element which surrounds point
sources/sinks for obtaining more accurate Lagrangian concentrations with injection/extraction
wells in the region of interest;
MXELWS = maximum no. of subelements in any global element which surrounds point sources/sinks
for obtaining more accurate Lagrangian concentrations with injection/extraction wells in the
region of interest.
MXNPFGK = maximum no. of forward tracked nodal points over the region of interest or maximum
no. of fine nodal points plus peak/valley nodal points;
MXKGLK = maximum no. of subelements in the Lagrangian step;
For flow simulations only, to demonstrate how to specify the above maximum control-integers with
PARAMETER statement in the MAIN, an example is given in the following.
Assume that a region of interest is discretized by 30 x 20 x 10 nodes and 29 x 19 x 9 hexahedral
elements. In other words, the region is discretized with 30 nodes along the longitudinal or x-direction, 20
nodes along the lateral or y-direction, and 10 nodes along the vertical or z-direction. Since we have a total of
30 x 20 x 10 = 6,000 nodes, the maximum number of nodes is MAXNPK = 6000. The total number of
elements is 29 x 19 x 9 = 4,959, i.e, MAXELK = 4959. For this simple discretization problem, the maximum
connecting number of nodes to any of the 6,000 nodes in the region of interest is 27, i.e., MXJBDK = 27. and
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the maximum connecting number of elements to any of the 6,000 nodes is 8, i.e. MXKBDK = 8. There will
be29x!9 = 551 element surfaces each on the bottom and top faces of the region, 29 x 9 = 261 element-
surfaces each on the front and back faces of the region, and 19x9=171 element-surfaces each on the left and
right faces of the region. Thus, there will be a total of 1966 element-surfaces, i.e., MXBESK = 1966.
Similarly, we can compute the surface-boundary nodes to be 1968, i.e., MXBNPK = 1968. Because no
transport simulation is involved in this problem, MXADNK = MAXNPK = 6000.
In order to specify maximum control-integers related to subregion data, one has to know how the
region of interest is subdivided into subregions. Assume that the region of interest is subdivided into 20
subregions, each subregion has 30 x 10 nodes. It is seen, in fact, a vertical slice is taken as a subregion. For
this subregionalization, one has MXRGNK = 20. Each subregion has 30 x 10 = 300 nodes, resulting
LMXNPK = 300. It is also seen that there will be 600 intraboundary nodes, 300 nodes each on the two
neighboring slices of a subregion. Thus, one has LTMXNK = 900. For each subregion, the maximum
bandwidth can be computed as LMXBWK = 23 if the nodes are labelled along the z-directions consecutively.
Assume that there will be a maximum of 11 elements that have the distributed sources/sinks (i.e.,
MXSELh =11) and a maximum of 10 nodal points that can be considered as well sources/sinks (i.e.,
MXWNPh =10). Also assume that there will be three different distributed source/sink profiles and five
distinct point source/sink profiles. Then one will have MXSPRh = 3 and MXWPRh = 5. Further assume that
four data points are needed to describe the distributed source/sink profiles as a function of time and that 8 data
points are required to describe point source/sink profiles (i.e., MXSDPh = 4 and MXWDPh = 8).
To specify maximum control-integers for boundary conditions, it is assumed that the top face is a
variable boundary (i.e., on the air-soil interface, either ponding, infiltration, or evapotranspiration may take
place). On the left face, fluxes from the adjacent aquifer are known. On the right face, the total head is
assumed known. On the bottom face, natural drainage is assumed to occur (i.e., the gradient of the pressure
head can be assumed zero).
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There are 20 x 10 = 200 nodes on the left face and 19x9= 171 element surfaces; thus MXCNPh =
200 and MXCESh= 171. It is further assumed that there are two different fluxes going into the region through
the left face and that each flux can be described by four data points as a function of time (i.e., MXCPRh = 2,
and MXCDPh = 4). On the bottom surface, there are 30 x 20 = 600 nodes and 29 x 19 = 551 surface
elements. Since the gradient of pressure head on the bottom surface is zero, there is only one Neumann flux
profile, and two data points, one at zero time and the other at infinite time, are sufficient to describe the
constant value of zero. Hence, one has MXNNPh = 600. MXNESh = 551. MXNPRh= 1. and MXNDPh =
2. On the top face, there will be 30 x 20 = 600 nodes and 29 x 19 = 551 surface elements. Assume that there
are three different rainfall intensities that might fall on the air-soil interface, and that each rainfall intensity is
a function of time and can be described by 24 data points. With these descriptions, one has MXVNPh = 600.
MXVESh = 551. MXVPRh = 3. and MXVDPh = 24. On the right face, there are 20 x 10 = 200 nodes.
Assume that there are twenty different values of the total head, one each on a vertical line of the right face.
It is further assumed that each of these twenty total head can be described by 8 data points as function of time.
One then has MXDNPh = 200. MXDPRh = 20. and MXDDPh = 8.
In this example, one has six material properties (six saturated hydraulic conductivity components) per
material. Assume that the whole region of interest is made of three different kinds of materials. The
characteristic curves of each material are assumed to be described by four parameters. One then has
MXMATK = 3. MXMPMK = 6. and MXSPMK = 4. Assume that a 500-time-step simulation will be made
and reinitiation of the change on the time-step size will be made for 20 times during the simulation, then one
has MXNTIK = 500 and MXDTCK = 20. The other PARAMETER settings for transport part can be set to
be 1.
From the above discussion, the following PARAMETER statements can be used to specify the
maximum control-integers in the MAIN for the problem at hand:
PARAMETER(MAXNPK=6000,MAXELK=4959,MXBNPK=1968,MXBESK=1966,
> MXTUBK=1,MXADNK=MAXNPK+0)
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PARAMETER(MXJBDK=27,MXKBDK=8,MXNTIK=500,MXDTCK=20)
PARAMETER(LTMXNK=900,LMXNPK=3 00,LMXBWK=23 ,MXRGNK=20)
PARAMETER(MXMATK=4,MXSPMK=6,MXMPMK=6)
PARAMETER(MXSELh= 11 ,MXSPRh=3,MXSDPh=4,MXWNPh= 10,MXWPRh=5 ,MXWDPh=8)
PARAMETER(MXCNPh=200,MXCESh= 171 ,MXCPRh=2,MXCDPh=4)
PARAMETER(MXNNPh=600,MXNESh=5 51 ,MXNPRh= 1 ,MXNDPh=2)
PARAMETER(MXVNPh=600,MXVESh=5 51 ,MXVPRh=3 ,MXVDPh=24)
PARAMETER(MXDNPh=200,MXDPRh=20,MXDDPh=8)
PARAMETER(MXSELc= 1 ,MXSPRc=1 ,MXSDPc=1 ,MXWNPc= 1 ,MXWPRc= 1 ,MXWDPc= 1)
PARAMETER(MXCNPc= 1 ,MXCESc= 1 ,MXCPRc= 1 ,MXCDPc= 1)
PARAMETER(MXNNPc= 1 ,MXNESc= 1 ,MXNPRc= 1 ,MXNDPc= 1)
PARAMETER(MXVNPc= 1 ,MXVESc= 1 ,MXVPRc= 1 ,MXVDPc= 1)
PARAMETER(MXDNPc= 1 ,MXDPRc= 1 ,MXDDPc= 1)
PARAMETER(MXNCCK= 1)
PARAMETER(MXLS VK= 1 ,MXMS VK= 1 ,MXKGLDK= 1 ,MXNDBK= 1)
PARAMETER(MXNEPK= 1 ,MXEPWK= 1)
PARAMETER(MXNPWK= 1 ,MXELWK= 1 ,MXNPWS= 1 ,MXELWS=1)
PARAMETER(MXNPFGK= 1 ,MXKGLK= 1)
In the following, for transport simulations only, it is demonstrated how to specify the maximum
control-integers with PARAMETER statements in the MAIN with an example.
Assume that a region of interest is discretized by 30 x 20 x 10 nodes and 29 x 19 x 9 hexahedral
elements. In other words, the region is discretized with 30 nodes along the longitudinal or x-direction, 20
nodes along the lateral or y-direction, and 10 nodes along the vertical or z-direction. In order to make sure that
every element surface is on the same plane, the region of interest is re-discretized to triangular prism elements.
Therefore, four triangular prisms are generated in each hexahedral element. As a matter of fact, 5,510 more
nodes are installed and there are 19,836 elements in total. Since one has a total of 6,000+5,510 = 11,510
nodes, the maximum number of nodes is MAXNPK= 11.510. The total number of elements is 29 x 19 x 9
x 4 = 19,836, i.e, MAXELK = 19.836. For this simple discretization problem, the maximum connecting
number of nodes to any of the 11,510 nodes in the region of interest is 15, i.e., MXJBDK= 15. The maximum
number of elements connecting to any node is 8, thus MXKBDK = 8. There will be 29 x 19 x 4 = 2,204
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element surfaces each on the bottom and top faces of the region, 29 x 9 = 261 element-surfaces each on the
front and back faces of the region, and 19x9 = 171 element-surfaces each on the left and right faces of the
region. Thus, there will be a total of 5,272 element-surfaces; i.e., MXBESK = 5.272. Similarly, one can
compute the surface-boundary nodes to be 1968, i.e., MXBNPK = 3,302. Because this simulation selects the
Lagrangian approach, MXTUBK value needs to be specified. For sake of safety one can assume that the
maximum number of accumulated unspecified boundary element sides is equal to 4 times of the maximum
number of boundary nodes; i.e. MXTUBK = 4 x MXBNPK = 13.208. According to the description of
boundary conditions below, the front and back surfaces are not specified. Therefore, the total number of nodes
with unspecified boundary conditions is 30x 10x2 = 600. The maximum number of elements connected to
each point on these two surfaces is 4. Actually, MXTUBK = 2.400 which saves a lot of storage in comparison
to setting MXTUBK = 13,208. Assume that the number of imbedded diffusion fine grids in each rough
element is NXD = 2, NYD =3, and NZD = 2. Then there are 2x2^2 = 8 fine grids imbedded in a triangular
prism element. It is further assumed that 25 rough elements at the most existing through the whole simulation;
i.e., MXADNK=11.510 + 8 x25 = 11.710.
In order to specify maximum control-integers related to subregion data, one has to know how the
region of interest is subdivided into subregions. Assume one has subdivided the region of interest into 39
subregions. Twenty of them have 30 x 10 nodes, the other 19 subregions have 29 x 10 nodes each. It is seen,
in fact, one has taken a vertical slice as a subregion. For this subregionalization, we have MXRGNK = 39.
Each subregion has 300 or 290 nodes, resulting LMXNPK = 300. It is also seen that there will be 600
intraboundary nodes, 300 nodes each on the two neighboring slices of a subregion. Thus, one has LTMXNK
= 890. For each subregion, the maximum bandwidth can be computed as LMXBWK = 23 if the nodes are
labelled along the z-direction consecutively.
Assume that there will be a maximum of 11 elements that have the distributed sources/sinks (i.e.,
MXSELc =11) and a maximum of 10 nodal points that can be considered as well sources/sinks (i.e.,
76
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MXWNPc =10). Also assume that there will be three different distributed source/sink profiles and five
distinct point source/sink profiles. Then one will have MXSPRc = 3 and MXWPRc = 5. It is further assumed
that four data points are needed to describe the distributed source/sink profiles as a function of time and that
8 data points are required to describe point source/sink profiles (i.e., MXSDPc = 4 and MXWDPc = 8).
To specify maximum control-integers for boundary conditions, assume that the top and right faces are
variable boundaries. On the left face, fluxes from the adjacent aquifer are known. On the bottom face, the
natural gradient is zero. The other faces are unspecified.
There are 20 x 10 = 200 nodes on the left face and 19 x 9 = 171 element surfaces; thus MXCNPc =
200 and MXCESc= 171. It is further assumed that there are two different fluxes going into the region through
the left face and that each flux can be described by four data points as a function of time (i.e., MXCPRc = 2.
and MXCDPc = 4). On the bottom surface, there are 30 x 20 + 29 x 19 = 1,151 nodes and 29 x 19 x 4 =
2,204 surface elements. Since the gradient of concentration on the bottom surface is zero, there is only one
Neumann flux profile, and two data points, one at zero time and the other at infinite time, are sufficient to
describe the constant value of zero. Hence, one has MXNNPc = 1.151. MXNESc = 2.204. MXNPRc = 1.
and MXNDPc = 2. On the top face, there will be 30 x 20 + 29 x 19 = 1,151 nodes and 29 x 19 x 4 = 2,204
surface elements. The discretization on the right surface is the same as that on the left. Assume that there are
three different mass intensities that might fall on the top and right faces, and that each concentration profile
is a function of time and can be described by 24 data points. With these descriptions, one has MXVNPc =
1.351. MXVESc = 2.375. MXVPRc = 3. and MXVDPc = 24.
In this example, one has eight material properties per material. Assume that the whole region of
interest is made of three different kinds of materials. One then has MXMATK = 3. and MXMPMK = 6. If
one assumes that he will make a 500-time-step simulation and he will reinitiate the change on the time step
size for 20 times during our simulation, then he has MXNTIK = 500 and MXDTCK = 20.
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There are seven components, say microbe #1, microbe #2, microbe #3, substrate, oxygen, nitrate, and
nutrient, involved in this system; i.e.. MXNCCK = 7.
Assume that there are 25 rough elements and all of them are disconnected. It is further assumed that
each element will be refined by 2x2X2 = 8 subelements in the Eulerian step to solve the diffusion problem.
The maximum number of subelements for assembling in the diffusion step is MXKGLDK = 25 x 8 = 200.
Since each rough triangular prism has 5 sides, the number of global element sides located on the intra-
boundaries between rough and smooth regions is MXMSVK = 25 x 5 = 125. Each global element side is
refined by 4 subelement sides, hence MXLSVK = 125 x 4 = 500. Assume that 5 out of the 25 rough elements
have at least one side as the global boundary. It is further assumed that Rough Element 1 has two sides
coinciding with the global boundary: one side has three global nodes and the other side has four global nodes.
The 3-node side has 3 fine nodes and the 4-node side has 5 fine nodes. Thus, Rough Element 1 has 8 fine
nodes on the global boundary. Also assume that Rough Elements 2 through 5 each has its 3-node side
coinciding with the global boundary. For these 4 rough elements, one has 4x3 = 12 fine nodes on the global
boundary. Hence, the number of diffusion fine nodal points on the global boundary is MXNDBK = 8 + 12
= 20.
The numerical schemes for solving transport equations are LEZOOMPC plus keeping EPCOF points
in the Lagrangian step. For practical problems, EPCOF points will not be kept; thus, MXNEPK = 1,
MXEPWK = 1. In the Lagrangian step, each element is assumed to be refined by 8 subelements (NXA = 2,
NYA = 2, and NZA = 2) for accurate tracking. With this assumption, one has MXNPWK = (2+1) x (2+2) x
(2+D/2 = 18. MXELWK = (NXA x NYA x NZA) = 4. For each element connected to the sources/sinks,
assume to it is refined with 3x3x2 elements for accurate computation of Lagrangian concentrations to yield
MXNPWS = (3+1) x ( 3+2) x (2+1V2 = 30 and MXELWS = 3x3x2=18.
The specification of MXNPFGK and MXKGLK is much more involved. These two control integers
depend on many things: (1) how all the nodal points (including global nodes and fine nodal points) at the
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beginning of a time-step simulation are forwardly tracked, (2) how many elements are rough at the end of the
time-step computation, (3) how each rough element is refined, and (4) how many peak/valley points are kept.
A detailed discussion on how to specify these two integers is given in Appendix C. For the time being, assume
that MXNPFGK = 20000. MXKGLK = 10000.
From the above discussion, the following PARAMETER statements can be used to specify the
maximum control-integers in the MAIN for the problem at hand:
PARAMETER(MAXNPK=11510,MAXELK=19836,MXBNPK=3302,MXBESK=5272,
> MXTUBK=2400,MXADNK=MAXNPK + 200)
PARAMETER(MXJBDK= 15 ,MXKBDK=8,MXNTIK=5 00,MXDTCK=20)
PARAMETER(LTMXNK=890,LMXNPK=3 00,LMXBWK=23 ,MXRGNK=3 9)
PARAMETER(MXMATK=3,MXSPMK=6,MXMPMK=6)
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh= 1 ,MXWPRh= 1 ,MXWDPh= 1)
PARAMETER(MXCNPh= 1 ,MXCESh= 1 ,MXCPRh= 1 ,MXCDPh= 1)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 1 ,MXVESh= 1 ,MXVPRh= 1 ,MXVDPh= 1)
PARAMETER(MXDNPh= 1 ,MXDPRh= 1 ,MXDDPh= 1)
PARAMETER(MXSELc= 11 ,MXSPRc=3 ,MXSDPc=4,MXWNPc=10,MXWPRc=5 ,MXWDPc=8)
PARAMETER(MXCNPc=200,MXCESc= 171 ,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=l 15 l,MXNESc=2204,MXNPRc=l,MXNDPc=2)
PARAMETER(MXVNPc=1351 ,MXVESc=23 75 ,MXVPRc=3 ,MXVDPc=24)
PARAMETER(MXDNPc=1 ,MXDPRc= 1 ,MXDDPc= 1)
PARAMETER(MXNCCK=7)
PARAMETER(MXLSVK=500,MXMSVK=125,MXKGLDK=200,MXNDBK=20)
PARAMETER(MXNEPK= 1 ,MXEPWK= 1)
PARAMETER(MXNPWK= 18,MXELWK=8,MXNPWS=30,MXELWS= 18)
PARAMETER(MXNPFGK=20000,MXKGLK= 10000)
3.2 Soil Property Function Specifications
Analytical functions are used to describe the relationships of water content, water capacity, and relative
hydraulic conductivity with pressure head. Therefore, the user must supply three functions to compute the
water content, water capacity, and relative hydraulic conductivity based on the current value of pressure head.
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The parameters needed to specify the functional form are read and stored in SPP. One example is shown in
the subroutine SPFUNC in the source code. In this example, the water content, water capacity, and relative
hydraulic conductivity are given by (van Genuchten 1980):
6 -6
6 = 6r + ^ (3.1)
[l+(ah)T
-^ = a(n-i)[i-f(e)]m[f(6)](es-er) (3.2)
Kr =[(e-er)/(es-er)]2{i-[i-f(6)]m}2
in which
f(6) = [e-er]/[6s-er]1/m
and
m = 1 - - (3.5)
To further demonstrate how one should modify the subroutine SPFUNC in Appendix A to
accommodate the material property functions that are different from those given by Eqs. (3.1) through (3.5),
assume that the following Fermi types of functions are used to represent the unsaturated hydraulic properties
(Yeh, 1987):
e = er + (6s-er)/{i+exP[-a(h-he)]} (3.6)
dO/dh = a(6s-er)exp[-a(h-he)]/{l+exp[-a(h-he)]}2 , (3.7)
and
log10(K) = e/{l+exp[-p(h-hk)]} - e , (3.8)
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where 6S, 6r, a, and he are the parameters for computing the water content and water capacity; and P, e, and
hk are the parameters for computing the relative hydraulic conductivity. The source code must be changed,
for this example, to the following form for computing the moisture content and water capacity
WCR=SPP(1,MTYP,1)
WCS=SPP(2,MTYP,1)
ALPHA=SPP(3,MTYP, 1)
HTHETA=SPP(4,MTYP, 1)
EPS=SPP(1,MTYP,2)
BETA=SPP(2,MTYP,2)
HSUBK=SPP(3 ,MTYP,2)
C
C SATURATED CONDITION
C
IF(HNP.LE.O.O) THEN
TH=WCS
IF(ISP .EQ. 1) GOTO 900
DTH=O.ODO
USKFCT=1.0DO
C
ELSE
C
C UNSATURATED CASE
C
EXPAH=DEXP(-ALPHA* (HNP-HTHETA))
TH=WCR+(WCS-WCR)/(1 .ODO+EXPAH)
DTH=ALPHA* (WCS-WCR) *EXPAH/( 1 .ODO+EXPAH) * * 2
AKRLOG=EPS/(1.0DO+DEXP(-BETA*(HNP-HSUBK))) - EPS
USKFCT=10.0DO**AKRLOG
ENDIF
3.3 Input and Output Devices
Five logical units are needed to execute 3DFATMIC. Units 15 and 16 are standard card input and line
printer devices, respectively. Unit 11 must be specified to store the flow simulation results, which can be used
for plotting purposes. Unit 12 must be specified to store the transport simulation results, which can be used
for plotting purposes. Unit 13 is used to store the boundary arrays for later uses, if these arrays are computed
for the present job. Unit 14 is used to store pointer arrays for later uses, if these arrays are generated for the
81
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present job. For large problems, experience has indicated that it would take too much time to process the
boundary arrays and to generate pointer arrays. Hence, it is advisable that for multi-job executions, these
boundary and pointer arrays should be computed only once and written on units 13 and 14, respectively. Once
they are stored on units 13 and 14, the IGEOM described in Appendix A should be properly identified for the
new job so they can be read via units 13 and 14, respectively. Finally, Unit 21 is used to print any variable
for debugging purpose.
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4. SAMPLE PROBLEMS
To verify 3DFATMIC, eight illustrative examples are used. Examples one, two, and three, originally
designed for 3DFEMWATER (Yeh, 1993a), are the flow only problems. Examples four and five, originally
designed for 3DLEWASTE (Yeh, 1993b), are the transport only problems. Example six is a two-dimensional
biodegradation problem which is used to verify the flow and transport coupling loop and show the effects of
biodegradation. Examples seven and eight illustrate the behavior of dissolved organic and oxygen plumes
undergoing natural biodegradation in a uniform flow field.
4.1 Example 1: One-Dimensional Column Flow Problem
This example is selected to represent the simulation of a one-dimensional flow problem with
3DFATMIC. The column is 200 cm long and 50 by 50 cm in crosssection (Figure 5). The column is
assumed to contain the soil with a saturated hydraulic conductivity of 10 cm/d, a porosity of 0.45 and a field
capacity of 0.1. The unsaturated characteristic hydraulic properties of the soil in the column are given as
6 = 6S - (6s-6r) (4.1)
and
e-e
,
(4.2)
where hb and ha are the parameters used to compute the water content and the relative hydraulic conductivity,
respectively.
The initial conditions assumed are a pressure head of -90 cm imposed on the top surface of the
column, 0 cm on the bottom surface of the column, and -97 cm elsewhere. The boundary conditions are
given as: no flux is imposed on the left, front, right, and back surfaces of the column; pressure head is held
at 0 cm on the bottom surface; and variable condition is used on the top surface of the column with a ponding
83
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depth of zero, minimum pressure of -90 cm, and a rainfall of 5 cm/d for the first ten days and a potential
evaporation of 5 cm/d for the second 10 days.
ฃ- 50cm -*S
50cm
200cm
Figure 4.1 Problem definition and sketch for Example 1.
The region of interest, that is, the whole column, will be discretized with 1 x 1 x 40 = 40 elements with
element size = 50 x 50 x 5 cm, resulting in 2 x 2 x 41 = 164 node points (Figure 6). For 3DFATMIC
simulation, each of the four vertical lines will be considered a subregion. Thus, a total of four subregions, each
with 41 node points, is used for the subregional block iteration simulation.
84
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41.
164
82
43
42
/' 123
oo
2. . . .
' ' 84
, ' CO 83
Figure 4.2 Finite element discretization for Example 1.
A variable time step size is used. The initial time step size is 0.05 days, and each subsequent time step
size is increased by 0.2 times with a maximum time step size not greater than 1.0 d. Because there is an abrupt
change in the flux value from 5 cm/d (infiltration) to -5 cm/d (evaporation) imposed on the top surface at day
10, the time step size is automatically reset to 0.05 d on the tenth day. A 20-day simulation will be made with
3DFATMIC. With the time step size described above, 44 time steps are needed.
The pressure head tolerance is 2 10"2 cm for nonlinear iteration and is 1 10"2 cm for block iteration.
The relaxation factors for both the nonlinear iteration and block iteration are set equal to 0.5.
To execute the problem, the maximum control-integers in the main program should be specified as
follows
85
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C For Example 1 through Example 5
C
PARAMETER(MAXNPK=2079,MAXELK= 1600,MXBNPK=999,MXBESK=999,
> MXTUBK=3008,MXADNK=maxnpk+0)
PARAMETER(MXJBDK=3 5 ,MXKBDK=8,MXNTIK= 100,MXDTCK=4)
PARAMETER(LTMXNK=693 ,LMXNPK=231 ,LMXBWK=49,MXRGNK=9)
PARAMETER(MXMATK=8,MXSPMK=5,MXMPMK=9)
C 2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 110,MXCESh=90,MXCPRh= 1 ,MXCDPh=2)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 198,MXVESh= 170,MXVPRh=2,MXVDPh=4)
PARAMETER(MXDNPh= 165 ,MXDPRh= 11 ,MXDDPh=2)
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=5)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=63 8,MXVESc=5 60,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=70,MXDPRc=6,MXDDPc=2)
C
PARAMETER(MXNCCK=2)
C
PARAMETER(MXLSVK=500,MXMSVK=500,MXKGLDK=2000,MXNDBK=2000)
PARAMETER(MXNEPK=20,MXEPWK=20)
PARAMETER(MXNPWK=99,MXELWK=27, mxnpws=133 l,mxelws=1000)
PARAMETER(MXNPFGK=2900,MXKGLK=2800)
C
To reflect the soil property function given by Eqs. (4.1) and (4.2), one has to modify the subroutine
SPFUNC. A segment of the code in the subroutine SPFUNC must be modified as follows:
WCR=SPP(1,MTYP,1)
WCS=SPP(2,MTYP,1)
HAA=SPP(3,MTYP,1)
HAB=SPP(4,MTYP,1)
C
C SATURATED CONDITION
C
IF(HNP.LE.O) THEN
TH=WCS
86
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IF(ISP .EQ. 1) GOTO 900
DTH=O.ODO
USKFCT=1.0DO
ELSE
C
C UNSATURATED CASE
C
TH=WCS-(WCS-WCR)*(-HNP-HAA)/(HAB-HAA)
IF(ISP.EQ.l) GOTO 900
USKFCT=(TH-WCR)/(WCS-WCR)
DTH=-(WCS-WCR)/(HAB-HAA)
ENDIF
C
Figure 5 depicts the pressure profiles along the z-axis at various times.
4.2 Input and output for Example 1
With the above descriptions, the input data can be prepared according to the instructions given
Appendix A. The input parameters are shown in Table 4.1 and the input data file content is given in Table
4.2. To save space, the output is available in electronic form.
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200 ,
160
120
80
40
-100
-20
200 i
160
120
80
40
-100
-20
Figure 4.3 Pressure head profiles at various times.
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Table 4.1 The list of input parameters for Example 1
Parameters
number of points
AX
Ay
AZ
K_
0r
0S
h,
\
no. of subregion
no. of points in a subregion
initial time step size
time step size increment
maximum time step size
no. of times to reset time step size
time to reset time step size
Total simulation time
no. of time steps
tolerance for nonlinear iteration
relaxation factor for nonlinear
iteration
Pw
Uw
g
Notation in the data
input guide
NNP
XAD
YAD
ZAD
PROPf(l,3)
SPP(1,U)
SPP(2,1,1)
SPP(3,1,1)
SPP(4,1,1)
NREGN
NODES
DELT
CHNG
DELMAX
NDTCHG
TDTCH(l)
TMAX
NTI
TOLBf
OMEf
RHO
vise
GRAY
Value
164
50
50
5
10
0.15
0.45
0
-100
4
41
0.05
0.2
1
1
10
22
44
2xlO-2
0.5
1.0
9483.26
7.32xl012
Unit
Dimensionless
cm
cm
cm
cm/day
dimensionless
dimensionless
cm
cm
dimensionless
dimensionless
day
dimensionless
day
dimensionless
day
day
dimensionless
cm
dimensionless
g/cm3
g/cm/day
cm/day2
Data set
7. A.
7. B.
7. B.
7. B.
5.B.
6. B.
6. B.
6. B.
6. B.
8. A.
8. B.
4. B.
4. B.
4. B.
4. A.
5.E.
4. B.
4. A.
3. A.
2. C.
5.B. &
6.A.
5.B. &
6.A.
6.A.
89
-------
Table 4.2 Input Data Set for Example 1
1 One-Dimensional Column Flow Problem; L=CM, T=DAY, M=G
========= data set 2: option parameters
10 0 1 0
100 O.SdO l.Od-4
1100000001
0
O.SdO O.OdO
O.SdO
20111
l.OdO O.SdO l.OdO l.OdO
========= data set 3: iteration parameters
50 20 100 2.0d-2 2.0d-2
1 100 l.Od-3 l.Od-4
========= data set 4: time control parameters
44 2
O.OSdO 0.20dO l.OdO 22.0dO
333030300030003003000033303030003000300300003
000000000000000000000000000000000000000000000
1.0D01 2.0000D1
========= DATA SET 5: MATERIAL PROPERTIES
1701
O.ODO O.ODO 10.ODD O.ODO O.ODO
O.OdO
l.OdO
O.ODO l.OdO
= = =
0
0.
0.
164
1
42
83
124
0
4
1
0
1
0
1
0
1
0
1
0
= = = =
4
150DO
OOODO
40
40
40
40
0
3
0
40
0
40
0
40
0
40
0
DATA
0
0.
0.
DATA
1
1
1
1
0
DATA
1
0
1
0
1
0
1
0
1
0
SET 6:
1 .
450DO
OOODO
SET 7 :
0.
0.
50
50
0.
SET 8 :
41
0
1
0
42
0
83
0
124
0
soil properties
OdO
0.
l.OdO l.OdO
.OODO
-1.
.002
O.OODO O.ODO
THPROP
AKPROP
NODE COORDINATES
ODO
ODO
.ODO
.ODO
50.
0.
0
50
ODO
ODO
.ODO
.ODO
0
0
.ODO
.ODO
O.ODO
O.ODO
0 0.0 0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO 5. ODO
O.ODO 5. ODO
O.ODO 5. ODO
O.ODO 5. ODO
0.0 0.0
SUBREGIONAL DATA
0
0
1
0
1
0
1
0
1
0
END
END
END
END
END
OF
OF
OF
OF
OF
NNPLR(K)
GNLR ( I
GNLR ( I
GNLR ( I
GNLR ( I
,1)
,2)
,3)
,4)
******* DATA SET 9:
40
1 39 1 42
0000
******* data setlO:
0
******* DATA SET 11
ELEMENT INCIDENCES
83 124 1 43
0000
material correction
84
0
125
0
END OF IE
1
2
43
84
125
41
0
3 41
38 1
38 1
38 1
38 1
3 41
0 0
: rial- a
INITIAL CONDITIONS
O.ODO O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
-9.70D1
-9.70D1
-9.70D1
-9.70D1
-9.00D1
0.0
0.0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
0.0
END OF 1C, flow
12: element(distributed) source/sink, flow
90
-------
0000
========= data set 13: point(well) source/sink, flow
0000
========= data set 16: rainfall/evaporation-seepage boundary conditions
14140
O.ODO 5.ODD 10.ODD 5.ODD 10.001DO -5.ODD 1.0D38 -5.ODD
10010
00000 END OF IRTYP
1 0 0 82 123 164 41 0 0 0 0
00000000000 END OF ISV(J,I) J=l,4
1 3 1 41 41
00000 END OF NPVB
131 O.ODO O.ODO 0.0
0 0 0 0.0 0.0 0.0 END OF HCON
131 -90.ODD O.ODO 0.0
0 0 0 0.0 0.0 0.0 END OF HMIN
C ******* DATA SET 17: DIRICHLET BOUNDARY CONDITIONS, flow
4120
O.ODO O.ODO 1.0D38 O.ODO
1 31 1 41
00000
13110
00000 END OF IDTYP
========= data set 18: cauchy boundary conditions, flow
00000
========= data set 19: neumann boundary conditions, flow
00000
0 END OF JOB
0000
4.3 Example 2: Two-dimensional Flow Drainage Problem
This example is selected to represent the simulation of a two-dimensional flow problem with
3DFATMIC. The region of interest is bounded on the left and right by parallel drains fully penetrating the
medium, on the bottom by an impervious aquifuge, and on the top by an air-soil interface (Figure 6). The
distance between the two drains is 20 m apart (Figure 6).
The medium is assumed to have a saturated hydraulic conductivity of 0.01 m/d, a porosity of 0.25, and
a field capacity of 0.05. The unsaturated characteristic hydraulic properties of the medium are given as
9 = e' + (e'-9')
91
-------
10m
~Pm
V
I = 0,006 m/day
V
X
10m
10m
Figure 4.4 Problem definition and sketch for Example 2.
and
e-e
(4.4)
where ha, A, and B are the parameters used to compute the water content and n is the parameter to compute
the relative hydraulic conductivity.
Because of the symmetry, the region for numerical simulation will be taken asO
-------
is set, h = 10-z.
The region of interest is discretized with 10x1x10=100 elements with element size = 1x10x1
cm, resulting in 11x2x11= 242 node points (Figure 7). For 3DFATMIC simulation, each of the two vertical
planes will be considered a subregion. Thus, the total of two subregions, each with 121 node points, is used
for the subregional block iteration simulation.
132 143 154 165 176 187 198 209 220 231
^^^^^^^^7^^^^^^^^^^^^^^^^7^^^^^^^7^^^^^^^^^^^^^^^^^^^^^^^^^7^^^^^^^7^^^^^^^^^^^~
11
10
9
8
7
6
5
41
3
2
1
23 34 45 56 67 78 89 100
Figure 4.5 Finite element discretization for Example 2.
