vvEPA
United States
Environmental Protection
Agency
Office of Research and
Development
Washington DC 20460
EPA/600/R-94/168b
September 1994
The Hydrologic
Evaluation of
Landfill Performance
(HELP) Model
Engineering
Documentation for
Version 3
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EPA/600/R-94/168b
September 1994
THE HYDROLOGIC EVALUATION OF LANDFILL
PERFORMANCE (HELP) MODEL
ENGINEERING DOCUMENTATION FOR VERSION 3
by
Paul R, Schroeder, Tamsen S. Dozier, Paul A. Zappi,
Bruce M. McEnroe, John W. Sjostrom and R. Lee Peyton
Environmental Laboratory
U.S. Army Corps of Engineers
Waterways Experiment Station
Vicksburg, MS 39180-6199
Interagency Agreement No. DW21931425
Project Officer
Robert E. Landreth
Waste Minimization, Destruction and Disposal Research Division
Risk Reduction Engineering Laboratory
Cincinnati, Ohio 45268
RISK REDUCTION ENGINEERING LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CINCINNATI, OHIO 45268
Printed on Recycled Paper
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Disclaimer
The information in this document has been funded wholly or in part by the United
States Environmental Protection Agency under Interagency Agreement No. DW21931425
to the U.S. Army Engineer Waterways Experiment Station (WES). It has been subjected
to WES peer and administrative review, and it has been approved for publication as an
EPA document. Mention of trade names or commercial products does not constitute
endorsement or recommendation for use.
i i
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FOREWORD
Today's rapidly developing and changing technologies and industrial products and
practices frequently carry with them the increased generation of materials that, if
improperly dealt with, can threaten both public health and the environment. Abandoned
waste sites and accidental releases of toxic and hazardous substances to the environment
also have important environmental and public health implications. The Risk Reduction
Engineering Laboratory assists in providing an authoritative and defensible engineering
basis for assessing and solving these problems. Its products support the policies,
programs and regulations of the Environmental Protection Agency, the permitting and
other responsibilities of State and local governments, and the needs of both large and
small businesses in handling their wastes responsibly and economically.
This report presents engineering documentation of the Hydrologic Evaluation of
Landfill Performance (HELP) model and its user interface. The HELP program is a
quasi-two-dimensional hydrologic model for conducting water balance analyses of
landfills, cover systems, and other solid waste containment facilities. The model accepts
weather, soil and design data and uses solution techniques that account for the effects of
surface storage, snow melt, runoff, infiltration, evapotranspiration, vegetative growth, soil
moisture storage, lateral subsurface drainage, leachate recirculation, unsaturated vertical
drainage, and leakage through soil, geomembrane or composite liners. Landfill systems
including various combinations of vegetation, cover soils, waste cells, lateral drain
layers, low permeability barrier soils, and synthetic geomembrane liners may be
modeled. The model facilitates rapid estimation of the amounts of runoff,
evapotranspiration, drainage, leachate collection and liner leakage that may be expected
to result from the operation of a wide variety of landfill designs. The primary purpose
of the model is to assist in the comparison of design alternatives. The model is a tool
for both designers and permit writers.
E. Timothy Oppelt, Director
Risk Reduction Engineering Laboratory
in
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ABSTRACT
The Hydrologic Evaluation of Landfill Performance (HELP) computer program is
a quasi-two-dimensional hydrologic model of water movement across, into, through and
out of landfills. The model accepts weather, soil and design data and uses solution
techniques that account for the effects of surface storage, snowmelt, runoff, infiltration,
evapotranspiration, vegetative growth, soil moisture storage, lateral subsurface drainage,
leachate recirculation, unsaturated vertical drainage, and leakage through soil,
geomembrane or composite liners. Landfill systems including various combinations of
vegetation, cover soils, waste cells, lateral drain layers, low permeability barrier soils,
and synthetic geomembrane liners may be modeled. The program was developed to
conduct water balance analyses of landfills, cover systems, and solid waste disposal and
containment facilities. As such, the model facilitates rapid estimation of the amounts of
runoff, evapotranspiration, drainage, leachate collection, and liner leakage that may be
expected to result from the operation of a wide variety of landfill designs. The primary
purpose of the model is to assist in the comparison of design alternatives as judged by
their water balances. The model, applicable to open, partially closed, and fully closed
sites, is a tool for both designers and permit writers.
This report documents the solution methods and process descriptions used in
Version 3 of the HELP model. Program documentation including program options,
system and operating requirements, file structures, program structure and variable
descriptions are provided in a separate report. Section 1 provides basic program
identification. Section 2 provides a narrative description of the simulation model.
Section 3 presents data generation algorithms and default values used in Version 3.
Section 4 describes the method of solution and hydrologic process algorithms. Section
5 lists the assumptions and limitations of the HELP model.
The user interface or input facility is written in the Quick Basic environment of
Microsoft Basic Professional Development System Version 7.1 and runs under DOS 2.1
or higher on IBM-PC and compatible computers. The HELP program uses an interactive
and a user-friendly input facility designed to provide the user with as much assistance as
possible in preparing data to run the model. The program provides weather and soil data
file management, default data sources, interactive layer editing, on-line help, and data
verification and accepts weather data from the most commonly used sources with several
different formats.
HELP Version 3 represents a significant advancement over the input techniques of
Version 2. Users of the HELP model should find HELP Version 3 easy to use and
should be able to use it for many purposes, such as preparing and editing landfill profiles
and weather data. Version 3 facilitates use of metric units, international applications, and
designs with geosynthetic materials.
IV
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This report should be cited as follows:
Schroeder, P. R., Dozier, T. S., Zappi, P. A., McEnroe, B. M.,
Sjostrom, J. W., and Peyton, R.L. (1994). "The Hydrologic Evaluation
of Landfill Performance (HELP) Model: Engineering Documentation for
Version 3," EPA/600/9-94/xxx, U.S. Environmental Protection Agency
Risk Reduction Engineering Laboratory, Cincinnati, OH.
This report was submitted in partial fulfillment of Interagency Agreement Number
DW21931425 between the U.S. Environmental Protection Agency and the U.S. Army
Engineer Waterways Experiment Station, Vicksburg, MS. This report covers a period
from November 1988 to August 1994 and work was completed as of August 1994.
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CONTENTS
page
DISCLAIMER ii
FOREWORD Ill
ABSTRACT iv
FIGURES viii
TABLES ix
ACKNOWLEDGMENTS x
1. PROGRAM IDENTIFICATION 1
2. NARRATIVE DESCRIPTION 3
3. DATA GENERATION AND DEFAULT VALUES 9
3.1 Overview 9
3.2 Synthetic Weather Generation 9
3.3 Moisture Retention and Hydraulic Conductivity Parameters 12
3.3.1 Moisture Retention Parameters 12
3.3.2 Unsaturated Hydraulic Conductivity 13
3.3.3 Saturated Hydraulic Conductivity for
Vegetated Materials 15
3.4 Evaporation Coefficient 16
3.5 Default Soil and Waste Characteristics 17
3.5.1 Default Soil Characteristics 17
3.5.2 Default Waste Characteristics 21
3.5.3 Default Geosynthetic Material Characteristics 25
3.6 Soil Moisture Initialization 25
3.7 Default Leaf Area Indices and Evaporative Zone Depths 26
4. METHOD OF SOLUTION 29
4.1 Overview 29
4.2 Runoff 30
4.2.1 Adjustment of Curve Number for Soil Moisture 34
4.2.2 Computation of Default Curve Numbers 36
4.2.3 Adjustment of Curve Number for Surface Slope 37
4.2.4 Adjustment of Curve Number for Frozen Soil 39
4.2.5 Summary of Daily Runoff Computation 39
4.3 Prediction of Frozen Soil Conditions 40
VI
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4.4 Snow Accumulation and Melt 41
4.4.1 Nonrain Snowmelt 42
4.4.2 Rain-on-Snow Melt Condition 43
4.4.3 Snowmelt Summary 45
4.5 Interception 47
4.6 Potential evapotranspiration 48
4.7 Surface Evaporation 51
4.7.1 No Snow Cover 51
4.7.2 Snow Cover Present 52
4.7.3 Remaining Evaporative Demand 54
4.8 Infiltration 55
4.9 Soil Water Evaporation 55
4.10 Plant Transpiration 59
4.11 evapotranspiration 60
4.12 Vegetative Growth 62
4.13 Subsurface Water Routing 68
4.14 Vertical Drainage '*-
4.15 Soil Liner Percolation 73
4.16 geomembrane Liner Leakage 74
4.16.1 Vapor Diffusion Through Intact geomembranes 75
4.16.2 Leakage Through Holes ingeomembranes 76
4.17 geomembraneand Soil Liner Design Cases 93
4.18 Lateral Drainage 98
4.19 Lateral Drainage Recirculation 103
4.20 Subsurface Inflow 104
4.21 Linkage of Subsurface Flow Processes 104
5. ASSUMPTIONS AND LIMITATIONS 106
5.1 Methods of Solution 106
5.2 Limits of Application 109
REFERENCES in
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FIGURES
No. Page
1 Schematic Profile View of a Typical Hazardous Waste Landfill 6
2 Relation Among Moisture Retention Parameters
and Soil Texture Class 13
3 Geographic Distribution of Maximum Leaf Area Index 26
4 Geographic Distribution of Minimum Evaporative Depth 27
5 Geographic Distribution of Maximum Evaporative Depth 27
6 Relation Between Runoff, Precipitation, and Retention 31
7 SCS Rainfall-Runoff Relation Normalized
on Retention Parameter s 33
8 Relation Between SCS Curve Number and Default
Soil Texture Number for Various Levels of Vegetation 37
9 Leakage with Interracial Flow Below Flawed geomembrane 84
10 Leakage with Interracial Flow Above Flawed geomembrane 84
llgeomembrane Liner Design Case 1 94
12 geomembrane Liner Design Case 2 95
13 geomembrane Liner Design Case 3 95
14 geomembrane Liner Design Case 4 96
15 geomembrane Liner Design Case 5 98
16 geomembrane Liner Design Case 6 99
17 Lateral Drainage Definition Sketch 100
Vlll
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TABLES
No. Page
1 Default Low Density Soil Characteristics 19
2 Moderate and High Density Default Soils 21
3 Default Soil Texture Abbreviations 22
4 Default Waste Characteristics 23
5 Saturated Hydraulic Conductivity of Wastes 24
6 Default Geosynthetic Material Characteristics 25
7 Constants for Use in Equation 32 38
8 geomembrane Diffusivity Properties 77
9 Needle-Punched, Non-Woven Geotextile Properties 79
IX
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ACKNOWLEDGMENTS
The support of the project by the Waste Minimization, Destruction and Disposal
Research Division, Risk Reduction Engineering Laboratory, U.S. Environmental
Protection Agency, Cincinnati, OH and the Headquarters, U.S. Army Corps of
Engineers, Washington, DC, through Interagency Agreement No. DW21931425 is
appreciated. In particular, the authors wish to thank the U.S. EPA Project Officer,
Mr. Robert Landreth, for his long standing support.
This document was prepared at the U.S. Army Corps of Engineers Waterways
Experiment Station. The final versions of this document and the HELP program were
prepared by Dr. Paul R. Schroeder. Dr. Paul R. Schroeder directed the development of
the HELP model and assembled the simulation code. Ms. Tamsen S. Dozier revised,
tested and documented the evapotranspiration, snowmelt and frozen soil processes.
Mr. Paul A. Z.appi revised, assembled and documented the geomembrane leakage
processes and the default soil descriptions. Dr. Bruce M. McEnroe revised, tested and
documented the lateral drainage process for Version 2, which was finalized in this
version. Mr. John W. Sjostrom assembled, tested and documented the synthetic weather
generator and the vegetative growth process for Version 2, which was finalized in this
version. Dr. R, Lee Peyton developed, revised, tested and documented the slope and soil
moisture effects on runoff and curve number. Dr. Paul R. Schroeder developed, tested
and documented the remaining processes and the output. Ms. Cheryl Lloyd assisted in
the final preparation of the report. The figures used in the report were prepared by
Messrs. Christopher Chao, Jimmy Farrell, and Shawn Boelman.
The documentation report and simulation model were reviewed by Dr. Jim Ascough,
USDA-ARS-NPA, and Dr. Ragui F. Wilson-Fahmy, Geosynthetic Research Institute,
Drexel University. This report has not been subjected to the EPA review and, therefore,
the contents do not necessarily reflect the views of the Agency, and no official
endorsement should be inferred.
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SECTION 1
PROGRAM IDENTIFICATION
PROGRAM TITLE: Hydrologic Evaluation of Landfill Performance (HELP) Model
WRITERS: Paul R. Schroeder, Tamsen S. Dozier, John W. Sjostrom and Bruce M. McEnroe
ORGANIZATION: U.S. Army Corps of Engineers, Waterways Experiment Station (WES)
DATE: September 1994
UPDATE: None Version No.: 3.00
SOURCE LANGUAGE: The simulation code is written in ANSI FORTRAN 77 using Ryan-
McFarland Fortran Version 2.44 with assembly language and Spindrift Library
extensions for Ryan-McFarland Fortran to perform system calls, and screen operations.
The user interface is written in BASIC using Microsoft Basic Professional Development
System Version 7.1. Several of the user interface support routines are written in ANSI
FORTRAN 77 using Ryan-McFarland Fortran Version 2.44, including the synthetic
weather generator and the ASCII data import utilities.
HARDWARE: The model was written to run on IBM-compatible personal computers under
the DOS environment. The program requires an IBM-compatible 8088, 80286, 80386
or 80486-based CPU (preferably 80386 or 80486) with an 8087, 80287, 80387 or 80486
math co-processor. The computer system must have a monitor (preferably color EGA
or better), a 3.5- or 5.25-inch floppy disk drive (preferably 3.5-inch double-sided, high-
density), a hard disk drive with 6 MB of available storage, and 400k bytes or more of
available low level RAM. A printer is needed if a hard copy is desired.
AVAILABILITY: The source code and executable code for IBM-compatible personal comput-
ers are available from the National Technical Information Service (NTIS). Limited
distribution immediately following the initial distribution will be available from the
USEPA Risk Reduction Engineering Laboratory, the USEPA Center for Environmental
Research Information and the USAE Waterways Experiment Station.
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ABSTRACT: The Hydrologic Evaluation of Landfill Performance (HELP) computer program
is a quasi-two-dimensional hydrologic model of water movement across, into, through
and out of landfills. The model accepts weather, soil and design data and uses solution
techniques that account for surface storage, snowmelt, runoff, infiltration, vegetative
growth, evapotranspiration, soil moisture storage, lateral subsurface drainage, leachate
recirculation, unsaturated vertical drainage, and leakage through soil, geomembrane or
composite liners. Landfill systems including combinations of vegetation, cover soils,
waste cells, lateral drain layers, barrier soils, and synthetic geomembrane liners may be
modeled. The program was developed to conduct water balance analyses of landfills,
cover systems, and solid waste disposal facilities. As such, the model facilitates rapid
estimation of the amounts of runoff, evapotranspiration, drainage, leachate collection, and
liner leakage that may be expected to result from the operation of a wide variety of
landfill designs. The primary purpose of the model is to assist in the comparison of
design alternatives as judged by their water balances. The model, applicable to open,
partially closed, and fully closed sites, is a tool for both designers and permit writers.