The pressure head tolerance is 2 10"3 m for nonlinear iteration and is 10"3 m for block iteration. The
relaxation factors for both the nonlinear iteration and block iteration are set equal to 0.5.
To execute the problem, the maximum control-integers in the MAIN should be specified as follows
C For Example 1 through Example 5
c
PARAMETER(MAXNPK=2079,MAXELK= 1600,MXBNPK=999,MXBESK=999,
> MXTUBK=3008,MXADNK=maxnpk+0)
93
-------
PARAMETER(MXJBDK=3 5 ,MXKBDK=8,MXNTIK= 100,MXDTCK=4)
PARAMETER(LTMXNK=693 ,LMXNPK=231 ,LMXBWK=49,MXRGNK=9)
PARAMETER(MXMATK=8,MXSPMK=5,MXMPMK=9)
C 2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 110,MXCESh=90,MXCPRh= 1 ,MXCDPh=2)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 198,MXVESh= 170,MXVPRh=2,MXVDPh=4)
PARAMETER(MXDNPh= 165 ,MXDPRh= 11 ,MXDDPh=2)
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=5)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=63 8,MXVESc=5 60,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=70,MXDPRc=6,MXDDPc=2)
C
PARAMETER(MXNCCK=2)
C
PARAMETER(MXLSVK=500,MXMSVK=500,MXKGLDK=2000,MXNDBK=2000)
PARAMETER(MXNEPK=20,MXEPWK=20)
PARAMETER(MXNPWK=99,MXELWK=27, mxnpws=133 l,mxelws=1000)
PARAMETER(MXNPFGK=2900,MXKGLK=2800)
C
To reflect the soil property function given by Eqs. (4.3) and (4.4), one has to modify the subroutine
SPFUNC given the source code.
wcr=spp( 1 ,mtyp, 1)
wcs=spp(2,mtyp, 1)
haa=spp(3 ,mtyp, 1)
thaa=spp(4,mtyp, 1)
thbb=spp(5 ,mtyp, 1)
power=spp( 1 ,mtyp,2)
C
C SATURATED CONDITION
C
IF(HNP.LE.O) THEN
TH=WCS
IF(ISP .EQ. 1) GOTO 900
DTH=O.ODO
USKFCT=1.0DO
94
-------
C UNSATURATED CASE
C
ELSE
th=wcr+(wcs-wcr)*thaa/(thaa+(DABS(-hnp-haa))**thbb)
IF(ISP.EQ.l) GOTO 900
dnom=thaa+(DABS(-hnp-haa))* *thbb
dth=(wcs-wcr)*thaa* (DAB S(-hnp-thaa))* * (thbb-1. OdO)/dnom* * 2
USKFCT=((th-wcr)/(wcs-wcr))**power
ENDIF
C
Figure 8 and Figure 9 depict the pressure distribution and the velocity field, respectively, from the
3DFATMIC simulation.
Figure 4.6 Pressure head distribution for Example 2.
95
-------
10.0
9.0
8.0
7.0
6.0
N 5-0
4.0
3.0
2.0
1.0
0.0
o.
ป*44l*4444
1 ป 1 1 t ป 1 i 1
- 1 1 1 t * 1 1 ป *
-
->ปปป % \ \ \
- v
- ป x
ซ ป * * *ป X N
- . . . . .......ป.ซ
0 2.5 5.0 7.5 10.
X
Figure 4.7 The velocity field for Example 2.
4.4 Input and Output for Example 2
With the above descriptions, the input data can be prepared according to the instructions in Appendix
A. The input parameters are listed in Table 4.3 and the input data file content are given in Table 4.4. To save
space, the output is available in electronic form.
96
-------
Table 4.3 The list of input parameters for Example 2
Parameters
number of points
AX
Ay
AZ
KS.XX
KSZZ
0r
9S
ha
A
B
n
no. of subregion
no. of points in
each subregion
steady-state
simulation
no. of times to
reset time step size
no. of time steps
tolerance for
nonlinear iteration
relaxation factor
for nonlinear
iteration
Pw
Uw
g
Notation in the data
input guide
NNP
XAD
YAD
ZAD
PROPf(l,l)
PROPf(l,3)
SPP(1,1,1)
SPP(2,1,1)
SPP(3,1,1)
SPP(4,1,1)
SPP(5,1,1)
SPP(1,1,2)
NREGN
NODES
KSSf
NDTCHG
NTI
TOLAf
OMEf
RHO
vise
GRAY
Value
242
1
10
1
0.01
0.01
0.05
0.25
0
10
4
4
2
121
0
0
0
2xlO'3
0.5
l.OxlO3
948.3264
7.316xl010
Unit
Dimensionless
m
m
m
m/day
m/day
dimensionless
dimensionless
m
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
m
dimensionless
Kg/m3
Kg/m/day
m/day2
Data set
7. A.
7. B.
7. B.
7. B.
5.B.
5.B.
6. B.
6. B.
6. B.
6. B.
6. B.
6. B.
8. A.
8. B.
2. C.
4. A.
4. A.
3. A.
2. C.
5.B., 6.A.
5.B. ,6.A.
6.A.
97
-------
Table 4.4 Input Data Set for Example 2
2 Two-dimensional Drainage Flow Problem; L=M, T=DAY, M=KG
===== DATA SET 2: OPTION PARAMETERS
10 0 1 0
1 0.5DO l.OD-4 NITRFT OMEFTF OMEFTT
0 00 0 000001 KSSF KSST ILUMP IMID IPNTSF IPNTST miconf nstrf nstrt
1 1.0 O.SdO O.SdO O.OdO KGRAV WF OMEF OMIF
10111 KVIT IWET IOPTIM ksorp Igran
l.OdO O.SdO l.OdO l.OdO WT WVT OMET OMIT
===== DATA SET 3: ITERATION PARAMETERS
50 20 100 2.0d-3 2.0d-3 NITERF NCYLF NPITRF TOLAF TOLBF
1 100 l.Od-3 l.Od-4 NITERT NPITRT TOLAT TOLBT
===== DATA SET 4: TIME CONTROL PARAMETERS
0 0 NTI NDTCHG
O.OSdO 0.20dO O.OSdO 22.0dO DELT CHNG DELMAX TMAX
55 KPRO KPR(1..NTI)
00 KDSKO KDSK(1..NTI)
0.0
===== DATA SET 5: MATERIAL PROPERTIES
1701
0.01DO O.ODO 0.01DO O.ODO
PROPF
0.0
RHOMU
l.OdO
===== DATA SET 6: SOIL PROPERTIES
050 l.OdO 7.316D10 1.
0.050DO 0.250DO O.OODO
4.000DO O.OOODO O.OODO
===== DATA SET 7: NODE COORDINATES
242 NNP
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
O.ODO 10.ODD
0.0 0.0 0.0
NMAT NMPPM
O.ODO O.ODO
l.OdO
1232d4 KSP NSPPM KCP GRAY
10.ODD 4.ODD THPROP
O.ODO O.ODO AKPROP
1 10 11
2 10 11
3 10 11
4 10 11
5 10 11
6 10 11
7 10 11
8 10 11
9 10 11
10 10 11
11 10 11
122 10 11
123 10 11
124 10 11
125 10 11
126 10 11
127 10 11
128 10 11
129 10 11
130 10 11
131 10 11
132 10 11
0
0
0
O.ODO
l.ODO
2. ODD
3 .ODD
4 . ODD
5. ODD
6. ODD
7. ODD
8. ODD
9. ODD
10.ODD
O.ODO
l.ODO
2. ODD
3 .ODD
4. ODD
5. ODD
6. ODD
7. ODD
8. ODD
9. ODD
10.ODD
0.0
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
0.0 0.0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
END OF COORDINATES
===== DATA SET 8: SUBREGIONAL DATA
2
1
0
1
1
0
120
121
0
1
END OF NNPLR(K)
98
-------
0
1
0
100
1
11
21
31
41
51
61
71
81
91
0
0
0 0
120 1
0 0
DATA SET 9
NEL
0 0 END OF GNLR(I,1)
122 1
0 0 END OF GNLR(I,2)
: ELEMENT INCIDENCES
1
12
23
34
45
56
67
78
100
0
12
23
34
45
56
67
78
89
100
111
0
133
144
155
166
177
188
199
210
221
232
0
122
133
144
155
166
177
188
199
210
221
0
2
13
24
35
46
57
68
79
90
101
0
13
24
35
46
57
68
79
90
101
112
0
134
145
156
167
178
189
200
211
222
233
0
123
134
145
156
167
178
189
200
211
222
0
END OF IE
DATA SET 10:
NCM
===== DATA SET 11:
1 10 11
2 10 11
3 10 11
4 10 11
5 10 11
6 10 11
7 10 11
8 10 11
9 10 11
10 10 11
11 10 11
122 10 11
123 10 11
124 10 11
125 10 11
126 10 11
127 10 11
128 10 11
129 10 11
130 10 11
131 10 11
132 10 11
000
= = = = = DATA SET 12 :
0000
= = = = = DATA SET 13 :
0000
===== DATA SET 16:
18 38 2 2 0
O.ODO
MATERIAL CORRECTION
INITIAL CONDITIONS
10.ODD O.ODO
9. ODD
8. ODD
7 . ODD
6. ODD
5. ODD
4 . ODD
3. ODD
2. ODD
l.ODO
O.ODO
10.ODD
9. ODD
8. ODD
7 . ODD
6. ODD
5. ODD
4. ODD
3. ODD
2. ODD
l.ODO
O.ODO
O.ODO
O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
ELEMENT(DISTRIBUTED) SOURCE/SINK OF FLOW
NSELF NSPRF NSDPF KSAIF
POINT(WELL) SOURCE/SINK OF FLOW
NWNPF NWPRF NWDPF KWAIF
RAINFALL/EVAPORATION-SEEPAGE BOUNDARY CONDITIONS OF FLOW
NVESF NVNPF NRPRF NRDPF KRAIF
END OF 1C FOR FLOW
1
11
0
1
11
0
1
12
20
31
0
1
O.ODO
9
7
0
9
7
0 0
10
7
10
7
0
37
6.0D-3
O.ODOO
1
2
0
11
120
000
11
120
132
241
0
1.0D38
1.0D38
0
0
0
22 143
241 242
000
11
-1
11
-1
0
O.ODO
6
0
132
121
0 0
EN
6.0D-3
O.ODOO
11
-1
TQVBFF QVBFF
TQVBFF QVBFF
END OF IVTYPF
11
-1
11
-1
11
-1
END OF ISVF(J,I) J=l,4
END OF NPVBF
O.ODO O.ODO
99
-------
0 0 0 0.0 0.0 0.0 END OF HCON
1 37 1 -90.0D2 O.ODO 0.ODD
0 0 0 0.0 0.0 0.0 END OF HMIN
===== DATA SET 17: DIRICHLET BOUNDARY CONDITIONS OF FLOW
6120 NDNPF NDPRF NDDPF KDAIF
O.ODO 2.ODD 1.0D38 2.ODD THDBFF HDBFF
121 111 1
421 232 1
0000 0
15110
00000 END OF IDTYPF
===== DATA SET 18: CAUCHY BOUNDARY CONDITIONS OF FLOW
00000 NCESF NCNPF NCPRF NCDPF KCAIF
===== DATA SET 19: NEUMANN BOUNDARY CONDITIONS, FLOW
00000 NNESF NNNPF NNPRF NNDPF KNAIF
0 ====== END OF JOB ======
4.5 Example 3: Three-Dimensional Pumping Flow Problem
This example is selected to represent the simulation of a three-dimensional problem with 3DFATMIC.
The problem involves the steady state flow to a pumping well. The region of interest is bounded on the left
and right by hydraulically connected rivers; on the front, back, and bottom by impervious aquifuges; and on
the top by an air-soil interface (Figure 10.1). A pumping well is located at (x,y) = (540,400) (102). Initially,
the water table is assumed to be horizontal and is 60 m above the bottom of the aquifer. The water level at the
well is then lowered to a height of 30 m. This height is held until a steady state condition is reached. The
medium in the region is assumed to be anisotropic and have saturated hydraulic conductivity components K^
= 5 m/d, Kyy = 0.5 m/d, and Kzz = 2 m/d. The porosity of the medium is 0.25 and the field capacity is 0.0125.
The unsaturated characteristic hydraulic properties of the medium are given as
6-6.
6=6
r l+(ocha-h|)P
and
re-e
(4.5)
r
(4.6)
where ha, a, and P are the parameters used to compute the water content and the relative hydraulic
conductivity.
100
-------
Because of the symmetry, the region for numerical simulation will be taken as 0 < x < 1000 m, 0 <
y < 400 m, and 0 < z < 72 m. The boundary conditions are given as: pressure head is assumed hydrostatic on
two vertical planes located at x = 0 and 0 < z < 60, and x = 1000 and 0 < z < 60, respectively; no flux is
imposed on all other boundaries of the flow regime. A steady state solution will be sought. A pre-initial
condition is set as h = 60 - z.
400m
12m
60m
O
-540m-
-460m
Figure 4.8 Problem definition and sketch for Example 3.
The region of interest is discretized with 20 x 8 x 10 = 1600 elements resulting in 21x9x11= 2079
node points (Figure 10.2). The nodes are located at x = 0, 70, 120, 160, 200, 275, 350, 400, 450, 500, 540,
570, 600, 650, 700, 750, 800, 850, 900, 950, and 1000 in the x-direction, and at z = 0, 15, 30, 35, 40, 45, 50,
55, 60, 66, and 72 m in the z-direction as reported by Huyakorn et al. (1986). In the y-direction, nodes are
spaced evenly at Ay = 50 m. For 3DFATMIC simulation, the matrix solver, incomplete Cholesky
preconditioned conjugate gradient method, is selected to solve the assembled global matrix equation.
101
-------
The pressure head tolerance is 10~2 m for nonlinear iteration and is 5 10~3 m for matrix solver. The
relaxation factors for nonlinear iteration is set equal to 1.0.
1859 1870
1628y
2068
24;
11
10
1C
x- -> V
<"lr
> >47-
MXTUBK=3008,MXADNK=maxnpk+0)
PARAMETER(MXJBDK=35,MXKBDK=8,MXNTIK=100,MXDTCK=4)
PARAMETER(LTMXNK=693 ,LMXNPK=231 ,LMXBWK=49,MXRGNK=9)
PARAMETER(MXMATK=8,MXSPMK=5,MXMPMK=9)
2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 110,MXCESh=90,MXCPRh= 1 ,MXCDPh=2)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 198,MXVESh= 170,MXVPRh=2,MXVDPh=4)
PARAMETER(MXDNPh= 165 ,MXDPRh= 11 ,MXDDPh=2)
102
-------
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=5)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=63 8,MXVESc=5 60,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=70,MXDPRc=6,MXDDPc=2)
C
PARAMETER(MXNCCK=2)
C
PARAMETER(MXLSVK=500,MXMSVK=500,MXKGLDK=2000,MXNDBK=2000)
PARAMETER(MXNEPK=20,MXEPWK=20)
PARAMETER(MXNPWK=99,MXELWK=27, mxnpws=133 l,mxelws=1000)
PARAMETER(MXNPFGK=2900,MXKGLK=2800)
C
To reflect the soil property function given by Eqs. (4.5) and (4.6), one has to modify the subroutine
SPFUNC in the source code as follows.
WCR=SPP(1,MTYP,1)
WCS=SPP(2,MTYP,1)
HAA=SPP(3,MTYP,1)
ALPHA=SPP(4,MTYP, 1)
BETA=SPP(5,MTYP,1)
C
C SATURATED CONDITION
C
IF(HNP.LE.O.O) THEN
TH=WCS
IF(ISP .EQ. 1) GOTO 900
DTH=O.ODO
USKFCT=1.0DO
ELSE
C
C UNSATURATED CASE
C
TH=WCR+(WCS-WCR)/(1.0DO+(ALPHA*DABS(-HNP-HAA))**BETA)
IF(ISP.EQ.l) GOTO 900
USKFCT=((TH-WCR)/(WCS-WCR))* *2
DNOM=1.0DO+(ALPHA*DABS(-HNP-HAA))**BETA
DTH=(WCS-WCR)* (ALPHA* DAB S(-HNP-HAA))* * (BETA-1 .ODO)/DNOM* * 2
ENDIF
Figure 4.10 and Figure 4.11 depict the pressure distribution and the velocity field in 3-D perspective
103
-------
view (top figure) and along the x-z crosssection through the well (bottom figure) as simulated by 3DFATMIC.
Figure 4.1 Oa Water table for problem No. 3.
Figure 4. lOb Water table on the x-z crosssection through the pumping well for Problem 3.
104
-------
Figure 4.1 la Velocity distribution throughout the domain for Problem 3.
Figure 4.1 Ib Velocity distribution on the x-z crosssection through the well for Problem 3.
4.6 Input and Output for Example 3
With the above descriptions, the input data can be prepared according to the instructions in Appendix
A. The input parameters are listed in Table 4.5 and the data input file content is given in Table 4.6. To save
space, the output isavailable in electronic form.
105
-------
Table 4.5 The list of input parameters for Example 3.
Parameters
number of points
no.of elements
KS.XX
KS.W
KSZZ
0r
0S
ha
a
P
ICP solver
steady-state
simulation
no. of times to
reset time step size
no. of time steps
tolerance for
nonlinear iteration
relaxation factor
for nonlinear
iteration
Pw
Uw
g
Notation in the data
input guide
NNP
NEL
PROPf(l,l)
PROPf(l,2)
PROPf(l,3)
SPP(1,1,1)
SPP(2,1,1)
SPP(3,1,1)
SPP(4,1,1)
SPP(5,1,1)
IPNTSf
KSSf
NDTCHG
NTI
TOLAf
OMEf
RHO
vise
GRAY
Value
2079
1600
5
0.5
2
0.0125
0.25
0
0.5
2
o
6
0
0
0
io-2
1.0
l.OxlO3
948.3264
7.316xl010
Unit
Dimensionless
dimensionless
m/day
m/day
m/day
dimensionless
dimensionless
m
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
m
dimensionless
Kg/m3
Kg/m/day
m/day2
Data set
7. A.
9. A.
5.B.
5.B.
5. B.
6. B.
6. B.
6. B.
6. B.
6. B.
2. B.
2. C.
4. A.
4. A.
3. A.
2. C.
5.B.&6.A.
5.B. &6.A.
6.A.
106
-------
Table 4.6 Input Data Set for Example 3
3 Three-Dimensional Pumping Flow Problem; L=M, T=DAY, M=KG
===== DATA SET 2: OPTION PARAMETERS
10 0 1 0
1 0.5DO l.Od-4 NITRFT OMEFTF OMEFTT
0000000001 KSSF KSST ILUMP IMID IPNTSF IPNTST miconf nstrf nstrt
1 1.0 l.OdO 1.5dO O.OdO KGRAV WF OMEF OMIF
101111 KVIT IWET IOPTIM ksorp Igrn
l.OdO O.SdO l.OdO l.OdO WT WVT OMET OMIT
===== DATA SET 3: ITERATION PARAMETERS
50 20 100 l.Od-2 l.Od-2 NITERF NCYLF NPITRF TOLAF TOLBF
1 100 l.Od-3 l.Od-4 NITERT NPITRT TOLAT TOLBT
===== DATA SET 4: TIME CONTROL PARAMETERS
NTI NDTCHG
DELT CHNG DELMAX TMAX
O.OOdO l.OdO 20.0dO
KPRO KPR(1..NTI)
KDSKO KDSK(1..NTI)
0 0
O.OSdO
55
00
O.OdO
===== DATA SET 5: MATERIAL PROPERTIES
1701 NMAT NMPPM
5.ODD 0.500 2.ODD O.ODO O.ODO O.ODO
l.OdO
PROPF
0.0 RHOMU
l.OdO
===== DATA SET 6: SOIL PROPERTIES
050 l.OdO 7.316D10 1.1232d4
0.01250DO 0.250DO O.OODO 0.500
O.OOODO O.OOODO O.OODO O.ODO
===== DATA SET 7: NODE COORDINATES
KSP NSPPM KCP GRAY
2. ODD
O.ODO
THPROP
AKPROP
2079
1 (.
2 I
3 ฃ
4 I
5 (.
6 I
7 ฃ
8 (.
9 (.
10 (.
11 ฃ
12 I
13 (.
14 I
15 ฃ
16 (.
17 I
18 (.
19 ฃ
20 (.
21 I
22 I
23 ฃ
24 I
25 (.
26 (.
27 ฃ
28 (.
29 (.
30 (.
31 ฃ
32 I
NNP
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
0.
0.
0.
0.
0.
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.000+00
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.120+03
. 120+03
. 120+03
.120+03
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. 120+03
. 120+03
.120+03
.120+03
. 120+03
0.
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.000+00
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.660+02
.720+02
.000+00
.150+02
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.400+02
.450+02
.500+02
.550+02
.600+02
.660+02
.720+02
.000+00
.150+02
.300+02
.350+02
.400+02
.450+02
.500+02
.550+02
.600+02
.660+02
0
0
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0
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0
0
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0
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.500+02
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0
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.000+00
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.000+00
.000+00
.000+00
.000+00
.000+00
.000+00
107
-------
33 ฃ
34 I
35 (.
36 (.
37 ฃ
38 (.
39 ฃ
40 (.
41 ฃ
42 ฃ
43 (.
44 ฃ
45 ฃ
46 ฃ
47 ฃ
48 ฃ
49 ฃ
50 (.
51 ฃ
52 ฃ
53 ฃ
54 ฃ
55 ฃ
56 ฃ
57 ฃ
58 (.
59 ฃ
60 (.
61 ฃ
62 ฃ
63 (.
64 ฃ
65 ฃ
66 ฃ
67 ฃ
68 ฃ
69 ฃ
70 (.
71 ฃ
72 ฃ
73 ฃ
74 ฃ
75 ฃ
76 ฃ
77 ฃ
78 (.
79 ฃ
80 (.
81 ฃ
82 ฃ
83 (.
84 ฃ
85 ฃ
86 ฃ
87 ฃ
88 ฃ
89 ฃ
90 (.
91 ฃ
92 ฃ
93 ฃ
94 ฃ
5 231
5 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
5 231
3 231
5 231
5 231
3 231
0.
0.
0.
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0.
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0.
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.12D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.16D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.20D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.28D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.35D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.40D+03
.45D+03
.45D+03
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.45D+03
.45D+03
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.150+02
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.550+02
.600+02
.660+02
.720+02
.000+00
.150+02
.300+02
.350+02
.400+02
.450+02
.500+02
.550+02
.600+02
.660+02
.720+02
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.150+02
.300+02
.350+02
.400+02
.450+02
.500+02
.550+02
.600+02
.660+02
.720+02
.000+00
.150+02
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.350+02
.400+02
.450+02
.500+02
.550+02
.600+02
.660+02
.720+02
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.150+02
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.450+02
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.600+02
.660+02
.720+02
.000+00
.150+02
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.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
.500+02
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
.OOD+00
.000+00
110
-------
219 8
220 8
221 8
222 8
223 8
224 8
225 8
226 8
227 8
228 8
229 8
230 8
231 8
0
9
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
= = = = = D
1600
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
201
211
221
231
231 0.95D+03 O.OOD+00 0.66D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.95D+03 O.OOD+00 0.72D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 O.OOD+00 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.15D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.30D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.35D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.40D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.45D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.50D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.55D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.60D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.66D+02 O.OOD+00 0.50D+02 O.OOD+00
231 0.10D+04 O.OOD+00 0.72D+02 O.OOD+00 0.50D+02 O.OOD+00
0 0 0.0 0.0 0.0 0.0 0.0 0.0 END OF COORDINATES
ATA SET 8 : SUBREGIONAL DATA
NREGN
8 1 231 0
0000 END OF NNPLR(9)
230 1 1 1
0000 END OF GNLR(I,1)
230 1 232 1
0000 END OF GNLR(I,2)
230 1 463 1
0000 END OF GNLR(I,3)
230 1 694 1
0000 END OF GNLR(I,4)
230 1 925 1
0000 END OF GNLR(I,5)
230 1 1156 1
0000 END OF GNLR(I,6)
230 1 1387 1
0000 END OF GNLR(I,7)
230 1 1618 1
0000 END OF GNLR(I,8)
230 1 1849 1
0000 END OF GNLR(I,9)
ATA SET 9 : ELEMENT INCIDENCES
NEL
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
188
199
210
232
243
254
265
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
188
199
210
221
243
254
265
276
243
254
265
276
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
452
474
485
496
507
232
243
254
265
276
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
463
474
485
496
2
13
24
35
46
57
68
79
90
101
112
123
134
145
156
167
178
189
200
211
233
244
255
266
13
24
35
46
57
68
79
90
101
112
123
134
145
156
167
178
189
200
211
222
244
255
266
277
244
255
266
277
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
453
475
486
497
508
233
244
255
266
277
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
464
475
486
497
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111
-------
241
251
261
271
281
291
301
311
321
331
341
351
361
371
381
391
401
411
421
431
441
451
461
471
481
491
501
511
521
531
541
551
561
571
581
591
601
611
621
631
641
651
661
671
681
691
701
711
721
731
741
751
761
771
781
791
801
811
821
831
841
851
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
276
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
463
474
485
496
507
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
694
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
925
936
947
958
969
980
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
452
474
485
496
507
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
683
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
914
936
947
958
969
980
991
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
683
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
914
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1145
1167
1178
1189
1200
1211
1222
507
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
694
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
925
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1156
1167
1178
1189
1200
1211
277
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
464
475
486
497
508
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
695
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
926
937
948
959
970
981
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
453
475
486
497
508
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
684
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
915
937
948
959
970
981
992
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
684
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
915
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1146
1168
1179
1190
1201
1212
1223
508
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
695
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
926
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1157
1168
1179
1190
1201
1212
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
112
-------
861
871
881
891
901
911
921
931
941
951
961
971
981
991
1001
1011
1021
1031
1041
1051
1061
1071
1081
1091
1101
1111
1121
1131
1141
1151
1161
1171
1181
1191
1201
1211
1221
1231
1241
1251
1261
1271
1281
1291
1301
1311
1321
1331
1341
1351
1361
1371
1381
1391
1401
1411
1421
1431
1441
1451
1461
1471
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1156
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1387
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1618
1629
1640
1651
1662
1673
1684
1695
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1145
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1376
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1607
1629
1640
1651
1662
1673
1684
1695
1706
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1376
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1607
1629
1640
1651
1662
1673
1684
1695
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
1838
1860
1871
1882
1893
1904
1915
1926
1937
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1387
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1618
1629
1640
1651
1662
1673
1684
1695
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
1849
1860
1871
1882
1893
1904
1915
1926
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1157
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1388
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1619
1630
1641
1652
1663
1674
1685
1696
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1146
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1377
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1608
1630
1641
1652
1663
1674
1685
1696
1707
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1377
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1608
1630
1641
1652
1663
1674
1685
1696
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
1839
1861
1872
1883
1894
1905
1916
1927
1938
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1388
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1619
1630
1641
1652
1663
1674
1685
1696
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
1850
1861
1872
1883
1894
1905
1916
1927
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
113
-------
1481
1491
1501
1511
1521
1531
1541
1551
1561
1571
1581
1591
0
0
= = = = =
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
9
9
9
9
9
9
9
9
9
9
9
9
0
DATA
DATA
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
1 1706 1717 1948
1 1717 1728 1959
1 1728 1739 1970
1 1739 1750 1981
1 1750 1761 1992
1 1761 1772 2003
1 1772 1783 2014
1 1783 1794 2025
1 1794 1805 2036
1 1805 1816 2047
1 1816 1827 2058
1 1827 1838 2069
0
SET 10:
NCM
SET 11:
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0
0 0
1937 1707 1718 1949 1938 1
1948 1718 1729 1960 1949 1
1959 1729 1740 1971 1960 1
1970 1740 1751 1982 1971 1
1981 1751 1762 1993 1982 1
1992 1762 1773 2004 1993 1
2003 1773 1784 2015 2004 1
2014 1784 1795 2026 2015 1
2025 1795 1806 2037 2026 1
2036 1806 1817 2048 2037 1
2047 1817 1828 2059 2048 1
2058 1828 1839 2070 2059 1
0 0
0
000
MATERIAL CORRECTION
INITIAL CONDITIONS
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
.600+02
.450+02
.300+02
.250+02
.200+02
. 150+02
. 10D+02
.500+01
.000+00
.600+01
. 120+02
.600+02
.450+02
.300+02
.250+02
.200+02
.150+02
. 100+02
.500+01
.000+00
.600+01
. 120+02
.600+02
.450+02
.300+02
.250+02
.200+02
.150+02
.100+02
.500+01
.000+00
.600+01
.120+02
.600+02
.450+02
.300+02
.250+02
.200+02
. 150+02
.100+02
.500+01
.000+00
.600+01
.120+02
.600+02
.450+02
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
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0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
.000+00
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0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
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0
0
0
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.0
.0
.0
.0
.0
.0
.0
.0
END OF IE
114
-------
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
.300+02
.250+02
.200+02
.150+02
.100+02
.500+01
.000+00
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0.
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115
-------
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
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-0.
-0.
0.
0.
0.
0.
0.
0.
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0.
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116
-------
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
0
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
0
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
-0.
-0.
O.ODO
.150+02
. 10D+02
.500+01
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.600+01
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000
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O.ODO
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0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
END OF 1C FOR FLOW
117
-------
===== DATA
000
===== DATA
000
===== DATA
000
===== DATA
165 2
O.ODO
O.ODO
81
1
10
19
28
37
46
55
64
73
82
91
100
109
118
127
136
145
154
163
0
1 161
163 2
00
===== DATA
000
===== DATA
000
0
SET 12
0
SET 13
0
SET 16
00
SET 17
2 0
60.
30.
11
232
463
694
925
1156
1387
1618
1849
221
452
683
914
1145
1376
1607
1838
2069
111
0
1
1
0
SET 18
00
SET 19
00
: ELEMENT (DISTRIBUTED) SOURCE/SINK OF FLOW
SELF NSPRF NSDPF KSAIF
: POINT (WELL) SOURCE/SINK OF FLOW
NWNPF NWPRF NWDPF KWAIF
: Rainfall/Evaporation-Seepage Boundary Conditions of Ffow
NVESF NVNPF NRPRF NRDPF KRAIF
: DIRICHLET BOUNDARY CONDITIONS OF FLOW
NDNPF NDPRF NDDPF KDAIF
ODD 1.0D38 60. ODD THDBFF HDBFF
ODD 1.0D38 30. ODD THDBFF HDBFF
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
10
20
00 END OF IDTYPF
: CAUCHY BOUNDARY CONDITIONS OF FLOW
NCESF NCNPF NCPRF NCDPF KCAIF
: NEUMANN BOUNDARY CONDITIONS, FLOW
NNESF NNNPF NNPRF NNDPF KNAIF
===== END OF JOB ======
4.7 Example 4: One-Dimensional Single Component Transport Problem
A simple problem is presented here to illustrate the application of this model and show the
improvement of results with the local grid refinement approach, LEZOOMPC. This is a one-dimensional
transport problem between z = 0 and z = 200.0 (Figure 4.12). Initially, the concentration is zero throughout
the region of interest. The concentration at x = 0.0 is maintained at C = C0 = 1.0 (Figure 4.12). The natural
condition of zero gradient flux is imposed at z = 200.0 (Figure 4.12). A bulk density of 1.2, a dispersivity of
5.0, an effective porosity of 0.4 (not used in the program) are assumed.
118
-------
8000
Pb=1'2
ne=0,4
aL=50
\
X
V=0.2
50
C=1,0
Figure 4.12 Problem definition and sketch for Example 4.
119
-------
A specific discharge (Darcy velocity) of 2.0 is assumed and a moisture content of 0.4 is used. For
numerical simulation the region is divided into 40 elements of equal size with 5.0 (Figure 4.13). A time step
size of 0.5 is used and 44 time-step simulation is made. No adsorption is allowed. For this discretization,
mesh Peclet number is P. = 1 and Courant number Cr = 0.5.
164
43
42
C
84
125
Figure 4.13 Finite element discretization for Example 4.
120
-------
To execute the problem, the maximum control-integers in the MAIN must be specified as follows:
C For Example 1 through Example 5
c
PARAMETER(MAXNPK=2079,MAXELK= 1600,MXBNPK=999,MXBESK=999,
> MXTUBK=3008,MXADNK=maxnpk+0)
PARAMETER(MXJBDK=35,MXKBDK=8,MXNTIK=100,MXDTCK=4)
PARAMETER(LTMXNK=693 ,LMXNPK=231 ,LMXBWK=49,MXRGNK=9)
PARAMETER(MXMATK=8,MXSPMK=5,MXMPMK=9)
C 2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 110,MXCESh=90,MXCPRh= 1 ,MXCDPh=2)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh=198,MXVESh=170,MXVPRh=2,MXVDPh=4)
PARAMETER(MXDNPh= 165 ,MXDPRh= 11 ,MXDDPh=2)
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=5)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=63 8,MXVESc=5 60,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=70,MXDPRc=6,MXDDPc=2)
C
PARAMETER(MXNCCK=2)
C
PARAMETER(MXLSVK=500,MXMSVK=500,MXKGLDK=2000,MXNDBK=2000)
PARAMETER(MXNEPK=20,MXEPWK=20)
PARAMETER(MXNPWK=99,MXELWK=27, mxnpws=133 l,mxelws=1000)
PARAMETER(MXNPFGK=2900,MXKGLK=2800)
Figure 4.14 depicts the concentration profiles along the z-axis at various times. It illustrates migration
of the contamination with time. In the meantime, it shows the results obtained by the implementation of
LEZOOMPC are almost the same as the exact solution. However, the Lagrangian-Eulerian approach, which
is much better than conventional finite element scheme, still generates numerical dispersion even though the
Courant number is less than 1 and Peclet number is only equal to one.