The HELP model uses many process descriptions that were previously developed,
reported in the literature, and used in other hydrologic models. The optional synthetic
weather generator is the WGEN model of the U.S. Department of Agriculture (USDA)
Agricultural Research Service (ARS) (Richardson and Wright, 1984). Runoff modeling
is based on the USDA Soil Conservation Service (SCS) curve number method presented
in Section 4 of the National Engineering Handbook (USDA, SCS, 1985). Potential
evapotranspiration is modeled by a modified Penman method (Penman, 1963).
Evaporation from soil is modeled in the manner developed by Ritchie (1972) and used
in various ARS models including the Simulator for Water Resources in Rural Basins
(SWRRB) (Arnold et al., 1989) and the Chemicals, Runoff, and Erosion from
Agricultural Management System (CREAMS) (Knisel, 1980). Plant transpiration is
computed by the Ritchie's (1972) method used in SWRRB and CREAMS. The
vegetative growth model was extracted from the SWRRB model. Evaporation of
interception, snow and surface water is based on an energy balance. Interception is
modeled by the method proposed by Horton (1919). Snowmelt modeling is based on the
SNOW-17 routine of the National Weather Service River Forecast System (NWSRFS)
Snow Accumulation and Ablation Model (Anderson, 1973). The frozen soil sub model
is based on a routine used in the CREAMS model (Knisel et al., 1985). Vertical
drainage is modeled by Darcy's (1856) law using the Campbell (1974) equation for
unsaturated hydraulic conductivity based on the Brooks-Corey (1964) relationship.
Saturated lateral drainage is modeled by an analytical approximation to the steady-state
solution of the Boussinesq equation employing the Dupuit-Forchheimer (Forchheimer,
1930) assumptions. Leakage through geomembranes is modeled by a series of equations
based on the compilations by Giroud et al. (1989, 1992). The processes are linked
together in a sequential order starting at the surface with a surface water balance; then
evapotranspiration from the soil profile; and finally drainage and water routing, starting
at the surface with infiltration and then proceeding downward through the landfill profile
to the bottom. The solution procedure is applied repetitively for each day as it simulates
the water routing throughout the simulation period.
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SECTION 2
NARRATIVE DESCRIPTION
The HELP program, Versions 1, 2 and 3, was developed by the U.S. Army Engineer
Waterways Experiment Station (WES), Vicksburg, MS, for the U.S. Environmental
Protection Agency (EPA), Risk Reduction Engineering Laboratory, Cincinnati, OH, in
response to needs in the Resource Conservation and Recovery Act (RCRA) and the
Comprehensive Environmental Response, Compensation and Liability Act (CERCLA,
better known as Superfund) as identified by the EPA Office of Solid Waste,
Washington, DC. The primary purpose of the model is to assist in the comparison of
landfill design alternatives as judged by their water balances.
The Hydrologic Evaluation of Landfill Performance (HELP) model was developed
to help hazardous waste landfill designers and regulators evaluate the hydrologic
performance of proposed landfill designs. The model accepts weather, soil and design
data and uses solution techniques that account for the effects of surface storage,
snowmelt, runoff, infiltration, evapotranspiration, vegetative growth, soil moisture
storage, lateral subsurface drainage, leachate recirculation, unsaturated vertical drainage,
and leakage through soil, geomembrane or composite liners. Landfill systems including
various combinations of vegetation, cover soils, waste cells, lateral drain layers, low
permeability barrier soils, and synthetic geomembrane liners may be modeled. Results
are expressed as daily, monthly, annual and long-term average water budgets.
The HELP model is a quasi-two-dimensional, deterministic, water-routing model for
determining water balances. The model was adapted from the HSSWDS (Hydrologic
Simulation Model for Estimating Percolation at Solid Waste Disposal Sites) model of the
U.S. Environmental Protection Agency (Perrier and Gibson, 1980; Schroeder and
Gibson, 1982), and various models of the U.S. Agricultural Research Service (ARS),
including the CREAMS (Chemical Runoff and Erosion from Agricultural Management
Systems) model (Knisel, 1980), the SWRRB (Simulator for Water Resources in Rural
Basins) model (Arnold et al., 1989), the SNOW-17 routine of the National Weather
Service River Forecast System (NWSRFS) Snow Accumulation and Ablation Model
(Anderson, 1973), and the WGEN synthetic weather generator (Richardson and Wright,
1984).
HELP Version 1 (Schroeder et al., 1984a and 1984b) represented a major advance
beyond the HSSWDS program (Perrier and Gibson, 1980; Schroeder and Gibson, 1982),
which was also developed at WES. The HSSWDS model simulated only the cover
system, did not model lateral flow through drainage layers, and handled vertical drainage
only in a rudimentary manner. The infiltration, percolation and evapotranspiration
routines were almost identical to those used in the Chemicals, Runoff, and Erosion from
Agricultural Management Systems (CREAMS) model, which was developed by Knisel
(1980) for the U.S. Department of Agriculture (USDA). The runoff and infiltration
3
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routines relied heavily on the Hydrology Section of the National Engineering Handbook
(USDA, Soil Conservation Service, 1985), Version 1 of the HELP model incorporated
a lateral subsurface drainage model and improved unsaturated drainage and liner leakage
models into the HSSWDS model. In addition, the HELP model provided simulation of
the entire landfill including leachate collection and liner systems.
Version 1 of the HELP program was tested extensively using both field and
laboratory data. HELP Version 1 simulation results were compared to field data for
20 landfill cells from seven sites (Schroeder and Peyton, 1987a). The lateral drainage
component of HELP Version 1 was tested against experimental results from two large--
scale physical models of landfill liner/drain systems (Schroeder and Peyton, 1987b). The
results of these tests provided motivation for some of the improvements incorporated into
HELP Version 2.
Version 2 (Schroeder et al., 1988a and 1988b) presented a great enhancement of the
capabilities of the HELP model. The WGEN synthetic weather generator developed by
the USDA Agricultural Research Service (ARS) (Richardson and Wright, 1984) was
added to the model to yield daily values of precipitation, temperature and solar radiation.
This replaced the use of normal mean monthly temperature and solar radiation values and
improved the modeling of snow and evapotranspiration. Also, a vegetative growth model
from the Simulator for Water Resources in Rural Basins (SWRRB) model developed by
the ARS (Arnold et al., 1989) was merged into the HELP model to calculate daily leaf
area indices. Modeling of unsaturated hydraulic conductivity and flow and lateral
drainage computations were improved. Default soil data were improved, and the model
permitted use of more layers and initialization of soil moisture content.
In Version 3, the HELP model has been greatly enhanced beyond Version 2. The
number of layers that can be modeled has been increased. The default soil/material
texture list has been expanded to contain additional waste materials, geomembranes,
geosynthetic drainage nets and compacted soils. The model also permits the use of a
user-built library of soil textures. Computations of leachate recirculation and
groundwater drainage into the landfill have been added. Moreover, HELP Version 3
accounts for leakage through geomembranes due to manufacturing defects (pinholes) and
installation defects (punctures, tears and seaming flaws) and by vapor diffusion through
the liner based on the equations compiled by Giroud et al. (1989, 1992). The estimation
of runoff from the surface of the landfill has been improved to account for large landfill
surface slopes and slope lengths. The snowmelt model has been replaced with an energy-
based model; the Priestly-Taylor potential evapotranspiration model has been replaced
with a Penman method, incorporating wind and humidity effects as well as long wave
radiation losses (heat loss at night). A frozen soil model has been added to improve
infiltration and runoff predictions in cold regions. The unsaturated vertical drainage
model has also been improved to aid in storage computations. Input and editing have
been further simplified with interactive, full-screen, menu-driven input techniques.
The HELP model requires daily climatologic data, soil characteristics, and design
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specifications to perform the analysis. Daily rainfall data may be input by the user,
generated stochastically, or taken from the model's historical data base. The model
contains parameters for generating synthetic precipitation for 139 U.S. cities. The
historical data base contains five years of daily precipitation data for 102 U.S. cities.
Daily temperature and solar radiation data are generated stochastically or may be input
by the user. Necessary soil data include porosity, field capacity, wilting point, saturated
hydraulic conductivity, and Soil Conservation Service (SCS) runoff curve number for
antecedent moisture condition IL The model contains default soil characteristics for 42
material types for use when measurements or site-specific estimates are not available.
Design specifications include such things as the slope and maximum drainage distance
for lateral drainage layers, layer thicknesses, leachate recirculation procedure, surface
cover characteristics and information on any geomembranes.
Figure 1 is a definition sketch for a somewhat typical closed hazardous waste landfill
profile. The top portion of the profile (layers 1 through 4) is the cap or cover. The
bottom portion of the landfill is a double liner system (layers 6 through 11), in this case
composed of a geomembrane liner and a composite liner. Immediately above the bottom
composite liner is a leakage detection drainage layer to collect leakage from the primary
liner, in this case, a geomembrane. Above the primary liner area geosynthetic drainage
net and a sand layer that serve as drainage layers for leachate collection. The drain
layers composed of sand are typically at least 1-ft thick and have suitably spaced
perforated or open joint drain pipe embedded below the surface of the liner. The
leachate collection drainage layer serves to collect any leachate that may percolate
through the waste layers. In this case where the liner is solely a geomembrane, a
drainage net may be used to rapidly drain leachate from the liner, avoiding a significant
buildup of head and limiting leakage. The liners are sloped to prevent pending by
encouraging leachate to flow toward the drains. The net effects are that very little
leachate should leak through the primary liner and virtually no migration of leachate
through the bottom composite liner to the natural formations below. Taken as a whole,
the drainage layers, geomembrane liners, and barrier soil liners may be referred to as the
leachate collection and removal system (drain/liner system) and more specifically a
double liner system.
Figure 1 shows eleven layers—four in the cover or cap, one as the waste layers, three
in the primary leachate collection and removal system (drain/liner system) and three in
the secondary leachate collection and removal system (leakage detection). These eleven
layers comprise three subprofiles or modeling units. A subprofile consists of all layers
between (and including) the landfill surface and the bottom of the top liner system,
between the bottom of one liner system and the bottom of the next lower liner system,
or between the bottom of the lowest liner system and the bottom of the lowest soil layer
modeled. In the sketch, the top subprofile contains the cover layers, the middle
subprofile contains the waste, drain and liner system for leachate collection, and the
bottom subprofile contains the drain and liner system for leakage detection. Six
subprofiles in a single landfill profile may be simulated by the model.
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ffieC/HTATtOHf
(2) LATERAL DRAINAGE LAYER SAND
(3)
LATffUi OPMHA6E
(FROU COVER)
GEOMEMBRANE LINER
(4) BARRIER SOIL LAYER
SLOPf
CLAY
VERTICAL
(?) PERCOLATION
LAYER
WASTE
(6) LATERAL DRAINAGE LAYER SAND
(l\ LATERAL DRAINAGE NET-
LATfRAL DRAINAGE
(IEACHATE COUSCTIOt/l
BARRIER SOIL LINER
CLAY
UAXtUUU
DRAINAGE
OlSTAMCf
I PEfiCOLATIOM(LEAKAGE)
GEOMEMBRANE LINER
LATERAL DRAINAGE
Hi
Figure 1. Schematic Profile View of a Typical Hazardous Waste Landfill
6
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The layers in the landfill are typed by the hydraulic function that they perform. Four
types are of layers are available: vertical percolation layers, lateral drainage layers,
barrier soil liners and geomembrane liners. These layer types are illustrated in Figure 1.
The topsoil and waste layers are generally vertical percolation layers. Sand layers above
liners are typically lateral drainage layers; compacted clay layers are typically barrier soil
liners, geomembranes are typed as geomembrane liners. Composite liners are modeled
as two layers. Geotextiles are not considered as layers unless they perform a unique
hydraulic function.
Flow in a vertical percolation layer (e.g., layers 1 and 5 in Figure 1) is either
downward due to gravity drainage or extracted by evapotranspiration. Unsaturated
vertical drainage is assumed to occur by gravity drainage whenever the soil moisture is
greater than the field capacity (greater than the wilting point for soils in the evaporative
zone) or when the soil suction of the layer below the vertical percolation layer is greater
than the soil suction in the vertical percolation layer. The rate of gravity drainage
(percolation) in a vertical percolation layer is assumed to be a function of the soil
moisture storage and largely independent of conditions in adjacent layers. The rate can
be restricted when the layer below is saturated and drains slower than the vertical
percolation layer. Layers, whose primary hydraulic function is to provide storage of
moisture and detention of drainage, should normally be designated as vertical percolation
layers. Waste layers and layers designed to support vegetation should be designated as
vertical percolation layers, unless the layers provide lateral drainage to collection
systems.
Lateral drainage layers (e.g., layers 2, 6, 7 and 9 in Figure 1) are layers that
promote lateral drainage to collection systems at or below the surface of liner systems.
Vertical drainage in a lateral drainage layer is modeled in the same manner as for a
vertical percolation layer, but saturated lateral drainage is allowed. The saturated
hydraulic conductivity of a lateral drainage layer generally should be greater than 1 x 10~3
cm/sec for significant lateral drainage to occur. A lateral drainage layer may be
underlain by only a liner or another lateral drainage layer. The slope of the bottom of
the layer may vary from 0 to 40 percent.
Barrier soil liners (e.g., layers 4 and 11 in Figure 1) are intended to restrict vertical
flow. These layers should have hydraulic conductivities substantially lower than those
of the other types of layers, typically below 1 x 10"6 cm/sec. The program allows only
downward flow in barrier soil liners. Thus, any water moving into a liner will
eventually percolate through it. The leakage (percolation) rate depends upon the depth
of water-saturated soil (head) above the base of the layer, the thickness of the liner and
the saturated hydraulic conductivity of the barrier soil. Leakage occurs whenever the
moisture content of the layer above the liner is greater than the field capacity of the
layer. The program assumes that barrier soil liner is permanently saturated and that its
properties do not change with time.
geomembrane liners (e.g., layers 3, 8 and 10 in Figure 1) are layers of nearly
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impermeable material that restricts significant leakage to small areas around defects.
Leakage (percolation) is computed to be the result from three sources: vapor diffusion,
manufacturing flaws (pinholes) and installation defects (punctures, cracks, tears and bad
seams). Leakage by vapor diffusion is computed to occur across the entire area of the
liner as a function of the head on the surface of the liner, the thickness of the
geomembrane and its vapor diffusivity. Leakage through pinholes and installation defects
is computed in two steps. First, the area of soil or material contributing to leakage is
computed as a function of head on the liner, size of hole and the saturated hydraulic
conductivity of the soils or materials adjacent to the geomembrane liner. Second, the
rate of leakage in the wetted area is computed as a function of the head, thickness of soil
and membrane and the saturated hydraulic conductivity of the soils or materials adjacent
to the geomembrane liner.