121
-------
TIME = 5
)^ No Local Grid Refinement
O ' LEZOOMPC (NZG=3, NZD=2)
I
10
Figure 4.14a The concentration profile of
Example 4 (1 of 6)
-f
25
0.0
Exact
No Local Grid Refinement
LEZOOMPC (NZG=2, NZD=2)
Figure 4.14b The concentration profile of
Example 4 (2 of 6)
Exact
No Local Grid Refinement
LEZOOMPC (NZG=2, NZD=2)
Exact
No Local Grid Refinement
LEZOOMPC (NZG=2, NZD=2)
\
120
Figure 4.14c The concentration profile of
Example 4 (3 Of 6)
Figure4.14d The concentration profile of
Example 4 (4 of 6)
122
-------
TIME = 20
TIME = 22
O
Exact
No Local Grid Refinement
LEZOOMPC (NZG=2, NZD=2)
Figure 4.14e The concentration profile of
Example 4 (5 of 6)
O
Exact
No Local Grid Refinement
LEZOOMPC (NZG=2, NZD=2)
\
120
Figure 4.14f The concentration profile of
Example 4 (6 of 6)
4.8 Input and Output for Example 4
Table 4.7 lists the input parameters and Table 4.8 shows the input data set for the sample problem
described in the above section. The output is given in the attached floppy disk.
Table 4.7 The list of input parameters for Example 4
Parameters
number of points
AX
Ay
AZ
Kd
Ph
Notation in the data
input guide
NNP
XAD
YAD
ZAD
RKD(l)
PROPt(U)
Value
164
50
50
5
0
1.2
Unit
Dimensionless
cm
cm
cm
cmVg
S/cm3
Data set
7. A.
7. B.
7. B.
7. B.
5.F.
5.E.
123
-------
ซL
U0(1)
un(2)
Hoฎ
un(3)
no. of elements
0
vz
no. of subregion
no. of points in each
subregion
initial time step size
time step size increment
percentage
maximum time step size
no. of times to reset time step
size
Total simulation time
no. of time steps
tolerance for nonlinear
iteration
relaxation factor for nonlinear
iteration
PROPt(l,2)
GRATE(l)
GRATE(2)
GRATE(3)
GRATE(4)
NEL
TH
VZ
NREGN
NODES
DELT
CHNG
DELMAX
NDTCHG
TMAX
NTI
TOLBt
OMEt
5.0
0.0
0.0
0.0
0.0
40
0.4
2.0
4
41
0.5
0
0.5
0
22
44
IxlO'4
1.0
cm
I/day
I/day
I/day
I/day
dimensionless
dimensionless
cm/day
dimensionless
dimensionless
day
dimensionless
day
dimensionless
day
dimensionless
dimensionless
dimensionless
5.E.
5. H.
5. H.
5. H.
5. H.
9. A.
25. B.
25. A.
8. A.
8. B.
4. B.
4. B.
4. B.
4. A.
4. B.
4. A.
3.B.
2. E.
Table 4.8 Input Data Set for Example 4
4 One-Dimensional Single Component Transport Problem; L=CM,T=DAY, M=G
========= data set 2: option parameters
1011
100 O.SdO l.Od-4
1110000002
1 1.0 O.SdO O.SdO O.OdO
-101111
l.OdO O.SdO l.OdO l.OdO
========= data set 3: iteration parameters
50 20 100 2.0d-2 2.0d-2
50 100 l.Od-3 l.Od-4
========= data set 4: time control parameters
44 2
124
-------
O.SdO O.OdO O.SdO 22.0dO
33 030003 0 0030 0 0 030 000 00 30003
010000000010000000001000000000100000000010001
1.0D01 2.0000D1
========= DATA SET 5: MATERIAL PROPERTIES
1 7
0.0
0.4
1.8D-
3 .00-
3.0d-
1.0
0.0
1
1
0
0
2
5
4
0
1
.2dO
.OdO
.OdO
0.0
0.17
1.
2.
3.
.375
0.0
0
0
0.
0.
8D-
OD-
OD-
1
5
.0
.0
0
4
2
5
4
.0
0
.OdO
0
0
1
3
3
.0
0.0
0.0
.0
.17
0
0
.8D-2
.OD
.OD
0.
-5
-4
375
0.
O.OdO
.0
.0
1.
2.
3.
0
0.
0.
.80-
.00-
.00-
0
0
2
5
4
O.OdO
0.0 0.0
0.0 0.0
GR
YC
RT
RT
RT
SC
EC
l.OdO
O.OdO O.OdO
0.0 0.0 0.0 0.0
3.0D-5 1.013D-4 3.0D-5 1.013D-4
0.0 0.0 0.0 0.0
l.ld-4
========= DATA SET 6
0401
0.150DO 0.450DO
O.OOODO O.OOODO
C
0
0
C
C
C
1
1
*
1
0
1
0
1
0
1
*
*
*
1
64
1
42
83
24
0
* * * *
4
3
0 0
40
0 0
40
0 0
40
0 0
1
0 0
* * * *
40
1
0
* * * *
0
* * * *
1
2
43
84
25
41
0
40
40
40
40
0
* *
1
0
1
0
1
0
1
0 0
40
0 0
* *
39
0
* *
* *
3
38
38
38
38
3
0
DATA
DATA
41
0
1
0
42
0
83 1
DATA
data
DATA
4
4
ria1~a
1
1
1
1
0
1
1
1
1
0
1
1
1
1
1
1
0
SET 7: N
O.ODO
O.ODO
50. ODO
50. ODO
0.0
SET 8: S
0
124
SET 9: E
42 8
0
setlO : ma
SET 11:
apt- 1 4- P
DCOEFF
SATURC
PCOEFF
COFK
soil properties
OdO 7.316dl2 1
O.OODO -1.0D2
O.OODO O.ODO
NODE COORDINATES
Kso, Ksn
Ko, Kn
Kpo, Kpn
gammao, gamman
alphao, alphan
lambdao, lambdan
GAMMAo, GAMMAn
Epsilon
1232d2
THPROP
AKPROP
50.ODO
O.ODO
O.ODO
50.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
5. ODO
5. ODO
5. ODO
5. ODO
0.0
SUBREGIONAL DATA
END OF NNPLR(K)
END OF GNLR(I,i;
END OF GNLR(I,2;
END OF GNLR(I,3;
END OF GNLR(I,4;
ELEMENT INCIDENCES
125
0
3 124 1 43 84
00000
material correction
INITIAL CONDITIONS
l.OdO O.OdO 0.0
O.OdO O.OdO O.OdO
O.OdO O.OdO O.OdO
O.OdO O.OdO O.OdO
O.OdO O.OdO O.OdO
O.OdO O.OdO O.OdO
O.OdO O.OdO O.OdO
END OF IE
end of ic, transport
15:
element(distributed) source/sink, transport
point(well) source/sink, transport
0000
========= data set
0000
========= data set 20: run-in/seep-out boundary
125
-------
14120
O.OdO O.OdO 1.0d38 0.OdO
10010
00000 end of irtyp
1 0 0 82 123 164 41 0 0 0 0
0 0 0 0 0 0 00000 end fof isvt(j,i),j=1,4
1 3 1 41 41
00000 end of npvbt
========= data set 21: dirichlet boundary conditions, transport
4120
O.OdO l.OdO 1.0d38 1.OdO
1 3 1 1 41
00000
13110
00000 end of idtyp
========= data set 22: Cauchy boundary condition, transport
00000
========= data set 23: Neumann boundary condition, transport
00000
C ***** DATA SET 24 : PARAMETERS CONTROLLING TRACKING SCHEME
1102121112122
l.Od-4 l.Od-4
C ******* DATA SET 25: HYDROLOGICAL BOUNDARY CONDITIONS
1 163 1 O.ODO O.ODO 2.ODO O.ODO O.ODO O.ODO
0 00 0.0 0.0 0.0 0.0 0.0 0.0 END OF VELOCITY
1 39 1 0.4DO 0.0
0 0 0 0.0 0.0 END OF TH
0 END OF JOB
0000
4.9 Example 5: Two-Dimensional Single Component Transport Problem
This is a two-dimensional transport problem in a rectangular region of (x,y,z) = (0.0, 0.0, 0.0) and
(x,y,z) = (540.0, 270.0, 1.0) (Figure 4.15). Initially, the concentration is zero throughout the region of interest.
The concentration of 1.0 is maintained at x = 0.0 and 90 < y < 180 (Figure 4.15).A concentration of 0.0 is
maintained atx = 0.0 and 0.0 < y < 90.0 or 180.0 < y < 270.0 (Figure 4.15). A natural condition is imposed
at x = 540. A bulk density of 1.2, a longitudinal dispersivity of 10.0, and a lateral dispersivity of 1.0 are
assumed. A specific discharge (Darcy's velocity) of 2.0 is used and a moisture content of 0.2 is assumed. The
region is divided into 9x9x1 = 81 elements resulting in 10 x 10 x 2 = 200 nodes (Figure 4.16). The element
size is 60.0 x 30.0 x 1.0. A time-step size of 4.5 is used and a 40 time-step simulation is made to illustrate how
to use 3DFATMIC. No adsorption is allowed.
126
-------
c = o,o
V = 2.0
X
c = o,o
=1.2
270 6=0.2
3L=10.0
a =1.0
1
z
XY
Figure 4.15 Problem definition and sketch for Example 5.
110
107 / ,//
120 130 140 150 160 170 180 190 200
oo
109/ / /I /\ _/\ /\ A _/\ _/\ /
108 A i / / \ / \ / / , / / , / A
/ , / / //\
106/ ,// x/ / XX // '/ // /
105 A ,/> XX XX XX XX XX XX XX . . ,
104_/i i/X xx XX xx XX xx XX xx //ri/s?
103_^1>> XX XX XX XX XX XX XX XX.-
102 / i/X 7X XX 7X XX 7X XX XX 7XI K95
1Q1A / / // // // // // // // //IT/94
I/
1
10
>^
19
28
37
46
55
^64V^
73
'93
11 21 31 41 51 61 71 81 91
Figure 4.16 Finite element discretization for Example 5.
C-
c
To execute the problem, the maximum control-integers in the MAIN should be specified as:
For Example 1 through Example 5
PARAMETER(MAXNPK=2079,MAXELK= 1600,MXBNPK=999,MXBESK=999,
> MXTUBK=3008,MXADNK=maxnpk+0)
PARAMETER(MXJBDK=35,MXKBDK=8,MXNTIK=100,MXDTCK=4)
PARAMETER(LTMXNK=693 ,LMXNPK=231 ,LMXBWK=49,MXRGNK=9)
PARAMETER(MXMATK=8,MXSPMK=5,MXMPMK=9)
2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 110,MXCESh=90,MXCPRh= 1 ,MXCDPh=2)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 198,MXVESh= 170,MXVPRh=2,MXVDPh=4)
PARAMETER(MXDNPh= 165 ,MXDPRh= 11 ,MXDDPh=2)
- 3. For transport source/sink, boundary conditions, and materials
127
-------
c
c
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=5)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=63 8,MXVESc=5 60,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=70,MXDPRc=6,MXDDPc=2)
PARAMETER(MXNCCK=2)
PARAMETER(MXLSVK=500,MXMSVK=500,MXKGLDK=2000,MXNDBK=2000)
PARAMETER(MXNEPK=20,MXEPWK=20)
PARAMETER(MXNPWK=99,MXELWK=27, mxnpws=133 l,mxelws=1000)
PARAMETER(MXNPFGK=2900,MXKGLK=2800)
Figure 4.17 depicts the 50% concentration contours at various times. It illustrates how the pollutant
is moving through the medium with time.
250
200
150
100
50
0
0
0.5
p
tf
100
t=4.5
t=45
t=90
t=135
t=180
200 300
X
'--- 0.5 -
0-5
400
0.5
0.5
0.5
500
Figure 4.17 Contours of 50% concentration at various times.
128
-------
4.10 Input and Output for Example 5
Table 4.9 lists the input parameters and Table 4.10 shows the input data set for the problem described
in the above section. To save space, the output is available in electronic form.
129
-------
Table 4.9 The list of input parameters for Example 5
Parameters
number of points
AX
Ay
AZ
Kd
Pb
ซL
(XT
"o(1)
Hn(2)
Ho(3)
u(3)
H-n
no. of elements
0
vx
no. of subregion
no. of points in each subregion
initial time step size
time step size increment percentage
maximum time step size
no. of times to reset time step size
Total simulation time
no. of time steps
tolerance for nonlinear iteration
relaxation factor for nonlinear iteration
Notation in the data
input guide
NNP
XAD
YAD
ZAD
RKD(l)
PROPt(l,l)
PROPt(l,2)
PROPt(l,3)
GRATE(l)
GRATE(2)
GRATE(3)
GRATE(4)
NEL
TH
VZ
NREGN
NODES
DELT
CHNG
DELMAX
NDTCHG
TMAX
NTI
TOLBt
OMEt
Value
200
60
30
1
0
1.2
10.0
1.0
0.0
0.0
0.0
0.0
81
0.2
2.0
2
100
4.5
0
4.5
0
180
40
io-4
1.0
Unit
Dimensionless
cm
cm
cm
cmVg
g/cm3
cm
cm
I/day
I/day
I/day
I/day
dimensionless
dimensionless
cm/day
dimensionless
dimensionless
day
dimensionless
day
dimensionless
day
dimensionless
dimensionless
dimensionless
Data set
7. A.
7. B.
7. B.
7. B.
5.F.
5.E.
5.E.
5.E.
5. H.
5. H.
5. H.
5. H.
9. A.
25. B.
25. A.
8. A.
8. B.
4. B.
4. B.
4. B.
4. A.
4. B.
4. A.
3. A.
2. E.
130
-------
Table 4.10 Input Data Set for Example 5
5 Two-Dimensional Single Compoent Transport Problem; L=CM,T=DAY,M=G
===== DATA SET 2: OPTION PARAMETERS
1010
50 0.5DO l.Od-4 NITRFT OMEFTF OMEFTT
1110000002 KSSF KSST ILUMP IMID IPNTSF IPNTST
1 1.0 O.SdO O.SdO O.OdO KGRAV WF OMEF OMIF
-101110 KVIT IWET IOPTIM ksorp Igrn miconf
l.OdO O.SdO l.OdO l.OODO WT WVT OMET OMIT
===== DATA SET 3: ITERATION PARAMETERS
50 20 100 2.0d-2 2.0d-2 NITERF NCYLF NPITRF TOLAF TOLBF
50 900 l.Od-3 l.Od-4 NITERT NPITRT TOLAT TOLBT
===== DATA SET 4: TIME CONTROL PARAMETERS
40 0 NTI NDTCHG
4.50dO 0.0OdO 4.5dO 1.8d2 DELT CHNG DELMAX TMAX
55000000005000000000500000000050000000005 KPRO KPR(1..NTI)
00000000000000000000000000000000000000000
O.OdO
===== DATA SET 5: MATERIAL PROPERTIES
1
.OdO l.OdO O.OdO 1.OdO
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0
0.17
1.8D-2 1.8D-2
3.0D-5 2.0D-5
3.0D-4 3.0D-4
0.375
0.0
DSKO KDSKd. .NTI;
0.0
0.0
0.0
171
1.2dO 10
0.0 0.0
0.0 0.0
0.0 0.0
0.4 0.17 0.4
1.8D-2 1.8D-2
3.0D-5 2.0D-5
3.0d-4 3.0D-4
1.0 0.375 1.0
0.0 0.0 0.0
NMAT NMPPM
O.OdO l.OdO
O.OdO PROPT
0.0 0.0 0.0 0.0
3.0D-5 1.013D-4 3.0D-5 1.013D-4
0.0 0.0 0.0 0.0
l.ld-4
===== DATA SET 6
0101
GRATE
YCOEFF
RTARDS
RTARDO
RTARDN
SCOEFF
ECOEFF
DCOEFF
SATURC
PCOEFF
COFK
Kso, Ksn
Ko, Kn
Kpo, Kpn
gammao, gamman
alphao, alphan
lambdao, lambdan
GAMMAo, GAMMAn
Epsilon
SOIL PROPERTIES
ODO l.OdO l.OdO
KSP NSPPM KCP GRAY
0.0(
0.0(
200
1 9
2 9
3 9
4 9
5 9
6 9
7 9
8 9
9 9
10 9
101 9
102 9
103 9
104 9
105 9
106 9
107 9
108 9
109 9
110 9
0 0
)ODO
)ODO
3ATA
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
0
THPROP
AKPROP
SET 7 : NODE COORDINATES
NNP
O.ODO O.ODO O.ODO 6 . 001 O.ODO O.ODO
O.ODO 3.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 6.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 9.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 12.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 15.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 18.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 21.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 24.001 O.ODO 6.001 O.ODO O.ODO
O.ODO 27.001 O.ODO 6.001 O.ODO O.ODO
O.ODO O.ODO l.ODO 6.001 O.ODO O.ODO
O.ODO 3.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 6.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 9.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 12.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 15.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 18.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 21.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 24.001 l.ODO 6.001 O.ODO O.ODO
O.ODO 27.001 l.ODO 6.001 O.ODO O.ODO
0.0 0.0 0.0 0.0 0.0 0.0 END OF COORDINATES
131
-------
2
1 1
0 0
1 99
0 0
1 99
0 0
1
0
1
0
1
0
100 0
0 0
1 1
0 0
101 1
0 0
===== DATA SET 9
===== DATA SET 8: SUBREGIONAL DATA
NREGN
END OF NNPLR(K)
END OF GNLR(I,1)
END OF GNLR(I,2)
ELEMENT INCIDENCES
112 102
122 112
132 122
142 132
152 142
162 152
172 162
182 172
192 182
0 0
===== DATA SET 10: MATERIAL CORRECTION
81
1
10
19
28
37
46
55
64
73
0
NEL
8
8
8
8
8
8
8
8
8
0
1
1
1
1
1
1
1
1
1
0
1
11
21
31
41
51
61
71
81
0
11
21
31
41
51
61
71
81
91
0
12
22
32
42
52
62
72
82
92
0
2
12
22
32
42
52
62
72
82
0
101
111
121
131
141
151
161
171
181
0
111
121
131
141
151
161
171
181
191
0
END OF IE
0
NCM
0
SET 11:
199 1
0 0
DATA SET 14 :
000
DATA SET 15:
000
DATA SET 20 :
END OF 1C FOR TRANSPORT
9
O.ODO
1
0
1
0
1
11
0
20
0
0
1
5
9
12
15
18
0
1
9
0
0
20 1
O.ODO
8
0
8
0
9
9
0
0
.0038
1
0
91
0
91
191
0
INITIAL CONDITIONS
O.ODO O.ODO 0.0
0.0 0.0 0.0
ELEMENT(DISTRIBUTED) SOURCE/SINK OF TRANSPORT
NSELT NSPRT NSDPT KSAIT
POINT(WELL) SOURCE/SINK OF TRANSPORT
NWNPT NWPRT NWDPT KWAIT
VARIABLE BOUNDARY CONDITIONS OF TRANSPORT
NVEST NVNPT NRPRT NRDPT KRAIT
O.ODO TCVBFT CVBFT
0
END OF IVTYPT
192 191 1111
0
92
0
1
1
0
000000 END OF ISVT(J,I) J=l,4
0
(2 * * * *
1 1
l.Od
DATA SET 21:
2 2 0
.ODO
.OdO
3
3
2
2
2
2
0
71
11 1
00
DATA SET 22
0000
DATA SET 23
0000
* DATA SET 24
l.ODO
O.OdO
4 1
104
1
8
101
108
0
1
2
0
END OF NPVBT
DIRICHLET BOUNDARY CONDITIONS OF TRANSPORT
NDNPT NDPRT NDDPT KDAIT
1.0D38 l.ODO TCDBFT CDBFT
1.0d38 O.OdO
NPDBT(1..NDNPT)
1
1
1
1
1
0
0 END OF IDTYPT
CAUCHY BOUNDARY CONDITIONS OF TRANSPORT
NCEST NCNPT NCPRT NCDPT KCAIT
NEUMANN BOUNDARY CONDITIONS, TRANSPORT
NNEST NNNPT NNPRT NNDPT KNAIT
PARAMETERS CONTROLLING TRACKING SCHEME
2212
0221111
-4 l.Od-4
==== DATA SET 25: HYDROLOGICAL VARIABLES
1 199 1 2.ODD O.ODO 0.0 0.0 0.0 0.0
132
-------
0 00 0.0 0.0 0.0 0.0 0.0 0.0 END OF X-VELOCITY
1 80 1 0.2DO 0.0 0.0
0 00 0.0 0.0 0.0 END OF TH
0 ====== END OF JOB ======
4.11 Example 6: Two-Dimensional Multicomponent Transport in a Uniform Flow Field
This problem is used to illustrate the behavior of a dissolved organic plume undergoing natural
biodegradation in a uniform ground-water flow field. The kinetic and microbial parameters for the simulation
are the same as those published by Macquarrie et al. (1990). But with the different setup of governing
equations in the system, the equivalent parameters in 3DFATMIC are adjusted and shown in Table 4.11.
Figure 4.18 shows the x-z cross section of the region of interest and the remaining transport parameters.
Substrate and oxygen are assumed to be at 0 and 3.5 mg/L everywhere in the domain at time zero, respectively.
The initial condition is comprised of a square patch, which is placed far enough from the domain limits to
avoid boundary effects, and shown in Figure 4.18. The concentrations in the initial patch are 3 mg/L for
substrate and 1 mg/L for oxygen. The total background concentration of microbial #1 population is 0.23 mg/L
and the retardation factor associated with microbes is 1000. Although the nitrate, nutrient, microbe #2, and
microbe #3 are included in the input data, the simulations for these four components are not performed.
Therefore, the initial and boundary conditions for these four components are set to zero in the input data file.
Because of the implementation of the developed Lagrangian-Eulerian finite element numerical scheme with
adapted local refinement, the Courant and Peclet criteria are not needed. Therefore, the nodal spacing is
greater than that specified by Macquarrie et al. and shown in Figure 4.19.
133
-------
6 m
S=0
S = 3 mg/L
^2m
\vv\
2m
\
Dd
= 0.09 m/d >
= 0.81m , aT= 0.005m
.Oj x 10 m/day, Ks 1.4
free - exit
boundary
6 m
O = 3.5 mg/L
45m
O = 1 mg/L
Xl2"
2m
Vx = 0.09 m/d >
0^= 0.81m , OCT= 0.005m
Dd - 8.05 x 10 m/day . RO 1-0
Oback =3.5 mg/L
free - exit
boundary
45m
Figure 4.18 The x-z crosssection of region of interest and the associated physical parameters.
6.0 m 13
,o m 12
11
10
9
8
i n m o
i ,u m o
0 5 m 2
0 m 1
i
^
ฉ
ฉ
ฉ
x*v
UJ
>21
>^
ฉ
r"^
234
^
247
^
260
^
273
//
286
^
299
V
312
>-
325
/
338
>-
351
'/
364
X
377
X
M /JQ1
nog
390
x
(18(1
ฉ
ฉ
d^9
403 4
V
/
^
/-
/
^
^
/-
/
^
^
/
/
16
415
414
413
412
411
410
409
408
407
406
405
404
14 27 40 53 66 79 92 105 118 131 144 157 170 183 196
0 3m 5m 9m 12m15m 45m
Figure 4.19 The discretization of the region of interest
134
-------
c-
c
c-
To execute the problem, the maximum control-integers in the MAIN should be specified as
For Example 6 &Example 8
PARAMETER(MAXNPK=2288,MAXELK=1800,MXBNPK=1999,MXBESK=1999,
> MXTUBK=2640,MXADNK=maxnpk+14000)
PARAMETER(MXJBDK=85,MXKBDK=8,MXNTIK= 100,MXDTCK=4)
PARAMETER(LTMXNK=693,LMXNPK=231,LMXBWK=49,MXRGNK=11)
PARAMETER(MXMATK= 1 ,MXSPMK=2,MXMPMK=7)
2. For flow source/sink, boundary conditions, and materials
C
C
C
C
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh= 1 ,MXWPRh= 1 ,MXWDPh= 1)
PARAMETER(MXCNPh= 1 ,MXCESh= 1 ,MXCPRh= 1 ,MXCDPh= 1)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 1 ,MXVESh= 1 ,MXVPRh= 1 ,MXVDPh= 1)
PARAMETER(MXDNPh= 1 ,MXDPRh= 1 ,MXDDPh= 1)
- 3. For transport source/sink, boundary conditions, and materials
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc= 1 ,MXWPRc= 1 ,MXWDPc= 1)
PARAMETER(MXCNPc= 1 ,MXCESc=1 ,MXCPRc=1 ,MXCDPc= 1)
PARAMETER(MXNNPc=1 ,MXNESc=1 ,MXNPRc= 1 ,MXNDPc= 1)
PARAMETER(MXVNPc=143 ,MXVESc= 120,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=143 ,MXDPRc=2,MXDDPc=2)
PARAMETER(MXNCCK=7)
PARAMETER(MXLSVK=5000,MXMSVK=5000,MXKGLDK=29999,MXNDBK=9999)
PARAMETER(MXNEPK= 1 ,MXEPWK= 1)
PARAMETER(MXNPWK=4 8 ,MXELWK= 15, mxnpws= 1 ,mxelws=1)
PARAMETER(MXNPFGK=260000,MXKGLK= 140000)
4.12 Input and Output for Example 6
Table 4.11 lists the input parameters and Table 4.12 shows the input data set for the sample problem
described in the above section. The output isavailable in electronic form.
Table 4.11 The list of input parameters for Example 6
Parameters
number of points
Notation in the data
input guide
NNP
Value
416
Unit
Dimensionless
Data set
7. A.
135
-------
AX
Ay
AZ
ซL
CXT
Dm
Kdi
K*
Un(1)
Hn(2)
Un(3)
Hn(3)
V C1)
1 0
V (2)
-1 n
V (3)
1 0
v (3)
-1- n
Kso(1)
Ksn(2)
Kso(3)
Ksn(3)
K0ซ
Kn(2)
K0(3)
Kn(3)
K (1)
ฑVDO
K (2)
-"-DII
K (3)
ฑVDO
K (3)
-"-DII
V (D
10
vn(2)
XAD
YAD
ZAD
PROP(1,2)
PROP(1,3)
PROP(1,4)
RKD(l)
RKD(4)
GRATE(l)
GRATE(2)
GRATE(3)
GRATE(4)
YCOEFF(l)
YCOEFF(2)
YCOEFF(3)
YCOEFF(4)
RTARDS(l)
RTARDS(2)
RTARDS(3)
RTARDS(4)
RTARDO(l)
RTARDO(2)
RTARDO(3)
RTARDO(4)
RTARDN(l)
RTARDN(2)
RTARDN(3)
RTARDN(4)
SCOEFF(l)
SCOEFF(2)
3.0 (except
arround x =
5.0)
0.5
0.5
0.81
5.0X1Q-3
8.05 xlO'5
1000
0.4
0.21
0..0
0.0
0.0
0.426
0.17
0.4
0.17
654
0.018
0.018
0.018
l.OxlO2
2.0xlO'5
3.0X10'5
2.0xlO'5
3.0X10'4
0.0
0.0
0.0
7.044
0.0
m
m
m
m
m
mVday
mVmg
mVmg
I/day
I/day
I/day
I/day
mg/mg
mg/mg
mg/mg
mg/mg
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
dimensionless
dimensionless
7. B.
7. B.
7. B.
5.E.
5.E.
5.E.
5.F.
5.F.
5. H.
5.H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5. H.
136
-------
V (3)
lo
v ฎ
in
a(1)
UQ
a (2)
i*n
a (3)
UQ
a (3)
ซ.n
A0(1)
V2)
V3)
V3)
(1)
J. 0
(2)
A. n
(3)
J. 0
(3)
x. n
e(1)
co
e(2)
t;n
e(3)
co
,z (3)
cn
Kc
no. of elements
no. of subregion
no. of points in each
subregion
Velocity
transient-state for transport
initial time step size
time step size increment
percentage
maximum time step size
no. of times to reset time step
size
Total simulation time
SCOEFF(3)
SCOEFF(4)
ECOEFF(l)
ECOEFF(2)
ECOEFF(3)
ECOEFF(4)
DCOEFF(l)
DCOEFF(2)
DCOEFF(3)
DCOEFF(4)
SATURC(l)
SATURC(2)
SATURC(3)
SATURC(4)
PCOEFF(l)
PCOEFF(2)
PCOEFF(3)
PCOEFF(4)
COFK
NEL
NREGN
NODES
vx
KSSt
DELT
CHNG
DELMAX
NDTCHG
TMAX
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
180
2
208
0.09
1
2.0
0
2.0
0
200
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
I/day
I/day
I/day
I/day
mg/m3
mg/m3
mg/m3
mg/m3
dimensionless
dimensionless
dimensionless
dimensionless
mg/m3
dimensionless
dimensionless
dimensionless
m/day
dimensionless
day
dimensionless
day
dimensionless
day
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5. H.
5. H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
9. A.
8. A.
8. B.
25. A.
2. B.
4. B.
4. B.
4. B.
4. A.
4. B.
137
-------
no. of time steps
tolerance for transport
nonlinear iteration
relaxation factor for transport
nonlinear iteration
Pw
Uw
g
NTI
TOLBt
OMEt
RHO
vise
GRAY
100
IxlO'4
1.0
109
94832640
7.316xl010
dimensionless
dimensionless
dimensionless
mg/m3
mg/m/day
m/day2
4. A.
3.B.
2. E.
5.B. &6.A.
5.B. & 6.A.
6.A.
Table 4.12 Input Data Set for Example 6
6 Two-D Multicomponent Transport in a Uniform Flow Field: mg,m,day
===== DATA SET 2: OPTION PARAMETERS
1010
50 0.5DO l.OD-4 NITRFT OMEFTF OMEFTT
1110010011 KSSF KSST ILUMP IMID IPNTSF IPNTST
1 1.0 l.OdO l.OdO O.OdO KGRAV WF OMEF OMIF
-1 1 0 1 1 KVIT IWET IOPTIM KSORP LGRAN
l.OdO l.OdO l.OdO l.OdO WT WVT OMET OMIT
===== DATA SET 3: ITERATION PARAMETERS
50 20 100 l.Od-2 l.Od-2 NITERF NCYLF NPITRF TOLAF TOLBF
500 100 l.Od-2 l.Od-4 NITERT NPITRT TOLAT TOLBT ALLOW
===== DATA SET 4: TIME CONTROL PARAMETERS
100 0 NTI NDTCHG
.OdO 2.Od2 DELT CHNG DELMAX TMAX
2.0dO O.OOdO
55 0
0 0
1
0
O.OdO
===== DATA SET 5
1771
l.OOdO 8.1d-l
1.0d3 O.OdO 0.0
O.OdO O.OdO 0.0
0.21 0.0 0.0
0.426 0.17 0.4
6.54D2 1.8D-2
1.0D2 2.0D-5
O.OdO O.ODO
7.044 0.0
0.0 0.0
000
5
1
MATERIAL PROPERTIES
NMAT NMPPM
5.0d-3 8.05D-5 1.OdO O.OdO
0.4DO O.ODO O.ODO O.ODO
0.0 0.0
0.0
0.17
1.8D-2
0
l.OdO
PROPT
3.0D-5
O.ODO
0.0 0.0
0.0 0.0
1.8D-2
2.0D-5
O.ODO
0.0 0.0 0.0 0.0
O.ODO O.ODO O.ODO O.ODO SATURC
0.0 0.0 0.0 0.0
O.OdO
===== DATA SET 6: SOIL PROPERTIES
120 l.OdO 9.8DO l.ODO
-1000.0 1000.0
0.1 0.1
1.0 1.0
0.0 0.0
GRATE
YCOEFF
RTARDS
RTARDO
RTARDN
SCOEFF
ECOEFF
DCOEFF
GAMMAo,
PCOEFF
COFK
Kso, Ksn
Ko, Kn
Kpo, Kpn
gammao, gamman
alphao, alphan
lambdao, lambdan
GAMMAn
Epsilon
KSP NSPPM KCP GRAY
PRESSURE
WATER CONTENT
RELATIVE CONDUCTIVITY
0
138
-------
(
41(
1
27
40
2
28
41
3
29
42
4
30
43
5
31
44
6
32
45
7
33
46
8
34
47
9
35
48
10
36
49
11
37
50
12
38
51
13
39
52
209
235
248
210
236
249
211
237
250
212
238
251
213
239
252
214
240
253
215
241
3.0
= DA
3
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
0.0
TA SET 7 :
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
NODE
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
WAT:
COORDINATES
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3.
3.
3 .
3 .
3.
3.
4.
4.
4 .
4 .
4.
4.
5.
5.
5.
5.
5.
5.
6.
6.
6.
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3 .
3.