-------
SECTION 3
DATA GENERATION AND DEFAULT VALUES
3.1 OVERVIEW
The HELP model requires general climate data for computing potential
evapotranspiration; daily climatologic data; soil characteristics; and design specifications
to perform the analysis. The required general climate data include growing season,
average annual wind speed, average quarterly relative humidities, normal mean monthly
temperatures, maximum leaf area index, evaporative zone depth and latitude. Default
values for these parameters were compiled or developed from the "Climates of the
States" (Ruffner, 1985) and "Climatic Atlas of the United States" (National Oceanic and
Atmospheric Administration, 1974) for 183 U.S. cities. Daily climatologic (weather)
data requirements include precipitation, mean temperature and total global solar radiation.
Daily rainfall data may be input by the user, generated stochastically, or taken from the
model's historical data base. The model contains parameters for generating synthetic
precipitation for 139 U.S. cities. The historical data base contains five years of daily
precipitation data for 102 U.S. cities. Daily temperature and solar radiation data are
generated stochastically or may be input by the user.
Necessary soil data include porosity, field capacity, wilting point, saturated hydraulic
conductivity, initial moisture storage, and Soil Conservation Service (SCS) runoff curve
number for antecedent moisture condition II. The model contains default soil
characteristics for 42 material types for use when measurements or site-specific estimates
are not available. The porosity, field capacity, wilting point and saturated hydraulic
conductivity are used to estimate the soil water evaporation coefficient and Brooks-Corey
soil moisture retention parameters. Design specifications include such items as the slope
and maximum drainage distance for lateral drainage layers; layer thicknesses; layer
description; area; leachate recirculation procedure; subsurface inflows; surface
characteristics; and geomembrane characteristics.
3.2 SYNTHETIC WEATHER GENERATION
The HELP program incorporates a routine for generating daily values of
precipitation, mean temperature, and solar radiation. This routine was developed by the
USDA Agricultural Research Service (Richardson and Wright, 1984) based on a
procedure described by Richardson (1981). The HELP user has the option of generating
synthetic daily precipitation data rather than using default or user-specified historical
data. Similarly, the HELP user has the option of generating synthetic daily mean
temperature and solar radiation data rather than using user-specified historical data. The
generating routine is designed to preserve the dependence in time, the correlation
between variables and the seasonal characteristics in actual weather data at the specified
-------
location. Coefficients for weather generation are available for up to 183
cities in the United States.
Daily precipitation is generated using a Markov chain-two parameter gamma
distribution model. A first-order Markov chain model is used to generate the occurrence
of wet or dry days. In this model, the probability of rain on a given day is conditioned
on the wet or dry status of the previous day. A wet day is defined as a day with 0.01
inch of rain or more. The model requires two transition probabilities: P|(W/W), the
probability of a wet day on day i given a wet day on day i-1; and Pj(W/D), the
probability of a wet day on day i given a dry day on day i-1.
When a wet day occurs, the two-parameter gamma distribution function, which
describes the distribution of daily rainfall amounts, is used to generate the precipitation
amount. The density function of the two-parameter gamma distribution is given by
«-i- -M 0)
^p* r(a)
where
f(p) = density function
p = the probability
a and /3 = distribution parameters
F = the gamma function of a
e = the base of natural logarithms
The values of P(W/W), P(W/D), a and /3 vary continuously during the year for most
locations. The precipitation generating routine uses monthly values of the four parame-
ters. The HELP program contains these monthly values for 139 locations in the United
States. These values were computed by the Agricultural Research Service from 20 years
(195 1-1970) of daily precipitation data for each location.
Daily values of maximum temperature, minimum temperature and solar radiation are
generated using the equation
where
ti(j) = daily value of maximum temperature (j=l)? minimum
temperature (j=2), or solar radiation (j=3)
10
-------
nifd) mean value on day i
c{(j) ~ coefficient of variation on day i
XiO) = stochastically generated residual element for day i
The seasonal change in the means and coefficients of variation is described by the
harmonic equation
Wj = « + Ccos
where
365
(3)
«,. value of w,(/J or ct(j) on day i
u = mean value of «,
C = amplitude of the harmonic
T = position of the harmonic in days
The Agricultural Research Service computed values of these parameters for the three
variables on wet and dry days from 20 years of weather data at 31 locations. The HELP
model contains values of these parameters for 184 cities. These values were taken from
contour maps prepared by Richardson and Wright (1984).
The residual elements for Equation 2 are generated using a procedure that preserves
important serial correlations and cross-correlations. The generating equation is
XjW = (A -Xj-iC/)) + (B '€,-(/)) (4)
where
Xi(j) =3x1 matrix for day i whose elements are residuals of maximum
temperature (j= 1), minimum temperature (J=2), and solar
radiation (J= 3)
e,.^ =3x1 matrix of independent random components for item j
A and B = 3x3 matrices whose elements are defined such that the new
sequences have the desired serial correlation and
cross-correlation coefficients
Richardson (198 1) computed values of the relevant correlation coefficients from 20
years of weather data at 3 1 locations. The seasonal and spatial variation in these
11
-------
correlation coefficients were found to be negligible. The elements of the A and B matri-
ces are therefore treated as constants.
3.3 MOISTURE RETENTION AND HYDRAULIC CONDUCTIVITY PARAMETERS
The HELP program requires values for the total porosity, field capacity, wilting
point, and saturated hydraulic conductivity of each layer that is not a liner. Saturated
hydraulic conductivity is required for all liners. Values for these parameters can be
specified by the user or selected from a list of default values provided in the HELP
program. The values are used to compute moisture storage, unsaturated vertical
drainage, head on liners and soil water evaporation.
3.3.1 Moisture Retention Parameters
Relative moisture retention or storage used in the HELP model differs from the water
contents typically used by engineers. The soil water storage or content used in the HELP
model is on a per volume basis (6), volume of water (VJ per total (bulk—soil, water and
air) soil volume (V, = Vi + Vw + Va), which is characteristic of practice in agronomy
and soil physics. Engineers more commonly express moisture content on a per mass
basis (w), mass of water (AfJ per mass of soil (AQ. The two can be related to each
other by knowing the dry bulk density (p^) and water density (pj, the dry bulk specific
gravity (F^) of the soil (ratio of dry bulk density to water density), (0 = w • F^), or
the wet bulk density (p^), wet bulk specific gravity (F^) of the soil (ratio of wet bulk
density to water density), (6 = [w • F,^] / [1 + w]).
Total porosity is an effective value, defined as the volumetric water content (volume
of water per total volume) when the pores contributing to change in moisture storage are
at saturation. Total porosity can be used to describe the volume of active pore space
present in soil or waste layers. Field capacity is the volumetric water content at a soil
water suction of 0.33 bars or remaining after a prolonged period of gravity drainage
without additional water supply. Wilting point is the volumetric water content at a
suction of 15 bars or the lowest volumetric water content that can be achieved by plant
transpiration (See Section 4. 11). These moisture retention parameters are used to define
moisture storage and relative unsaturated hydraulic conductivity.
The HELP program requires that the wilting point be greater than zero but less than
the field capacity. The field capacity must be greater than the wilting point and less than
the porosity. Total porosity must be greater than the field capacity but less than 1. The
general relation among moisture retention parameters and soil texture class is shown in
Figure 2,
The HELP user can specify the initial volumetric water contents of all non-liner
layers. Soil liners are assumed to remain saturated at all times. If initial water contents
12
-------
I
8
oc
I
0.60
0.50
0.40
0.30
0.20
0.10
0.00
ponosjEC
SAND SANDY
LOAM
LOAM SILTY CLAY SILTY CLAY
LOAM LOAM CLAY
Figure 2. Relation Among Moisture Retention Parameters and Soil Texture Class
are not specified, the program assumes values near the stead y-state values (allowing no
long-term change in moisture storage) and runs a year of simulation to initialize the
moisture contents closer to steady state. The soil water contents at the end of this year
are substituted as the initial values for the simulation period. The program then runs the
complete simulation, starting again from the beginning of the first year of data. The
results of the volumetric water content initialization period are not reported in the output.
3.3.2 Unsaturated Hydraulic Conductivity
Darcy's constant of proportionality governing flow through porous media is known
quantitatively as hydraulic conductivity or coefficient of permeability and qualitatively
as permeability. Hydraulic conductivity is a function of media properties, such as
particle size, void ratio, composition, fabric, degree of saturation, and the kinematic
viscosity of the fluid moving through the media. The HELP program uses the saturated
and unsaturated hydraulic conductivities of soil and waste layers to compute vertical
drainage, lateral drainage and soil liner percolation. The vapor diffusivity for
geomembranes is specified as a saturated hydraulic conductivity to compute leakage
through geomembranes by vapor diffusion.
13
-------
Saturated Hydraulic Conductivity
Saturated hydraulic conductivity is used to describe flow through porous media where
the void spaces are filled with a wetting fluid (e.g., water). The saturated hydraulic
conductivity of each layer is specified in the input. Equations for estimating the
hydraulic conductivity for soils and other materials are presented in Appendix A of the
HELP Program Version 3 User's Guide.
Umatumted Hydraulic Conductivity
Unsaturated hydraulic conductivity is used to describe flow through a layer when the
void spaces are filled with both wetting and non-wetting fluid (e.g., water and air). The
HELP program computes the unsaturated hydraulic conductivity of each soil and waste
layer using the following equation, reported by Campbell (1974):
[6 - 81
HI)
(5)
where
Ku = unsaturated hydraulic conductivity, cm/sec
Ks = saturated hydraulic conductivity, cm/sec
6 = actual volumetric water content, vol/vol
6r = residual volumetric water content, vol/vol
4> = total porosity, vol/vol
X = pore-size distribution index, dimensionless
Residual volumetric water content is the amount of water remaining in a layer under
infinite capillary suction. The HELP program uses the following regression equation,
developed using mean soil texture values from Rawls et al. (1982), to calculate the
residual volumetric water content:
f 0.014 + 0.25 WP for WP > 0.04
er = { (6)
I 0.6 WP for WP < 0.04
where
WP = volumetric wilting point, vol/vol
14
-------
The residual volumetric water content and pore-size distribution index are constants in
the Brooks-Corey equation relating volumetric water content to matrix potential (capillary
pressure and adsorptive forces) (Brooks and Corey, 1964):
(7)
(b — o
where
^ = capillary pressure, bars
i/-t = bubbling pressure, bars
Bubbling pressure is a function of the maximum pore size forming a continuous network
of flow channels within the medium (Brooks and Corey, 1964). Brakensiek et al. (1981)
reported that Equation 7 provided a reasonably accurate representation of water retention
and matrix potential relationships for tensions greater than 50 cm or 0.05 bars
(unsaturated conditions).
The HELP program solves Equation 7 for two different capillary pressures
simultaneously to determine the bubbling pressure and pore-size distribution index of
volumetric moisture content for use in Equation 7. The total porosity is known from the
input data. The capillary pressure-volumetric moisture content relationship is known at
two points from the input of field capacity and wilting point. Therefore, the field
capacity is inserted in Equation 7 as the volumetric moisture content and 0.33 bar is
inserted as the capillary pressure to yield one equation. Similarly, the wilting point and
15 bar are inserted in Equation 7 to yield a second equation. Having two equations and
two unknowns (bubbling pressure and pore-size distribution index), the two equations are
solved simultaneously to yield the unknowns. This process is repeated for each layer to
obtain the parameters for computing moisture retention and unsaturated drainage.
3.3.3 Saturated Hydraulic Conductivity for Vegetated Materials
The HELP program adjusts the saturated hydraulic conductivities of soils and waste
layers in the top half of the evaporative zone whenever those soil characteristics were
selected from the default list of soil textures. This adjustment, developed for the model
from changes in runoff characteristics and minimum infiltration rates as function of
vegetation, is made to account for channeling due to root penetration. These adjustments
for vegetation are not made for user-specified soil characteristics; they are made only for
default soil textures, which assumed that the soil layer is unvegetated and free of
continuous root channels that provide preferential drainage paths. The HELP program
calculates the vegetated saturated hydraulic conductivity as follows:
15
-------
= ( 1.0 + 0.5966 LAI + 0.132659 LAI2 + 0.1123454 LAI3
- 0.04777627 LAI4 + 0.004325035 LAI5)
where
(KJV = saturated hydraulic conductivity of vegetated material in
top half of evaporative zone, cm/sec
LAI = leaf area index, dimensionless (described in Section 4. 1 1)
= saturated hydraulic conductivity of unvegetated material
in top half of evaporative zone, cm/sec
3.4 EVAPORATION COEFFICIENT
The evaporation coefficient indicates the ease with which water can be drawn upward
through the soil or waste layer by evaporation. Using laboratory soil data Ritchie (1972)
indicated that the evaporation coefficient (in mm/day0-5) can be related to the unsaturated
hydraulic conductivity at 0.1 bar capillary pressure (calculated using Equations 5 and 7).
The HELP program uses the following form of Ritchie's equation to compute the
evaporation coefficient:
CON =
3.30 (Ku)ol ^ < 0.05 cm/day
2.44 + 17.19 (/goi 0.05 cm/day < (Jgo.i ^ < °-178 cm/day
5.50 (Ku)Ql ^ > 0.178 cm/day
where
CON = evaporation coefficient, mm/day05
(KJo.i bar = unsaturated hydraulic conductivity at 0.1 bar
capillary pressure, cm/sec
The HELP program imposes upper and lower limits on the evaporation coefficient
so as not to yield a capillary flux outside of the range for soils reported by Knisel (1980).
If the calculated value of the evaporation coefficient is less than 3.30, then it is set equal
to 3.30, and if the evaporation coefficient is greater than 5.50, then it is set equal to
5.50. The user cannot enter the evaporation coefficient independently.
Since Equation 9 was developed for soil materials, the HELP program imposes
16
-------
additional checks on the evaporation coefficient based on the relative field capacity and
saturated hydraulic conductivity of each soil and waste layer. Relative field capacity is
calculated using the following equation:
where
FCrd = relative field capacity, dimensionless
FC = field capacity, vol/vol
If the relative field capacity is less than 0.20 (typical of sand), then the evaporation
coefficient is set equal to 3.30. Additionally, if the saturated hydraulic conductivity is
less than 5x10"* cm/sec (the range of compacted clay), the evaporation coefficient is set
equal to 3.30.
3.5 DEFAULT SOIL AND WASTE CHARACTERISTICS
The total density of soil and waste layers can be defined as the mass of solid and
water particles per unit volume of the media. The total density of these layers is
dependent on the density of the solid particles, the volume of pore space, and the amount
of water in each layer. As previously discussed, total porosity can be used to describe
the volume of pore space in a soil or waste layer. Therefore, total porosity can be used
to indicate the density of soil and waste layers.