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
NNP
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
WATER CAPACITY
139
-------
254 12 13 9.0
216 1 13 0.0
242 0 0 5.0
255 12 13 9.0
217 1 13 0.0
243 0 0 5.0
256 12 13 9.0
218 1 13 0.0
244 0 0 5.0
257 12 13 9.0
219 1 13 0.0
245 0 0 5.0
258 12 13 9.0
220 1 13 0.0
246 0 0 5.0
259 12 13 9.0
221 1 13 0.0
247 0 0 5.0
260 12 13 9.0
0 00 0.0 0
===== DATA SET 9 :
180
1
2
3
4
5
6
7
8
9
10
11
12
0
14
14
14
14
14
14
14
14
14
14
14
14
0
===== DATA
0
T~\7\
1 415
0 0
1 415
0 0
1 415
0 0
1 16
18 4
23 7
31 4
36 189
226 4
231 7
239 4
244 172
0 0
1 16
18 4
23 7
31 4
36 189
226 4
231 7
239 4
TA
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
12
12
12
12
12
12
12
12
12
12
12
12
0
SET 10:
SET 11
2.3D-2
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
3.0d3
O.OdO
3 .Od3
O.OdO
3.0d3
O.OdO
3 .Od3
O.OdO
O.OdO
3.5D3
1.0d3
3 .5d3
1.0d3
3.5d3
1.0d3
3 .5d3
1.0d3
0.5 3.0
0.5 3.5
0.5 3.5
0.5 3.5
0.5 4.0
0.5 4.0
0.5 4.0
0.5 4.5
0.5 4.5
0.5 4.5
0.5 5.0
0.5 5.0
0.5 5.0
0.5 5.5
0.5 5.5
0.5 5.5
0.5 6.0
0.5 6.0
0.5 6.0
.0 0.0
ELEMENT
1
2
3
4
5
6
7
8
9
10
11
12
0
14
15
16
17
18
19
20
21
22
23
24
25
0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
3.0 0.0 0.0
0.0 0.0 0.0
INCIDENCES
222
223
224
225
226
227
228
229
230
231
232
233
0
209
210
211
212
213
214
215
216
217
218
219
220
0
2
3
4
5
6
7
8
9
10
11
12
13
0
15
16
17
18
19
20
21
22
23
24
25
26
0
END OF COORDINATES
NEL
223
224
225
226
227
228
229
230
231
232
233
234
0
210
211
212
213
214
215
216
217
218
219
220
221
0
13
13
13
13
13
13
13
13
13
13
13
13
0 END OF IE
MATERIAL CORRECTION
NCM
: INITIAL CONDIDTIONS
0
0
0
0
0
0
0
0.
0
0.
0
0.
0
0.
0
0
0.
0.
0.
0.
0.
0.
0.
0.
.ODD
.ODD
.ODD
.ODD
.ODD
.ODD
.ODD
OdO 0
.OdO
OdO 0
.OdO
OdO 0
.OdO
OdO 0
.OdO
.OdO
ODO 0
OdO 0
OdO 0
OdO 0
OdO 0
OdO 0
OdO 0
OdO 0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
.OdO
O.OdO
.OdO
O.OdO
.OdO
O.OdO
.OdO
O.OdO
O.OdO
.ODO
.OdO
.OdO
.OdO
.OdO
.OdO
.OdO
.OdO
140
-------
244 172 1
000
415 1
0 0
415 1
0 0
===== DATA
000
===== DATA
000
===== DATA
12 26
O.ODO
1 11
0 0
11
0
11
0
11
0
11
0
11
0
11
0
1 11
0 0
1 12
14 12
0 0
===== DATA
26 2
O.ODO
O.ODO
1 12
14
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
12
0
25
0
25
0
25
0
25
0
25
0
25
0
25
0
DATA
0 0
===== DATA
000
===== DATA
110
l.Od-4 1
===== DATA
1 415 1
0
3 .5d3
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
SET 14:
0
SET 15:
0
SET 20:
1 2
O.ODO
1 1
0 0
1
0
1
0
1
0
1
0
1
0
1
0
196
0
196
404
0
SET 21:
0
.ODD 1
.503 1
1 1
209
0
1
0
1
0
1
0
1
0
2
0
1
0
1
0
SET 22:
0 0
SET 23:
0 0
SET 24
315
.Od-4
SET 25
.00-
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
OdO
OdO
OdO
OdO
ELEMENT(DISTRIBUTED) SOURCE/SINK OF TRANSPORT
NSELT NSPRT NSDPT KSAIT
POINT(WELL) SOURCE/SINK OF TRANSPORT
NWNPT NWPRT NWDPT KWAIT
VARIABLE BOUNDARY CONDITIONS OF TRANSPORT
0 NVEST NVNPT NRPRT NRDPT KRAIT
1.0D38 O.ODO
405
0
197
0
1
1
000
404
0
1
1
0
DIRICHLET BOUNDARY CONDITIONS OF TRANSPORT
NDNPT NDPRT NDDPT KDAIT
.0038 O.ODO
.0038 3.5D3
1
CAUCHY BOUNDARY CONDITIONS OF TRANSPORT
NCEST NCNPT NCPRT NCDPT KCAIT
NEUMANN BOUNDARY CONDITIONS, TRANSPORT
NNEST NNNPT NNPRT NNDPT KNAIT
: PARAMETERS CONTROLLOING TRACKING SCHEME
111 315 2
: VELOCITY AND MOISTURE CONTENT
2 O.ODO O.ODO O.ODO O.ODO O.ODO
141
-------
000 O.ODO O.ODO O.ODO 0.ODD 0.ODD 0 . ODD
1 179 1 l.ODO O.ODO
0 0 0 O.ODO O.ODO
0 ====== END OF JOB ======
142
-------
Figure 4.20, Figure 4.21, and Figure 4.22 show the simulation results of substrate, oxygen, and total microbial
mass distributions at 100 days and 200 days, respectively.
(a)
Substrate at Time = 100 Days (NXG=3,NZG=5)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
6-
5-
4-
3-
2-
1-
0-
10 15
20
\
25
X
Oxygen at Time = 100 Days (NXG=3,NZG=5)
3000
10 15
20
\
25
30 35
X
40
30 35 40 45
\
45
Figure 4.20 Dissolved plumes at 100 days: (a) substrate and (b) oxygen. Concentration isolines
are in micrograms per liter
143
-------
(a)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
6-
5-
4-
3-
2-
1-
0-
Substrate at Time = 200 Days (NXG=3,NZG=5)
10
15
20
\
25
30
35
X
Oxygen at Time = 200 Days (NXG=3,NZG=5)
3000
10
15
20
\
25
30
35
X
40
40
\
45
\
45
Figure 4.21 Dissolved plumes at 200 days: (a) substrate and (b) oxygen. Concentration isolines
are in micrograms per liter
144
-------
(a)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
6-
5-
4-
3-
2-
1-
0-
Microbe at Time = 100 Days (NXG=3,NZG=5)
10 15 20 25 30 35 40
X
Microbe at Time = 200 Days (NXG=3,NZG=5)
10 15 20
\
25
30 35 40
X
\
45
\
45
Figure 4.22 Total microbial mass distributions: (a) 100 and (b) 200 days. Concentration
isolines are in mg/1000 cm3 of aquifer material
145
-------
4.13 Example 7: Two-Dimensional Coupled Flow and Multicomponent Transport Problem
This problem is presented in "Denitrification in nonhomogeneous laboratory scale aquifers: 5: user's
manual for the mathematical model LT3VSI" by G.A. Bachelor et al. reported in 1990. The example aquifer
used for this problem is 1.4 meters long with 15 nodes in the X direction, 1.6 meters thick with IVnodesin
the Z direction, 1 meter wide in the Y direction, and shown in Figure 4.23 and Figure 4.24. This aquifer has
8 different materials, two injection wells at (0.1,0.0,0.1) and (0.1,1.0,0.1), and two extraction wells at
(1.3,0.0,1.5) and (1.3,1.0,1.5). The hydrological and microbial dynamical data are all from this report. The
initial condition of the flow field is obtained by simulating steady state of flow field without sources and sinks.
Then the flow field and concentration distribution are updated at each time step. There are two type of
microbes included in the system, say microbe #1 and microbe #3 with 1.77* 10"4 Kg/m3 initially. The initial
concentrations of the chemicals are 5><10"3 Kg/m3 of substrate, 5><10"3 Kg/m3 of oxygen, 5><10"3 Kg/m3 of
nitrate, and 3><10"3 Kg/m3 of nutrient. The daily injection and withdrawal rates of water are 3.75><10"3 and
3.75xlO~3m3, respectively. The total hydraulic head is 1.0 mat the upstream boundary AB (Figure 4.23) and
0.0 m at the downstream boundary CD (Figure 4.23). For transport simulation, variable boundary condition
is implemented at the downstream boundary CD (Figure 4.23) and 1.77xlO~4 Kg/m3 of microbe #1, 1.77><10~4
Kg/m3 of microbe #3, l.SxlO'2 Kg/m3 of substrate, 5.0xlQ-3 Kg/m3 of oxygen, S.OxlO'3 Kg/m3 of nitrate, and
3.Ox 10"3 Kg/m3 of nutrient influents through the upstream boundary. Because microbe #2 does not exist in
this environment, the initial and boundary conditions for this component is set to zero in this simulation. This
problem is set up for 4 days simulation.
To execute the problem, the maximum control-integers in the MAIN should be specified as
C For Example7
c
PARAMETER(MAXNPK=510,MAXELK=224,MXBNPK=510,MXBESK=508,
> MXTUBK=3552,MXADNK=MAXNPK+20000)
PARAMETER(MXJBDK=45,MXKBDK=8,MXNTIK=80,MXDTCK=1)
PARAMETER(LTMXNK= 1 ,LMXNPK= 1 ,LMXBWK= 1 ,MXRGNK= 1)
PARAMETER(MXMATK=8,MXSPMK=2,MXMPMK=8)
146
-------
C 2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh=4,MXWPRh=2,MXWDPh=3)
PARAMETER(MXCNPh= 1 ,MXCESh= 1 ,MXCPRh= 1 ,MXCDPh= 1)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 1 ,MXVESh= 1 ,MXVPRh= 1 ,MXVDPh= 1)
PARAMETER(MXDNPh=68,MXDPRh=2,MXDDPh=2)
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc=4,MXWPRc=2,MXWDPc=3)
PARAMETER(MXCNPc=55,MXCESc=40,MXCPRc=2,MXCDPc=4)
PARAMETER(MXNNPc=11 ,MXNESc=4,MXNPRc= 1 ,MXNDPc=2)
PARAMETER(MXVNPc=34,MXVESc= 16,MXVPRc= 1 ,MXVDPc=2)
PARAMETER(MXDNPc=34,MXDPRc=5,MXDDPc=2)
C
PARAMETER(MXNCCK=7)
C
PARAMETER(MXLSVK=5 000,MXMSVK=5 000,MXKGLDK=3 9900,MXNDBK=9600)
PARAMETER(MXNEPK= 1 ,MXEPWK= 1)
PARAMETER(MXNPWK=27,MXELWK=8,mxnpws=27,mxelws=8)
PARAMETER(MXNPFGK= 190000,MXKGLK=99999)
C
147
-------
-No flux for both flow
B 1 .6
1 .2
.0
=
0.6
0.4
A n r
and transport
(1)
(6)
x(3)
(0.1,0.1
(5)
(4)
)
(7)
(6)
(5)
(7)
(8)
(1.3,1.5
(3)
(4)
(2)
D
h = 0.0 for
flow
Variable
boundary
for
transport
0.0
0.3 0.5
0.9 1.1 1.4
No flux for both flow
and transport
(1) : material type 1
x: injection well
: extraction well
Figure 4.23 The x-z crosssection of the region of interest.
148
-------
496 497 498
508 509 510
31
16
'21*
^ j
'19?
^ >
>f<
? V
J,
Sl2S
<. _>
:*;
223
^^/
&
221
s /
216
^ y
'28S
s x
^ s
O4/
495
480
300
285
270
1 2 3
13 14 15
Figure 4.24 The Discretization of Example 7.
Because the soil properties are input by tabular form, the specificcation of soil property functions in
subroutine SPFUNC is not needed.
4.14 Input and Output for Example 7
Table 4.13 lists the input parameters and Table 4.14 shows the input data set for the sample problem
described in the above section. The output is available in electronic form.
Table 4.13 The list of input parameters for Example 7
Parameters
number of points
Notation in the data
input guide
NNP
Value
510
Unit
Dimensionless
Data set
7. A.
149
-------
AX
Ay
AZ
no. of materials
Uo(1)
un(2)
Uo(3)
^
V C1)
1 0
v &
*- n
V (3)
1 0
V (3)
-1- n
Kso(1)
Ksn(2)
Kso(3)
Ksn(3)
K0ซ
Kn(2)
Koฎ
Kn(3)
K (1)
ฑVDO
K (2)
A^Dn
K (3)
ฑVDO
K (3)
A^Dn
y (D
lo
v ฎ
in
v 0)
Io
v ฎ
in
a (1)
UQ
a (2)
ซ.n
an(3)
XAD
YAD
ZAD
NMAT
GRATE(l)
GRATE(2)
GRATE(3)
GRATE(4)
YCOEFF(l)
YCOEFF(2)
YCOEFF(3)
YCOEFF(4)
RTARDS(l)
RTARDS(2)
RTARDS(3)
RTARDS(4)
RTARDO(l)
RTARDO(2)
RTARDO(3)
RTARDO(4)
RTARDN(l)
RTARDN(2)
RTARDN(3)
RTARDN(4)
SCOEFF(l)
SCOEFF(2)
SCOEFF(3)
SCOEFF(4)
ECOEFF(l)
ECOEFF(2)
ECOEFFC3)
0.1
1.0
0.1
8
4.0
0.0
4.0
2.5
0.4
0.17
0.4
0.17
0.018
0.018
0.018
0.018
3.0X10'5
2.0xlO'5
3.0X10'5
2.0xlO'5
3.0X10'4
3.0xlO'4
3.0X10'4
3.0xlO'4
1.0
0.375
1.0
0.375
0.004
0.002
0.004
m
m
m
dimensionless
I/day
I/day
I/day
I/day
Kg/Kg
Kg/Kg
Kg/Kg
Kg/Kg
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
Kg/m3
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
7. B.
7. B.
7. B.
5. A.
5. H.
5. H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5. H.
5.H.
5. H.
5.H.
5.H.
5.H.
5.H.
150
-------
a (3)
un
A0(1)
1 (2)
An
A0(3)
1 (3)
An
r o)
-1- o
r P)
A n
r (3)
-1- o
r (3)
A n
e0(1)
en(2)
- (3)
t;0
enฎ
Kc
no. of elements
no. of materials to be
corrected
ICP solver
steady-state for flow
transient-state for transport
initial time step size
time step size increment
percentage
maximum time step size
no. of times to reset time step
size
Total simulation time
no. of time steps
tolerance for flow nonlinear
iteration
relaxation factor for flow
nonlinear iteration
ECOEFF(4)
DCOEFF(l)
DCOEFF(2)
DCOEFF(3)
DCOEFF(4)
SATURC(l)
SATURC(2)
SATURC(3)
SATURC(4)
PCOEFF(l)
PCOEFF(2)
PCOEFF(3)
PCOEFF(4)
COFK
NEL
NCM
IPNTSf
IPNTSt
KSSf
KSSt
DELT
CHNG
DELMAX
NDTCHG
TMAX
NTI
TOLAf
OMEf
0.002
0.02
0.02
0.02
0.02
S.OxlO'5
2.0X10'5
S.OxlO'5
2.0X10'5
0.05
0.021
0.05
0.021
l.lxlO'4
224
134
o
J
0
1
0.05
0
0.05
0
4
80
IxlO'2
1.0
dimensionless
I/day
I/day
I/day
I/day
Kg/m3
Kg/m3
Kg/m3
Kg/m3
dimensionless
dimensionless
dimensionless
dimensionless
Kg/m3
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
day
dimensionless
day
dimensionless
day
dimensionless
m
dimensionless
5.H.
5. H.
5. H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
9. A.
10. A.
2. B.
2. B.
2. B.
4. B.
4. B.
4. B.
4. A.
4. B.
4. A.
3. A.
2. C
151
-------
tolerance for transport
nonlinear iteration
relaxation factor for transport
nonlinear iteration
Pw
Uw
g
TOLBt
OMEt
RHO
vise
GRAY
IxlO'4
1.0
1000.0
948.3264
7.316xl010
dimensionless
dimensionless
Kg/m3
Kg/m/day
m/day2
3.B.
2. E.
5.B. & 6.A.
5.B. &6.A.
6.A.
Table 4.14 Input Data Set for Example 7
7 2-D Coupled
= = = = = DATA SET 2 :
11 0 1 0
1 0.5DO l.OD-4
011033000
1 1.0 l.OdO 0.
21011
l.OdO l.OdO l.OdO
1 l.OdO
===== DATA SET 3:
50 20 999 l.Od-2
50 200 l.Od-3 1.
===== DATA SET 4:
0
Od-2 O.OOdO 5
5
Flow and Multicomponent Transport, L= M, M=KG, T=DAY
OPTION PARAMETERS
IMOD,IGEOM,IBUG,ICHNG
NITRFs,OMEFTs,ALLOW
1 KSSf,KSSt,ILUMP,IMID,IPNTSf,IPNTSt,NSTRf,NSTRt,MICONF,IQUAR
5dO O.OdO KGRAV,Wf,OMEf,OMIf
KVIt,IWET,IOPTIM,KSORP,LGRAN
l.OdO Wt,WVt,OMEt,OMIt WT WVT OMET OMIT
IEIGEN,GG
80
5.
55
5
1
1
ITERATION PARAMETERS
l.Od-2
Od-4
TIME CONTROL PARAMETERS
.Od-2 4.0dO
555
NITERf,NCYLf,NPITER,TOLAf,TOLBf
NITERt,NPITERt,TOLAt,TOLBt
NTI,NDTCHG
DELT,CHNG,DELMAX,TMAX
555
=== DATA
3 7 7
1.
1.
1 .
3.
5.
3 .
1.
3.
1.
0.
1.
0.
1.
1.
0.
0.
1.
0.
1.
1.
0.
Od-2
OdO
Od-1
16d-l
62d-2
16d-2
Od-1
16d-l
Od3
0 0.
414d3 0
OdO 0.
Od-2 1.
9019d3
OdO 0.
OdO 0.
6895d3
OdO 0.
Od-4 1.
7558d3
OdO 0.
1
SET 5
1
.OdO
.OdO
.OdO
.OdO
.OdO
.OdO
.OdO
.OdO
Od3
OdO
.OdO
OdO
Od-2
O.OdO
OdO 0
OdO 0
O.OdO
OdO 0
Od-4 1
O.OdO
OdO 0
: MATERIAL PROPERTIES
.Od-2 O.ODO O.ODO O.ODO 1.OD3
.OdO O.ODO O.ODO O.ODO 1.OD3
.Od-1 O.ODO O.ODO O.ODO 1.OD3
.16d-l O.ODO O.ODO O.ODO 1.OD3
.62d-2 O.ODO O.ODO O.ODO 1.OD3
.16d-2 O.ODO O.ODO O.ODO 1.OD3
.OOd-1 O.ODO O.ODO O.ODO 1.OD3
.16d-l O.ODO O.ODO O.ODO 1.OD3
Od3 1.0d3
OdO O.OdO
O.OdO 0.0
OdO O.OdO
1.0d3 1.0d3 1.0d3
O.OdO O.OdO O.OdO
l.OdO O.OdO l.OdO
O.OdO O.OdO O.OdO
NMAT,NMPPM,NCC,IRXN
PROPf
PROPf
PROPf
PROPf
PROPf
PROPf
PROPf
PROPf
DINTS
RHOMU
PROPt
Od-2 l.Od-2 l.Od-2 l.Od-2 l.Od-2
O.OdO 0.0 l.OdO O.OdO
OdO O.OdO O.OdO O.OdO O.OdO
OdO O.OdO O.OdO O.OdO O.OdO
O.OdO 0.0 l.OdO O.OdO
OdO O.OdO O.OdO O.OdO O.OdO
Od-4 l.Od-4 l.Od-4 l.Od-4 l.Od-4
O.OdO 0.0 l.OdO O.OdO
OdO O.OdO O.OdO O.OdO O.OdO
1
RKD
TRANC
OdO
RKD
TRANC
l.OdO
RKD
TRANC
l.OdO
RKD
PROPT
PROPT
PROPT
152
-------
O.OdO O.OdO 0
1.4728d3 O.OdO
O.OdO O.OdO 0
O.OdO O.OdO 0
1.5158d3 O.OdO
O.OdO O.OdO 0
l.Od-3 l.Od-3 1
1.5124d3 O.OdO
O.OdO O.OdO 0
O.OdO O.OdO 0
1.7061d3 O.OdO
O.OdO O.OdO 0
O.OdO O.OdO 0
4.0 0.0 4.0
0.4 0.4 0.4
1.8D-2 1.8d-2
3.0D-5 3.0d-5
3.0d-4 3.0d-4
1.0 0.375 1.0
0.004 0.002 0.
0.02 0.02 0.02 0
3.0D-5 2.0d-5
0.05 0.021 0.05
l.ld-4
===== DATA SET 6
12 01.
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
-1000.0 1000.0
0.465 0.465
0.285 0.285
0.365 0.365
0.323 0.323
0.387 0.387
0.412 0.412
0.364 0.364
0.322 0.322
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
1.0 1.0
0.0 0.0
0.0 0.0
0.0 0.0
0.0 0.0
0.0 0.0
0.0 0.0
0.0 0.0
0.0 0.0
===== DATA SET 7
510
1 16 15 0.0
2 16 15 0.1
.OdO O.OdO O.OdO
O.OdO 0.0
.OdO O.OdO O.OdO
.OdO O.OdO O.OdO
O.OdO 0.0
.OdO O.OdO O.OdO
.Od-3 l.Od-3 l.Od-3
O.OdO 0.0
.OdO O.OdO O.OdO
.OdO O.OdO O.OdO
O.OdO 0.0
.OdO O.OdO O.OdO
.OdO O.OdO O.OdO
2.5
0.17
1.8D-2 1.8D-2
3.0D-5 2.0D-5
3.0D-4 3.0D-4
0.375
004 0.002
.02
3 .OD-5 2.0D-5
0.021
: SOIL PROPERTIES
OdO 9.8DO l.ODO
O.OdO
l.OdO
O.OdO
O.OdO
l.OdO
O.OdO
l.Od-3 1.
l.OdO 0.
O.OdO
O.OdO
l.OdO
O.OdO
O.OdO
GRATE
YCOEFF
RTARDS
RTARDO
RTARDN
SCOEFF
ECOEFF
DCOEFF
SATURC
PCOEFF
COFK
OdO
OdO
OdO
OdO
.OdO
OdO
Od-3
OdO
OdO
OdO
OdO
OdO
OdO
TRANC
l.OdO
RKD
TRANC
l.OdO
RKD
TRANC
l.OdO
RKD
TRANC
l.OdO
RKD
TRANC
PROPT
PROPT
PROPT
PROPT
Kso, Ksn
Ko, Kn
Kpo, Kpn
gammao, gamman
alphao, alphan
lambdao, lambdan
GAMMAo, GAMMAn
Epsilon
KSP,NSPPM,KCP,RHO,GRAY,VISC
PRESUURE
MOISTURE CONTENT
RELATIVE HYDRAULIC CONDUCTIVITY
D(THETA)/DH
NODE COORDINATES
0.0 0.0 0.0 0.
0.0 0.0 0.0 0.
NNP
0 0.1
0 0.1
153
-------
3
4
5
6
7
8
9
10
11
12
13
14
15
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
0
224
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0
134
1 3
2 3
3 3
6 8
20 8
34 8
48 8
68 5
69 5
70 5
180
181
182
152 2
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
16 15
0 0
DATA
15
15
15
15
15
15
15
15
15
15
15
15
15
15
0
DATA
14 3
14 3
14 3
1 2
1 2
1 2
1 2
14 2
14 2
14 2
3 14
3 14
3 14
1 4
0.
0.
0.
0.
0.
0.
0.
0.
1.
1 .
1 .
1.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
1.
1 .
1 .
1.
0.
SET
14
14
14
14
14
14
14
14
14
14
14
14
14
14
0
SET
0
0
0
0
0
0
0
0
0
0
3 0
3 0
3 0
0
2
3
4
5
6
7
8
9
0
1
2
3
4
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
0
9
1
2
3
4
5
6
7
8
9
10
11
12
13
14
10
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0 0.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
: ELEMENT
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
257
258
259
260
261
262
263
264
265
266
267
268
269
270
0
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.0
INCIDENCES
256
257
258
259
260
261
262
263
264
265
266
267
268
269
0
16
17
18
19
20
21
22
23
24
25
26
27
28
29
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
NEL
272 271
273 272
274 273
275 274
276 275
277 276
278 277
279 278
280 279
281 280
282 281
283 282
284 283
285 284
0 0
15
15
15
15
15
15
15
15
15
15
15
15
15
15
0
: MATERIAL CORRECTION
NCM
END OF IE
154
-------
166 214
4 3 14 4
5 3 14 4
57 2 1 6
2 1
1 1
1 1
3 1
3 1
71
60
74
62
76
66
80
94
95
11
11
3 14 7
3 14 7
153 1 1 5
167 1 1 5
181 3 14 6
182 3 14 6
0000
===== DATA
0 IHTR
509
0
509
0
509
0
509
0
509
0
509
0
509
0
509
0
===== DATA
000
===== DATA
423
O.OdO
O.OdO
17 239
112
1 2
0 0
DATA
0 0
== DATA
423
O.OdO
O.OdO
17 239
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
SET 11
O.OdO
O.ODO
1.770
O.ODO
O.ODO
O.ODO
1 .770
O.ODO
5.0D-
O.ODO
5.0D-
O.ODO
5.0D-
O.ODO
3.0D-
O.ODO
SET 12
0
SET 13
0
0.0 1.
0.0 1.
272
1 0
2 0
0 0
SET 14
0
SET 15
0
0.0 0.
0.0 0.
272
INITIAL CONDIDTIONS
O.OdO O.OdO
O.ODO O.ODO
-4 O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
-4 O.ODO O.ODO
O.ODO O.ODO
3 O.ODO O.ODO
O.ODO O.ODO
3 O.ODO O.ODO
O.ODO O.ODO
3 O.ODO O.ODO
O.ODO O.ODO
3 O.ODO O.ODO
I.C. FOR FLOW
I.C. FOR MICROBE 1
I.C. FOR MICROBE 2
I.C. FOR MICROBE 3
I.C. FOR SUBSTRATE
I.C. FOR OXYGEN
I.C. FOR NITRATE
I.C.
FOR NUTRIENT
O.ODO O.ODO
: ELEMENT(DISTRIBUTED) SOURCE/SINK OF FLOW
NSELF,NSPRF,NSDPF,KSAIF
: POINT(WELL) SOURCE/SINK OF FLOW
NWNPF,NWPRF,NWDPF,KWAIF
Od-6 3.75d-3 1.0d38 3.75d-3
Od-6 -2.0d-3 1.0d38 -2.0d-3
494
: ELEMENT(DISTRIBUTED) SOURCE/SINK OF TRANSPORT
NSELT,NSPRT,NSDPT,KSAIT
: POINT(WELL) SOURCE/SINK OF TRANSPORT
NWNPF,NWPRF,NWDPF,KWAIF
OdO l.Od-6 3.75d-3 O.OdO 1.0d38 3.75d-3 O.OdO
OdO l.Od-6 -2.0d-3 O.OdO 1.0d38 -2.0d-3 O.OdO
494
155
-------
= = =
1
2
0
1
2
0
1
2
0
1
2
0
= =
0
68
0
0
1
18
35
52
0
1
35
0
= = =
= = =
1
1
0
1
1
0
1
1
0
1
1
0
DATA
0
DATA
2
.OdO
.OdO
0
= =
0
16
0.
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
18
= =
34
0
0
0
0
0
1
18
0
.0
.0
.0
.0
.0
16
16
16
16
0 0
33
33
0
DATA
0
DATA
0
DATA
34 1
OdO
15
0
15
0
15
0
15
0
15
0
15
0
15
0
15
0
16
16
0
DATA
5
2
2
0
2
2
0
2
2
0
2
2
0
1
2
0
1
2
0
1
2
0
1
2
0
SET
0
0
SET
2
1
0
1
1
1
1
0
0
1
1
0
0
0
SET
0
SET
0
0
SET
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
16:
0
17 :
1.
1.
1
256
15
270
1
2
0
18:
0
19:
0
20:
.OdO 1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
0
SET
2
1 .77d-
0.0
1.5d-
5.0d-
3.0d-
16
16
0 0
1
1
2
3
3
0
4
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
15
0
15
270
0
21:
RAINFALL/EVAPORATION- SEEPAGE
DIRICHLET
Od38
Od38
15
15
15
15
0
0
0
0
1.0
0.0
NVESF,NVNPF,
BOUNDARY CONDITIONS OF FLOW
NRPRF , NRDPF , KRAI F
BOUNDARY CONDITIONS OF FLOW
NDNPF,NDPRF,
NDDPF,KDAIF
CAUCHY BOUNDARY CONDITIONS OF FLOW
0
NCESF,NCNPF,
NEUMANN BOUNDARY CONDITIONS,
NNESF,NNNPF,
VARIABLE BOUNDARY CONDITIONS
.Od38
0
0
0
0
0
0
0
0
0
0
0
0
0
0
270
0
15
15
0
O.OdO
end of
end of
end of
end of
end of
end of
end of
285 30
0 0
NVESF,NVNPF,
irtyp
irtyp
irtyp
irtyp
irtyp
irtyp
irtyp
15 15 15 15
0000 end
NCPRF,NCDPF,KCAIF
FLOW
NNPRF , NNDPF , KNAI F
OF TRANSPORT
NVPRF , NVDPF , KVAIF
fof isvt ( j , i) , j =1 , 4
NVEST , NVNPT , NVPRT , NVDPT , NVAIT
DIRICHLET
BOUNDARY CONDITIONS OF TRANSPORT
NDNPT , NDPRT , NDDPT , KDAIT
1.
1.
1.
1.
1.
1
256
Od38
Od38
Od38
Od38
Od38
15
15
0
1 .77d-
0.0
1.5d-2
5.0d-3
3.0d-3
4
156
-------
1
0
1
0
1
0
1
0
1
0
1
0
1
0
= = = =
1 1
1.
33
0
33
0
33
0
33
0
33
0
33
0
33
0
T~\7\ TTA
0 0
= DATA
0 0
T~\7\ TTA
021
Od-4
n
1 1
0 0
1 2
0 0
1 1
0 0
1 3
0 0
1 4
0 0
1 4
0 0
1 5
0 0
C "PT1 O O .
OIL i z z :
0 0
SET 23 :
000
C "PT1 O A
o Ji 1 Z ^
2212
l.Od-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PTV T-
L-AL
0
NEU
DS
; Jr/i
2 1
4
CAUCHY BOUNDARY CONDITIONS OF TRANSPORT
NCEST,NCNPT,NCPRT,NCDPT,KCAIT
TN BOUNDARY CONDITIONS, TRANSPORT
NNEST,NNNPT,NNPRT,NNDPT,KNAIT
PARAMETERS CONTROLLOING TRACKING SCHEME
2 1 IZOOM,IDZOOM,IEPC,NXA,NYA,NZA,NXW,NYW,NZW,NXD,NYD,NZD
ADPEPS,ADPARM
END OF JOB ======
Figure 4.25 depicts the simulation results of velocity field and Figures 4.26a through 4.26c show
concentration contours of microbes, substrate and oxygen, and nitrate and nutrient, respectively.
157
-------
o
Figure 4.25 The velocity field at (a) time = 2 days and (b) time = 4 days
158
-------
Microbe 1 at Time = 2 Days
Microbe 1 at Time = 4 Days
N
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
N
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
Microbe 3 at Time = 2 Days
Microbe 3 at Time = 4 Days
N
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
N
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X X
Fig. 4.26a Concentration contours of microbe #1 and microbe #3 at time = 2 days and 4 days, respectively.
159
-------
Substrate at Time = 2 Days
Substrate at Time = 4 Days
N
N
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
N
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
Oxygen at Time = 2 Days
1.6-
1.4
1.2
1.0
0.8
0.6
0.4
0.2
-0.0
N
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
Oxygen at Time = 4 Days
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X X
Fig. 4.26b Concentration contours of substrate and oxygen at time = 2 days and 4 days, respectively.
160
-------
Nitrate at Time = 2 Days
Nitrate at Time = 4 Days
N
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
N
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
-0.0-
1 I I I I I I
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
Nutrient at Time = 2 Days
Nutrient at Time = 4 Days
N
1.6-
1.4-
1.2-
1.0-
0.8-
0.6-
0.4-
0.2-
o.o-
III
0.0024
0.0024
^ 0.0018
1.6-
1.4-
1.2-
1.0-
N 0.8-
0.6-
0.4-
0.2-
-o.o-
0.
1
III
0.0024
0024
1
L^\ /-
^0.0024 >^^
_^^ 0.0018'
- o.ooi
i i r
0.0 0.2 0.4 0.6 0.*
X
1.0 1.2 1.4
i i r
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
X
Fig. 4.26c Concentration contours of nitrate and nutrient at time = 2 days and 4 days, respectively.
161
-------
4.15 Example 8: Three-Dimensional Multicomponent Transport in a Uniform Flow Field
This problem is used to demonstrate the 3-D multicomponent transport behavior. The kinetic and
microbial parameters for the simulation are the same as those adopted by the previous example. The region
is taken asO
/'''
/ ^
y
u
/
A
v/
J / '.
'/ / / '
' '. //
'///'
' / ', /
' / / /
'///'.