The density of soil and waste layers can be increased by compaction, static loading,
and/or dewatering of soil and waste layers. Compaction increases density through the
application of mechanical energy. Static loading increases density by the application of
of the weight of additional soil, barrier, or waste layers. Dewatering increases density
by removing pore water and/or reducing the pore pressures in the layer. Dewatering can
be accomplished by installing horizontal and/or vertical drains, trenches, water wells,
and/or the application of electrical currents. The HELP program provides default values
for the total porosity, field capacity, wilting point, and saturated hydraulic conductivity
of numerous soil and waste materials as well as geosynthetic materials.
3.5.1 Default Soil Characteristics
Information on default soil moisture retention values for low, moderate, and high-
density soil layers is provided in the following sections. High-density soil layers are also
described as soil liners. Application of the default soil properties should be limited to
17
-------
planning level studies and are not intended to replace design level laboratory and field
testing programs.
Low-Density Soil Layers
Rawls et al. (1982) reported mean values for total porosity, residual volumetric water
content, bubbling pressure, and pore-size distribution index, for the major US
Department of Agriculture (USDA) soil texture classes. These values were compiled
from 1,323 soils with about 5,350 horizons (or layers) from 32 states. The geometric
mean of the bubbling pressure and pore-sire distribution index and the arithmetic mean
of total porosity and residual volumetric water content for each soil texture class were
substituted into Equation 7 to calculate the field capacity (volumetric water content at a
capillary pressure of 1/3 bar) and wilting point (volumetric water content at a capillary
pressure of 15 bars) of each soil texture class. Rawls et al. (1982) also reported
saturated hydraulic conductivity values for each major USDA uncompacted soil texture
class. These values were derived from the results of numerous experiments and
compared with similar data sets. Default characteristics for the coarse and fine sands (Co
and F) were developed by interpolating between Rawls' data.
Freeze and Cherry (1979) reported that typical unconsolidated clay total porosities
range from 0.40 to 0.70. Rawls' sandy clay, silty clay, and clay had total porosities of
0.43, 0.48, and 0.47, respectively. Therefore, Rawls' loam and clay soils data are
considered to represent conditions typical of minimal densification efforts or low-density
soils. Default characteristics for Rawls et al. (1982) low-density soil layers
are summarized in Table 1. The USDA soil textures reported in Table 1 were converted
to Unified Soil Classification System (USCS) soil textures using a soil classification
triangle provided in McAneny et al. (1985). Applicable USDA and USCS soil texture
abbreviations are provided in Table 3.
Moderate-Density Soil Layers
Rawls et al. (1982) presented the following form of Brutsaert's (1967) saturated
hydraulic conductivity equation:
K - a _ _ (ID
2 u + na+2)
where
Ks = saturated hydraulic conductivity, cm/sec
18
-------
TABLE 1. DEFAULT LOW DENSITY SOIL CHARACTERISTICS
Soil Texture Class
HELP
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
21
USDA
CoS
S
FS
LS
LFS
SL
FSL
L
SiL
SCL
CL
SiCL
SC
SiC
C
G
uses
SP
SW
SW
SM
SM
SM
SM
ML
ML
SC
CL
CL
SC
CH
CH
GP
Total
Porosity
vol/vol
0.417
0.437
0.457
0.437
0.457
0.453
0.473
0.463
0.501
0.398
0.464
0.471
0.430
0.479
0.475
0.397
Field
Capacity
vol/vol
0.045
0.062
0.083
0.105
0.131
0.190
0.222
0.232
0.284
0.244
0.310
0.342
0.321
0.371
0.378
0.032
Wilting
Point
vol/vol
0.018
0.024
0.033
0.047
0.058
0.085
0.104
0.116
0.135
0.136
0.187
0.210
0.221
0.251
0.251
0.013
Saturated
Hydraulic
Conductivity
cm/sec
l.OxlO-2
5.8xlO-3
3.1xlO-3
1.7xlO-3
l.OxlO-3
7.2x10^
5.2x10^
3.7x10^
1.9x10^
1.2xl04
6.4xlO-5
4.2xlO-5
3.3xlO-5
2.5xlO-5
2.5xlO-5
3.0x10-'
a = constant representing the effects of various
fluid constants and gravity, 21 cmYsec
= total porosity, vol/vol
9, = residual volumetric water content, vol/vol
^b = bubbling pressure, cm
X = pore-size distribution index, dimensionless
A more detailed explanation of Equation 11 can be found in Appendix A of the HELP
program Version 3 User's Guide and the cited references.
19
-------
Since densification is known to decrease the saturated hydraulic conductivity of a soil
layer, the total porosity, residual volumetric water content, bubbling pressure, and pore-
size distribution index data reported in Rawls et al. (1982) were adjusted by a fraction
of a standard deviation and substituted into Equation 11 to reflect this decrease.
Examination of Equation 11 and various adjustments to Rawls' reported data indicated
that a reasonable representation of moderate-density soil conditions can be obtained by
a 0.5 standard deviation decrease in the total porosity and pore-size distribution index and
a 0.5 standard deviation increase in the bubbling pressure and residual saturation of
Rawls' compressible soils (e.g. loams and clays). These adjustments were substituted
into Equations 7 and 11 to determine the total porosity, field capacity, wilting point, and
hydraulic conductivity of these soils. The values obtained from these adjustments are
thought to represent moderate-density soil conditions typical of compaction by vehicle
traffic, static loading by the addition of soil or waste layers, etc. Default characteristics
for moderate-density, compressible loams and clays are summarized in Table 2. The
USDA soil textures reported in Table 2 were converted to Unified Soil Classification
System (USCS) soil textures using information provided in McAneny et al. (1985).
Applicable USDA and USCS soil texture abbreviations are provided in Table 3.
High-Density Soil Layers
Similar to moderate-density soil layers, densification produces a high-density, low
saturated hydraulic conductivity soil layer or soil liner. Due to the geochemical and low
saturated hydraulic conductivity properties of clay, soil liners are typically constructed
of compacted clay. Elsbury et al. (1990) indicated that the hydraulic conductivity of clay
liners can be impacted by the soil workability, gradation, and swell potential; overburden
stress on the liner; liner thickness; liner foundation stability; liner desiccation and/or
freeze and thawing; and degree of compaction. Compaction should destroy large soil
clods and provide interlayer bonding. The process can be impacted by the lift thickness;
soil water content, dry density, and degree of saturation; size of soil clods; soil
preparation; compactor type and weight; number of compaction passes and coverage; and
construction quality assurance. The HELP program provides default characteristics for
clay soil liners with a saturated hydraulic conductivity of Ix 10"7 and Ix 10"9 cm/sec.
Similar to the procedure used to obtain the default moderate-density clay soil
properties, Rawls et al.'s (1982) reported total porosity, pore-size distribution index,
bubbling pressure, and residual saturation for clay soil layers were adjusted to determine
the field capacity and wilting point of the Ix 107 cm/sec clay liner. A hydraulic
conductivity of 6.8xlO~8 cm/sec was obtained by substituting a 1 standard deviation
decrease in Rawls' reported total porosity and pore-size distribution index and a 1
standard deviation increase in Rawls' reported bubbling pressure and residual saturation
into Equation 11. These adjustments were substituted into Equation 7 to obtain a field
capacity and wilting point representative of the Ix 10"7 cm/sec soil liner.
20
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TABLE 2. MODERATE AND HIGH DENSITY DEFAULT SOILS
Soil Texture Class
HELP
22
23
24
25
26
27
28
29
16
17
USDA
L
(Moderate)
SiL
(Moderate)
SCL
(Moderate)
CL
(Moderate)
SiCL
(Moderate)
SC
(Moderate)
SiC
(Moderate)
C
(Moderate)
uses
ML
ML
SC
CL
CL
SC
CH
CH
Liner Soil
(High)
Bentonite
(High)
Total
Porosity
vol/vol
0.419
0.461
0.365
0.437
0.445
0.400
0.452
0.451
0.427
0.750
Field
Capacity
vol/vol
0.307
0.360
0.305
0.373
0.393
0.366
0.411
0.419
0.418
0.747
Wilting
Point
vol/vol
0.180
0.203
0.202
0.266
0.277
0.288
0.311
0.332
0.367
0.400
Saturated
Hydraulic
Conductivity
cm/sec
1.9x10*
9.0x10^
2.7x10^
3.6x10^
1.9x10*
7.8xlO-7
1.2x10-*
6.8xlO-7
l.OxlO-7
3.0xlO-9
3.5.2 Default Waste Characteristics
Table 4 provides a summary of default moisture retention values for various waste
layers. Municipal waste properties provided in Tchobanoglous et al. (1977) and
Equations 6 and 7 were used to determine the total porosity, field capacity, and wilting
point of a well compacted municipal waste. The field capacity and wilting point were
calculated using Tchobanoglous et al.'s high and low water content values, respectively.
Oweis et al. (1990) provided information on the in-situ saturated hydraulic conductivity
of municipal waste. Zeiss and Major (1993) described the moisture flow through
21
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TABLE 3. DEFAULT SOIL TEXTURE ABBREVIATIONS
US Department of Agriculture
G
S
Si
C
L
Co
F
Unified Soil Classification System
G
S
M
C
P
W
H
L
Definition
Gravel
Sand
Silt
Clay
Loam (sand, silt, clay, and humus mixture)
Coarse
Fine
Definition
Gravel
Sand
Silt
Clay
Poorly Graded
Well Graded
High Plasticity or Compressibility
Low Plasticity or Compressibility
municipal waste and the effective moisture retention of municipal waste, providing
information on waste with dead zones and channeling. In addition Toth et al. (1988)
provided information on compacted coal-burning electric plant ash, Poran and Ahtchi-Ali
(1989) provided information on compacted municipal solid waste ash, and Das et al.
(1983) provided information on fine copper slag.
The total porosities of the ash and slag wastes were determined using a phase
relationship at maximum dry density. The field capacities and wilting points of the ash
and slag wastes were calculated using the following empirical equations reported by
Brakensiek et al. (1984):
22
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TABLE 4. DEFAULT WASTE CHARACTERISTICS
Waste Identification
HELP
18
19
30
31
32
33
Waste Material
Municipal Waste
Municipal Waste with
Channeling
High-Density Electric
Plant Coal Fly Ash*
High-Density Electric
Plant Coal Bottom Ash"
High-Density Municipal
Solid Waste Incinerator
Fly Ash*'
High-Density Fine
Copper Slag"
Total
Porosity
vol/vol
0.671
0.168
0.541
0.578
0.450
0.375
Field
Capacity
vol/vol
0.292
0.073
0.187
0.076
0,116
0.055
Wilting
Point
vol/vol
0.077
0.019
0.047
0.025
0,049
0.020
Saturated
Hydraulic
Conductivity
cm/ sec
l.OxlO-3
l.OxlO-3
5.0xlO-5
4.1xlO-3
l.OxlO-2
4.1xlO-2
* All values, except saturated hydraulic conductivity, are at maximum dry density.
Saturated hydraulic conductivity was determined in-situ.
** All values are at maximum dry density. Saturated hydraulic conductivity was
determined by laboratory methods.
Field Capacity = 0.1535 - (0.0018) (% Sand) + (0.0039) (% Clay)
+ (0.1943) (Total Porosity)
(12)
Wilting Point = 0.0370 - (0.0004) (% Sand) + (0.0044) (% Clay)
+ (0.0482) (Total Porosity)
(13)
where 0.05 mm < Sand Particles < 2 mm and Clay Particles < 0.002 mm (McAneny
et al. 1985). These equations were developed for natural soils having a sand content
between 5 and 70 percent and a clay content between 5 and 60 percent. While the
particle size distribution of some of the ash and slag wastes fell outside this
range, the effects of this variation on water retention were thought to be minimal. The
applicability of these equations to waste materials has not been verified.
23
-------
The saturated hydraulic conductivities of the ash and slag wastes were taken directly
from the references. The saturated hydraulic conductivities of the coal burning electric
plant ashes at maximum dry density were determined in-situ and the maximum dry
density municipal solid waste incinerator ash and fine copper slag values were determined
by laboratory methods. The saturated hydraulic conductivities of various other waste
materials are provided in Table 5. Similar to default soils, the HELP program uses
Equation 8 to adjust the saturated hydraulic conductivities of the default wastes in the top
half of the evaporative zone to account for root penetration.
A more detailed explanation of the calculation procedure used for the ash and slag
wastes can be found in Appendix A of the HELP program Version 3 User's Guide. Like
the soil properties, the default waste properties were determined using empirical
equations developed from soil data. Therefore, these values should not be used in place
of a detailed laboratory and field testing program.
TABLE 5. SATURATED HYDRAULIC CONDUCTIVITY OF WASTES
Waste Material
Stabilized Incinerator Fly
Ash
High-Density Pulverized
Fly Ash
Solidified Waste
Electroplating Sludge
Nickel/Cadmium Battery
Sludge
Inorganic Pigment Sludge
Brine Sludge - Chlorine
Production
Calcium Fluoride Sludge
High Ash Papermill Sludge
Saturated
Hydraulic
Conductivity
cm/ sec*
8.8xlO-5
2.5xlO-5
4.0xlO-2
1.6xlO-5
3.5x10-*
S.QxlO-6
8.2xlO'5
3.2xlO'5
1.4xlO-6
Reference
Poran and Ahtchi-Ali (1989)
Swain (1979)
Rushbrook et al. (1989)
Bartos and Palermo (1977)
ti
11
it
tt
Perry and Schultz (1977)
* - Determined by laboratory methods.
24
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3.5.3 Default geosynthetic Material Characteristics
Table 6 provides a summary of default properties for various geosynthetic materials.
The values were extracted from Geotechnical Fabrics Report— 1992 Specifiers Guide
(Industrial Fabrics Association International, 1991) and Giroud and Bonaparte (1985).
3.6 SOIL MOISTURE INITIALIZATION
The soil moisture of the layers may be initialized by the user or the program. If
initialized by the program, the soil moisture is initialized near steady-state using a three
step procedure. The first step sets the soil moisture of all liners to porosity or saturation
and the moisture of all other layers to field capacity.
In the second step the program computes a soil moisture for each layer below the top
liner system. These soil moistures are computed to yield an unsaturated hydraulic
conductivity equal to 85% of the lowest effective saturated hydraulic conductivity of all
TABLE 6. DEFAULT geosynthetic MATERIAL CHARACTERISTICS
Geosynthetic Material Description
HELP
20
34
35
36
37
38
39
40
41
42
Geosynthetic Material
Drainage Net (0.5 cm)
Drainage Net (0.6 cm)
High Density Polyethylene (HDPE) Membrane
Low Density Polyethylene (LDPE) Membrane
Polyvinyl Chloride (PVC) Membrane
Butyl Rubber Membrane
Chlorinated Polyethylene (CPE) Membrane
Hypalon or Chlorosulfonated Polyethylene (CSPE)
Membrane
Ethylene-Propylene Diene Monomer (EPDM)
Membrane
Neoprene Membrane
Saturated
Hydraulic
Conductivity
cm/sec
1.0xlO+l
3.3xKT'
2.0xlO'13
4.0xlO-13
2.0xlO-n
l.OxlO-12
4.0xlO-12
3.0xlO'12
2.0xlO-12
3.0xlO-12
25
-------
liner systems above the layer, including consideration for geomembrane liners. If the
unsaturated hydraulic conductivity is greater than 5 x 10"7 cm/sec or if the computed soil
moisture is less than field capacity, the soil moisture is set to equal the field capacity.