,v/S
-------
C For Example 6 & Examples
c
PARAMETER(MAXNPK=2288,MAXELK=1800,MXBNPK=1999,MXBESK=1999,
> MXTUBK=2640,MXADNK=maxnpk+14000)
PARAMETER(MXJBDK=85,MXKBDK=8,MXNTIK= 100,MXDTCK=4)
PARAMETER(LTMXNK=693,LMXNPK=231,LMXBWK=49,MXRGNK=11)
PARAMETER(MXMATK= 1 ,MXSPMK=2,MXMPMK=7)
C 2. For flow source/sink, boundary conditions, and materials
PARAMETER(MXSELh= 1 ,MXSPRh= 1 ,MXSDPh= 1 ,MXWNPh= 1 ,MXWPRh= 1 ,MXWDPh= 1)
PARAMETER(MXCNPh= 1 ,MXCESh= 1 ,MXCPRh= 1 ,MXCDPh= 1)
PARAMETER(MXNNPh= 1 ,MXNESh= 1 ,MXNPRh= 1 ,MXNDPh= 1)
PARAMETER(MXVNPh= 1 ,MXVESh= 1 ,MXVPRh= 1 ,MXVDPh= 1)
PARAMETER(MXDNPh= 1 ,MXDPRh= 1 ,MXDDPh= 1)
C 3. For transport source/sink, boundary conditions, and materials
C
PARAMETER(MXSELc= 1 ,MXSPRc= 1 ,MXSDPc= 1 ,MXWNPc= 1 ,MXWPRc= 1 ,MXWDPc= 1)
PARAMETER(MXCNPc= 1 ,MXCESc=1 ,MXCPRc=1 ,MXCDPc= 1)
PARAMETER(MXNNPc=1 ,MXNESc=1 ,MXNPRc= 1 ,MXNDPc= 1)
PARAMETER(MXVNPc=143 ,MXVESc= 120,MXVPRc=1 ,MXVDPc=2)
PARAMETER(MXDNPc=143 ,MXDPRc=2,MXDDPc=2)
C
PARAMETER(MXNCCK=7)
C
PARAMETER(MXLSVK=5000,MXMSVK=5000,MXKGLDK=29999,MXNDBK=9999)
PARAMETER(MXNEPK= 1 ,MXEPWK= 1)
PARAMETER(MXNPWK=4 8 ,MXELWK= 15, mxnpws= 1 ,mxelws=1)
PARAMETER(MXNPFGK=260000,MXKGLK= 140000)
C
4.16 Input and Output for Example 8
Table 4.15 lists the input parameters and Table 4.16 shows the input data set for the sample problem
described in the above section. The output isavailable in electronic form.
163
-------
Table 4.15 The list of input parameters for Example 8
Parameters
number of points
AX
Ay
AZ
ซL
ซT
Dm
Kdl
K*
un(1)
un(2)
un(3)
un(3)
Y W
A o
Y C)
-"-n
Y (3)
-1- o
Y (3)
-"-n
Kso(1)
Ksn(2)
Kso(3)
Ksn(3)
K0(1)
Kn(2)
K0ฎ
K(3)
n
Kno(1)
K<2)
Notation in the data
input guide
NNP
XAD
YAD
ZAD
PROP(1,2)
PROP(1,3)
PROP(1,4)
RKD(l)
RKD(4)
GRATE(l)
GRATE(2)
GRATE(3)
GRATE(4)
YCOEFF(l)
YCOEFF(2)
YCOEFF(3)
YCOEFF(4)
RTARDS(l)
RTARDS(2)
RTARDS(3)
RTARDS(4)
RTARDO(l)
RTARDO(2)
RTARDO(3)
RTARDO(4)
RTARDN(l)
RTARDW2)
Value
2288
3.0 (except
arround x =
5.0)
0.5
0.5
0.81
5.0xlO'3
8.05 xlO'5
1000
0.4
0.21
0..0
0.0
0.0
0.426
0.17
0.4
0.17
654
0.018
0.018
0.018
l.OxlO2
2.0X10'5
3.0xlO'5
2.0X10'5
3.0xlO'4
0.0
Unit
Dimensionless
m
m
m
m
m
mVday
mVmg
mVmg
I/day
I/day
I/day
I/day
mg/mg
mg/mg
mg/mg
mg/mg
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
mg/m3
Data set
7. A.
7. B.
7. B.
7. B.
5.E.
5.E.
5.E.
5.F.
5.F.
5. H.
5.H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
164
-------
K (3)
ฑVDO
K (3)
A^Dn
v (1)
Io
v ฎ
in
V (3)
Ifo
v ฎ
in
a(1)
UQ
a (2)
ซ.n
a (3)
UQ
a (3)
ซ.n
A0(1)
V2)
V3)
V3)
(1)
J. 0
(2)
x. n
(3)
J. 0
(3)
x. n
e(1)
co
e(2)
cn
e(3)
co
,z (3)
cn
Kc
no. of elements
no. of subregion
no. of points in each
subregion
Velocity
transient-state for transport
initial time step size
time step size increment
percentage
RTARDN(3)
RTARDN(4)
SCOEFF(l)
SCOEFF(2)
SCOEFF(3)
SCOEFF(4)
ECOEFF(l)
ECOEFF(2)
ECOEFF(3)
ECOEFF(4)
DCOEFF(l)
DCOEFF(2)
DCOEFF(3)
DCOEFF(4)
SATURC(l)
SATURC(2)
SATURC(3)
SATURC(4)
PCOEFF(l)
PCOEFF(2)
PCOEFF(3)
PCOEFF(4)
COFK
NEL
NREGN
NODES
vx
KSSt
DELT
CHNG
0.0
0.0
7.044
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1800
11
208
0.09
1
2.0
0
mg/m3
mg/m3
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
dimensionless
I/day
I/day
I/day
I/day
mg/m3
mg/m3
mg/m3
mg/m3
dimensionless
dimensionless
dimensionless
dimensionless
mg/m3
dimensionless
dimensionless
dimensionless
m/day
dimensionless
day
dimensionless
5.H.
5.H.
5.H.
5. H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5. H.
5. H.
5. H.
5. H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
5.H.
9. A.
8. A.
8. B.
25. A.
2. B.
4. B.
4. B.
165
-------
maximum time step size
no. of times to reset time step
size
Total simulation time
no. of time steps
tolerance for transport
nonlinear iteration
relaxation factor for transport
nonlinear iteration
Pw
Uw
g
DELMAX
NDTCHG
TMAX
NTI
TOLBt
OMEt
RHO
vise
GRAY
2.0
0
200
100
IxlO'4
1.0
109
94832640
7.316xl010
day
dimensionless
day
dimensionless
dimensionless
dimensionless
mg/m3
mg/m/day
m/day2
4. B.
4. A.
4. B.
4. A.
3.B.
2. E.
5.B. &6.A.
5.B. & 6.A.
6.A.
Table 4.16 Input Data Set for Example 8
8 3-d multicomponent transport in uniform flow field,mg,m,day
===== DATA SET 2: OPTION PARAMETERS
1010
50 0.5DO l.OD-4 NITRFT OMEFTF OMEFTT
KSSF KSST ILUMP IMID IPNTSF IPNTST
KGRAV WF OMEF OMIF
KVIT IWET IOPTIM KSORP LGRAN
WT WVT OMET OMIT
00011
l.OdO O.OdO
11100
1 1.0 l.OdO
-11011
l.OdO l.OdO l.OdO l.OdO
===== DATA SET 3: ITERATION PARAMETERS
50 20 100 l.Od-2 l.Od-2 NITERF NCYLF NPITRF TOLAF TOLBF
50 100 l.Od-2 l.Od-4 NITERT NPITRT TOLAT TOLBT ALLOW
===== DATA SET 4: TIME CONTROL PARAMETERS
NTI NDTCHG
DELT CHNG DELMAX TMAX
100
OdO
2
55
0
1
0
1.0d5
O.OOdO
0
0
.OdO
0
5
2.0d2
0
0
0
1
2.0d4 1.0D38
TDTCH
===== DATA SET 5: MATERIAL PROPERTIES
1771
l.OOdO 8.1d-l 5.0d-3
1.0d3 O.OdO 0.0 0.4DO
O.OdO O.OdO 0.0 0.0
0.21 0.0 0.0 0.0
0.426 0.17 0.4 0.17
6.54D2 1.8D-2 1.8D-2
1.0D2 2.0D-5 3.0D-5
O.OdO O.ODO O.ODO
7.044 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
O.ODO O.ODO O.ODO O.ODO
NMAT NMPPM
8.05D-5 l.OdO O.OdO
O.ODO O.ODO O.ODO
0.0 0.0 0.0
GRATE
YCOEFF
1.8D-2 RTARDS
2.0D-5 RTARDO
O.ODO RTARDN
SCOEFF
ECOEFF
DCOEFF
l.OdO PROPT
Kso, Ksn
Ko, Kn
Kpo, Kpn
gammao, gamman
alphao, alphan
lambdao, lambdan
SATURC GAMMAo, GAMMAn
166
-------
0.0
0.(
1
-100(
(
(
0.0 0.0
)dO
= DATA SET
2 0 I
3.0 100(
3.1 (
L.O I
3.0 (
= DATA SET
0.0
6: SOIL
L.OdO 9
3.0
3.1
L.O
3.0
7: NODE
2288
1
27
40
2
28
41
3
29
42
4
30
43
5
31
44
6
32
45
7
33
46
8
34
47
9
35
48
10
36
49
11
37
50
12
38
51
13
39
52
209
235
248
210
236
249
211
237
250
212
238
251
213
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
5.
9.
0.
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PROPERTIES
.800 l.ODO
COORDINATES
PCOEFF
COFK
NNP
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3 .
3.
3.
3 .
3 .
3.
4 .
4.
4.
4 .
4 .
4.
5.
5.
5.
5.
5.
5.
6.
6.
6.
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
.0
.0
.0
.0
.0
.5
.5
.5
.0
.0
.0
.5
.5
.5
.0
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
Epsilon
KSP NSPPM KCP GRAY
PRESSURE
WATER CONTENT
RELATIVE CONDUCTIVITY
WATER CAPACITY
167
-------
239
252
214
240
253
215
241
254
216
242
255
217
243
256
218
244
257
219
245
258
220
246
259
221
247
260
417
443
456
418
444
457
419
445
458
420
446
459
421
447
460
422
448
461
423
449
462
424
450
463
425
451
464
426
452
465
427
453
466
428
454
467
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
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12
1
0
12
1
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12
1
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1
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1
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13
13
0
13
13
0
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13
0
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13
0
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13
0
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13
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13
0
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13
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13
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13
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9.
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2.
2.
2.
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2.
3.
3.
3 .
3 .
3.
3.
4.
4.
4 .
4 .
4.
4.
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6.
0.
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3.
3.
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3 .
3.
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3 .
3 .
0.
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3 .
0.
3.
3.
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3 .
3.
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3 .
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0.
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3.
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.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
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0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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.0
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.0
.0
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.0
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.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
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0
0
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0
0
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0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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168
-------
429
455
468
625
651
664
626
652
665
627
653
666
628
654
667
629
655
668
630
656
669
631
657
670
632
658
671
633
659
672
634
660
673
635
661
674
636
662
675
637
663
676
833
859
872
834
860
873
835
861
874
836
862
875
837
863
876
838
864
877
839
865
1
0
12
1
0
12
1
0
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1
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12
1
0
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1
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1
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12
1
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1
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0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0
0
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0
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169
-------
878
840
866
879
841
867
880
842
868
881
843
869
882
844
870
883
845
871
884
1041
1067
1080
1042
1068
1081
1043
1069
1082
1044
1070
1083
1045
1071
1084
1046
1072
1085
1047
1073
1086
1048
1074
1087
1049
1075
1088
1050
1076
1089
1051
1077
1090
1052
1078
1091
1053
1079
1092
1249
1275
1288
1250
12
1
0
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1
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1
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1
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2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
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170
-------
1276
1289
1251
1277
1290
1252
1278
1291
1253
1279
1292
1254
1280
1293
1255
1281
1294
1256
1282
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1296
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1457
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1458
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1460
1486
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1461
1487
1500
1462
1488
1501
1463
1489
1502
1464
1490
1503
1465
1491
1504
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171
-------
1466
1492
1505
1467
1493
1506
1468
1494
1507
1469
1495
1508
1665
1691
1704
1666
1692
1705
1667
1693
1706
1668
1694
1707
1669
1695
1708
1670
1696
1709
1671
1697
1710
1672
1698
1711
1673
1699
1712
1674
1700
1713
1675
1701
1714
1676
1702
1715
1677
1703
1716
1873
1899
1912
1874
1900
1913
1875
1901
1914
1876
1902
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0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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.0
.0
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.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
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.0
.0
.0
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.0
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.0
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172
-------
1915
1877
1903
1916
1878
1904
1917
1879
1905
1918
1880
1906
1919
1881
1907
1920
1882
1908
1921
1883
1909
1922
1884
1910
1923
1885
1911
1924
2081
2107
2120
2082
2108
2121
2083
2109
2122
2084
2110
2123
2085
2111
2124
2086
2112
2125
2087
2113
2126
2088
2114
2127
2089
2115
2128
2090
2116
2129
2091
2117
2130
2092
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
0
12
1
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
0
13
13
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13
13
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13
13
9.
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5.
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4.
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4 .
4.
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4 .
4.
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4 .
4 .
4.
4.
4 .
4 .
4.
4.
4 .
4 .
4.
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4 .
4 .
4.
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4 .
4.
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1.
2.
2.
2.
2.
2.
2.
3 .
3 .
3.
3.
3 .
3 .
4 .
4 .
4.
4.
4 .
4 .
5.
5.
5.
5.
5.
5.
6.
6.
6.
0.
0.
0.
0.
0.
0.
1.
1.
1.
1.
1.
1.
2.
2.
2.
2.
2.
2.
3.
3 .
3 .
3.
3.
3 .
4.
4 .
4 .
4.
4.
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5.
5.
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3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
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3 .
3.
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3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
0.
3 .
3.
0.
3 .
3 .
0.
3.
3 .
0.
3.
3.
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3.
.0
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.0
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.0
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.0
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0
0
0
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0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
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.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
173
-------
2118 0
2131 i;
2093 1
2119 0
2132 i;
0 0
11
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
= = = = = i
1800
1
2
3
4
5
6
7
8
9
10
11
12
181
182
183
184
185
186
187
188
189
190
191
192
361
362
363
364
0 5.0
I 13 9.0
13 0.0
0 5.0
2 13 9.0
0 0.0
3ATA SET 8
10 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
207 1
0 0
3ATA SET 9
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
5.0 5.5 0.0 0.0 0.
5.0 5.5 3.0 0.0 0.
5.0 6.0 3.0 0.0 0.
5.0 6.0 0.0 0.0 0.
5.0 6.0 3.0 0.0 0.
0.0 0.0 0.0 0.0
: SUBREGIONAL DATA
208 0
0 0
1 1
0 0
209 1
0 0
417 1
0 0
625 1
0 0
833 1
0 0
1041 1
0 0
1249 1
0 0
1457 1
0 0
1665 1
0 0
1873 1
0 0
2081 1
0 0
: ELEMENT INCIDENCES
1
2
3
4
5
6
7
8
9
10
11
12
209
210
211
212
213
214
215
216
217
218
219
220
417
418
419
420
NEL
14
15
16
17
18
19
20
21
22
23
24
25
222
223
224
225
226
227
228
229
230
231
232
233
430
431
432
433
222
223
224
225
226
227
228
229
230
231
232
233
430
431
432
433
434
435
436
437
438
439
440
441
638
639
640
641
209
210
211
212
213
214
215
216
217
218
219
220
417
418
419
420
421
422
423
424
425
426
427
428
625
626
627
628
2
3
4
5
6
7
8
9
10
11
12
13
210
211
212
213
214
215
216
217
218
219
220
221
418
419
420
421
0
0
0
0
0
0 . 0 END OF
NREGN
END OF
END OF
END OF
END OF
END OF
END OF
END OF
END OF
END OF
END OF
END OF
END OF
15
16
17
18
19
20
21
22
23
24
25
26
223
224
225
226
227
228
229
230
231
232
233
234
431
432
433
434
223
224
225
226
227
228
229
230
231
232
233
234
431
432
433
434
435
436
437
438
439
440
441
442
639
640
641
642
COORDINATES
NNPLR(K)
GNLR (1,1)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
GNLR (1,2)
210
211
212
213
214
215
216
217
218
219
220
221
418
419
420
421
422
423
424
425
426
427
428
429
626
627
628
629
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
174
-------
365
366
367
368
369
370
371
372
541
542
543
544
545
546
547
548
549
550
551
552
721
722
723
724
725
726
727
728
729
730
731
732
901
902
903
904
905
906
907
908
909
910
911
912
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1261
1262
1263
1264
1265
1266
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
421
422
423
424
425
426
427
428
625
626
627
628
629
630
631
632
633
634
635
636
833
834
835
836
837
838
839
840
841
842
843
844
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1457
1458
1459
1460
1461
1462
434
435
436
437
438
439
440
441
638
639
640
641
642
643
644
645
646
647
648
649
846
847
848
849
850
851
852
853
854
855
856
857
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1470
1471
1472
1473
1474
1475
642
643
644
645
646
647
648
649
846
847
848
849
850
851
852
853
854
855
856
857
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1678
1679
1680
1681
1682
1683
629
630
631
632
633
634
635
636
833
834
835
836
837
838
839
840
841
842
843
844
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1665
1666
1667
1668
1669
1670
422
423
424
425
426
427
428
429
626
627
628
629
630
631
632
633
634
635
636
637
834
835
836
837
838
839
840
841
842
843
844
845
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1458
1459
1460
1461
1462
1463
435
436
437
438
439
440
441
442
639
640
641
642
643
644
645
646
647
648
649
650
847
848
849
850
851
852
853
854
855
856
857
858
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1471
1472
1473
1474
1475
1476
643
644
645
646
647
648
649
650
847
848
849
850
851
852
853
854
855
856
857
858
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1679
1680
1681
1682
1683
1684
630
631
632
633
634
635
636
637
834
835
836
837
838
839
840
841
842
843
844
845
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1666
1667
1668
1669
1670
1671
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
175
-------
1267
1268
1269
1270
1271
1272
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
0
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
14
0
===== DATA
0
T~\7\
1 2287
0 0
1 2287
0 0
1 2287
0 0
1 848
850 4
855 7
863 4
868 189
1058 4
1063 7
1071 4
1076 189
1266 4
1271 7
1279 4
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
0
SET 10
1463 1476
1464 1477
1465 1478
1466 1479
1467 1480
1468 1481
1665 1678
1666 1679
1667 1680
1668 1681
1669 1682
1670 1683
1671 1684
1672 1685
1673 1686
1674 1687
1675 1688
1676 1689
1873 1886
1874 1887
1875 1888
1876 1889
1877 1890
1878 1891
1879 1892
1880 1893
1881 1894
1882 1895
1883 1896
1884 1897
0 0
1684
1685
1686
1687
1688
1689
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
0
1671
1672
1673
1674
1675
1676
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
0
1464
1465
1466
1467
1468
1469
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
0
1477
1478
1479
1480
1481
1482
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
0
1685
1686
1687
1688
1689
1690
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
0
1672
1673
1674
1675
1676
1677
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
0
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
0 END OF IE
: MATERIAL CORRECTION
NCM
TA
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1284 1004 1
0 0
1 848
850 4
855 7
863 4
868 189
1058 4
1063 7
1071 4
0
1
1
1
1
1
1
1
1
SET 11
2
0
0
0
0
0
0
3
0
3
0
3
0
3
0
3
0
3
0
0
3
1
3
1
3
1
3
1
.30-
.ODO
.ODD
.ODD
.ODD
.ODD
.ODD
.Od3
.OdO
.Od3
.OdO
.Od3
.OdO
.Od3
.OdO
.Od3
.OdO
.Od3
.OdO
.OdO
.5D3
.Od3
.5d3
.Od3
.5d3
.Od3
.5d3
.Od3
: INITIAL CONDIDTIONS
2 O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.OdO 0
O.OdO
O.OdO 0
O.OdO
O.OdO 0
O.OdO
O.OdO 0
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.ODO
O.OdO 0
O.OdO
O.OdO 0
O.OdO
O.OdO 0
O.OdO
O.OdO 0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
.OdO
O.OdO
.OdO
O.OdO
.OdO
O.OdO
.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.OdO
O.ODO
.OdO
O.OdO
.OdO
O.OdO
.OdO
O.OdO
.OdO
176
-------
1076 189 1 3.5d3 O.OdO 0.OdO
1266 4 1 1.0d3 O.OdO 0.OdO
1271 7 1 3.5d3 O.OdO O.OdO
1279 4 1 1.0d3 O.OdO O.OdO
1284 1004 1 3.5d3 O.OdO O.OdO
0 0 0 O.OdO O.OdO O.OdO
1 2287 1 O.OdO O.OdO O.OdO
000 O.OdO O.OdO O.OdO
1 2287 1 O.OdO O.OdO O.OdO
0 0 0 O.OdO O.OdO O.OdO
===== DATA SET 14: ELEMENT(DISTRIBUTED) SOURCE/SINK OF TRANSPORT
0000 NSELT NSPRT NSDPT KSAIT
===== DATA SET 15: POINT(WELL) SOURCE/SINK OF TRANSPORT
0000 NWNPT NWPRT NWDPT KWAIT
===== DATA SET 20: VARIABLE BOUNDARY CONDITIONS OF TRANSPORT
120 143 120 NVEST NVNPT NRPRT NRDPT KRAIT
O.ODO O.ODO 1.0D38 O.ODO
1 119 110
00 000
1 119 110
00 000
1 119 110
00 000
1 119 110
00 000
1 119 110
00 000
1 119 110
00 000
1 119 110
00 000
1 11 1 196 404 405 197 1 111
13 11 1 404 612 613 405 1 111
25 11 1 612 820 821 613 1 111
37 11 1 820 1028 1029 821 1 111
49 11 1 1028 1236 1237 1029 1 111
61 11 1 1236 1444 1445 1237 1 111
73 11 1 1444 1652 1653 1445 1 111
85 11 1 1652 1860 1861 1653 1 111
97 11 1 1860 2068 2069 1861 1 111
109 11 1 2068 2276 2277 2069 1 111
0 0000 0 00000
1 12 1 196 1
14 12 1 404 1
27 12 1 612 1
40 12 1 820 1
53 12 1 1028 1
66 12 1 1236 1
79 12 1 1444 1
92 12 1 1652 1
105 12 1 1860 1
118 12 1 2068 1
131 12 1 2276 1
000 00
===== DATA SET 21: DIRICHLET BOUNDARY CONDITIONS OF TRANSPORT
143 220 NDNPT NDPRT NDDPT KDAIT
.0038 O.ODO
.0038 3.5D3
O.ODO
O.ODO
1 12
14 12
27 12
40 12
O.ODO 1
3.5D3 1
111
1 209 1
1 417 1
1 625 1
177
-------
53 12 1 833 1
66 12 1 1041 1
79 12 1 1249 1
92 12 1 1457 1
105 12 1 1665 1
118 12 1 1873 1
131 12 1 2081 1
00000
1 142 110
00 000
1 142 110
00 000
1 142 110
00 000
1 142 110
00 000
1 142 120
00 000
1 142 110
00000
1 142 110
00000
===== DATA SET 22: CAUCHY BOUNDARY CONDITIONS OF TRANSPORT
00 000 NCEST NCNPT NCPRT NCDPT KCAIT
===== DATA SET 23: NEUMANN BOUNDARY CONDITIONS, TRANSPORT
00000 NNEST NNNPT NNPRT NNDPT KNAIT
===== DATA SET 24 : PARAMETERS CONTROLLOING TRACKING SCHEME
110222111 222 2
l.Od-2 l.Od-2
===== DATA SET 25 : VELOCITY AND MOISTURE CONTENT
1 2287 1 9.0D-2 0.ODD 0.ODD 0.ODD 0.ODD 0.ODD
000 O.ODO O.ODO O.ODO 0.ODD 0.ODD 0.ODD
1 1799 1 l.ODO O.ODO
0 0 0 O.ODO O.ODO
0 ====== END OF JOB ====== 00000
Figure 4.28, Figure 4.29, and Figure 4.30 show the simulation results of substrate, oxygen, and total microbial
mass distributions at 100 days and 200 days on x-y crosssection, respectively. Figure 4.31, Figure 4.32, and
Figure 4.33 are the results of substrate, oxygen, and microbe at 100 days and 200 days on x-z crosssection,
respectively.
178
-------
(a)
Substrate at Time = 100 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
\
0
10
15
20
\
25
30
\
35
40
45
(b)
X
Oxygen at Time = 100 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
2000 3000
\
0
10
15
20
\
25
30
\
35
40
45
X
Figure 4.28 Dissolved plumes at 100 days: (a) substrate and (b) oxygen on x-y
crosssection. Concentrations isolines are in micrograms per liter
179
-------
(a)
Substrate at Time = 200 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
10
15
20
25
30
35
40
45
(b)
X
Oxygen at Time = 200 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
3000
\
0
10
15
20
\
25
30
\
35
40
45
X
Figure 4.29 Dissolved plumes at 200 days: (a) substrate and (b) oxygen on x-y
crosssection. Concentrations isolines are in micrograms per liter
180
-------
(a)
Microbe at Time = 100 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
10
15
20
25
30
35
40
45
(b)
X
Microbe at Time = 200 Days (NXG=NYG=NZG=2)
5.0-
4.5-
4.0-
3.5-
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
\
0
10
15
20
\
25
30
\
35
40
45
X
Figure 4.30 Total microbial mass distributions: (a) 100 and (b) 200 days on x-y
crosssection. Concentrations isolines are in mg/liter of aquifer materials
181
-------
(a)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
\
0
6-
5-
4-
3-
2-
1-
0-
\
0
Substrate at Time = 100 Days (NXG=NYG=NZG=2)
10
15
20
25
30
35
X
Oxygen at Time = 100 Days (NXG=NYG=NZG=2)
10
15
20
25
30
35
X
\
40
\
40
Figure 4.31 Dissolved plumes at 100 days: (a) substrate and (b) oxygen on x-z
crosssection. Concentrations isolines are in micrograms per liter
45
45
182
-------
(a)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
\
0
6-
5-
4-
3-
2-
1-
0-
\
0
Substrate at Time = 200 Days (NXG=NYG=NZG=2)
10
15
20
25
30
35
X
Oxygen at Time = 200 Days (NXG=NYG=NZG=2)
3000
10
15
20
25
30
35
X
\
40
\
40
Figure 4.32 Dissolved plumes at 200 days: (a) substrate and (b) oxygen on x-z
crosssection. Concentrations isolines are in micrograms per liter
45
45
183
-------
(a)
N
(b)
N
6-
5-
4-
3-
2-
1-
0-
\
0
6-
5-
4-
3-
2-
1
0-
\
0
Microbe at Time = 100 Days (NXG=NYG=NZG=2)
10
15
20
25
30
35
X
Microbe at Time = 200 Days (NXG=NYG=NZG=2)
10
15
20
25
30
35
X
\
40
\
40
Figure 4.33 Total microbial mass distributions: (a) 100 and (b) 200 days on x-z
crosssection. Concentrations isolines are in mg/liter of aquifer materials
45
45
184
-------
REFERENCES
Bachelor, G. A., D. E. Cawlfield, F. T. Lindstrom, and L. Boersma, Denitrification in nonhomogeneous
laboratory scale aquifers: 5: User's manual for the mathematical model LT3VSI, A draft report, 1990.
Benefield, L. D., and F. J. Molz, A model for the activated sludge process which considers wastewater
characteristics, flux behavior, and microbial population, Biotechnol. Bioeng., 26, 352-361, 1984.
Freeze, R. A., Role of subsurface flow in generating surface runoff: 1. Base flow contribution to channel flow,
Water Resour. Res., 8, 609-623, 1972a.
Freeze, R. A., Role of subsurface flow in generating surface runoff: 2. Upstream source areas, Water Resour.
Res., 8, 1272-1283, 1972b.
Frind, E. O., Simulation of long-term transient density-dependent transport in groundwater, Adv.Water
Res., Vol. 5, No. 2, 73-88, 1982
Herbert, D., Some principles of continuous culture, in Recent Progress in Microbiology, edited by G.
Tunevall, Blackwell Scientific Publishers, Oxford, England, 1958.
Huyakorn, P. S., E. P. Springer, V. Guvanasen, and T. D. Wadsworth, A three-dimensional finite-element
model for simulating water flow in variably saturated porous media, Water Resources Research, Vol.
22, No. 13, 1790-1808, 1986.
MacQuarrie, K. T. B. and E. A. Sudicky, Simulation of biodegradable organic contaminants in groundwater,
2. plume behavior in uniform and random flow fields, Water Resour. Res., 26(2), 207-222, 1990.
Molz, F. J. M. A. Widdowson, and L. D. Benfield, Simulation of microbial growth dynamics coupled to
nutrient and oxygen transport in porous media, Water Resour. Res., 22(8), 1207-1216, 1986.
van Genuchten, M. Th., A closed form equation for predicting the hydraulic conductivity of unsaturated soils,
Soil Science Society of Ameirca Journal, 44, 892-898, 1980.
Widdowson, M. A., F. J. Molz, and L. D. Benfield, A numerical transport model for oxygen- and nitrate-based
respiration linked to substrate and nutrient availability in porous media, Water Resour. Res., 24(9),
1553-1565, 1988.
Yeh, G. T., and D. S. Ward, FEMWATER: A finite-element model of water flow through saturated-
unsaturated porous media, Rep. ORNL-5567, Oak Ridge Nat. Lab., Oak Ridge, Tenn., 37831, 137
pp., 1980.
Yeh, G. T., and D. S. Ward, FEMWASTE: A finite-element model of waste transport through saturated-
unsaturated porous media, Rep. ORNL-5601, Oak Ridge Nat. Lab., Oak Ridge, Tenn., 37831, 137
pp., 1981.
Yeh, G. T., FEMWATER: A finite element model of water flow through saturated-unsaturated porous media,
First Revision, Rep. ORNL-5567/R1, Oak Ridge Nat. Lab., Oak Ridge, Tenn., 37831, 258 pp., 1987.
185
-------
Yeh, G. T., A Lagrangian-Eulerian method with zoomable hidden fine mesh approach to solving advection-
dispersion equations, Water Resources Research Vol. 26, No. 6, 1133-1144, 1990.
Yeh, G. T., J. R. Chang, and T. E. Short, An exact peak capturing and oscillation-free scheme to solve
advection-dispersion transport equations, Water Resour. Res., 28(11), 2937-2951, 1992a.
Yeh, G. T., Class notes: CE597C: Computational Subsurface Hydrology Part II, The Pennsylvania State
University, University Park, Pa., 16802, Spring semester 1992b.
Yeh, G. T., 3DFEMWATER: A Three-Dimensional Finite Element Model of WATER Flow through
Saturated-Unsaturated Media: Version 2.0, Short course notes of Simulation of Subsurface Flow and
Contaminant Transport by Finite Element and Analytical Methods, The Pennsylvania State University,
University Park, Pa., 16802, May 1993a.
Yeh, G. T., 3DLEWASTE: A Three-Dimensional Hybrid Lagrangian-Eulerian Finite Element Model of
WASTE Transport through Saturated-Unsaturated Media: Version 2.0, Short course notes of
Simulation of Subsurface Flow and Contaminant Transport by Finite Element and Analytical
Methods, The Pennsylvania State University, University Park, Pa., 16802, May 1993b.
Yeh, G. T., J. R. Chang, J. P. Gwo, H. C. Lin, D. R. Richards, and W. D. Martin, 3DSALT: A three-
dimensional finite element model of density-dependent flow and transport through saturated-
unsaturated media, Instruction Report HL-94-1, US Army Corps of Engineers, 1994.
186
-------
APPENDIX A: Data Input Guide
* * * Data sets 2 through 25 must be preceded by a record * * *
*** containing description of the data set ***
1. TITLE
One record with FORMAT(I5,A70) per problem. This record contains the following variables.
1. NPROB = Problem number.
2. TITLE = Title of the problem. It may contain up to 70 characters.
2. OPTION PARAMETERS
Seven lines of free-formatted data records are required for this data set.
A. Line 1:
1.1. IMOD = Integer indicating the simulation modes to be carried on. 0 = Do the initial variable
computation ONLY, for both flow and transport simulations. The purpose for this mode is
to verify the input data. 10 = Do the flow simulation ONLY; 1 = Do the transport simulation
only; 11 = Do both flow and transport simulations.
1.2. IGEOM = Integer indicating if (1) the geometry, boundary and pointer arrays are to be printed;
(2) the boundary and pointer arrays are to be computed or read via logical units. If to be
computed, they should be written on logical units. If IGEOM is an even number, (1) will not
be printed. If IGEOM is an odd number, (1) will be printed. If IGEOM is less than or equal
to 1, boundary arrays will be computed.
1.3. IBUG = Integer indicating if the diagnostic output is desired? 0 = No, nonzero = Yes.
1.4. ICHNG = Integer control number indicating if the cyclic change of rainfall-seepage nodes is to
be printed, =0, no = 1, yes.
B. Line 2:
2.1. NITFTS = Iteration numbers allowed for solving the coupled nonlinear equations for the steady-
state solutions. If the steady-state simulation is for either flow or transport only, then
the value of NITFTS must be set to 1.
2.2. OMEFTS = Iteration parameter for solving the coupled nonlinear equations for the steady-state
solutions.
2.3. ALLOW = The allowed factor for neglecting concentrations in a convergence test.
A-1
-------
C. Line 3:
3.1. KSSf = Flow steady-state control, 0 = steady-state solution desired, 1 = transient state or transient
solutions.
3.2. KSSt = Transport steady-state control, 0 = steady-state solution desired, 1 = transient state or
transient solutions.
3.3. ILUMP = Is mass lumping? 0 = no, 1 = yes.
3.4. IMID = Is mid-difference? 0 = no, 1 = yes.
3.5. IPNTSf = matrix solver indicator for flow simulation:
0 = block iteration solver,
1 = successive iteration methods,
2 = polynomial preconditioned conjugate gradient methods,
3 = incomplete Cholesky preconditioned conjugate gradient methods.
3.6. IPNTSt = matrix solver indicator for transport simulation:
0 = block iteration solver,
1 = successive iteration methods,
2 = polynomial preconditioned conjugate gradient methods,
3 = incomplete Cholesky preconditioned conjugate gradient methods.
3.7. NSTRf = No. of logical records to be read via logical unit 11 for restarting calculation.
0 = No restart.
3.8. NSTRt = No. of logical records to be read via logical unit 12 for restarting calculation.
0 = No restart.
3.9. MICONF = Index of the simulation of microbial configuration:
0 = mobile microbes
1 = immobile microbes
3.10. IQUAR = Index of using quadrature for numerical integration:
1 = Nodal quadrature for surface integration, Gaussian quadrature for element
integration,
2 = Nodal quadrature for surface integration, Nodal quadrature for element
integration,
3 = Gaussian quadrature for surface integration, Gaussian quadrature for element
integration,
4 = Gaussian quadrature for surface integration, Nodal quadrature for element
integration.
D. Line 4:
4.1. KGRAV = Gravity term control: 0 = no gravity term, 1 = with gravity term.
A-2
-------
4.2. Wf = Time derivative weighting factor for flow simulations:
0.5 = Crank-Nicolson central,
1.0 = backward difference and/or mid-difference.
4.3. OMEf = Iteration parameter for solving the nonlinear flow equation:
0.0 - 1.0 = under-relaxation,
1.0 - 1.0 = exact relaxation,
1.0 - 2.0 = over-relaxation.
4.4. OMIf = Relaxation parameter for solving the linearized flow matrix equation pointwisely or
blockwisely:
0.0 - 1.0 = under relaxation,
1.0 - 1.0 = exact relaxation,
1.0 - 2.0 = over relaxation.
4.5. CNSTKR = constraint on relative hydraulic conductivity:
0 = no constraint,
0.0001, 0.001, or 0.01 should be tried when nonconvergency occurs in solving the
nonlinear flow equation.
E. Line 5:
5.1. KVIt = Velocity input control:
-1 = card input for velocity and moisture content,
1 = steady-state velocity and moisture content will be calculated from steady-state
flow simulations,
2 = transient velocity and moisture content will be obtained from transient-flow
simulations.
5.2. IWET = Weighting function control which is used only if the conventional FEM is employed
to solve transport equations:
0 = Galerkin weighting,
1 = Upstream weighting.
5.3. IOPTIM = Optimization factor computing indicator which is used only if the conventional FEM
is employed to solve transport equations:
1 = Optimization factor is to be computed,
0 = optimization factor is to be set to -1.0 or 0.0 or 1.0 depending on the velocity.
5.4. KSORP = Sorption model control:
1 = linear isotherm, the only option used in this model.
5.5. LGRN = Lagrangian approach control: 0 = no, 1 = yes.
F. Line 6:
6.1. Wt = Time derivative weighting factor for transport simulations:
0.5 = Crank-Nicolson central,
A-3
-------
1.0 = backward difference and/or mid-difference.
6.2. WVt = Integration factor for velocity used only if the conventional FEM is employed to solve
transport equations; should be between 0.0 to 1.0.
6.3. OMEt = Iteration parameter for solving the nonlinear transport equation; always used 1.0
because KSORP = 1.
6.4. OMIt = Relaxation parameter for solving the linearized transport matrix equation pointwisely
or blockwisely; used only the block iteration or the pointwise solver:
0.0 - 1.0 = under relaxation,
1.0 - 1.0 = exact relaxation,
1.0 - 2.0 = over relaxation.
G. Line 7: This line is needed if and only if IPNTSf or IPNTSt is greater than 0.
7.1. IEIGEN = signal of parameter estimation for GG in the polynomial preconditioned conjugate
gradient method:
zero = not requested,
non-zero = requested.
7.2. GG = the upper bound on the maximum eigenvalue of the coefficient matrix used in the
polynomial preconditioned conjugate gradient method. When requested, GG is
usually set to 1.0.
3. ITERATION PARAMETERS
Two subsets of free-formatted data records are required for this data set, one for flow simulations, the
other for transport simulations.
A. subset 1: For flow simulations -
1.1. NITERf = Number of iterations allowed for solving the non-linear flow equation.
1.2. NCYLf = No. of cycles permitted for iterating rainfall-seepage boundary conditions per time step.
1.3. NPITERf = No. of iterations permitted for solving the linearized flow equation using block or
pointwise iterative matrix solver.
1.4. TOLAf = Steady-state convergence criteria for flow simulations, (L).
1.5. TOLBf = Transient-state convergence criteria for flow simulations, (L).
B. subset 2: For transport simulations -
2.1. NITERt = Number of iterations allowed for solving the non-linear transport equation.
2.2. NPITERt = No. of iterations for block or pointwise iteration to solve the linearized transport
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equation.
2.3. TOLAt = Steady-state convergence criteria for transport simulations.
2.4. TOLBt = Transient-state convergence criteria for transport simulations.
4. TIME CONTROL PARAMETERS
Five subsets of data records are required for this data set.
A. subset 1: free format
1.1. NTI = Number of time steps or time increments.
1.2. NDTCHG = No. of times to reset time-step size to initial time-step size.
B. subset 2: free format
2.1. BELT = Initial time step size, (T).
2.2. CHNG = Percentage of change in time-step size in each of the subsequent time increments,
(dimensionless in decimal point).
2.3. DELMAX = Maximum value of DELT, (T).
2.4. TMAX = Maximum simulation time, (T).
C. subset 3: format = 8011
3.1. KPRO = Printer control for steady state and initial conditions;
0 = print nothing,
1 = print FLOW, FRATE, and TFLOW,
2 = print above (1) plus pressure head H,
3 = print above (2) plus total head,
4 = print above (3) plus moister content,
5 = print above (4) plus Darcy velocity.
3.2. KPR(I) = Printer control for the I-th (I = 1,2, ..., NTI) time step similar to KPRO.
D. subset 4: format = 8011
4.1. KDSKO = Auxiliary storage control for steady state and initial condition:
0 = no storage, 1 = store on Logical Unit 11 (for flow output) or 12 (for transport output).
4.1. KDSK(I) = Auxiliary storage control for the I-th time step similar to KDSKO.
E. subsetS: free format
5.1. TDTCH(I,1) = Time when the I-th (I = 1, 2, ..., NOTCH) step-size-resetting is needed.
A-5
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5.2. TDTCH(I,2) = Time-step size of the first step of the I-th (I = 1, 2, ..., NOTCH)
step-size-resetting.
**** NOTE: Two ways to terminate the execution: either NTI is reached first or TMAX is reached
first.
5. MATERIAL PROPERTIES
Four subsets of free-formatted data records are required for this data set.
A. subset 1:
1.1. NMAT = Number of material types.
1.2. NMPPM = No. of material properties per material. > 7 for the present version.
1.3. NCC = No. of components in the system. Since the kinetic reaction model is built in the program
according to Eq. (2.9) through Eq. (2.15), NCC is assigned to 7 and IRXN is set to 1 if
the microbial-chemical Monod type reactions are involved. NCC can be 1 for the single
component simulation and NCC is equal to 2 by using stochiometric model which results
in IRXN = -1. NCC can be any value if users modify the kinetic model in the program
(Subroutine ADVRX).
1.4. IRXN = the index indicating the chemical-microbial kinetic reaction type. -1 refers to
stochiometric reaction; 1 indicates Monod type reaction.
The following three subdata sets ( B ~ D) are needed only if IMOD = 10 or IMOD = 11.
B. subset 2: A total of NMAT records are needed per problem, one each for one material.
2.1.1. PROPf(1,1) = Saturated xx-conductivity or permeability of the medium I, (L/T or L* * 2).
2.1.2. PROPf(I,2) = Saturated yy-conductivity or permeability of the medium I, (L/Tor L**2).
2.1.3. PROPf(I,3) = Saturated zz-conductivity or permeability of the medium I, (L/TorL**2).
2.1.4. PROPf(I,4) = Saturated xy-conductivity or permeability of the medium I, (L/Tor L**2).
2.1.5. PROPf(I,5) = Saturated xz-conductivity or permeability of the medium I, (L/TorL**2).
2.1.6. PROPf(I,6) = Saturated yz-conductivity or permeability of the medium I, (L/T or L* *2).
2.1.7. PROPf(I,7) = Fluid density of the medium I, (L/T or L* *2).
C. subset 3: The intrinsic density for each component used in Eq. (2.2b).
A-6
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A total number of NCC parameters appears in this record. NCC is the total number of
components in the system.
3.1. DINTS(I) = intrinsic density (M/L**3) of the I-th component.
D. subset 4: Coefficient for calculating dynamic viscosity used in Eq. (2.2c).
A total number of NCC parameters appears in this record. NCC is the total number of
components in the system.
4.1. RHOMU(I) = coefficient for calculating dynamic viscosity as a function of concentration, (L3/M).
The following three subdata sets ( E~ H) are needed only if IMOD = 1 or IMOD = 11.
Subdata sets E to G should be repeated NMAT times.
E. subsetS: A total of NMAT records are needed per problem, one each for one material.
5.1. PROPt(I,l) = Bulk density, (M/L**3) for medium I.
5.2. PROPt(I,2) = Longitudinal dispersivity, (L), for medium I.
5.3. PROPt(I,3) = Lateral dispersivity, (L), for medium I.
5.4. PROPt(I,4) = Molecular diffusion coefficient, (L**2/T), for medium I.
5.5. PROPt(I,5) = Tortuosity, (Dimensionless) for medium I.
5.6. PROPt(I,6) = Decay constant, (1/L) in medium I.
5.7. PROPt(I,7) = 0.0.
F. subset 6: A total number of NCC parameters appears in this record. NCC is the total number of
components in the system.
6.1. RKD(I,J) = distribution coefficient of the J-th component in the I-th material.
G. subset 7: A total number of NCC parameters appears in this record. NCC is the total number of
components in the system.
7.1. TRANC(I,J) = chemical transformation rate of the J-th component in the I-th material.
H. subset 8: MICROBE-CHEMICAL INTERACTION CONSTANTS
Eleven records of FREE-FORMATTED data are needed.
8.1. Record 1: Four parameters describing the specific growth rate of microbes (1/T) are needed in
this record. If there are no microbes in this system, the following four numbers have
A-7
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to be set to zeros.
(1) GRATE(l) = Maximum specific oxygen-based growth rate for microbe #1. (\i0m in Eqs. (2.9) ~
(2.15)).
(2) GRATE(2) = Maximum specific nitrate-based growth rate for microbe #2. (^2) in Eqs. (2.9) ~
(2.15)).
(3) GRATE(3) = Maximum specific oxygen-based growth rate for microbe #3. (|4(3) in Eqs. (2.9) ~
(2.15)).
(4) GRATE(4) = Maximum specific nitrate-based growth rate for microbe #3. ((jj-3-1 in Eqs. (2.9) ~
(2.15)).
8.2. Record 2 : Four yield coefficients (M/M) are needed in this record and these four values cannot
be zeros.
(1) YCOEFF(l) = Yield coefficient for microbe #1 utilizing Oxygen. (Y0(1) in Eqs. (2.9) & (2. 12)).
(2) YCOEFF(2) = Yield coefficient for microbe #2 utilizing Nitrate. (Yn(2) in Eqs. (2.9) & (2. 12)).
(3) YCOEFF(3) = Yield coefficient for microbe #3 utilizing Oxygen. (Y0(3) in Eqs. (2.9) & (2. 12)).
(4) YCOEFF(4) = Yield coefficient for microbe #3 utilizing Nitrate. (Yn(3) in Eqs. (2.9) & (2. 12)).
8.3. Record 3: Four retarded substrate saturation constants (M/L3) are needed in this record.
(1) RTARDS(l) = Retarded substrate saturation constant under aerobic conditions with respect to
microbe #1. (Kj1' in Eqs. (2.9) ~ (2.15)).
(2) RTARDS(2) = Retarded substrate saturation constant under anaerobic conditions with respect to
microbe #2. (Kj2' in Eqs. (2.9) ~ (2.15)).
(3) RTARDS(3) = Retarded substrate saturation constant under aerobic conditions with respect to
microbe #3. (Kj3' in Eqs. (2.9) ~ (2.15)).
(4) RTARDS(4) = Retarded substrate saturation constant under anaerobic conditions with respect to
microbe #3. (Kj3' in Eqs. (2.9) ~ (2.15)).
8.4. Record 4: Four retarded Oxygen or Nitrate saturation constants (M/L3) are needed in this record.
(1) RTARDO(l) = Retarded Oxygen saturation constant under aerobic conditions with respect to
microbe #1. (K0(1) in Eqs. (2.9) ~ (2.15)).
(2) RTARDO(2) = Retarded Nitrate saturation constant under anaerobic conditions with respect to
microbe #2. (K in Eqs. (2.9) ~ (2.15)).
(3) RTARDO(3) = Retarded Oxygen saturation constant under aerobic conditions with respect to
microbe #3. (K0(3) in Eqs. (2.9) ~ (2.15)).
(4) RTARDO(4) = Retarded Nitrate saturation constant under anaerobic conditions with respect to
microbe #3. (K^ in Eqs. (2.9) ~ (2.15)).
8.5. Record 5: Four retarded nutrient saturation constants (M/L3) are needed in this record.
(1) RTARDN(l) = Retarded nutrient saturation constant under aerobic conditions with respect to
microbe #1. (Kp0(1) in Eqs. (2.9) ~ (2.15)).
(2) RTARDN(2) = Retarded nutrient saturation constant under anaerobic conditions with respect to
microbe #2. (Kpn(2) in Eqs. (2.9) ~ (2.15)).
(3) RTARDN(3) = Retarded nutrient saturation constant under aerobic conditions with respect to
microbe #3. (Kp0(3) in Eqs. (2.9) ~ (2.15)).
(4) RTARDN(4) = Retarded nutrient saturation constant under anaerobic conditions with respect to
microbe #3. (Kpn(3) in Eqs. (2.9) ~ (2.15)).
8.6. Record 6: Four Oxygen-use or Nitrate-use coefficients for synthesis are needed in this record.
A-8
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(1) SCOEFF(l) = Oxygen-use coefficient for synthesis by microbe #1. (y0(1) in Eq. (2.10)).
(2) SCOEFF(2) = Nitrate-use coefficient for synthesis by microbe #2. (yn(2) in Eq. (2.11)).
(3) SCOEFF(3) = Oxygen-use coefficient for synthesis by microbe #3. (y0(3) in Eq. (2.10)).
(4) SCOEFF(4) = Nitrate-use coefficient for synthesis by microbe #3. (yn(3) in Eq. (2.11)).
8.7. Record 7: Four Oxygen-use or Nitrate-use coefficients for energy are needed in this record.
(1) ECOEFF(l) = Oxygen-use coefficient for energy by microbe #1. (ce0(1) in Eq. (2.10)).
(2) ECOEFF(2) = Nitrate-use coefficient for energy by microbe #2. (cen(2) in Eq. (2.11)).
(3) ECOEFF(3) = Oxygen-use coefficient for energy by microbe #3. (ce0(3) in Eq. (2.10)).
(4) ECOEFF(4) = Nitrate-use coefficient for energy by microbe #3. (cen(3) in Eq. (2.11)).
8.8. Record 8: Four microbial decay coefficients (1/T) are needed in this record.
(1) DCOEFF(l) = Microbial decay coefficient of aerobic respiration of microbe #1. (A0(1) in Eqs.
(2.13) & (2.15)).
(2) DCOEFF(2) = Microbial decay coefficient of anaerobic respiration of microbe #2. (An(2) in Eqs.
(2.14) & (2.15)).
(3) DCOEFF(3) = Microbial decay coefficient of aerobic respiration of microbe #3. (A0(3) in Eqs.
(2.13) & (2.15)).
(4) DCOEFF(4) = Microbial decay coefficient of anaerobic respiration of microbe #3. (An(3) in Eqs.
(2.14) & (2.15)).
8.9. Record 9: Four Oxygen or Nitrate saturation constants (M/L3) for decay are needed in this record.
(1) SATURC(l) = Oxygen-saturation constant for decay with respect to microbe #1. (ro(1) in Eq.
(2.10)).
(2) SATURC(2) = Nitrate-saturation constant for decay with respect to microbe #2. (Fn(2) in Eq.
(2.H)).
(3) SATURC(3) = Oxygen-saturation constant for decay with respect to microbe #3. (F0(3) in Eq.
(2.10)).
(4) SATURC(4) = Nitrate-saturation constant for decay with respect to microbe #3. (Fn(3) in Eq.
(2.H)).
8.10. Record 10: Four nutrient-use coefficients for the production are needed in this record.
(1) PCOEFF(l) = Nutrient-use coefficient for the production of microbe #1 with aerobic respiration.
(e0ซmEq.(2.12)).
(2) PCOEFF(2) = Nutrient-use coefficient for the production of microbe #2 with anaerobic
respiration. (en(2) in Eq. (2.12)).
(3) PCOEFF(3) = Nutrient-use coefficient for the production of microbe #3 with aerobic respiration.
(e0ซmEq.(2.12)).
(4) PCOEFF(4) = Nutrient-use coefficient for the production of microbe #3 with anaerobic
respiration. (en(3) in Eq. (2.12)).
8.11. Record 11: One variable (M/L3) is included in this record.
(1) COFK = Inhibition coefficient. (Kc in inhibition function, I(C0))
6. SOIL PROPERTIES
Three or five subsets of free-formatted data records are required for this data set depending on the
forms of the soil property functions given.
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A. subset 1: Soil property control parameters
1.1. KSP = Soil property input control: 0 = analytical input, 1 = Tabular data input.
1.2. NSPPM = Number of points in tabular soil property functions or number of parameters to
specify analytical soil functions per material.
1.3. KCP = Permeability input control:
0 = input saturated hydraulic conductivity,
1 = input saturated permeability.
1.4. RHO = Referenced density of water, (M/L**3).
1.5. GRAY = Acceleration of gravity, (L/T**2).
1.6. VISC = Referenced dynamic viscosity of water, (M/L/T).
B. subset 2a: Analytical soil parameters - This sub-data-set is needed if and only if KSP is 0. Two
sets of records are required, one for moisture-content parameters and the other for
conductivity (permeability) parameters and each set should be repeated NMAT times.
2.1. SPP(J,I,1) = Analytical moisture-content parameter J of material I, J = 1..NSPPM. NMAT sets
of these parameters are required for I = 1..NMAT. That is, if SPP(J,I,1) for J =
1..NSPPM can be put on a single line, NMAT consecutive lines are needed for the
sets of parameters.
2.2. SPP(J,I,2) = Analytical relative conductivity parameter J of material I. Similar input data setting
is required for these parameters as for SPP(J,I,1).
C. subset 2b: Soil properties in tabular form - This sub-data-set is needed if and only if KSP is not 0.
Four sets of records are needed ~ for pressure, water-content, relative conductivity
(or relative permeability), and water capacity, respectively.
3.1. SPP(J,I,4) = Tabular value of pressure head of the J-th point for material I. NMAT sets of these
parameters are required for I = 1..NMAT. That is, if SPP(J,I,4) for J = 1..NSPPM can
be put on a single line, NMAT consecutive lines are needed for the sets of
parameters.
3.2. SPP(J,I,1) = Tabular value of moisture-content of the J-th point in material I. Similar input data
setting is required for these parameters as for SPP(J,I,4).
3.3. SPP(J,I,2) = Tabular value of relative conductivity of the J-th point in material I. Similar input
data setting is required for these parameters as for SPP(J,I,4).
3.4. SPP(J,I,3) = Tabular value of moisture-content capacity of the J-th point in material I. Similar
input data setting is required for these parameters as for SPP(J,I,4).
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7. NODAL COORDINATE
Two subsets of free-formatted data records are required if NSTRf = 0 and NSTRt = 0.
A. subset 1:
1.1. NNP = Number of nodes.
B. subset 2: nodal coordinates - Coordinates for NNP nodes are needed if KVI .LE. 0. Usually a total
of NNP records (KVI records are required. However, if a group of subsequent nodes
appear in a regular pattern, automatic generation can be made. Each record contains
the following variables and is FREE-FORMATTED.
2.1. NI = Node number of the first node in the sequence.
2.2. NSEQ = NSEQ subsequent nodes will be automatically generated.
2.3. NAD = Increment of node number for each of the NSEQ subsequent nodes.
2.4. XNI = x-coordinate of node NI, (L).
2.5. YNI = y-coordinate of node NI, (L).
2.6. ZNI = z-coordinate of node NI, (L).
2.7. XAD = Increment of x-coordinate for each of the NSEQ subsequent nodes, (L).
2.8. YAD = Increment of y-coordinate for each of the NSEQ subsequent nodes, (L).
2.9. ZAD = Increment of z-coordinate for each of the NSEQ subsequent nodes, (L).
**** NOTE: A record with 9 O's must be used to signal the end of this data set.
SUBREGION DATA
This data set is required if either IPNTSf or IPNTSt is 0. Three subsets of free-formatted data records
are required.
A. subset 1: One free format data record is needed for this sub-data-set.
1.1. NREGN = No. of subregions.
B. subset 2: No. of Nodes for each Subregion - Normally, NREGN records are required. However,
if regular pattern appears, automatic generation can be made. Each record contains the
5 variables and is FREE-FORMATTED.
2.1. NK = Subregion number of the first subregion region in a sequence.
A-11
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2.2. NSEQ = NSEQ subsequent subregions will have their no. of nodes automatically generated.
2.3. NKAD = Increment of NK in each of the NSEQ subsequent subregions.
2.4. NODES = No. of nodes for the subregion NK.
2.5. NOAD = Increment of NODES in each of the NSEQ subsequent subregions.
**** NOTE: A record with 5 O's must be used to end the input of this subdata set.
C. subset 3: Mapping between Global nodes and Subregion Nodes - This subdata set should be
repeated NREGN times, one for each subregion. For each subregion, normally, LNNP
records are needed. However, automatic generation can be made if subregional node
number appears in regular pattern. Each record contains 5 variables and is FREE-
FORMATTED.
3.1. LI = Local node number of the first node in a sequence.
3.2. NSEQ = NSEQ subsequent local nodes will be generated automatically.
3.3. LIAD = Increment of LI for each of the NSEQ subsequent nodes.
3.4. NI = Global node number of local node LI.
3.5. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's must be used to signal the end of this subdata set.
9. ELEMENT DATA
Two subsets of free-formatted data records are required for this data set.
A. subset 1:
1.1. NEL = Number of elements.
B. subset 2: Element incidence for NEL elements is needed if NSTRt = 0 and NSTRf = 0. Usually,
atotal ofNEL records are needed. However, if agroup of elements appear in aregular
pattern, automatic generation is made. Each record contains the following variable and
is FREE-FORMATTED.
2.1. MI = Global element number of the first element in a sequence.
2.2. NSEQ = NSEQ subsequent elements will be automatically generated.
2.3. MIAD = Increment of MI for each of the NSEQ subsequent elements.
A-12
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2.4. IE(MI,1) = Global node number of the first node of element MI.
2.5. IE(MI,2) = Global node number of the second node of element MI.
2.6. IE(MI,3) = Global node number of the third node of element MI.
2.7. IE(MI,4) = Global node number of the fourth node of element MI.
2.8. IE(MI,5) = Global node number of the fifth node of element MI.
2.9. IE(MI,6) = Global node number of the sixth node of element MI.
2.10. IE(MI,7) = Global node number of the seventh node of element MI.
2.11. IE(MI,8) = Global node number of the eighth node of element MI.
2.12. IEMAD = Increment of IE(MI,1) through IE(MI,8) for each of the NSEQ elements.
**** Note: IE(MI,1) - IE(MI,8) are numbered according to the convention shown in following
diagram. The first four nodes start from the front, lower, left corner and progress around the bottom
element surface in a counterclockwise direction. The other four nodes begin from the front, upper,
left corner and progress around the top element surface in a counterclockwise direction.
A-13
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(A)
(C)
Figure A. 1 Global Node Number Index of (A) a Hexahedral, (B) a Triangular Prism,
and (C) a Tetrahedral Element.
A-14
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10. MATERIAL TYPE CORRECTION
Two subsets of free-formatted data records are required for this data set.
A. subset 1:
1.1. NCM = Number of elements with material corrections.
B. subset 2: This set of data records is required only if NCM > 0. Normally, NCM records are
required. However, if a group of elements appear in a regular pattern, automatic
generation may be made. Each record contains the following variables.
2.1. MI = Global element number of the first element in the sequence.
2.2. NSEQ = NSEQ subsequent elements will be generated automatically.
2.3. MAD = Increment of element number for each of the NSEQ subsequent elements.
2.4. MITYP = Type of material correction for element MI.
2.5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent elements.
**** NOTE: A line with 5 O's must be used to signal the end of this data set.
11. CARD INPUT FOR INITIAL OR PRE-INITIAL CONDITIONS
Two subsets of free-formatted data records are required for this data set, one for initial pressure head,
the other for initial concentration. Generally, for each subset NNP record, one record for each node
is needed. However, if a group of nodes appears in regular pattern, auto-generation is made.
A. subset 1: Initial pressure head -The first record contains one variable and each of subsequent
records contains 6 variables. This subset is needed if IMOD = 10 or IMOD =11.
1.1 IHTR = Is total head to be read as the initial condition? 0 = No, 1 = yes.
2.1. NI = Global node number of the first node in the sequence.
2.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
2.3. NAD = Increment of node number for each of the NSEQ nodes.
2.4. HNI = Initial or pre-initial pressure head of node NI, (L).
2.5. HAD = Increment of initial or pre-initial head for each of the NSEQ nodes, (L).
2.6. HRD = 0.0.
**** NOTE: A line with 6 O's must be used to signal the end of this data set.
A-15
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NOTE ON INITIAL CONDITIONS AND RESTARTING: The initial condition for a transient
calculation may be obtained in two different ways: from card input, or steady-state calculation using
time-invariant boundary conditions that are different from those for transient computation. In the
latter case a card input of the pre-initial conditions is required as the zero-th order iterate of the steady-
state solution.
NOTE ON STEADY-STATE INPUT: Steady-state option may be used to provide either the final
state of a system under study or the initial conditions for a transient state calculation. In the former
case KSSf = 0, KSSt = 0, and NTI = 0, and in the latter case KSSf = 0 or KSSt > 0 and NTI > 0. If
KSSf > 0, there will be no steady-state calculation for flow part.
B. subset 2: Initial concentration for microbe #1 - each record contains the following variables.
This subset is needed if IMOD = 1 or IMOD = 11.
2.1. NI = Global node number of the first node in the sequence.
2.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
2.3. NAD = Increment of node number for each of the NSEQ nodes.
2.4. CNI = Initial or pre-initial concentration of node NI, (M/L**3).
2.5. CAD = Increment of CNI for each of the NSEQ nodes, (M/L**3).
2.6. CRD = Geometrical increment of CNI for each of the NSEQ subsequent nodes.
**** NOTE: A record with 6 O's must be used to signal the end of this data set.
C. subset3 -subsetS: Initial concentration for microbe #2, microbe #3, substrate, oxygen,nitrate, and
nutrient, respectively. The input format is the same as subset 2.
NOTE ON INITIAL CONDITIONS: The initial condition for a transient calculation may be obtained
in two different ways: from card input or steady-state calculation using time-invariant boundary
conditions that are different from those for transient computation. In the latter case a card input of
the pre-initial conditions is required as the zero-th order iterate of the steady-state solution.
NOTE ON STEADY-STATE INPUT: Steady-state option may be used to provide either the final
state of a system under study or the initial conditions for a transient state calculation. In the former
case KSSt = 0, KSSf > and NTI = 0, and in the latter case KSSt = 0, KSSf > and NTI > 0. If KSSt
> 0, there will be no steady-state calculation for transport part.
12. ELEMENT (DISTRIBUTED) SOURCE/SINK FOR FLOW SIMULATIONS
This data set is needed if IMOD = 10 or IMOD =11.
Four subsets of free-formatted data records are required in this data set.
A. subset 1: control parameters
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1.1. NSEL = No. of source/sink elements.
1.2. NSPR = No. of source/sink profiles.
1.3. NSDP = No. of data points in each of the NSPR source/sink profiles.
1.4. KSAI = Is element-source/sink profile to be input analytically, 0 = no, 1 = yes.
B. subset 2: source/sink profiles - This group of data is needed if and only if NSEL .GT. 0. For each
sub-data-record, NSDP of the data pair (TSOSF(J,I),SOSF(J,I)) are required. If this
sub-data-record can be fitted in a line, NSPR lines are needed.
2.1. TSOSF(J,I) = Time of the J-th data point in the I-th profile, (T).
2.2. SOSF(1,I) = Source/sink value of the J-th data point in the I-th profile, (L**3/T/L**2/L).
C. subset 3: global source/sink element number - This group of data is needed if and only if NSEL
.GT. 0. NSEL data points are required for this record.
3.1. MSEL(I) = Global element number of the I-th compressed distributed source/sink element.
D. subset 4: Source type assigned to each element - Usually one record per element. However,
automatic generation can be made. For I-th (I = 1, 2, ..., ) record, it contains the
following.
4.1. MI = Global element number of the first element in the sequence.
4.2. NSEQ = NSEQ elements will be generated automatically.
4.3. MAD = Increment of element number for each of the NSEQ elements.
4.4. MITYP = Source type in element MI.
4.5. MTYPAD = Increment of MITYP for each of the NSEQ elements.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
13. POINT (WELL) SOURCE/SINK DATA FOR FLOW SIMULATION
This data set is needed if IMOD = 10 or IMOD =11.
Four subsets of free-formatted data records are required for this data set.
A. subset 1: control parameters
1.1. NWNP = No. of well or point source/sink nodal points.
1.2. NWPR = No. of well or point source/sink strength profiles.
A-17
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1.3. NWDP = No. of data points in each of the NWPR profiles.
1.4. KWAI = Is well-source/sink profile to be input analytically, 0 = no, 1 = yes.
B. subset 2: source/sink profiles - This group of data is needed if and only if NWNP .GT. 0. For each
sub-data-record, NWDP of the data pair (TWSSF(J,I),WSSF(J,I)) are required. If this
sub-data-record can be fitted in a line, only NWPR lines are needed.
2.1. TWSSF(J,I) = Time of the J-th data point in the I-th profile, (T).
2.2. WSSF(J,I) = Source/sink value of the J-th data point in the I-th profile, (L**3/T/L).
C. Record 3: global source/sink nodal number - This group of data is needed if and only if NWNP
.GT. 0. NWNP data points are required for this record.
3.1. NPW(I) = Global node number of the I-th compressed well source/sink node.
D. subset 4: Source type assigned to each well - Usually one record per well. However, automatic
generation can be made. For I-th (I = 1,2, ...,) record, it contains the following.
4.1. NI = Compressed well node number of the first node in the sequence.
4.2. NSEQ = NSEQ nodes will be generated automatically.
4.3. NAD = Increment of well node number for each of the NSEQ nodes.
4.4. NITYP = Source type in node NI.
4.5. NTYPAD = Increment of NITYP for each of the NSEQ nodes.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
14. ELEMENT (DISTRIBUTED) SOURCE/SINK FOR TRANSPORT SIMULATIONS
This data set is needed if IMOD = 1 or IMOD =11.
Ten subsets of free-formatted data records are required in this data set.
A. subset 1: control parameters
1.1. NSEL = No. of source/sink elements.
1.2. NSPR = No. of source profiles, should be .GE. 1.
1.3. NSDP = No. of data points in each profile, should be .GE. 2.
1.4. KSAI = Is element-source/sink profile to be input analytically? 0 = no, 1 = yes.
A-18
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B. Subset 2: source/sink profile - This sub-data-set is needed if and only if NSEL .GT. 0. For each
sub-data-record, NSDP of the data group (TSOSF(J,I), SOSF(J,I,1), SOSF(J,I,2)) are
required. If this sub-data-record can be fitted in a line, only NSPR lines are needed.
2.1. TSOSF(J,I) = Time of J-th data point in I-th profile, (T).
2.2. SOSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th profile, (L**3/T/L**3);
positive for source and negative for sink.
2.3. SOSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th profile, (M/L**3).
C. subset 3: global source/sink element number. NSEL data points are required for this record.
3.1. LES(I) = Global element number of the I-th compressed distributed source/sink element.
D. subset 4: Source type assigned to each element for microbe # 1 - Usually one record per element.
However, automatic generation can be made. For I-th (1=1,2,...,) record, it contains
the following.
4.1. MI = Global element number of the first element in the sequence.
4.2. NSEQ = NSEQ elements will be generated automatically.
4.3. MAD = Increment of element number for each of the NSEQ elements.
4.4. MITYP = Source type in element MI.
4.5. MTYPAD = Increment of MITYP for each of the NSEQ elements.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
E. subset 5 ~ Subset 10: Source type assigned to each element for microbe #2, microbe #3, substrate,
Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
15. POINT (WELL) SOURCE/SINK DATA FOR TRANSPORT SIMULATION
This data set is needed if IMOD = 1 or IMOD =11.
Ten subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NWNP = No. of well or point source/sink nodes.
1.2. NWPR = No. of well or point source/sink strength profiles.
1.3. NWDP = No. of data points in each of the NWPR profiles.
1.4. KWAI = Is well-source/sink profile to be input analytically? 0 = no, 1 = yes.
A-19
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B. subset 2: source/sink profiles - This group of data is needed if and only if NWNP .GT. 0. For each
sub-data-record, NWDP of the data group (TWSSF(J,I), WSSF(J,I,1), WSSF(J,I,2))
are required. If this sub-data-record can be fitted in a line, only NWPR lines are
needed.