In all other cases, the computed soil moistures are used.
The third step in the initialization consists of running the model for one year of
simulation using the first year of climatological data and the initial soil moistures selected
in step 2. At the end of the year of initialization, the soil moistures existing at that point
are reported as the initial soil moistures. The simulation is then started using the first
year of climatological data again.
3.7 DEFAULT LEAF AREA INDICES AND EVAPORATIVE ZONE DEPTHS
Recommended default values for leaf area index and evaporative depth are given in
the program. Figures 3, 4 and 5 show the geographic distribution of the default values
for minimum and maximum evaporative depth and maximum leaf area index. The
evaporative zone depths are based on rainfall, temperature and humidity data for the
climatic regions. The estimates for minimum depths are based loosely on literature
values (Saxton et al., 197 1) and unsaturated flow model results for bare loamy soils
(Thompson and Tyler, 1984; Fleenor, 1993), while the maximum depths are for loamy
soils with a very good stand of grass, assuming rooting depths will vary regionally with
Maximum Leaf Area Index
Figure 3. Geographic Distribution of Maximum Leaf Area Index
26
-------
Minimum Evaporative Depths
8
14
Figure 4. Geographic Distribution of Minimum Evaporative Depth
Maximum Evaporative Depths
36
42
Figure 5. Geographic Distribution of Maximum Evaporative Depth
27
-------
plant species and climate. The zones and values for the maximum leaf area index are
based on recommendations in the documentation for the Simulator for Water Resources
in Rural Basins (SWRRB) model (Arnold et al, 1989), considering both rainfall and
temperature.
28
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SECTION 4
METHOD OF SOLUTION
4.1 OVERVIEW
The HELP program simulates daily water movement into, through and out of a
landfill. In general, the hydrologic processes modeled by the program can be divided
into two categories: surface processes and subsurface processes. The surface processes
modeled are snowmelt, interception of rainfall by vegetation, surface runoff, and
evaporation of water, interception and snow from the surface. The subsurface processes
modeled are evaporation of water from the soil, plant transpiration, vertical unsaturated
drainage, geomembrane liner leakage, barrier soil liner percolation and lateral saturated
drainage. Vegetative growth and frozen soil models are also included in the program to
aid modeling of the water routing processes.
Daily infiltration into the landfill is determined indirectly from a surface-water
balance. Each day, infiltration is assumed to equal the sum of rainfall and snow melt,
minus the sum of runoff, surface storage and surface evaporation. No liquid water is
held in surface storage from one day to the next, except in the snow cover. The daily
surface-water accounting proceeds as follows. Snowfall and rainfall are added to the sur-
face snow storage, if present, and then snowmelt plus excess storage of rainfall is
computed. The total outflow from the snow cover is then treated as rainfall in the
absence of a snow cover for the purpose of computing runoff. A rainfall-runoff
relationship is used to determine the runoff. Surface evaporation is then computed. Sur-
face evaporation is not allowed to exceed the sum of surface snow storage and inter-
cepted rainfall. Interception is computed only for rainfall, not for outflow from the snow
cover. The snowmelt and rainfall that does not run off or evaporate is assumed to
infiltrate into the landfill. Computed infiltration in excess of the storage and drainage
capacity of the soil is routed back to the surface and is added to the runoff or held as
surface storage.
The first subsurface processes considered are evaporation from the soil and plant
transpiration from the evaporative zone of the upper subprofile. These are computed on
a daily basis. The evapotranspiration demand is distributed among the seven modeling
segments in the evaporative zone.
The other subsurface processes are modeled one subprofile at a time, from top to
bottom, using a design dependent time step, varying from 30 minutes to 6 hours.
Unsaturated vertical drainage is computed for each modeling segment starting at the top
of the subprofile, proceeding downward to the liner system or bottom of the subprofile.
The program performs a water balance on each segment to determine the water storage
and drainage for each segment, accounting for infiltration or drainage from above,
subsurface inflow, leachate recirculation, moisture content and material characteristics.
29
-------
If the subprofile contains a liner, water-routing or drainage from the segment directly
above the liner is computed as leakage or percolation through the liner, and lateral
drainage to the collection system, if present. The sum of the lateral drainage and
leakage/percolation is first estimated to compute the moisture storage and head on the
liner. Using the head, the leakage and lateral drainage is computed and compared to
their initial guesses. If the sum of these two outflows is not sufficiently close to the
initial estimate, new estimates are generated and the procedure is repeated until
acceptable convergence is achieved. The moisture storage in liner systems is assumed
to be constant; therefore, any drainage into a liner results in an equal drainage out of the
liner. If the subprofile does not contain a liner, the lateral drainage is zero and the
vertical drainage from the bottom subprofile is computed in the same manner as the
upper modeling segments.
4.2 RUNOFF
The rainfall-runoff process is modeled using the SCS curve-number method, as
presented in Section 4 of the National Engineering Handbook (USDA, SCS, 1985). This
procedure was selected for four reasons: (1) it is widely accepted, (2) it is
computationally efficient, (3) the required input is generally available and (4) it can
conveniently handle a variety of soil types, land uses and management practices.
The SCS procedure was developed from rainfall-runoff data for large storms on small
watersheds. The development is as follows (USDA, SCS, 1985). Runoff was plotted
as a function of rainfall on arithmetic graph paper having equal scales, yielding a curve
that becomes asymptotic to a straight line with a 1:1 slope at high rainfall as shown in
Figure 6. The equation of the straight-line portion of the runoff curve, assuming no lag
between the times when rainfall and runoff begin, is
Q = P> - S' (14)
where
Q = actual runoff, inches
P' = maximum potential runoff (actual rainfall after runoff starts or
actual rainfall when initial abstraction does not occur), inches
S' = maximum potential retention after runoff starts, inches
The following empirical equation was found to describe the relationship among
precipitation, runoff and retention (the difference between the rainfall and runoff) at any
point on the runoff curve:
30
-------
where
S' P'
F = actual retention after runoff starts, inches
= P'-Q
(15)
Substituting for F,
P' - Q _ _Q_
S' P'
(16)
If initial abstraction is considered, the runoff curve is translated to the right, as
shown in Figure 6, by the amount of precipitation that occurs before runoff begins. This
amount of precipitation is termed the initial abstraction, la. To adjust Equation 16 for
initial abstraction, this amount is subtracted from the precipitation,
RAINFALL, P
Figure 6. Relation Between Runoff, Precipitation, and Retention
31
-------
P1 = P - Ia (17)
Equation 16 becomes
P "Ia ~ Q- —2_ (is)
s1 P - ia
where
P = actual rainfall, inches
Ia = initial abstraction, inches
Figure 6 shows that the two retention parameters, 5' and S, are equal:
S - 5' (19)
Rainfall and runoff data from a large number of small experimental watersheds
indicate that, as a reasonable approximation (USDA, SCS, 1985),
/„ = 0.2 S (20)
Substituting Equations 19 and 20 into Equation 18 and solving for Q,
n = CP__0.2S)2 (21)
(P + 0.8 5)
Performing polynomial division on Equation 21 and dividing both sides of the
equation by S,
Q - I -12 -—M_
~S S ' /> no (22)
— + U.s
5
Equation 22 is the normalized rainfall-runoff relationship for any S and is plotted in
Figure 7.
32
-------
2.0
Q/S
1.0
Q/S - P/S
.8
0.2
1.0 1.2
2.0
I./S • 0.2
1.0
1.2
P/S
3.0
Figure 7. SCS Rainfall-Runoff Relation Normalized on Retention Parameter S
The retention parameter, S, is transformed into a so-called runoff curve number, CN,
to make interpolating, averaging and weighting operations more nearly linear. The
relationship between CN and 5 is
CN =
1000
S + 10
(23)
5 =
1000
CN
10
(24)
The HELP program computes the runoff, Qi on day i, from Equation 21 based on
the net rainfall, P(, on this day. The net rainfall is zero when the mean temperature is
less than or equal to 32 'F; is equal to the precipitation when the mean temperature is
above 32 *F and no snow cover is present; or is equal to the outflow from the snow
33
-------
cover when a snow cover is present and the mean temperature is above 32 °F:
where
P, =
0.0
Oi - EMELTt
for i; s 32 °F
for Tt > 32 °F,
/or T, > 32 °F,
= 0.0 (25)
> 0.0
Pt = net rainfall and snowmelt available for runoff on day i, inches
R( = rainfall on day i, inches
Ot = outflow from snow cover subject to runoff on day i, inches
= evaporation of snowmelt on day i, inches
SNO^j = water equivalence of snow cover at end of day i-1, inches
4.2.1 Adjustment of Curve Number for Soil Moisture
The value of the retention parameter, 5, for a given soil is assumed to vary with soil
moisture as follows:
5 =
where
1 -
SM - [(FC + WP)I2]
UL - [(FC +
for SM > (FC + WP){2
for SM < (FC + WP)}2
(26)
S^ = maximum value of 5, inches
SM = soil water storage in the vegetative or evaporative zone, inches
UL = soil water storage at saturation, inches
FC = soil water storage at field capacity (the water remaining following
gravity drainage in the absence of other losses), inches
WP = soil water storage at wilting point (the lowest naturally occurring
soil water storage), inches.
S^ is the retention parameter, S, for a dry condition. It is assumed that the soil water
content midway between field capacity and wilting point is characteristic of being dry.
Since soil water is not distributed uniformly through the soil profile, and since the
soil moisture near the surface influences infiltration more strongly than soil moisture
34
-------
located elsewhere, the retention parameter is depth-weighted. The soil profile of the
vegetative or evaporative zone depth is divided into seven segments, The thickness of
the top segment is set at one thirty-sixth of the thickness of the vegetative or evaporative
depth. The thickness of the second segment is set at five thirty-sixths of the thickness
of the vegetative or evaporative zone depth. The thickness of each of the bottom five
segments is set at one-sixth of the thickness of the vegetative or evaporative zone depth.
The user-specified evaporative depth is the maximum depth from which moisture can be
removed by evapotranspiration. This depth cannot exceed the depth to the top of the
uppermost barrier soil layer. The depth-weighted retention parameter is computed using
the following equation (Knisel, 1980):
s - E
(27)
SJ =
, SM>-
} - {(FCj
for SMj > (FCj
for SMj <
(28)
where
Wj = weighting factor for segment j
= soil water storage in segment], inches
= storage at saturation in segment], inches
= storage at field capacity in segment], inches
= storage at wilting point in segment], inches
The weighting factors decrease with the depth of the segment in accordance with the
following equation from the CREAMS model (Knisel, 1980):
FCj
W] = 1.0159
-4.16
EZD
D,
-4.16 —L
- e
EZD
(29)
where
= depth to bottom of segment j, inches
35
-------
EZD = vegetative or evaporative zone depth, inches
For the assumed segment thicknesses, this equation gives weighting factors of 0. 1 1 1,
0.397, 0.254, 0.127, 0.063, 0.032 and 0.016 for segments 1 through 7. The top
segment is the highest weighted in a relative sense since its thickness is 1/36 of the
evaporative zone depth while the thickness of the second segment is 5/36 and the others
are 1/6.
The runoff curve number required as input to the HELP program is that
corresponding to antecedent moisture condition II (AMC-II) in the SCS method. AMC-II
represents an average soil-moisture condition. The corresponding curve number is
denoted CN,,. The HELP user can either input a value of CNU directly; input a curve
number and have the program adjust it for surface slope conditions; or have the program
compute a value based on the vegetative cover type, the default soil type and surface
slope conditions.
The value of the maximum moisture retention parameter, S^, is assumed to equal
the value of 5 for a dry condition, antecedent moisture condition I (AMC-I) in the SCS
method (USDA, SCS, 1985). It is assumed that the soil moisture content for this dry
condition (a condition where the rainfall in the last five days totaled less than 0.5 inches
without vegetation and 1 .4 inches with vegetation) is midway between field capacity and
wilting point. S^ is related to the curve number for AMC-I, CN,, as follows:
- 10 (30)
CNf
CN, is related to CNU by the following polynomial (Knisel, 1980):
<} = 3.751 x 1CT1 CNa + 2.757 x 10' „.„
- 1.639 x 10'5 CN? + 5.143 x 1(T
4.2.2 Computation of Default Curve Numbers
When the user requests the program to generate and use a default curve number, the
program first computes the AMC-II curve number for the specified soil type and
vegetation for a mild slope using the following equation:
CN = C + C
*n0 ^o ' ^i
36
-------
where
CNa = AMC-II curve number for mild slope (unadjusted for slope)
C0 = regression constant for a given level of vegetation
Cj ~ regression constant for a given level of vegetation
C2 = regression constant for a given level of vegetation
IR = infiltration correlation parameter for given soil type
The relationship between CNU , the vegetative cover and default soil texture is shown
graphically in Figure 8. Table 7 gives values of C0, C, and C2 for the five types of
vegetative cover built into the HELP program.
4.2.3 Adjustment of Curve Number for Surface Slope
A regression equation was developed to adjust the AMC-II curve number for surface
slope conditions. The regression was developed based on kinematic wave theory where
100
80 -
cc
01
SB
Z
LU
SL
3
O
60 -
40 -
20
1
CoS
3
FS
5
LFS
7
FSL
9
SIL
11
CL
13
sc
15
c
SOIL TEXTURE NUMBER
Figure 8. Relation between SCS Curve Number and Default Soil Texture
Number for Various Levels of Vegetation
37
-------
TABLE 7. CONSTANTS FOR USE IN EQUATION 32
Vegetative Cover
Bare Ground
Poor Grass
Fair Grass
Good Grass
Excellent Grass
C0
96.77
93.51
90.09
86.72
83.83
c.
-20.80
-24.85
-23.73
-43.38
-26.91
C2
-54.94
-71.92
-158.4
-151.2
-229.4
the travel time of runoff from the top of a slope to the bottom of the slope is computed
as follows:
*1/3
a - /)1/3
1.49 V2'3
n J
(33)
where
/„„ = runoff travel time (time of concentration), minutes
/ = steady-state rainfall intensity (rate), inches/hour
I = steady-state infiltration rate, inches/hour
L = slope length, feet
S = surface slope, dimensionless
n = Manning's roughness coefficient, dimensionless
A decrease in travel time results in less infiltration because less time is available for
infiltration to occur.
Using the KINEROS kinematic runoff and erosion model (Woolhiser, Smith, and
Goodrich, 1990), hundreds of runoff estimates were generated using different
combinations of soil texture class, level of vegetation, slope, slope length, and rainfall
depth, duration and temporal distribution. Using these estimates, the curve number that
would yield the estimated runoff was calculated from the rainfall depth and the runoff
estimate. These curve numbers were regressed with the slope length, surface slope and
the curve number that would be generated for the soil texture and level of vegetation
placed at a mild slope. The four soil textures used included loamy sand, sandy loam,
38
-------
loam, and clayey loam as specified by saturated hydraulic conductivity, capillary drive,
porosity, and maximum relative saturation, Two levels of vegetation were described—a
good stand of grass (bluegrass sod) and a poor stand of grass (clipped range). Slopes of
0.04,0.10,0.20,0.35, and 0.50 ft/ft and slope lengths of 50, 100, 250, and 500 ft were
used. Rainfalls of 1.1 inches, 1-hour duration and 2nd quartile Huff distribution and of
3.8 inches, 6-hour duration and balanced distribution were modeled.
The resulting regression equation used for adjusting the AMC-II curve number
computed for default soils and vegetation placed at mild slopes, CNa , is:
(T.A)
CNa = 100 - (100 - CNnJ-\Z—\ ~° Vj
V S* )
where
L* = standardized dimensionless length, (L/500 ft)
5* = standardized dimensionless slope, (S/0.04)
This same equation is used to adjust user-specified AMC-II curve numbers for surface
slope conditions by substituting the user value for CN,, in Equation 34.
4.2.4 Adjustment of Curve Number for Frozen Soil
When the HELP program predicts frozen conditions to exist, the value of CNn is
increased, resulting in a higher calculated runoff. Knisel et al. (1985) found that this
type of curve number adjustment in the CREAMS model resulted in improved predictions
of annual runoff for several test watersheds. If theCNu for unfrozen soil is less than or
equal to 80, the CN,, for frozen soil conditions is set at 95. When the unfrozen soil CN,,
is greater than 80, the CN,, is reset to be 98 on days when the program has determined
the soil to be frozen. This adjustment results in an increase in CN, and consequently a
decrease in S^ and 5' (Equations 19, 26, and 30).
From Equations 19 and 21, it is apparent that as S' approaches zero, Q approaches
P. In other words, as S' decreases, the calculated runoff becomes closer to being equal
to the net rainfall which is most often, when frozen soil conditions exist, predominantly
snowmelt. This will result in a decrease in infiltration under frozen soil conditions,
which has been observed in numerous studies.
4.2.5 Summary of Daily Runoff Computation
The HELP model determines daily runoff by the following procedure:
39
-------
1) Given CNa from input or calculated by Equations 32 or 34, CN, and 5TO are
computed once using Equations 31 and 30, respectively.
2) 5 is computed daily using Equations 27 and 28.
3) The daily runoff resulting from the daily rainfall and snowmelt is computed using
Equation 21.
4.3 PREDICTION OF FROZEN SOIL CONDITIONS
In cold regions, the effects of frozen soil on runoff and infiltration rates are
significant. Because of the necessary complexity and the particular data requirements of
any approach to estimating soil temperatures, the inclusion of a theoretically-based frozen
soil model in the HELP program is prohibitive for the purposes of the program.
However, for some regions, it is desirable to have some method for predicting the occur-
rence of frozen soil and the resulting increase in runoff.
Knisel et al. (1985) proposed a rather simple procedure for predicting the existence
of frozen soils in the CREAMS model. A modification of that approach has been
incorporated into HELP. In the HELP modification, the soil is assumed to enter a frozen
state when the average temperature of the previous 30 days first drops below 32 'F.
During the time in which the soil is considered to be frozen, the infiltration capacity of
the soil is reduced by increasing the calculated runoff. As explained earlier, this is done
by increasing the curve number. In addition, other processes are affected such as soil
evaporation, vertical drainage in the evaporative zone and groundmelt of snow.
The point in which the soil is no longer considered to be frozen is determined by
calculating the length of time required to thaw frozen soil; that is, the number of days
in which the soil is to remain frozen after the daily mean air temperature first rises above
freezing. The thaw period in days, DPS, is a constant for a particular set of climatic
data. The thaw period increases with latitude and decreases with solar radiation in the
winter at the site and is determined using the following relation.
DPS = 35.4 - 0.154 RS(Dec) (35)
where
= estimate of the normal total solar radiation in December (June in
the southern hemisphere) at the selected location, langleys
DFS = estimate of the number of days with mean temperatures above
freezing in excess of days with mean temperatures below freezing
required to thaw a frozen soil after a thaw is started
Rs
-------
December, R^^, (June in southern hemisphere) (Richardson and Wright, 1984) and the
mean daily solar radiation for December (June in southern hemisphere) from the first
year of the user's input data file, R^a />«>• This estimate is used to provide consistency
throughout the simulation and to limit the importance of the first year of solar radiation
data. RS(Dtc) is computed as follows:
where
°.75 -
= 7I1-38 DD UH sin
+ (sinff cos \1AT\ cosSD)]
<36>
where
DD
J
XT
LAT
SD
= average daily potential solar radiation at site in December (June in
the southern hemisphere), langleys
= 1 + 0.0335 sin [0.0172 (J + 88.2)]
= Julian date, 350 for northern hemisphere and 167 for southern
hemisphere
= arccos [( - tan | LAT | ) ( tan SD )]
= latitude of site, radians
=0.4102 sin [0.0172 (J- 80.25)]
In addition, a counter in the program keeps track of the number of days of below
freezing (one is subtracted for each day down to a minimum of zero) or above freezing
temperatures (one is added for each day up until a maximum of DPS is reached, at which
point the soil becomes unfrozen) since the soil became frozen. When the soil freezes for
the first time during the season, the counter is set to 0. When a thaw is completed, the
counter is reset to (DPS + 2)/3, but not less than 3 unless greater than DPS. When the
counter returns to 0, the soil is refrozen if the average temperature of the previous thirty
days is below freezing. As such, the value of the counter also limits the occurrence of
a refreeze after a thaw (i.e. the soil is prevented from refreezing immediately following
a thaw when the previous 30-day average temperature may not yet have increased to
above freezing) (Dozier, 1992).
4.4 SNOW ACCUMULATION AND MELT
Studies have shown that the temperature at which precipitation is equally likely to
be rain or snow is in the range of 32 to 36 'F. A delineation temperature of 32 'F is
used in the HELP model, that is, when the daily mean temperature is below this value,
41
-------
the program stores precipitation on the surface as snow, snowmelt is computed using
a procedure patterned after portions of the SNOW-17 routine of the National Weather
Service River Forecast System (NWSRFS) Snow Accumulation and Ablation Model
(Anderson, 1973). Using this approach, the melt process is divided into that which
occurs during nonrain periods and that occurring during rainfall. Rain-on-snow melt is
computed using an energy balance approach. To compute the nonrain melt, air
temperature is used as an index to energy exchange across the snow-air interface. This
is similar to the degree-day method of the Soil Conservation Service (used in Version 2),
which uses air temperature as an index to snow cover outflow. The SNOW-17 model
uses SI units in all calculations; therefore, the results are converted to English units for
compatibility with other HELP routines.
4.4.1 Nonrain Snowmelt
The nonrain snowmelt equation of the SNOW-17 model is computed using the
following equation (Knisel, 1980):
M. = — f AfF. / Tf - MBASE } - AS, - Fm } (38)
25.4 I \ ' ' ' J
where
Mi = surface melt discharged from the snow cover on day i, inches
MFt = melt factor for day i, millimeters per °C
Tc. = mean air temperature on day i, °C
MBASE = base temperature below which no melt is produced, 0 °C
A5, = change in storage of liquid water in the snow cover on day i,
millimeters
Fm, = portion of the surface melt refrozen during day i, millimeters
In the absence of rain,
where
Ot = Ml (39)
Ot = outflow from snow cover on day i available for evaporation,
runoff, and infiltration, inches
Unlike in version 2, the melt factor, MF, is not constant but varies seasonally due,
in large part, to the seasonal variation in solar radiation. In most areas, the variation in
42
-------
the melt factor can be represented by a sine function and is expressed as:
MF{ =
MFMAX - MFMIN
MFMAX + MFML
ty •
- ;sin
(40)
where
MFMAX = the maximum melt factor, millimeters per day per°C.
MFMIN = the minimum melt factor, millimeters per day per°C.
n, = number of days since March 21 in northern hemisphere, or
since September 21 in southern hemisphere
The maximum melt factor used in Version 3 is 5.2 mm/day-°C and is assumed to
occur on June 21 in the northern hemisphere and on December 21 in the southern hemi-
sphere, The minimum melt factor occurs on the reverse of the dates, and its value is 2.0
mm/day-°C. These melt factors are for open areas (Anderson, 1973). At latitudes
greater than 50 degrees, the seasonal variation of the melt factor becomes less sinusoidal.
Research has shown that at latitudes near 60 degrees the melt factor actually stays at its
minimum value for most of the snow season. Therefore, for sites at latitudes above 50
degrees, an adjustment is made to MFjto represent this gradually "flattening out" of the
melt factor during the prolonged winter (Dozier, 1992).
4.4.2 Rain-on-Snow Melt Condition
The rain-on-snow equation is an energy balance equation that makes use of the
following assumptions:
1) solar (short-wave) radiation is neglected due to assumed overcast con-
ditions,
2) the incoming long-wave radiation is equal to blackbody radiation at the tempera-
ture of the bottom of the cloud cover (assumed to be the mean air temperature).
3) a relative humidity of 90% is assumed.
The daily outflow from a snow cover available for runoff, infiltration and evaporation
during a rain-on-snow occurrence may be calculated as the sum of the melt by based on
air temperature, latent heat energy transfer based on vapor pressure differences, sensible
heat transfer based on air temperature, and advected heat transfer from the mass of rain,
less the surface melt that is refrozen by the cold snow cover or stored in the snow cover
as a liquid. Taking these assumptions into account and assuming typical values for
43
-------
barometric pressure and wind function, the four energy transfer components and outflow
from the snow cover are computed as follows:
where
0, = outflow from snow cover on day i available for evaporation, runoff,
and infiltration, inches
Qn. = net long-wave radiation transfer on day i, langleys
Qe = latent heat energy transfer based on vapor pressure differences,
langleys
Qh. = latent heat energy transfer based on temperature differences,
langleys
Qm. = advected heat transfer from the mass of rain, langleys
Lf = latent heat of fusion for water, 7.97 langleys/millimeters
Fm. = quantity of melt refrozen in snow cover on day i, millimeters
A57 = change in liquid storage in snow cover on day i, millimeters
25.4 = conversion from millimeters to inches
and
Qni = 1.171 xlO'7 (ICj + 273)4 - (Ts + 273)4 1 (42)
(43)
= Ls Y (TCf - TJ /(«) (44)
Qm = c ROSi Tc
f(u) = 0.262 + (0.0391 «) (46)
eai = 33.8639 RH [ ( 0.00738 TCi + 0.8072)8
- (0.000019) | 1.8 Tc< + 48 | + 0.001316]
44
-------
where
Tc = mean air temperature on day i, °C
Ts = snow temperature, 0°C
Ls = latent heat of sublimation for snow, 67.7 langleys/millimeters
ea = vapor pressure of the atmosphere on day i, millibars (Linsley et al,
1982)
e, = vapor pressure of the snow cover on day i, 6.11 millibars
f(u) = wind function, dimensionless
u = wind speed, in kilometers/hour (average annual wind speed used in
model)
7 = psychometric constant, 0.68 millibars/°C
c = specific heat of water, 0.1 langleys/millimeter- ° C
ROSt = rain on snow cover during day i, millimeters
RH = relative humidity, 0.9
In addition to the surface melt due to heat exchange at the snow-air interface, a small
amount of daily heat exchange occurs at the snow-soil interface. The soil heat exchange
is not considered directly because the model does not include a soil temperature model.
Therefore, this exchange is considered indirectly by the use of a constant daily
groundmelt when the ground is not predicted to be frozen. This daily groundmelt, GMf,
for typical landfills with biological activity is estimated to be:
GM. =
0.2 inches Nonfrozen soil conditions
(48)
| 0 Frozen soil conditions
All groundmelt is assumed to infiltrate and is not subject to runoff or evaporation prior
to infiltration. The groundmelt on day i may be limited by the quantity of snow available
(SM?wplus snowfall on day i).
4.4.3 Snowmelt Summary
The SNOW- 17 routine differs from the degree-day method in that it accounts for
refreezing of melt water due to any heat deficit of the cover and also for the retention
and transmission of liquid water in and through the cover. The liquid water in excess
of that held within the snow cover becomes outflow or runoff from the snow cover.
45
-------
When rain-on-snow occurs, the quantity of rain is added to the surface melt, from which
refreeze and retention in the snow cover may also occur. A positive value for AS in
Equations 38 and 41 indicates the amount of liquid water in storage 'within the snow
cover has increased. For further explanation concerning the calculation of AS and Fm
and the attenuation of excess liquid water, the user is referred to the SNOW-17
documentation (Anderson, 1973).
Naturally the amount of snowmelt is limited by the quantity of snow which is
present. The order in which the HELP program determines the amount of snowmelt and
remaining snow cover is as follows:
1. The amount of snow available for ablation (surface melt or evaporation) on day i,
AVLSNOj, is determined, recognizing that surface melt occurs only at mean daily
air temperatures above freezing and that groundmelt occurs only when the soil is
not frozen:
AVLSNOi =
^ + PREi - GMt
for Tc < 0°C
ci
for Tc > 0°C
(49)
where
^ = water storage in the snow cover at the end of day i-1, inches
= precipitation on day i, inches
2. The surface melt is calculated using Equations 38 or 41, but is limited to the
quantity of available snow:
o for TFt < 32° F
Ot from Equation 36 far PREt = 0 and Tpj > 32° F (5Q)
O. from Equation 38 for PREt > 0 and Tf > 32°F
A VLSNOf
for
i < °t (^- 36 or 38)
3. The quantity of snow remaining after considering all the types of melt is what is
available for evaporation (See Surface Evaporation section),
46
-------
4. The amount of water present in the snow cover at the end of day i, SNOh is
summed as follows:
SNO =
,.! + PREt - GMl - ESNOt
for Tc < 0°C
(51)
- Ot - ESNOf for TC( >
0°C
where
ESNO{ = evaporation of snow in excess of surface melt, Oit on day i, inches
4.5 INTERCEPTION
Initially during a rainfall event, nearly all rainfall striking foliage is intercepted.