2.1. TWSSF(J,I) = Time of J-th data point in I-th profile, (T).
2.2. WSSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th profile, (L**3/T/L**3);
positive for source and negative for sink.
2.3. WSSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th profile, (M/L**3).
C. subset 3: global source/sink element number - This group of data is needed if and only if NWNP
.GT. 0. NWNP data points are required for this record.
3.1. NPW(I) = Global node number of the I-th compressed point source/sink node.
D. subset 4: Source type assigned to each well for microbe #1 - Usually one record per element.
However, automatic generation can be made.
4.1. NI = Compressed point source/sink node number of the first node in a sequence.
4.2. NSEQ = NSEQ nodes will contain the source types that will be automatically generated.
4.3. NIAD = Increment of NI for each of the NSEQ nodes.
4.4. NITYP = Source type in node NI.
4.5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's must be used to signal the end of this data set.
E. subset 5 ~ Subset 10: Source type assigned to each well for microbe #2, microbe #3, substrate,
Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
16. RAINFALL/EVAPORATION-SEEPAGE BOUNDARY CONDITIONS
This data set is needed if IMOD = 10 or IMOD =11.
Seven subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NVES = No. of variable boundary element sides.
1.2. NVNP = No. of variable boundary nodal points.
1.3. NVPR = No. of rainfall profiles.
A-20
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1.4. NVDP = No. of rainfall data points in each of the NRPR rainfall profiles.
1.5. KVAI = Is rainfall profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: boundary profiles - This subset is required only when NVES is not 0. NRPR profiles are
needed. For each profile, NRDP of the data pair (TRF(J,I),RF(J,I)) are required. If
these data pairs can fit in a line, only NRPR lines are needed.
2.1. TRF(J,I) = Time of the J-th data point in the I-th profile, (T).
2.2. RF(J,I) = Rainfall/evaporation rate of the J-th data point in the I-th profile, (L/T).
C. subset 3: boundary profile types assigned to each element. At most NVES records are needed.
However, automatic generation can be made. For I-th (I = 1, 2, ...,) record, it contains
the following variables.
3.1. MI = Compressed VB element side of the first side in the sequence.
3.2. NSEQ = NSEQ sides will be generated automatically.
3.3. MIAD = Increment of NI for each of the NSEQ sides.
3.4. MITYP = Type of rainfall/evaporation profiles assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ sides.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
D. subset 4: Specification of Rainfall/evaporation-seepage sides. Normally, NVES records are
required, one each for a variable boundary (VB) element side. However, if a group of
rainfall/evaporation-seepage element sides appears in a regular pattern, automatic
generation may be made. For I-th (I = 1, 2, ..., ) record, it contains the following
variables.
4.1. MI = Compressed VB element side number of the first element side in a sequence.
4.2. NSEQ = NSEQ subsequent VB element sides will be generated automatically.
4.3. MIAD = Increment of MI for each of the NSEQ subsequent VB element sides.
4.4. II = Global node number of the first node of element side MI.
4.5. 12 = Global node number of the second node of element side MI.
4.6. 13 = Global node number of the third node of element side MI.
4.7. 14 = Global node number of the fourth node of element side MI.
A-21
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4.8. HAD = Increment of II for each of the NSEQ subsequent VB element sides.
4.9. I2AD = Increment of 12 for each of the NSEQ subsequent VB element sides.
4.10. BAD = Increment of 13 for each of the NSEQ subsequent VB element sides.
4.11. MAD = Increment of 14 for each of the NSEQ subsequent VB element sides.
**** NOTE: A blank with 11 O's must be used to signal the end of this subdata set.
E. subset 5: Global Node Number of All Compressed Variable Boundary (VB) Nodes. At most,
NVNP records are needed for this subset, one each for NVNP variable boundary nodes.
For I-th (I = 1, 2, ...,) Record, it contains the following 5 variables.
5.1. NI = Compressed VB node number of the first node in the sequence.
5.2. NSEQ = NSEQ nodes will be generated automatically.
5.3. NIAD = Increment of NI for each of the NSEQ nodes.
5.4. NODE = Global node number of node NI.
5.5. NODEAD = Increment of NODE for each of the NSEQ nodes.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
F. subset 6: Ponding Depth Allowed in Each of NVNP Variable Boundary Nodes. Normally, NVNP
records are needed, one for each of the NVNP nodes. However, if a group of nodes has
a regular pattern of ponding depth, automatic generation is made. For I-th (I = 1,2, ...,
) record, it contains the following variables.
6.1. NI = Compressed VB node number of the first node in a sequence.
6.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
6.3. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
6.4. HCONNI = Ponding depth of node NI, (L).
6.5. HCONAD = Increment of HCONNI for each of the NSEQ nodes, (L).
6.6. 0.0
**** NOTE: A line with 6 O's must be used to signal the end of this data set.
G. subset 7: Minimum Pressure Head Allowed in Each NVNP Variable Boundary Nodes. This subset
is read-in similar to the above subset. For I-th (I = 1,2, ..., ) record, it contains the
following variables.
A-22
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7.1. NI = Compressed VB node number of the first node in a sequence.
7.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
7.3. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
7.4. HMINNI = Minimum pressure head allow for node NI, (L).
7.5. HMINAD = Increment of HMINNI for each of the NSEQ nodes, (L).
7.6. 0.0
**** NOTE: A line with 6 O's must be used to signal the end of this data set.
17. DIRICHLET BOUNDARY CONDITIONS FOR FLOW SIMULATION
This data set is needed if IMOD = 10 or IMOD =11.
Four subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NDNP = No. of Dirichlet nodal points, should be .GE. 1.
1.2. NDPR = No. of total Dirichlet-head profiles, should be .GE. 1.
1.3. NDDP = No. of data points in each total head profiles, should be .GE. 1.
1.4. KDAI = Is Dirichlet boundary value profile to be input analytically? 0= no, 1= yes.
B. subset 2: Dirichlet-head profiles - This subset is required only if NDNP is not 0. NDPR of profiles
are needed. For each profile, NDDP of the data pair (THDBF(J,I),HDBF(J,I)) are
needed. If these data pairs can fit in a line, only NDPR lines are needed.
2.1. THDBF(J,I) = Time of the J-th data point in the I-th profile, (T).
2.2. HDBF(J,I) = Total head of the J-th data point in the I-th profile, (L).
C. subset 3: Dirichlet nodes - At most, NDNP records are needed for this subset, one each for NDNP
Dirichlet boundary nodes. However, if the Dirichlet nodes appear in a regular pattern,
automatic generation may be made. For I-th (I = 1, 2, ..., ) Record, it contains the
following 5 variables.
3.1. NI = Compressed DB node number of the first node in the sequence.
3.2. NSEQ = NSEQ nodes will be generated automatically.
3.3. NIAD = Increment of NI for each of the NSEQ nodes.
A-23
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3.4. NODE = Global node number of node NI.
3.5. NODEAD = Increment of NODE for each of the NSEQ nodes.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
D. subset 4: boundary profile type assign to each Dirichlet node - Normally one record per Dirichlet
node; i.e., a total of NDNP records. However, if the Dirichlet nodes appear in regular
pattern, automatic generation may be made. For I-th (I = 1,2, ...,) record, it contains
the following variables.
4.1. NI = Compressed Dirichlet node number of the first node in the sequence.
4.2. NSEQ = NSEQ subsequent Dirichlet nodes will be generated automatically.
4.3. NAD = Increment of NI for each of the NSEQ nodes.
4.4. NITYP = Type of total head profile for node NI and NSEQ subsequent nodes.
4.5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes.
**** NOTE: A line with 5 O's must be used to signal the end of this data set.
18. CAUCHY BOUNDARY CONDITIONS FOR FLOW SIMULATIONS
This data set is needed if IMOD = 10 or IMOD =11.
Five subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NCES = No. of Cauchy boundary element sides.
1.2. NCNP = No. of Cauchy nodal points.
1.3. NCPR = No. of Cauchy-flux profiles.
1.4. NCDP = No. of data points in each of the NCPR Cauchy-flux profiles.
1.5. KCAI = Is Cauchy flux profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: prescribed Cauchy-flux profiles - This set is required only if NCES is not 0. NCPR of
profiles are needed. For each profile, NCDP of the data pair (TQCBF(J,I),QCBF(J,I))
are needed. If these data pairs can fit in a line, only NDPR lines are needed.
2.1. TQCBF(J,I) = Time of the J-th data point in the I-th profile, (T).
A-24
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2.2. QCBF(J,I) = Normal Cauchy flux of the J-th data point in the I-th profile, (L**3/T/L**2);
positive out from the region, negative into the region.
C. subset 3: type of Cauchy flux profiles assigned to each of all NCES sides. At most NCES records
are needed. However, automatic generation can be made. For I-th (I = 1, 2, ..., )
record, it contains the following variables.
3.1. MI = Compressed Cauchy side number of the first side in the sequence.
3.2. NSEQ = NSEQ sides will be generated automatically.
3.3. MIAD = Increment of MI for each of the NSEQ sides.
3.4. MITYP = Type of Cauchy flux profile assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ sides.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
D. subset 4: Cauchy boundary element sides - Normally, NCES records are required, one each for a
Cauchy boundary element side. However, if a group of Cauchy boundary element sides
appears in a regular pattern, automatic generation may be made. For I-th (I = 1,2, ...,
) record, it contains the following variables.
4.1. MI = Compressed Cauchy element side number of the first element-side in a sequence.
4.2. NSEQ = NSEQ subsequent Cauchy element-sides will be generated automatically.
4.3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4.4. II = Global node number of the first node on the Cauchy element-side MI.
4.5.12 = Global node number of the second node on the Cauchy element-side MI.
4.6.13 = Global node number of the third node on the Cauchy element-side MI.
4.7.14 = Global node number of the fourth node on the Cauchy element-side MI.
4.8. HAD = Increment of II for each of the NSEQ subsequent element-sides.
4.9.12AD = Increment of 12 for each of the NSEQ subsequent element-sides.
4.10. BAD = Increment of 13 for each of the NSEQ subsequent element-sides.
4.11. MAD = Increment of 14 for each of the NSEQ subsequent element-sides.
**** NOTE: A line with 11 O's is used to end this data set input.
A-25
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E. subset 5: global node number of all compressed Cauchy nodes - Normally one record per
compressed Cauchy node; i.e., a total of NCNP records. However, if the Cauchy nodes
appear in a regular pattern, automatic generation may be made. For I-th (I = 1, 2, ...,
) record, it contains the following variables.
5.1. NI = Compressed Cauchy node number of the first node in the sequence.
5.2. NSEQ = NSEQ subsequent Cauchy nodes will be generated automatically.
5.3. NAD = Increment of NI for each of the NSEQ nodes.
5.4. NODE = Global node number for node NI and NSEQ subsequent nodes.
5.5. NODEAD = Increment of NODE for each of the NSEQ subsequent nodes.
**** NOTE: A line with 5 O's must be used to signal the end of this sub-data set.
19. NEUMANN BOUNDARY CONDITIONS FOR FLOW SIMULATIONS
This data set is needed if IMOD = 10 or IMOD =11.
Five subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NNES = No. of Neumann boundary element sides.
1.2. NNNP = No. of Neumann nodal points.
1.3. NNPR = No. of Neumann flux profiles.
1.4. NNDP = No. of data points in each of the NNPR Neumann-flux profiles.
1.5. KNAI = Is Neumann flux profile to be input analytically; 0 = no, 1 = yes.
B. subset 2: prescribed Neumann-flux profiles - This sub-data-set is required only if NNES is not 0.
NNPR of profiles are needed. For each profile, NNDP of the data pair
(TQNBF(J,I),QNBF(J,I)) are needed. If these data pairs can fit in a line, only NDPR
lines are needed.
2.1. TQNBF(J,I) = Time of the J-th data point in the I-th profile, (T).
2.2. QNBF(J,I) = Normal Neumann flux of the J-th data point in the I-th profile, (L**3/T/ L**2);
positive out from the region, negative into the region.
C. subset 3: type of Neumann flux profiles assigned to each of all NNES sides. At most NNES
records are needed. However, automatic generation can be made. For I-th (I = 1,2,...,
) record, it contains the following variables.
A-26
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3.1. MI = Compressed Neumann side number of the first side in the sequence.
3.2. NSEQ = NSEQ sides will be generated automatically.
3.3. MIAD = Increment of MI for each of the NSEQ sides.
3.4. MITYP = Type of Neumann flux profile assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ sides.
**** NOTE: A line with 5 O's is used to signal the end of this data set.
D. subset 4: Neumann boundary element sides - Normally, NNES records are required, one each for
a Neumann boundary element side. However, if a group of Neumann boundary
element sides appears in a regular pattern, automatic generation may be made. For I-th
(I = 1,2, ...,) record, it contains the following variables.
4.1. MI = Compressed Neumann side number of the first side in sequence.
4.2. NSEQ = NSEQ subsequent Neumann sides will be generated automatically.
4.3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4.4. II = Global node number of the first node on the Neumann element-side MI.
4.5. 12 = Global node number of the second node on Neumann element-side MI.
4.6. 13 = Global node number of the third node on the Neumann element-side MI.
4.7. 14 = Global node number of the fourth node on the Neumann element-side MI.
4.8. HAD = Increment of II for each of the NSEQ subsequent element-sides.
4.9. I2AD = Increment of 12 for each of the NSEQ subsequent element-sides.
4.10. BAD = Increment of 13 for each of the NSEQ subsequent element-sides.
4.11. MAD = Increment of 14 for each of the NSEQ subsequent element-sides.
**** NOTE: A line with 11 O's is used to end this data set input.
E. subset 5: global node number of all compressed Neumann nodes - Normally one record per
compressed Neumann node; i.e., a total of NNNP records. However, if the Neumann
nodes appear in a regular pattern, automatic generation may be made. For I-th (1=1,
2, ...,) record, it contains the following variables.
5.1. NI = Compressed Neumann node number of the first node in the sequence.
A-27
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5.2. NSEQ = NSEQ subsequent Neumann nodes will be generated automatically.
5.3. NAD = Increment of NI for each of the NSEQ nodes.
5.4. NITYP = Type of total head profile for node NI and NSEQ subsequent nodes.
5.5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes.
**** NOTE: A line with 5 O's must be used to signal the end of this data set.
20. RUN-IN/FLOW-OUT (VARIABLE) BOUNDARY CONDITIONS FOR TRANSPORT
SIMULATIONS
This data set is needed if IMOD = 1 or IMOD =11.
Eleven subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NVES = No. of variable boundary element sides.
1.2. NVNP = No. of variable boundary nodal points.
1.3. NVPR = No. of incoming fluid concentration profiles to be applied to variable boundary element
sides.
1.4. NVDP = No. of data points in each of the NRPR profiles.
1.5. KVAI = Is incoming concentration profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: variable boundary flux profile - NRPR records are needed. Each record contains NRDP
data points and is FREE-FORMATTED. Each data point has 2 numbers representing
the time and run-in flow-out concentrations, respectively as follows:
2.1. TCVSF(J,I) = Time of the J-th data point on the I-th run-in concentration profile, (T).
2.2. CVSF(J,I) = Concentration of the J-th data point on the I-th profile, (M/L**3).
C. subset 3: Run-in concentration type assigned to each of all NVES sides for microbe # 1. Usually
one record per variable element side. However, automatic generation can be made.
Each record contains 5 variables and is FREE-FORMATTED.
3.1. MI = Compressed VB element side of the first side in a sequence.
3.2. NSEQ = NSEQ subsequent sides will be generated automatically.
3.3. MIAD = Increment of MI for each of NSEQ subsequent sides.
A-28
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3.4. MITYP = Type of concentration profile assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
**** NOTE: A record with 5 O's must be used to signal the end of this data set.
D. subset 4 ~ Subset 9: Run-in concentration type assigned to each element for microbe #2, microbe
#3, substrate, Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
J. subset 10: Specification of run-in boundary element sides - Normally, NVES records are required,
one each for a VB element side. However, if a group of VB element sides appears in
a regular pattern, automatic generation may be made. Each record contains 11
variables and is FREE-FORMATTED.
10.1. MI = Compressed VB element side number of the first side in a sequence.
10.2. NSEQ = NSEQ subsequent VB element sides will be generated automatically.
10.3. MIAD = Increment of MI for each of the NSEQ subsequent Vb element sides.
10.4. II = Global node number of the first node of element side MI.
10.5. 12 = Global node number of the second node of element side MI.
10.6. 13 = Global node number of the third node of element side MI.
10.7. 14 = Global node number of the fourth node of element side MI.
10.8. HAD = Increment of II for each of the NSEQ subsequent element sides.
10.9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10.10. BAD = Increment of 13 for each of the NSEQ subsequent element sides.
10.11. MAD = Increment of 14 for each of the NSEQ subsequent element sides.
**** NOTE: A record with 11 O's is used to signal the end of this data set.
K. subset 11: global nodal number of all run-in flow-out boundary nodes. Usually NVNP records are
needed for this subdata set. However, automatic generation can be made. Each record
contains 5 variables and is FREE-FORMATTED.
11.1. NI = Compressed VB node number of the first node in a sequence.
11.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
11.3. NIAD = Increment for NI for each of the NSEQ nodes.
A-29
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11.4. NODE = Global nodal number of the node NI.
11.5. NODEAD = Increment of NODE for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's is used to signal end of this data set.
21. DIRICHLET BOUNDARY CONDITIONS FOR TRANSPORT SIMULATIONS
This data set is needed if IMOD = 1 or IMOD =11.
Ten subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NDNP = No. of Dirichlet nodes, should be .GE. 1.
1.2. NDPR = No. of Dirichlet profiles, should be .GE. 1.
1.3. NDDP = No. of data points in each profile, should be .GE. 2.
1.4. KDAI = Is Dirichlet boundary value profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: Dirichlet-concentration profiles - NDPR records are needed. Each record contains NDDP
data points and is FREE-FORMATTED. Each data point has 2 numbers representing
the time and Dirichlet concentrations, respectively as follows:
2.1. TCDBF(J,I) = Time of J-th data point in I-th Dirichlet-concentration profile, (T).
2.2. CDBF(J,I) = Concentration of J-th data point in I-th Dirichlet-concentration profile, (M/L**3).
C. subset 3: global node number of compressed Dirichlet nodes - Usually NDNP records are needed
for this subdata set. However, automatic generation can be made. Each record contains
5 variables and is FREE-FORMATTED.
3.1. NI = Compressed Dirichlet boundary node number of the first node in a sequence.
3.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
3.3. NIAD = Increment for NI for each of the NSEQ nodes.
3.4. NODE = Global nodal number of the node NI.
3.5. NODEAD = Increment of the global nodal number for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's is used to signal end of this data set.
D. subset 4: Dirichlet concentration types assigned to Dirichlet nodes for microbe #1. Normally one
record per Dirichlet node; i.e., a total of NDNP records, is needed. However, if the
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Dirichlet nodes appear in a regular pattern, automatic generation may be made. Each
record contains 5 variables and is FREE-FORMATTED.
4.1. NI = Compressed Dirichlet node number of the first node in the sequence.
4.2. NSEQ = NSEQ nodes will contain the Dirichlet concentration types that will be automatically
generated.
4.3. NIAD = Increment of NI for each of the NSEQ nodes.
4.4. NITYP = Dirichlet concentration type in node NI.
4.5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's must be used to signal the end of this data set.
E. subset 5 ~ Subset 10: Dirichlet concentration type assigned to each node for microbe #2, microbe
#3, substrate, Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
22. CAUCHY BOUNDARY CONDITIONS FOR TRANSPORT SIMULATION
This data set is needed if IMOD = 1 or IMOD =11.
Eleven subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NCES = No. of Cauchy element sides.
1.2. NCNP = No. of Cauchy nodal points.
1.3. NCPR = No. of Cauchy-flux profiles.
1.4. NCDP = No. of data points on each Cauchy-flux profile.
1.5. KCAI = Is Cauchy flux profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: Cauchy flux profiles - NCPR records are needed. Each record contains NCDP data
points and is FREE-FORMATTED. Each data point has 2 numbers representing the
time and Cauchy flux, respectively as follows:
2.1. TQCBF(J,I) = Time of the J-th data point in the I-th Cauchy flux profile, (T).
2.2. QCBF(J,I) = Value of Cauchy flux of the J-th data point in the I-th Cauchy-flux profile,
(M/T/L**2).
C. subset 3: Cauchy flux type assigned to each of all NCES sides for microbe #1 - Usually one record
per Cauchy element side. However, automatic generation can be made. Each record
contains 5 variables and is FREE-FORMATTED.
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3.1. MI = Compressed Cauchy boundary element side of the first side in a sequence.
3.2. NSEQ = NSEQ subsequent sides will be generated automatically.
3.3. MIAD = Increment of MI for each of NSEQ subsequent sides.
3.4. MITYP = Type of Cauchy flux profile assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
**** NOTE: A record with 5 O's must be used to signal the end of this data set.
D. subset 4 ~ Subset 9: Cauchy flux type assigned to each element for microbe #2, microbe #3,
substrate, Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
J. Subset 10: specification of Cauchy boundary element sides -Normally, NCES records are required,
one each for a Cauchy boundary element side. However, if a group of Cauchy element
sides appears in a regular pattern, automatic generation may be made. Each record
contains 11 variable and is FREE-FORMATTED.
10.1. MI = Compressed Cauchy boundary element side number of the first element side in a
sequence.
10.2. NSEQ = NSEQ subsequent Cauchy boundary element sides will be generated automatically.
10.3. MIAD = Increment of MI for each of the NSEQ subsequent Cauchy boundary element sides.
10.4. II = Global node number of the first node of element side MI.
10.5. 12 = Global node number of the second node of element side MI.
10.6. 13 = Global node number of the third node of element side MI.
10.7. 14 = Global node number of the fourth node of element side MI
10.8. HAD = Increment of II for each of the NSEQ subsequent element sides.
10.9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10.10. BAD = Increment of 13 for each of the NSEQ subsequent element sides.
10.11. MAD = Increment of 14 for each of the NSEQ subsequent element sides.
**** NOTE: A record with 11 O's is used to signal the end of this data set.
K. subset 11: global nodal number of all Cauchy boundary nodes - Usually NCNP records are needed
for this subdata set. However, automatic generation can be made. Each record
contains 5 variables and is FREE-FORMATTED.
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11.1. NI = Compressed Cauchy boundary node number of the first node in a sequence.
11.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
11.3. NIAD = Increment for NI for each of the NSEQ nodes.
11.4. NODE = Global nodal number of the node NI.
11.5. NODEAD = Increment of the global nodal number for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's is used to signal end of this data set.
23. NEUMANN BOUNDARY CONDITIONS FOR TRANSPORT SIMULATIONS
This data set is needed if IMOD = 1 or IMOD =11.
Eleven subsets of data records are required for this data set.
A. subset 1: control parameters
1.1. NNES = No. of Neumann element sides.
1.2. NNNP = No. of Neumann nodal points.
1.3. NNPR = No. of Neumann-flux profiles.
1.4. NNDP = No. of data points on each Neumann-flux profile.
1.5. KNAI = Is Neumann flux profile to be input analytically? 0 = no, 1 = yes.
B. subset 2: Neumann flux profiles - NNPR records are needed. Each record contains NNDP data
points and is FREE-FORMATTED. Each data point has 2 numbers representing the
time and Neumann flux, respectively, as follows:
2.1. TQNBF(J,I) = Time of the J-th data point in the I-th Neumann flux profile, (T).
2.2. QNBF(J,I) = Value of Neumann flux of the J-th data point in the I-th Neumann-flux profile,
(M/T/L**2).
C. subset 3: Neumann flux type assigned to each of all NNES sides for microbe # 1 - Usually one
record per Neumann element side. However, automatic generation can be made. Each
record contains 5 variables and is FREE-FORMATTED.
3.1. MI = Compressed Neumann boundary element side of the first side in a sequence.
3.2. NSEQ = NSEQ subsequent sides will be generated automatically.
3.3. MIAD = Increment of MI for each of NSEQ subsequent sides.
A-33
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3.4. MITYP = Type of Neumann flux profile assigned to side MI.
3.5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
**** NOTE: A record with 5 O's must be used to signal the end of this data set.
D. subset 4 ~ Subset 9: Neumann flux type assigned to each element for microbe #2, microbe #3,
substrate, Oxygen, Nitrate, and nutrient. The input format is the same as subset 4.
J. subset 10: specification of Neumann boundary element sides -Normally, NNES records are
required, one each for a Neumann boundary element side. However, if a group of
Neumann element sides appears in a regular pattern, automatic generation may be
made. Each record contains 11 variables and is FREE-FORMATTED.
10.1. MI = Compressed Neumann boundary element side number of the first element side in a
sequence.
10.2. NSEQ = NSEQ subsequent Neumann boundary element sides will be generated automatically.
10.3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
10.4. II = Global node number of the first node of element side MI.
10.5. 12 = Global node number of the second node of element side MI.
10.6. 13 = Global node number of the third node of element side MI.
10.7. 14 = Global node number of the fourth node of element side MI.
10.8. HAD = Increment of II for each of the NSEQ subsequent element sides.
10.9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10.10. BAD = Increment of 13 for each of the NSEQ subsequent element sides.
10.11. MAD = Increment of 14 for each of the NSEQ subsequent element sides.
**** NOTE: A record with 11 O's is used to signal the end of this data set.
K. subset 11: global nodal number of all Neumann boundary nodes - Usually NNNP records are
needed for this subdata set. However, automatic generation can be made. Each record
contains 5 variables and is FREE-FORMATTED.
11.1. NI = Compressed Neumann boundary node number of the first node in a sequence.
11.2. NSEQ = NSEQ subsequent nodes will be generated automatically.
11.3. NIAD = Increment for NI for each of the NSEQ nodes.
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11.4. NODE = Global nodal number of the node NI.
11.5. NODEAD = Increment of the global nodal number for each of the NSEQ subsequent nodes.
**** NOTE: A record with 5 O's is used to signal end of this data set.
24. PARAMETERS CONTROLLING TRACKING SCHEME
Two subdata sets are needed if IMOD * 10.
A. subset 1: Thirteen integers are typed by free format.
1.1. IZOOM = Is zooming needed for advection computation? 0 = No, 1 = Yes.
1.2. IDZOOM = Is zooming needed for dispersion computation? 0 = No, 1 = Yes.
1.3. IEPC = Is EPCOF scheme included? 0 = No, 1 = Yes. Note: 0 for this version.
1.4. NXA = No. of regularly refined subelements for the advection step in the X-direction in an
element.
1.5. NYA = No. of regularly refined subelements for the advection step in the Y-direction in an
element.
1.6. NZA = No. of regularly refined subelements for the advection step in the Z-direction in an
element.
1.7. NXW = The number of subelements in each global element for element tracking in x-direction.
1.8. NYW = The number of subelements in each global element for element tracking in y-direction.
1.9. NZW = The number of subelements in each global element for element tracking in z-direction.
1.10. NXD = No. of dispersion fine subelements in each global element in X-direction.
1.11. NYD = No. of dispersion fine subelements in each global element in Y-direction.
1.12. NZD = No. of dispersion fine subelements in each global element in Z-direction.
1.13. IDETQ = Index of particle tracking pattern:
1 = Average velocity is used (more accurate);
2 = Single velocity of the starting point is used (less computation).
B. Subset 2: It reads the following 2 variables (FREE FORMAT)
2.1. ADPEPS = Error tolerance of relative concentration and nonlinear convergence criteria.
A-35
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2.2. ADPARM = Error tolerance of concentration relative to maximum concentration.
25. HYDROLOGICAL VARIABLES
This data set is needed if and only if KVI .LE. 0. When KVI .LE. 0, two groups of data are needed,
one group for the velocity field and the other group for the moisture content.
A. subset 1: velocity field - Usually NNP records are needed. However, if velocity appears in regular
pattern, automatic generation can be made. Each record contains 9 variables and is
FREE-FORMATTED.
1.1. NI = Node number of the first node in a sequence.
1.2. NSEQ = NSEQ subsequent nodes will be automatically generated.
1.3. NIAD = Increment of node number in each of the NSEQ subsequent nodes.
1.4. VXNI = x-velocity component at node NI, (L/T).
1.5. VYNI = y-velocity component at node NI, (L/T).
1.6. VZNI = z-velocity component at node NI, (L/T).
1.7. VXAD = Increment of VXNI for each of the NSEQ subsequent nodes, (L/T).
1.8. VYAD = Increment of VYNI for each of the NSEQ subsequent nodes, (L/T).
1.9. VZAD = Increment of VZNI for each of the NSEQ subsequent nodes, (L/T).
**** NOTE: A record with 9 O's is used to signal the end of this data set.
B. subset 2: moisture content field - Usually, NEL records are needed. However, if moisture content
appears in regular pattern, automatic generation can be made. Each record contains
5 variables and is FREE-FORMATTED.
2.1. MI = Element number of the first element in a sequence.
2.2. NSEQ = NSEQ subsequent elements will be automatically generated.
2.3. MIAD = Increment of MI for each of NSEQ subsequent elements.
2.4. THNI = Moisture content of element NI, (Decimal point).
2.5. THNIAD = Increment of THNI for NSEQ subsequent elements, (Decimal point).
2.6. 0.0
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**** NOTE: A record with 6 O's is used to signal the end of this data set.
26. END OF JOB
If another problem is to be run, then input begins again with input data set 1. If termination of the job
is desired, a blank card must be inserted at the end of the data set.
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A-38
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APPENDIX B: Mathematical Formulation
B. 1 Governing Equations for Flow
From the notion for continuity of fluid, continuity of solid, consolidation of the media, and the
equation of state (Yeh, 1992), one obtains the starting equation for this derivation:
a(n Sp)
-V-(pneSVs) + p*q = 2 (B.I.I)
at
where p is the fluid density (M/L3), k is the intrinsic permeability tensor of the media (L2), /j, is the dynamic
viscosity of the fluid (M/L/T), p is the fluid pressure [(ML/T^/L2], g is the acceleration of gravity (L/T2), z is
the potential head (L), ne is the effective porosity (L3/L3), S is the degree of saturation (dimensionless), Vs is
the velocity of the deformable surface due to consolidation (L/T), p* is the density of the injected fluid (M/L3),
q is the internal source/sink [(L3/T)/L3], and t is the time (T).
Expanding the right hand side of Eq.(B.l.l):
+
at eat r at ei at
Expanding Eq.(B. 1.2) by the chain rule:
nep (B.I. 2)
(B.L3)
at eap at eac at at e at
where C is chemical concentration (M/L3). Rearranging Eq.(B.l.S), one obtains:
3(nesp) c ap ap c 5ne c ap ac as
- = Sn -t- ^ + pS - + Sn t- + no (B.I. 4)
at eap at at eac at e at
where the first and second terms represent the storativity term, the third term is the density-concentration
coupling term, and the fourth term is the unsaturated term. Substituting Eq.(B.1.4) into Eq.(B.l.l):
B-1
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APPENDIX C
The determination of Maximum Control Parameters for LEZOOMPC implementation
The example shown in this section is Example 6 with 510 global nodes and 224 hexahedral elements.
According to the input data file, the number of refined subelements in each global element is NXA=NXW=2,
NYA=NYW=1, NZA=NZW=2 in the Lagrangian step and NXD=2, NYD=1, NZD=2 in the Eulerian step.
Consider NX, NY, and NZ to represent the above values for both Lagrangian and Eulerian steps. Hence, there
are NX*NY*NZ, NX*NX*NZ, and NX*NX*NX regular refined subelements in each hexahedral global
element, triangular prism global element, and tetrahedral global element, respectively. The number of regular
fine grids is (NX+1)*(NY+1)*(NZ+1), -(NX+1)*(NX+2)*(NZ + 1) , and
(NX+l)*(NX+2)*(NX+3) for each hexahedral, triangular prism, and tetrahedral global element,
6
respectively. During the simulation, 54 rough elements are assumed to be zoomed. Therefore, there are
54x(NX+l)x(NY+l)x(NZ+l) regular fine grids and 54xNXxNYxNZ regular subelements in the region of
interest. The assumption that 340 peak and valley points in the associated subelements are captured increases
the total fine grids to 54x(NX+l)x(NY+l)x(NZ+l)+340. Because NCC components is included in the
system, MXNPFGK is equal to NCCx[ 54x(NX+l)x(NY+l)x(NZ+l)+340)]. Then, 2200 additional
subelements are assumed to be generated after tetrangulating the captured peak and valley points and
MXKGLK is assigned to 54xNXxNYxNZ+2200. The working array declaration of MXNPWK and
MXELWK are 18 and 4, respectively, for particle tracking computation. If there are injection/extraction wells
in the region of interest, then MXNPWS=(NXA+l)x(NYA+l)x(NZA+l)=18 and
MXELWS=NXAxNYAxNZA=4. For the Eulerian step, the calculation of diffusion fine grids and of refined
subelements is dependent on the number of extended rough elements which can be predicted from the number
of rough elements in the Lagrangian step. In total, 79 extended rough elements are assumed in this case.