However, the fraction of the rainfall intercepted decreases rapidly as the storage capacity
of the foliage is reached. The limiting interception storage is approached only after
considerable rainfall has reached the ground surface. This process is approximated by
the equation:
1 - e
RAIN
(52)
where
IWi = interception of rainfall by vegetation on day i, inches
INTn^u. = interception storage capacity of the vegetation on day i, inches
Ri = rainfall on day i (not including rainfall on snow), inches
Although INT^u depends upon vegetation type, growth stage, and wind speed, the data
of Horton (1919) and others indicate that 0.05 inches is a reasonable estimate of INTmM
for a good stand of most types of non-woody vegetation. The HELP program relates
INTma to the above ground biomass of the vegetation, CV. This empirical relationship
is:
0.05
0.05
14000
for CV{ < 14000
for CK( > 14000
(53)
47
-------
where CVf is the above ground biomass on day i in kilograms per hectare.
4.6 POTENTIAL EVAPOTRANSPIRATION
The method used in the HELP program for computing evapotranspiration was
patterned after the approach recommended by Ritchie (1972). This method uses the con-
cept of potential evapotranspiration as the basis for prediction of the surface and soil
water evaporation and the plant transpiration components. The term "potential
evapotranspiration" refers to the maximum quantity of evaporation rate that the
atmosphere may extract from a plot in a day. A modified Penman (1963) equation is
used to calculate the energy available for evapotranspiration.
LEi = PENRt + PENAt (54)
where
LEt = energy available on day i for potential evapotranspiration in the
absence of a snow cover, langleys
PEAT?, = radiative component of the Penman equation on day i, langleys
PENAt = aerodynamic component of the Penman equation on day i, langleys
The first term of Equation 54 represents that portion of the available evaporative
energy due to the radiation exchange between the sun and the earth. The second term
expresses the influence of humidity and wind on LE. These two components are
evaluated as follows:
A.
PENR, — /?„ (55)
( / * v n i
PENA = 1536 Y ( 1 + ' °, (56)
where
Rn. = net radiation received by the surface on day i, langleys
A, = slope of the saturation vapor pressure curve at mean air temperature
on day i, millibars per °C
48
-------
constant of the wet and dry bulb psychrometer equation, assumed to
be constant at 0.68 millibars per 'C
wind speed at a height of 2 meters, in kilometers/hour (average
annual wind speed used in model)
saturation vapor pressure at mean air temperature on day i,
millibars (computed using Equation 47, where RH = 1)
= mean vapor pressure of the atmosphere on day i, millibars
(computed using Equation 47, where RH is the quarterly average
dimensionless relative humidity on day i from the input data or on
days with precipitation, RH = 1)
The value of A,, is computed using an equation presented by Jensen (1973):
where
A ,. = 1.9993 [ (0.00738 T + 0.8072)7 - 0.0005793]
= mean air temperature on day i, °C
(57)
The net solar radiation received by the earth's surface, Rn., is the difference between
the total incoming and total outgoing radiation and is estimated as follows (Hillel, 1982;
Jensen, 1973):
(58)
where
Rs = incoming global (direct and sky) solar radiation on day i, langleys
a = albedo (reflectivity coefficient of the surface toward short-wave
radiation, a = 0.23 when there is no snow present; a = 0.6 when
a snow cover storing more than 5 mm of water exists)
Rb. = the long-wave radiation flux from soil on day i, langleys
Rb, decreases with increasing humidity and cloud cover and is calculated using the
following equations (Jensen, 1973):
= R
bo.
(59)
49
-------
where
Rty = the maximum outgoing long-wave radiation (assuming a clear day)
for day i
/?„. = the maximum potential global solar radiation for day i
a{ and bi = coefficients which are dependent upon the humidity on day i
(for RHt < 50%, 0, = 1.2, bt = 0.2;
for 50% <^RHt< 75%, a, = 1.1, bt = 0.1; and
for RHt >. 75%, a,, = 1.0, ft, = 0.0)
The outgoing long-wave radiation (heat loss) on a clear day, R^, is estimated as
follows:
Rb0i - 1.171 xlO-7(7C( + 273)4 (0.39 -0.05^)
(60)
The potential solar radiation for a given day, Rw, is calculated using a set of equations
from the WGEN model (Richardson and Wright, 1984).
R,OI = 711.38 DDi [(Ht sin \LAT\ sinSZ),) +(smHt cos \LAT\ cosSZ).)] (61)
where
LAT = latitude of the location, radians
DDt = 1 + {0.0335 sin [0.0172 (J, + 88.2)]}
SD, = 0.4102 sin [0.0172 (7, - 80.25)]
H{ = arc cosine [(- tan | LAT \ ) (tan SD)]
/, = the Julian date for day i in northern hemisphere and the Julian date
minus 182.5 in the southern hemisphere (negative latitudes)
The potential evapotranspiration is determined by dividing the available energy,LE,
by the latent heat of vaporization, 1^, (or the latent heat effusion, Lf, depending on the
state of the evaporated water). The latent heat of vaporization is a function of the water
temperature. In the HELP model, unless the evaporated water is snow or snowmelt, the
mean daily temperature is used to estimate the water temperature and potential
evapotranspiration is computed as:
50
-------
IE.
E0 = '— (62)
°< 25.4 Ly
59.7 - 0.0564 Tf for water
L - < ' (63)
v 67.67 - 0.0564 7, for snow
where
E0. = potential evapotranspiration on day i, in inches
Lv = latent heat of vaporization (for evaporating water) or latent heat of
fusion (for evaporating snow), langleys per millimeters
Ts = snow temperature, "C
25.4 = conversion from millimeters to inches
4.7 SURFACE EVAPORATION
4.7.1 No Snow Cover
The rate of evapotranspiration from a landfill cover is a function of solar radiation,
temperature, humidity, vegetation type and growth stage, water retained on the surface,
soil water content and other soil characteristics. evapotranspiration has three
components: evaporation of water or snow retained on foliage or on the landfill surface,
evaporation from the soil and transpiration by plants. In the HELP program, the
evapotranspiration demand is exerted first on water available at the landfill surface. This
available surface water may be either rainfall intercepted by vegetation, ponded water,
snowmelt or accumulated snow.
If there is no snow (SNO^ = 0) on the surface at the start of the day and no
snowfall (PREt = 0 and Tc. <_ 0°C) during the day or if there is no available snow
(AVLSNOf = 0) and no outflow from the snow cover (0, = 0) on day i, the potential
evapotranspiration (E0} is applied to any calculated interception (INT) from rainfall for
that day and, partially, to the ponded water. In this situation, the portion of the
evaporative demand that is met by the evaporation of surface moisture on day i is given
by
ESS; =
for E0t <; INTi + PW. ( 1 - PRF)
(64)
INT,
'. + PWi ( 1 - PRF) for E0 > INT{ + PWt ( 1 - PRF)
51
-------
where
ESSi = evaporation of surface moisture, inches
PWf = water ponded on surface that is unable to run off and is in excess
of infiltration capacity, inches
PRF = fraction of area where runoff can potentially occur
If the evaporative demand is less than the calculated interception, the amount of
interception is adjusted to equal the evaporative potential.
INT. = E0i for E0i < INT. from Equation 49 (65)
4.7.2 Snow Cover Present
If snow is present on the ground after calculating the melt for the day, the program
computes an estimated dew-point temperature based on the mean daily temperature for
the day, Tc., and the quarterly average humidity, RHL, or the existence of precipitation;
the dew-point temperature is assumed to equal to the mean daily temperature if
precipitation occurred (assumes RH( = 1 if PREi > 0). If the estimated dew-point
temperature is greater than or equal to the temperature of the snow cover, Ts, then the
evapotranspiration (evaporation of surface moisture, evaporation of soil moisture, and
plant transpiration) is assigned to be zero.
ESS. = 0 for AVLSNOi - Oi > 0 and 7^ * Ts (66)
Tdi = (112 - 0.9 Tc) RH}1* + 0.1 TCf - 112 (67)
where Td[ is the estimated dew-point temperature in °C for day i.
If a snow cover existed at the start of the day after discounting the groundmelt
(A VLSNOj > 0) and the estimated dew point is lower than the snow temperature, then
evaporation of the surface melt available for outflow, Oit from the snow cover is
computed. If the potential evapotranspiration, E0., exceeds the surface melt, then the
excess evaporative demand is exerted on the snow cover. When evaporating snow or
snowmelt, the estimation of the latent heat of vaporization, Lv, by Equation 63 is
modified. The temperature of the water is estimated to be 0 °C instead of the mean daily
air temperature, TCj. Therefore, Lv equals 59.7 langleys per millimeter of snowmelt and
67.67 langleys per millimeter of snow water,
52
-------
Under the conditions just described (A VLSNOt > 0 and Td, 0, the potential evaporative energy, LEit is reduced by the estimated
amount of energy consumed by melting the snow. This lower potential is then
exerted on the surface melt outflow. The portion of the surface melt that is
evaporated is calculated as:
LE\ = LEi - 202.4 Oi
(68)
EMELT =
where
far
25.4 L
O,
for LE\ < 0
for 0 < LE\ < 25.4 Lv O(
for LE'. ;> 25.4 LVO.
(69)
EMELTt = surface melt that is evaporated on day i, inches
LE'j = potential evaporative energy discounted for surface melt on day i,
langleys
2. After allowing for the energy dissipated by the melting of snow at the surface and
any evaporation of the melt, the remaining potential evaporative energy is
computed as follows:
IE", = LE\ - 25.4 Lv O.
(70)
where
"i = potential evaporative energy discounted for surface melt and
evaporation of surface melt on day i, langleys
3. If there is energy available after any evaporation of surface melt (££", > 0), this
remaining energy is applied to the snow cover. The amount of evaporated snow,
i, on day i is calculated as follows:
53
-------
ESNOt =
0 for IE", * 0
LE",
for 0 < LE\ < 25.4 Ly (AVLSNOt - Ot) (71)
25.4 Lv
AVLSNOt - Ol for IE", * 25.4 Lv (AVLSNO{ - O,)
4. The total amount of evaporation of surface moisture, ESSit on day i is then
calculated as the sum of the evaporated snow and evaporated outflow from the
snow cover:
ESS. = ESNOt + EMELTf (72)
where ESS, ESNO, and EMELT are water equivalent in inches.
4.7.3 Remaining Evaporative Demand
The amount of energy remaining available to be applied to subsurface
evapotranspiration (i.e., soil water evaporation and plant transpiration) is then the
original potential evaporative energy less the energy dissipated in the melt of snow and
evaporation of surface water. If snow was available for surface melt or evaporation
(A VLSNOi > 0), the remaining energy for subsurface evapotranspiration is:
LE = LEt - 25.4 (Lf Oi + Ly EMELTi + Ly ESNOf) (73)
In the absence of a snow cover or snowfall (A VLSNOi = 0), the remaining energy for
subsurface evapotranspiration is:
LESi = LE{ - 25.4 Lv ESS,. (74)
where LES. is the energy available for potential evapotranspiration of soil water.
The potential evapotranspiration from the soil column in inches is a function of the
energy available and the mean air temperature. The potential evapotranspiration from
the soil is:
LE
(75)
25.4 L
v
where ETS0. is the potential evapotranspiration of soil water.
54
-------
4.8 INFILTRATION
In the absence of a snow cover (AVLSNO( = 0), the infiltration is equal to the sum
of rainfall (precipitation at temperatures > 0° C) and ground melt less the sum of
interception evaporation of surface moisture) and runoff.
!NFt = PREi + GM(. - INT( - Qt (76)
In the presence of a snow cover, the infiltration is equal to the sum of outflow from the
snow cover and groundmelt less the sum of evaporation of the outflow from the snow
cover and runoff.
lNFi = Oi + GM , - EMELTt - Qt (77)
where 1NF( is the infiltration on day i in inches.
Since the runoff is computed using the total rainfall in Equation 21, it is possible for
the sum of the runoff and interception to exceed rainfall. Therefore, when- the sum of
the runoff, Qit and interception, /AT]-, exceeds the rainfall, PREit the computed runoff
is reduced by the excess and the infiltration is assigned the value of the groundmelt,GM,..
It is not possible for the sum of runoff and evaporation of the outflow from the snow
cover to exceed the outflow from the snow cover because the evaporation of melt is
subtracted from the outflow prior to computation of runoff by Equation 21.
4.9 SOIL WATER EVAPORATION
When the soil is not frozen, any demand in excess of the available surface water is
exerted on the soil column first through evaporation of soil water and then through plant
transpiration. When the soil is considered frozen, the program assumes that no soil
water evaporation or plant transpiration occurs.
The potential soil evaporation is estimated from the following equation based on the
work of Penman (1963) when evaporation is not limited by the rate at which water can
be transmitted to the surface:
= (PENRt + KE PENAJ e (- 0.000029 CK,) (7g)
°- 25.4 (59.7 - 0.0564 T )
where
55
-------
ES0. = potential evaporation of soil water on day i, inches
PENRi = radiative component of the Penman equation on day i, langleys
PENAj = aerodynamic component of the Penman equation on day i, langleys
CVj = above ground biomass on day i, kg/ha
Te. = mean air temperature on day i, "C
KE = fraction of aerodynamic component contributing to evaporation of
soil water
= 1 - 0.0000714 CVlt but not less than 0
This equation assumes that an above ground biomass of 14,000 kg/ha or more
defines a full cover canopy such that the effect of the wind and humidity term, PENA,
is negligible in the potential soil evaporation equation.
As patterned after Ritchie (1972), evaporation of soil water occurs in two stages.
Stage 1 evaporation demand is controlled only by the available energy, while stage 2
evaporation demand is limited by the rate at which water can be transmitted through the
soil to the surface. In stage 1, the rate of evaporation is equal to the potential
evaporation from the soil:
ESlt = ES0f (79)
where ESlt is the stage 1 soil water evaporation rate on day i in inches.
Stage 1 soil water evaporation will continue to occur as long as the cumulative value
of the soil water evaporation minus the infiltration is less than the upper limit for stage
1 evaporation. This limit represents the quantity of water that can be readily transmitted
to the surface. Cumulative soil water depletion by soil water evaporation is computed
as:
ESlTt = £ (ESk - INFk) (80)
k=m
where
ESI 7} = cumulative soil water depletion on day i by soil water evaporation,
inches
ESk = soil water evaporation on day k (computed by Equation 90), inches
m = the last day when ESI T was equal to 0
56
-------
The upper limit of stage 1 evaporation is (Knisel, 1980):
U = _i- (CON - 3)°-42 (81)
25.4
where
U = upper limit of stage 1 evaporation, inches
CON = evaporation coefficient (Equation 9), millimeters per day0-5
When ESlTf is greater than U, stage 1 evaporation ceases and stage 2 evaporation
begins. The following equation is used to compute the stage 2 evaporation rate Ritchie
(1972).