Because each rough element is refined by 2 (NXD) x 2 (NYD) x 1 (NZD) = 4 subelements, the total number
C-1
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Appendix D: Nomenclatures
t = time (T)
x, y, z = the coordinate in the x-, y-, and z-directions in the region of interest (L)
xb, yb, zb = the x, y, z on the boundary of the region of interest (L)
R = region of interest
B= boundary of the region of interest
n = outward unit normal vector
h = the referenced pressure head defined as p/pwg
p = pressure (M/LT2)
K = hydraulic conductivity tensor (L/T)
K,, = the saturated hydraulic conductivity tensor (L/T)
Kj = the relative hydraulic conductivity or relative permeability
z = the potential head (L)
q = flow rate of the source and/or sink (L3/T)
6 = the moisture content
p= density of the fluid (M/L3)
(i = dynamic viscosity (M/LT)
KjW = referenced saturated hydraulic conductivity tensor (L/T)
pw = referenced density of the fluid (M/L3),
(iw = referenced dynamic viscosity (M/LT)
p* = density of the injected fluid (M/L3)
V = Darcy flux (L/T)
hj = prescribed initial pressure head (L)
hd = prescribed Dirichlet pressure head (L)
D-1
-------
qn = Neumann flux (L/T)
qc = Cauchy flux (L/T)
qp = the throughfall of precipitation of the variable boundary (L/T)
hp = the allowed ponding depth on variable boundary (L)
hm = the allowed minimum pressure head (L)
qe = the allowed maximum evaporation rate on the variable boundary, i.e., the potential
evaporation (L/T)
Bd = the Dirichlet boundary
Bn = the Neumann boundary
Bc = the Cauchy boundary
Bv = the variable boundary
Cs = dissolved concentration of the substrate (M/L3)
ps = intrinsic density of substrate, (M/L3)
C0 = dissolved concentration of oxygen (M/L3)
p0 = intrinsic density of oxygen (M/L3)
Cn = dissolved concentration of nitrate (M/L3)
pn = intrinsic density of nitrate (M/L3)
Cp = dissolved concentration of nutrient (M/L3)
pp = intrinsic density of nutrient (M/L3)
Q = dissolved concentration of microbe #1 (M/L3)
PJ = intrinsic density of microbe #1 (M/L3)
C2 = dissolved concentration of microbe #2 (M/L3)
p2 = intrinsic density of microbe #2 (M/L3)
C3 = dissolved concentration of microbe #3 (M/L3)
D-2
-------
p3 = intrinsic density of microbe #3 (M/L3)
Ps = viscosity effecting factor associated with substrate (L2/T)
P0 = viscosity effecting factor associated with oxygen (L)
Pn = viscosity effecting factor associated with nitrate (L)
Pp = viscosity effecting factor associated with nutrient (L2/T)
P! = viscosity effecting factor associated with microbe #1 (L2/T)
P2 = viscosity effecting factor associated with microbe #2 (L2/T)
P3 = viscosity effecting factor associated with microbe #3 (L2/T)
pb = the bulk density of the medium (M/L3)
D = the dispersion coefficient tensor (L2/T)
As = transformation rate constant for substrate (1/T)
A0 = transformation rate constant for oxygen (1/T)
Ap = transformation rate constant for nutrient (1/T)
An= transformation rate constant for nitrate (1/T)
A! = transformation rate constant for microbe #1 (1/T)
A2 = transformation rate constant for microbe #2 (1/T)
A3 = transformation rate constant for microbe #3 (1/T)
Kds = distribution coefficient of substrate (L/M3)
Kdo = distribution coefficient of oxygen (L/M3)
Kjjj = distribution coefficient of nitrate (L/M3)
Kdp = distribution coefficient of nutrient (L/M3)
Kdl = distribution coefficient of microbe #1 (L/M3)
K^ = distribution coefficient of microbe #2 (L/M3)
Kfc = distribution coefficient of microbe #3 (L/M3)
D-3
-------
qm = source rate of water (L3/T)
Csm = concentration of substrate in the source (M/L3)
Com = concentration of oxygen in the source (M/L3)
Cnm = concentration of nitrate in the source (M/L3)
Cpm = concentration of nutrient in the source (M/L3)
Clm = concentration of microbe #1 in the source (M/L3)
C2m = concentration of microbe #2 in the source (M/L3)
C3m = concentration of microbe #3 in the source (M/L3)
(i0(1) = maximum specific oxygen-based growth rate for microbe #1 (1/T)
(in(2) = maximum specific nitrate-based growth rate for microbe #2 (1/T)
(i0(3) = maximum specific oxygen-based growth rate for microbe #3 (1/T)
(in(3) = maximum specific nitrate-based growth rate for microbe #3 (1/T)
Y0(1) = yield coefficient for microbe #1 utilizing oxygen in mass of microbe per unit mass of substrate (M/M)
Yn(2) = the yielding coefficient for microbe #2 utilizing nitrate in mass of microbe per unit mass of substrate
(M/M)
Y0(3) = the yielding coefficient for microbe #3 utilizing oxygen in mass of microbe per unit mass of substrate
(M/M)
Yn(3) = yielding coefficient for microbe #3 utilizing nitrate in mass of microbe per unit mass of substrate
(M/M)
I(C0) = an inhibition function which is under the assumption that denitrifying enzyme inhibition is
reversible and noncompetitive
Kc = inhibition coefficient (M/L3)
Kso(1) = retarded substrate saturation constants under aerobic conditions with respect to microbe #1 (M/L3)
Kso(3) = retarded substrate saturation constants under aerobic conditions with respect to microbe #3 (M/L3)
D-4
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Ksn(2) = retarded substrate saturation constants under anaerobic conditions with respect to microbe #2
(M/L3)
Ksn(3) = retarded substrate saturation constants under anaerobic conditions with respect to microbe #3 (M/L3)
Kp0(1) = retarded nutrient saturation constants under aerobic conditions with respect to microbe #1
(M/L3)
Kp0(3) = retarded nutrient saturation constants under aerobic conditions with respect to microbe #3
(M/L3)
Kpn(2) = retarded nutrient saturation constants under anaerobic conditions with respect to microbe #2
(M/L3)
Kpn(3) = retarded nutrient saturation constants under anaerobic conditions with respect to microbe #3
(M/L3)
K0(1) = retarded oxygen saturation constants under aerobic conditions with respect to microbe #1 (M/L3)
K0(3) = retarded oxygen saturation constants under aerobic conditions with respect to microbe #3 (M/L3)
Kn(2) = retarded nitrate saturation constant under anaerobic conditions with respect to microbe #2 (M/L3)
j^(3) _ retarded nitrate saturation constant under anaerobic conditions with respect to microbe #3 (M/L3)
A0(1) = microbial decay constant of aerobic respiration of microbe #1 (1/T)
A0(3) = microbial decay constant of aerobic respiration of microbe #3 (1/T)
An(2) = microbial decay constant of anaerobic respiration of microbe #3 (1/T)
An(3) = microbial decay constant of anaerobic respiration of microbe #3 (1/T)
y0(1) = oxygen-use for syntheses by microbe #1
y0(3) = oxygen-use for syntheses by microbe #3
yn(2) = nitrate-use for syntheses by microbe #2
yn(3) = nitrate-use for syntheses by microbe #3
ce0(1) = oxygen-use coefficient for energy by microbe # 1
D-5
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ce0(3) = oxygen-use coefficient for energy by microbe #3
cen(2) = nitrate-use coefficient for energy by microbe #2
cen(3) = nitrate-use coefficient for energy by microbe #3
F0(1) = oxygen saturation constants for decay with respect to microbe #1 (M/L3)
F0(3) = oxygen saturation constants for decay with respect to microbe #3 (M/L3)
Fn(2) = nitrate saturation constants for decay with respect to microbe #2 (M/L3)
Fn(3) = nitrate saturation constants for decay with respect to microbe #3 (M/L3)
e0(1) = nutrient-use coefficients for the production of microbe #1 with respect to aerobic respiration
e0(3) = nutrient-use coefficients for the production of microbe #3 with respect to aerobic respiration
en(2) = nutrient-use coefficients for the production of microbe #2 with respect to anaerobic
respiration
en(3) = nutrient-use coefficients for the production of microbe #3 with respect to anaerobic respiration
S = material concentration in the absorbed phase (M/M)
rsop(1) = the removal rate of substrate under aerobic respiration with respect to microbe #1 (M/M)
rsop(3) = the removal rate of substrate under aerobic respiration with respect to microbe #3 (M/M)
rsnP(2) = me removal rate of substrate under anaerobic respiration with respect to microbe #2 (M/M)
rsnp(3) = the removal rate of substrate under anaerobic respiration with respect to microbe #3 (M/M)
rs(1) = the removal rates of substrate by microbe #1
rs(2) = the removal rates of substrate by microbe #2
rs(3) = the removal rates of substrate by microbe #3
r0(1) = oxygen utilization rate per unit of biomass, microbe # 1
r0(2) = oxygen utilization rate per unit of biomass, microbe #2
r0(3) = oxygen utilization rate per unit of biomass, microbe #3
rn(1) = nitrate utilization rate per unit of biomass, microbe #1
D-6
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rn(2) = nitrate utilization rate per unit of biomass, microbe #2
rn(3) = nitrate utilization rate per unit of biomass, microbe #3
rpฐ) = nutrient utilization rate per unit of biomass, microbe # 1
rp(2) = nutrient utilization rate per unit of biomass, microbe #2
rp(3) = nutrient utilization rate per unit of biomass, microbe #3
6 = the Kronecker delta tensor
aj. = the lateral dispersivity (L)
aL = the longitudinal dispersivity (L)
a^ = the molecular diffusion coefficient (L2/T)
T = the tortuosity
CS1 = the prescribed initial concentrations of substrate (M/L3)
C01 = the prescribed initial concentrations of oxygen (M/L3)
Cm = the prescribed initial concentrations of nitrate (M/L3)
Cpl = the prescribed initial concentrations of nutrient (M/L3)
CH = the prescribed initial concentrations of microbe #1 (M/L3)
C2l = the prescribed initial concentrations of microbe #2 (M/L3)
C3l = the prescribed initial concentrations of microbe #3 (M/L3)
Csd = the prescribed Dirichlet boundary concentrations of substrate (M/L3)
Cod = the prescribed Dirichlet boundary concentrations of oxygen (M/L3)
Cnd = the prescribed Dirichlet boundary concentrations of nitrate (M/L3)
Cpd = the prescribed Dirichlet boundary concentrations of nutrient (M/L3)
Cld = the prescribed Dirichlet boundary concentrations of microbe #1 (M/L3)
C2d = the prescribed Dirichlet boundary concentrations of microbe #2 (M/L3)
C3d = the prescribed Dirichlet boundary concentrations of microbe #3 (M/L3)
D-7
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Csv =
Cov =
Cnv =
the prescribed concentrations of substrate (M/L3) on variable boundary
the prescribed concentrations of oxygen (M/L3) on variable boundary
the prescribed concentrations of nitrate (M/L3) on variable boundary
the prescribed concentrations of nutrient (M/L3) on variable boundary
the prescribed concentrations of microbe #1 (M/L3) on variable boundary
the prescribed concentrations of microbe #2 (M/L3) on variable boundary
the prescribed concentrations of microbe #3 (M/L3) on variable boundary
the prescribed total flux of substrate through Cauchy boundary
the prescribed total flux of oxygen through Cauchy boundary
the prescribed total flux of nitrate through Cauchy boundary
the prescribed total flux of nutrient through Cauchy boundary
the prescribed total flux of microbe #1 through Cauchy boundary
q2c = the prescribed total flux of microbe #2 through Cauchy boundary
the prescribed total flux of microbe #3 through Cauchy boundary
the prescribed gradient flux of substrate through Neumann boundary
the prescribed gradient flux of oxygen through Neumann boundary
the prescribed gradient flux of nitrate through Neumann boundary
the prescribed gradient flux of nutrient through Neumann boundary
the prescribed gradient flux of microbe #1 through Neumann boundary
the prescribed gradient flux of microbe #2 through Neumann boundary
the prescribed gradient flux of microbe #3 through Neumann boundary
N = the total number of nodes in the region of interest
Nj = the shape function at node j
hj = the pressure head at node j (L)
Cpv =
Clv =
C2v =
C3v =
qsc =
qoc =
qnc =
qpc =
qlc =
q3c =
qsn =
qon =
qm =
qpn =
qln =
q2n =
q3n =
D-8
-------
F = water capacity; F = d6/dh (1/L)
{dh/dt} = column vector containing the values of dh/dt (L/T)
{h} = column vector containing the values of h (L)
[M] = mass matrix
[S] = stiffness matrix
{G} = load vector resulting from the gravity force
{Q} = load vector due to sources/sinks
{B} = load vector by the implementation of boundary condition
R,, = the region of element e
Me = the set of elements that have a local side ce-p coinciding with the global side i-j
Nae = the a-th local basis function of element e
Npe = the p-th local basis function of element e
Be = the element surface of the boundary segment e
Nse = the set of boundary segments that have a local node a coinciding with the global node i
Vx, Vy, Vz = the Darcy flux components along the x-, y-, and z-directions (L/T)
i, j, k = the unit vectors along the x-, y-, and z-directions
{Bce} = boundary-element column vector for a Cauchy boundary side
{Bne} = boundary-element column vector for a Neumann boundary side
{Bve} = boundary-element column vector for a variable boundary side
{qce} = Cauchy boundary flux vector
{qne} = Neumann boundary flux vector
{qve} = variable boundary flux vector
Nne = number of Neumann boundary element sides
N-. = number of Cauchy boundary element sides
D-9
-------
Nve = number of variable boundary element sides
f(Q, C2, C3, Cs, C0, Cn, Cp) = a microbial-chemical interaction function
g(Q, C2, C3, Cs, C0, Cn, Cp) = a microbial growth function
Vd = retarded velocity (L/T)
Cj = the concentration at node j (M/L3)
[A] = stiffness matrix associated with the velocity term
[D] = stiffness matrix associated with the dispersion term
[K] = stiffness matrix associated with the decay term, density effect, and microbial-chemical
interaction
[Bv] = stiffness matrix resulting from boundary conditions
[V] = stiffness matrix associated with the convection term
Vn = normal Darcy flux with respect to the flow-in variable boundary
w = the derivative weighting factor
^ = previous time (T)
Cn = concentration at time tn (M/L3)
tn+j = current time (T)
Cn+1 = concentration at time t^ (M/L3)
x = position vector representing (x, y, z) (L)
AT (x) = the transport time associated with x
Nn = number of activated fine-grid nodes
At = time-step size (T)
Cjf = the concentration at location (xjf, y^, Zjf) (M/L3)
Cj1 = approximated concentration determined from finite element interpolation (M/L3)
CMf = the maximum concentration of Cjf (M/L3)
D-10
-------
En-j1, Err2r = the first and second relative errors
[Ae] = element coefficient matrix
{ Ce} = unknown vector of concentration
{Re} = element load vector
D-11
-------
D-12
-------
of subelements is MXKGLDK = 316. There are 584 imbedded diffusion fine grids in these 79 extended rough
elements. Therefore, MXADNK=MAXNPK+584. Because of the simulation of 2-D problem by using a 3-D
model, 830 of MXADNK nodes are located on the global boundaries. Hence, MXNDBK should not be less
than 830. In the rough region, there are 33 global element surfaces and 66 subelemental surface located on
the intra-boundaries. So MXMSVK should not be less than 33 and MXLSVK must be greater than or equal
to 66. The maximum number of nodes connected to each node is assumed to 35. Thus MXJBD=35.
C-2
-------
C-3
-------
V-
-(Vp + pgVz)
as
eap at eac at
an.
(E.I.5)
Making the approximation by neglecting the second-order term:
neV-V(Sp) 0
one has:
eap at
an.
^_ac
3ac at
,
'at
+ pS + SpV-n V
r- e s
Defining the compressibility of the fluid as:
where P is the compressibility of the fluid (LT2/M). Also defining the moisture content as:
(B.I.6)
(B.I.7)
(B.1.8)
(B.I.9)
where 6 is the moisture content (dimensionless). One may substitute Eqs.(B. 1.8) and (B. 1.9) into Eq.(B. 1.7)
and rewrite it to obtain:
B-2
-------
V-[ฃ--(Vp + pgVz)]
OR dp a dp dC d$
upp - + 01- + n p
at ac at e at
(B.I. 10)
pS
at
V-(n V)
Remembering that the continuity statement of incompressible solids but a compressible skeleton is
(Yeh, 1992):
at
Rearranging Eq.(B. 1.11) in the following form:
V-(l-ne)Vs =
(B.I.11)
at
V-neVs=V-Vs
Substituting Eq.(B.1.12) into Eq.(B.l.lO), one obtains:
V[ -(Vp + pgVz)]
no ap n ap ac as c,*-,*7
86 ฑ- + 8 t- + n p + pSV-V
| ",, ",/-( -,, e> ",, i s
at ac at at
(B.I.12)
(B.I.13)
Recalling that the flux of solid velocity is the divergence of V (Yeh, 1992):
V-V =a
at
(B.I.14)
where a is the coefficient of consolidation of the media (LI2 /M). Substituting Eq.(B.1.14) into Eq.(B.1.13)
and rewriting:
B-3
-------
V-[ฃ^-(Vp + pgVz] +p*q =
H
/QO o \ ap Q ap ac as
p(0p + Sec)- + 01- + n p
at ac at e at
(B.I.15)
Remembering Eq.(B.1.9) and substituting:
san 9 \ dp a dp dC 9S
p(0p + a)s- + 01- + n p
n at ac at e at
(B.I. 16)
Experimental evidence has shown that the degree of saturation is a function of pressure as:
S = S(p)
Substitution of Eq. (B.I.17) into Eq. (B.I.18) give:
n
-(Vp + pgVz)
pq
\^P- + B-^H- + dS ap
nl "aT+ ac^t"+ plledp~^t"
(B.I. 17)
(B.I.18)
Next, one needs to define the reference pressure head as:
h =
Pwง
where h is the reference pressure head (L) and pw is the reference water density (M/L3).
Eq.(B.1.19) into Eq.(B.l.lS), one obtains:
(B.I. 19)
Substituting
B-4
-------
'
pgVz)
ma 9 N dh a dp dC dS 5h
p(0p +cc)p g + 01- + pn
r \ r /rw0 s\j_ --\/^ --\. '611 s\j_
n, dt dC dt dh dt
(B.I.20)
Dividing Eq.(B.1.20) by pw and rearranging, one gets:
Pgk.
-i^-Vz
q
Pw
e
5h 0 dp dC p dS 5h
(B.1.21)
_^(0gp p +^gp a) + ^^L^L + _^n ^^i
\ Or \Yซ &' W ^^, -^/^^, Sii^,
Pw ne dt Pw dC dt Pw dh dt
Defining the modified compressibilities of the media and water as
= PPwg
(B.I.22)
(B.I.23)
where a' is the modified compressibility of the media (1/L) and P' is the modified compressibility of the water
(1/L). Substituting Eqs.(B.1.22) and (B.1.23) into Eq.(B.1.21) and rearranging:
V
I Pv
^q
Pw
e
ds.ah e d
(B.I.24)
Pw ne
Defining the storage coefficient as:
, / n/o .
^(a +60 +n ) + --
V e
dr/dt pwdCdt
= a' + p76 + n
n
dS
edh
(B.1.25)
where F is the storage coefficient. Substituting Eq.(B. 1.25) into Eq.(B. 1.24) and following Frind (1982) by
neglecting the second term on the right hand side of Eq.(B.I.24), one gets:
B-5
-------
Pgk.l
H I
Pw
-J- q = -!I
Pw
(B.I.26)
Defining the relation:
K
Pgk
(B.I.27)
where K is the hydraulic conductivity tensor. Substituting Eq.(B. 1.27) into Eq.(B. 1.26) and rearranging, one
gets the density-dependent flow equation:
Pw
Pw
(B.I.28)
From the Darcy's law
V = -1 P^.
P H
pgVz)
(B.I.29)
where V is the Darcy flux (L/T). Recalling Eq.(B.1.19) and substituting into Eq.(B.1.29), one obtains:
V = -1 P^.(p gvh + pgVz)
p ]i v 7
(B.I.30)
Rearranging Eq. (B. 1.3 0):
(B.1.31)
and substituting Eq.(B.1.27) into Eq.(B.1.31), one gets the Darcy flux equation for density-dependent flow
in its final form:
V = -K- Vh+Vz
V P )
(B.I.32)
B-6
-------
The density is a function of water, chemical, and microbial concentrations, Cw, Cs, C0, Cn, Cp, Q, Q
and C3, as the following form:
P=Cw + Cs + C0 + Cn + Cp + C1+C2 + C3 (B.1.33)
Physically, the following equation, Eq. (B.I.34), is valid.
C C C C C C, C9 C,
1=++ + +^+++ (B.1.34)
Pw PS Po Pn Pp Pi P2 P3
Eq. (B.I.35) is obtained from (B.1.33) divided by pw and substituting (B.1.34) into the term -t- .
Pw
Pw Pw Pw Pw Pw Pw Pw Pw Pw
(B.1.35)
The viscosity is assumed the following form
J- = 1 + P1CS + P0C0 + PnCb + PpCp + PA + P2C2 + P3C3 (B.I. 36)
MW
where C is the chemical concentration (M/L3) and PS, PO, Pn, Pp, P1; P2, and P3are the parameters (L3/M) that
are used to describe the concentration dependence of dynamic viscosity.
The initial conditions for the flow equations are stated as:
B-7
-------
h = hj(x,y,z) in R (B.I.37)
where R is the region of interest and hj is the prescribed initial condition for hydraulic head. The hj can either
be obtained by solving the steady-state version of Eq. (B.I.26) or alternatively by defining through field
measurements.
The specification of boundary conditions is probably the most critical and complex chore in flow
modeling. As explained by Yeh (1987), the boundary conditions of the region of interest can be examined
from a dynamic, physical, or mathematical point of view. From a dynamic standpoint, a boundary segment
can be either considered as impermeable or flow-through. On the other hand, from a physical point of view,
such a segment could be classified as a soil-soil interface, soil-air interface, or soil-water interface. Lastly,
from a mathematical point of view, the boundary segment can be classified as one of four types of boundary
conditions, namely as (1) Dirichlet, (2) Neumann, (3) Cauchy, or (4) variable boundary conditions. In
addition, a good numerical model must be able to handle these boundary conditions when they vary on the
boundary and are either abruptly or gradually time-dependent.
The Dirichlet boundary condition is usually applied to soil-water interfaces, such as streams, artificial
impoundments, and coastal lines, and involves prescribing the functional value on the boundary. The
Neumann boundary condition, on the other hand, involves prescribing the gradient of the function on the
boundary and does not occur very often in real-world problems. This condition, however, can be encountered
at the base of the media where natural drainage occurs. The third type of boundary condition, the Cauchy
boundary condition, involves prescribing the total normal flux due to the gradient on the boundary. Usually
surface water bodies with known infiltration rates through the layers of the bottom of their sediments or liners
into the subsurface media are administered this boundary condition. If there exists a soil-air interface in the
region of interest, a variable boundary condition is employed. In such a case, either Dirichlet or Cauchy
boundary conditions dominate, mainly depending on the potential evaporation, the conductivity of the media,
and the availability of water such as rainfall (Yeh, 1987).
B-8
-------
From the above discussion, four types of boundary conditions can be specified for the flow equations
depending on the physical location of the boundaries. These boundary conditions are stated as:
Dirichlet Boundary Conditions:
h = hd(xb>yb>V) on Bd
Neumann Boundary Conditions:
nK
Pw.
p
Cauchy Boundary Conditions:
B
(B.1.38)
(B.1.39)
-n.K- -Vh + Vz = qc(xb,yb,zb,t) on Bc
Variable Boundary Conditions - During Precipitation Period:
h = VWb'V) on Bv
or
Kl ' Wv
-\
= qp(xb,yb,zb,t) on
Variable Boundary Conditions - During Non-Precipitation Period:
h =
on
or
or
nK
Pw
k p
= qe(xb>yb>V) on Bv
(B.I.40)
where n is the outward unit vector normal to the boundary; (x^y^) is the spatial coordinate on the boundary;
B-9
-------
hd, QJJ, and qc are the Dirichlet functional value, Neumann flux, and Cauchy flux, respectively; Bd, Bn, Bc, and
Bv are the Dirichlet, Neumann, Cauchy, and variable boundaries, respectively; hp and qp are the allowed
ponding depth and the throughfall of precipitation, respectively, on the variable boundary; h,,, is the allowed
minimum pressure on the variable boundary; and qe is the allowed maximum evaporation rate (= potential
evaporation) on the variable boundary. Note that only one of Eqs. (B.1.41a) through (B.1.41e) is utilized at
any point on the variable boundary at any time.
B.2 Governing Equations for Transport
This section derives the governing equations for chemical and microbial transport and fate in
subsurface media. The assumptions, which form the basis for the transport and fate model and which hold for
each one of the four chemical compounds, i.e. substrate s, nutrient p, oxygen o, and nitrate n, and of three
microbial biomass, i.e., microbe #1, microbe #2, and microbe #3, are now listed.
(1) Mass transport is via advection and dispersion plus artificial sources and sinks - To simplify the
notation, let C stand for Cs, C0, Cn, Cp, Q, C2, and C3. The well known transport equation is derived
in (Yeh, et al, 1994) and written as
r>C r)S r)n r)f)
e-T + Pb^T+V-VC = ' a^(eC+PbS)--^C + m (B.2.1)
at at at at
where S is the material concentration in the absorbed phase (M/M), 6 is moisture content (L3/L3), a
is the compressibility of the medium, p is the pressure, and m is the artificial source/sink (which is
equal to qCm for the case of sources or equal to qC for the case sinks with Cm being the concentration
of the source).
(2) The porous medium follows the linear isotherm rule as follows.
S = KdC (B.2.2)
where Kd is the distribution coefficient. Substituting Eq. (B.2.2) into Eq. (B.2.1), the transport
B-10
-------
equation becomes
(6 + pbKd)+V-VC = V-6D-VC + m - C - oc^(6+pbKd)C (B.2.3)
dt dt dt
(3) Loss of chemicals and microbes can occur via first order irreversible loss processes, such as chemical
transformations and precipitation in both the free and sorbed phases, in addition to loss via microbial
degradation or growth. Then the governing equation is given as Eq. (B.2.4).
(6+pbKd)+V-VC = V-6D-VC - I a^+A| (6+pbKd)C-C + m (B.2.4)
dt \ dt ) dt
(4) To rewrite the above conservative form of the transport equation to the advective form, the governing
equation is obtained as
(6 + pbKd)+V-VC = V-6D-VC - cc^+A (6+pbKd)C-C + m-CV-V (B.2.5)
dt \ dt ) dt
The following relationship can be derived from the Darcy velocity Eq. (B. 1.32)
-V-K-(Vh + -P-Vz) = V-(-P-V) = V-V-P- + -P-V-V m 2 6)
Pw Pw Pw Pw
Substituting Eq. (B.2.6) into Eq. (B.I.28), one has
p p* p rdh ,TV7 p
-tLv-V = -^q--^F-V-V-^ (B27)
Pw Pw Pw at Pw
The transport governing equation is expressed as Eq. (B.2.8) after substituting Eq. (B.2.7) into Eq.
(B.2.5).
B-11
-------
pbKd)+V-VC = V-6D-VC
at at
(B.2.8)
(5) Microbiological processes are modeled using process laws described by Molz et al. [1986] and
Widdowson et al. [1988], who constructed biodegradation models using the carbon assimilation and
oxidation assumptions of Herbert [1958]. The model developed here includes three microbial popula-
tions, namely Q, C2, and C3. The utilization rate laws adapted from Benefield and Molz [1984] are
sop
Mi"
VW
1 0
f C> 1
k+cj
f cฐ 1
k+cj
f Cp 1
k+cj
(B.2.9)
r(2)
snp
c
pn
(B.2.10)
r(3)
sop
c
c
(B.2.11)
r(3)
snp
M?
c
pn
(1) (1) (2) (2) (3) (3) (3)
rv' = r Y = r r = r+r
s sop' s snp' s sop snp
(B.2.12)
(B.2.13)
where rsop(1), rsop(3), rsnp(2), and rsnp(3) (M/M) represent the removal rate of substrate under aerobic or
anaerobic conditions with respect to microbes #1, #2, or #3.
The growth of three microbial populations adapted from Widdowson et al. [1988] are
B-12
-------
microbe activity
o sop o
(B.2.14)
microbe activity
- |V(2)r(2)-'
~ ! * n rsnp '
(B.2.15)
at
microbe activity
(B.2.16)
(6) Expressions for the respective electron acceptor utilization rates are based on the assumptions that (i)
utilization resulting from the energy requirement for gross heterotrophic biomass production is
proportional to substrate utilization, and (ii) requirement for energy of maintenance follows a Monod-
type response with respect to the particular species. Thus the oxygen utilization rate per unit of
biomass (specific rate) is expressed as
r - vY
ro - Yo Yo
sop o o
c
r(3) _ v(3)Y(3
ro - Yo Yo
sop o o
Likewise, the expression for the specific rate of nitrate utilization is given by
r(2) _ (2) (2) (2) (2)^(2)
rn - Yn Yn rs + a A
snp "'n n
r(3) _ (3) (3) (3) (3)^(3)
rn - Yn Yn r + a A
snp "'n n
(B.2.17)
(B.2.18)
(B.2.19)
(B.2.20)
B-13
-------
The specific rate of nutrient removal for the synthesis of heterotrophic biomass is assumed
proportional to the specific rate of the substrate utilization rate and is expressed by
-o sop? p
= (2) (2) (3) = (3) (3) (3) (3)
n snp> p o sop n snp
(B.2.21)
After coupling the biodegradation and microbial growth, the governing equations of fate and transport
of chemicals and microbes are expressed as the following:
dC
=V-6D-VC
+As(e+PbKds)C
C
K
so ^- s
C
= V-6D-VC
)f CS + (-
PO ^p
u(2)
H-H
Yf}
\ Cs 1
Kf^Cs
\ C" 1
K?-Cn^
p
^pn + ^p
.(3)
F
(3)
G
c
C
(B.2.22)
B-14
-------
uu A I x/-v TT-\/~< M /~< /T-- L/V/ \ L/ii ,-. /i W\ซ 7- TT/ M \ /-<
- a-+A.o (9+PbKdo)C0 + mo - * qC0 + (F -)C0 + ()V-V(-^-) C
= V-6D-VC
Ao (6+pbKdo)C0 + mo-qC0 + (F- )
) p at
SO s O
c
C
P Pw
c
C
C
(B.2.23)
dC
-ซ +An(B+PbKdn)Cn
-f )f Cn + ()V.V()Cn
(2)
-(6+pbKd2)rnC2-(e
= V-6D-VC
"f^O^PA
V(^)V.^)Cn
(2)(2)
(2) + r
pn +Cp
K
K
(3)
pn
(B.2.24)
B-15
-------
dC
-f )f
*2UA (6-
at at
-(i)
(B.2.25)
ri21
c
u(3)
(3) ^o
Y
1
(3)
(3)
, c(3) ^n
11 Y(3)
cs
K (3) + r
. sn s.
cn
Kn(3)+Cn
cp
v- (3) , r
V +Lp.
B-16
-------
,
(e+PbKdl)
at
-f )f C,
= V-6D-VC,
microbial activity
(B.2.26)
f )f C,
"
e
(9 + PbKd2)-+V-VC2 =V-6D-
-a|+A2(e+PbKd2)C2
-f )f
= V-9D-VC,
microbial activity
(B.2.27)
c
I (2) I
n
B-17
-------
pbKd3)
dC,
^
at
= V-6D-VC
E +A3(6 +PbKd3)C3 + m3 - qC3 + (F - )C3 + ()V-V(-) C
dC,
I microbial activity
= V-6D-VC3
f
+ M(3)
tf
cs
so
Ks(n3) + Cs.
~CS^
co
Kf<
Kf+Cn.
C0.
cp
Kp(3) +
Kpn + Up
cp.
^(Cc
^0
)A(-3Vrr ">
An JWo''
(B.2.28)
+A3(6 +PbKd3)C3 + m3 - qC3 + (F - )C3 + ()V-V(-) C:
The specification of boundary conditions is a difficult and intricate task in transport modeling. From
the dynamic point of view, a boundary segment may be classified as either flow-through or impervious. From
the physical point of view, it is a soil-air interface, or soil-soil interface, or soil-water interface. From the
mathematical point of view, it may be treated as a Dirichlet boundary on which the total analytical
concentration is prescribed, Neumann boundary on which the flux due to the gradient of total analytical
concentration is known, or Cauchy boundary on which the total flux is given. An even more difficult
mathematical boundary is the variable conditions on which the boundary conditions are not known a priori but
are themselves the solution to be sought. In other words, on the mathematically variable boundary, either
Neumann or Cauchy conditions may prevail and change with time. Which condition prevails at a particular
time can be determined only in the cyclic processes of solving the governing equations (Freeze 1972a, 1972b;
Yeh and Ward 1980; Yeh and Ward, 1981).
B-18
-------
B.3 Simplification
The governing equations derived in Appendix B represents the density-dependent flow and the fate
and transport of microbes and chemicals in slightly deformable media as characterized by the modified
compressibilities, a' and P'. In the development of 3DFATMIC, it is assumed that the media are
non-deformable, i.e, a' = 0 and P' = 0 . As a result of this simplification, one has
T7 36 ft
F - = 0 (B.3.1)
To remove this restriction, it is as simple as making apple pie. Finally, it should be noted that the terms
associated with source/sinks can be reduced to source terms only. It is obvious that
r\ r\
m - qC = qinCin - qC for a source
*
m - qC = qinCin for a sink
(B.3.2)
P
because for a source m = qmCm and for a sink m = qC and p* = p. The governing equations used in Chapter
2 were obtained from the corresponding equations in this appendix using Eqs. (B.3.1) and (B.3.2) and setting
-------
B-20
-------
|