£52; - — CON [f,05 - (tt - I)05] (82)
where
£S2, = stage 2 soil water evaporation rate for day i, inches
ti = number of days since stage 1 evaporation ended
Since the daily total of surface and soil water evaporation cannot exceed the daily
potential evaporative demand, the evaporation from the soil is limited by the amount of
energy available after considering the evaporation of surface water, £,£,., (i. e., the actual
evaporative demand from the soil on day i, ESDt, cannot exceed the potential
evapotranspiration of soil water, E7S ). The following equation is used to determine the
daily soil water evaporative demand:
ESDi =
ESI. far ES1T. < U and ESli ETS0f
ETSa for ES1T. z U and ES2. > ETS0
(83)
The soil water evaporative demand is then distributed to the soils near the surface,
down to a maximum depth for soil water evaporation but not exceeding the evaporative
zone depth, The maximum depth is a function of the capillarity of the material,
increasing with smaller pore size, Pore size is related to the saturated hydraulic
57
-------
conductivity of a soil. Therefore, the following correlation was developed to estimate
the maximum depth of soil water evaporation based on empirical observations:
SEDMX = 4.6068 x 1.5952"
(84)
where
K = saturated hydraulic conductivity in the evaporative zone, cm/sec
SEDMX ~ estimated maximum depth of soil water evaporation, inches
Limits are placed on the depth of soil water evaporation as follows to confine the
capillary rise to the zone where a significant upward moisture flux is likely to occur (in
top 18 inches for sands and in the top 48 inches for clays):
SED =
EZD for EZD < SEDMX and EZD <. 48
48 for EZD $ SEDMX and EZD > 48
SEDMX for EZD > SEDMX and 18 <. SEDMX
18 for EZD > SEDMX and SEDMX < 18
48 for EZD > SEDMX and SEDMX > 48
where
48
(85)
SED = depth of soil water evaporation, inches
EZD = evaporative zone depth, inches
The soil water evaporation demand is distributed throughout the seven segments in
the soil water evaporative depth, SED, by the following equation (Knisel, 1980):
(86)
where
) = soil water evaporative demand on segment j on day i, inches
W(j) = weighting factor for segment], for j = 1 to 7
W(j) = 1.0159
-4.16
SED _
-4.16
SED
for D <: SED
(87)
58
-------
where
depth to bottom of segment j, inches; if D is greater than SED, then
then SED is substituted for D in Equation 87
4.10 PLANT TRANSPIRATION
The potential plant transpiration, EP0, is computed as follows when the mean daily
temperature is above 32 °F and the soil is not frozen:
EP. = E (88)
The actual plant transpiration demand equals the potential plant transpiration except
when the daily total of the soil water evaporative demand and potential plant transpiration
demand exceeds the potential evapotranspirative demand on the soil water for the day:
EPDi =
where EPDt is the actual plant transpiration demand in inches on day i.
The plant transpiration demand is distributed throughout the seven segments in the
evaporative zone, EZD, by the following equation (Knisel, 1980):
EPDtf) = EPDt • W(j) (90)
where
EPa for EPa + ES
(89)
- £5, for
jfl) = soil water evaporative demand on segment j on day i, inches
W(j) = weighting factor for segment], for j = 1 to 7
W = 1.0159
e
-4.16 ^LL -4.16 -^- (91)
EZD _ „ EZD ,„„ ,- _ i ,„ T V '
'j
where
DJ = depth to bottom of segment j, inches
59
-------
4.11 EVAPOTRANSPIRATION
The actual subsurface evapotranspiration on day i, ETSit is often less than the sum
of the soil water evaporative demand on day i, ESDt, and the plant transpiration demand
on day i, EPDit due to a shortage of soil water. The segment demands are then exerted
on the soil profile from the surface down. If there is inadequate water storage above the
wilting point in the segment to meet the demand, the soil water evapotranspiration from
the segment is limited to that storage and the excess (unsatisfied) demand is added to the
demand on the next lower segment within the evaporative zone.
The soil water evaporation demand is exerted first from the surface down. The
actual soil water evaporation from a segment is equal to the demand, ESD^), plus any
excess demand, ESX(j), but not greater than the available water, SMj - WPj. The soil
water evaporation is
f ESD.(j) H- ESX(j) for ESD. + ESX(j) ± SM(j) - WP(j)
ES,(j) -I (92)
SA/0') - WP(j) for ESDi + £5X0') > SM(j) - WP(j)
ESX(j + 1) = ESD&) + ESX(j) ~ ESt(j) (93)
where
ESt(j) = soil water evaporation from segment j on day i, inches
The plant transpiration demand is exerted next from the surface down. The actual
transpiration from a segment is equal to the demand, EPDfi), plus any excess demand,
EPX(j), but not greater than the available water, AWt(j), after extracting the soil water
evaporation. The plant transpiration from a segment is also limited to one quarter of the
plant available water capacity plus the available drainable water.
EPDf(j) + EPX(j) for EPDf(j) + EPX(j)
& EPDt(j) + EPX(j)
AW.(j) for EPD((J) + EPX(j) > AW.(j) (94)
& AWtf) <: EPLf(j)
EPLt(j) for EPDt(j) + EPX(j) > AWt(j)
& AWt(j) > EPLtf)
60
-------
EPX(j + 1) =
(96)
0.25 [FC(/) - WPO')1 /or SMt(j) - ESt(j) < FC(j)
else (97)
.(j) - [ £5,0') + FCO)1 + 0-25
EPLt(j) =
where
= plant transpiration from segment j on day i, inches
jd) = plant transpiration limit from segment j on day i, inches
WP(j) = wilting point of segment j, inches
FC(j) = field capacity of segment j, inches
The actual evapotranspiration from segment] on day i, ETt(/)t is the sum of the soil
water evaporation and the plant transpiration.
£T,0) = ES.(j) + £P.O) (98)
The water extraction profile agrees very well with profiles for permanent grasses
measured by Saxton et al. (1971).
The total subsurface evapotranspiration on day i, £TS,, is the sum of the
evapotranspiration from the top seven segments, the evaporative zone.
7
ETSi = Y, ETW
The total evapotranspiration on day i, £T,, is the sum of the subsurface
evapotranspiration and the surface evaporation.
£T. = £7S,. + ESSi
61
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4.12 VEGETATIVE GROWTH
The HELP program accounts for seasonal variation in leaf area index through a
general vegetative growth model. This model was extracted from the SWRRB program
(Simulator for Water Resources in Rural Basins) developed by the USDA Agricultural
Research Service (Arnold et al., 1989). The vegetative growth model computes daily
values of total and active above ground biomass based on the maximum leaf area index
value input by the user, daily temperature and solar radiation data, mean monthly tem-
peratures and the beginning and ending dates of the growing season. The maximum
value of leaf area index depends on the type of vegetation and the quality of the
vegetative stand. The program supplies typical values for selected covers; these range
from O for bare ground to 5.0 for an excellent stand of grass. The default weather data
files contain normal mean monthly temperatures and beginning and ending dates of the
growing season for 1 83 locations in the United States.
The vegetative growth model assumes that the vegetative species are perennial and
that the vegetation is not harvested. Phonological development of the vegetation is based
on the cumulative heat units during the growing season. Vegetative growth starts at the
beginning of the growing season and continues during the first 75 percent of the growing
season. Growth occurs only when the daily temperature is above the base temperature,
assumed to be O °C for winter tolerant crops and mixtures of perennial grasses. The
heat units for a day is computed as follows.
HU, = r - Tk (101)
I Ci v
where
= heat units on day i, °C-days
Tb = base temperature for plants, 5 °C
The fraction of the growing season that has occurred by day i is equal to the heat
unit index on day i. It is computed as follows.
HUL =Y - 1 for m ± i <. n, else HUL = 0 (102)
'
It
PHU = £ HUt (103)
where
HUI( = heat unit index or fraction of the growing season on day i,
dimensionless
62
-------
PHU = potential heat units in normal growing season, °C-days
m = Julian date of start of growing season
n = Julian date of end of growing season
The daily mean temperature is assumed to vary harmonically as follows for
computing PHU:
T = TM + 0.5 (TM - TM . ) cosflu k—~ 20Q-} (104)
ct ^ m« mm' ^ 3^5 I
where
Tc/c = estimate of normal mean daily temperature on day k, °C
TM = mean of the 12 normal mean monthly temperatures, °C
= maximum of the 12 normal mean monthly temperatures, °C
= minimum of the 12 normal mean monthly temperatures, °C
The potential increase in biomass for a day is a function of the interception of
radiation energy. The photosynthetic active radiation is estimated as follows.
PARt = 0.02092 R,t{l- exp[ -0,65 (LAIi_l + 0.05)] } (105)
where
PAR; = photosynthetic active radiation, MJ/m2
Rs. = global solar radiation, langleys
Iu = leaf area index of active biomass at end of day i-1, dimensionless
The leaf area index on day i, LAI{, is given by the equation
0 HUL = 0
LAI =
i
0 < HUI. <; 0.75 (106)
WLVi + 13360 exp ( -0.000608 WLVt)
16 (L4/0.75) (1 - HUlf HUI, > 0.75
63
-------
where
LAI^ = maximum leaf area index from input, dimensionless
WLV; = active above-ground biomass on day i, kg/ha
LAIj = LAI value on day d, (# is the day when vegetation starts declining
as estimated as the day when HVIt = 0.75), dimensionless
The potential increase in biomass during the growing season is
DDM0i = BE • PARt (107)
where
DDM0. = potential increase in total biomass on day i, kg/ha
BE = conversion from energy to biomass, 35 kg m2/ha MJ
The actual increase in biomass may be regulated by water or temperature stress.
DDM, = REG, • DDMa (108)
t l QI
REGt = min(^5., 7S,) (109)
where
DDMt = actual daily increase in total biomass (dry matter) on day i during
first 75% of growing season, kg/ha
REG = minimum stress factor for growth regulation (smaller of water stress
factor, WSj, and temperature stress factors, TS^), dimensionless
The water stress factor, WSit is the ratio of the actual transpiration to plant
transpiration demand.
(no)
t EPD4(j)
'
The temperature stress factor, TSt, is given by the equation
64
-------
25, =
exp
exp
0
T - Tb
To - T
\To - (Tb
for TCi * Tb
for Tb < Tf < To
J ci
for Tc 2: To
(111)
where
6 =
In (0.9)
To -\(To + Tb)l2}}2
(To + Tb)/2 j
(112)
Tb = base temperature for mixture of perennial winter and summer
grasses, 5 °C
To = optimal temperature for mixture of perennial winter and summer
grasses, 25 °C
As an additional constraint, the temperature stress factor is set equal to zero (and there-
fore growth ceases) when the daily mean temperature is more than 10 °C below the
average annual temperature.
TSt = 0 when T < (TM - 10 ° C)
(113)
The active live biomass is
DM. =
0
E DDM.
for HUIi = 0
for 0 < HUIt z 0.75
8 DMd (1 - HUIt)2 for HU^ > 0.75
(114)
where
DM,- = total active live biomass (dry matter) on day i, kg/ha
65
-------
= total active live biomass (dry matter) on day d (when #£//, = 0.75),
kg/ha
The root mass fraction of the total active biomass is a function of the heat unit index
where the fraction is greatest at the start of the growing season and decreases throughout
the season,
RFt = (0.4 - 0.2 HUI.) (115)
where
RFj = fraction of total active biomass partitioned to root system on day i,
dimensionless
The above ground photosynthetic active biomass is computed as follows:
WLVi = DMi (1.0 - RFJ (116)
where
WLVi = active above-ground biomass on day i, kg/ha
The program also accounts for plant residue (inactive biomass) in addition to active
biomass because inactive biomass also provides shading of the surface and reduces
evaporation of soil water. Plant residue is predicted to decay throughout the year as a
function of temperature and soil moisture. Plant residue is formed as the active plants
go into decline and at the end of the growing season. The decrease in active biomass
during the last quarter of the growing season is added to the plant residue. Similarly,
at the end of the growing season the active biomass is also added to the plant residue or
photosynthetic inactive biomass.
RSD; =
RSDi_l ( 1.0 - DECRJ for i <. d or i > n+1
.^ + DM.^ - DM.) (1.0 - DECRJ for d < i ± n (117)
(RSD.^ + DM._J (1.0 - DECR^ for i = n+1
where
RSDn = plant biomass residue on day i, kg/ha
DECRt = biomass residue decay rate on day i, dimensionless
66
-------
The plant residue decay rate decreases with moisture contents below field capacity
in the evaporative zone, becoming zero at the wilting point. Similarly, the decay rate
decreases as the soil temperature at the bottom of the evaporative zone falls below 35 "C
and becomes very slow at temperatures below 10 "C, approaching a rate of 0.005. The
maximum rate is 0.05. The decay rate is
DECRt = 0.05 [min (CDG., SUTJ 1 (118)
where
SUTt = soil moisture factor on plant residue decay, dimensionless
CDGj = soil temperature factor on plant residue decay, dimensionless
The soil moisture factor on plant residue decay is computed as follows:
1.0 for
SUTi = \
y=i
7
for £SM.(y') <
The soil temperature factor on plant residue decay is computed as follows:
CDG. -
-------
The total above ground biomass is the sum of the above ground active biomass and
the plant residue. This total is used to compute soil temperature for plant decay rate,
evaporation of soil water, and interception.
CV. = WLVt + RSD; (121)
where
CVf = total above ground biomass on day i, kg/ha
When the normal daily mean temperature is greater than 10 °C all year round, the
model assumes that grasses can grow all year. As such, there is no longer a winter
dormant period when the active biomass decreases to zero. Therefore, the model
assumes that biomass also decays all year in proportion to the quantity of biomass
present. Assuming that 20 percent of the biomass is in the root system, the above
ground active biomass is computed as follows:
WLVi = 0.8 BE - PARt - REGi + WLV.^ ( 1.0 - DECK,.) (122)
The growth and decay terms are computed the same as given in Equations 105, 107, 109,
116, and 118. The leaf area index is computed using WLVt from Equation 122 in
Equation 106.
4.13 SUBSURFACE WATER ROUTING
The subsurface water routing proceeds one subprofile at a time, from top to bottom.
Water is routed downward from one segment to the next using a storage routing
procedure, with storage evaluated at the mid-point of each time step. Mid-point routing
provides an accurate and efficient simulation of simultaneous incoming and outgoing
drainage processes, where the drainage is a function of the average storage during the
time step. Mid-point routing tends to produce relatively smooth, gradual changes in flow
conditions, avoiding the more abrupt changes that result from applying the full amount
of moisture to a segment at the beginning of the time step. The process is smoothed
further by using time steps that are shorter than the period of interest.
Mid-point routing is based on the following equation of continuity for a segment:
A Storage = Drainage In - DrainageOut - Evapotranspiration
+ Leachate Recirculation + Subsurface Inflow
68
-------
ASAftf) = 0.5 { [DRt(j) +
- [DRtf
A SM(j) = SM,(/) - SM^ (/)
(125)
(l)
where
ASMfl) = change in storage in segment j, inches
= drainage into segment j from above during time step i, inches
= soil water storage of segment j at the mid-point of time step i,
inches
= evapotranspiration from segment j during time step i, inches
= lateral drainage recirculated into segment j during time step i,
inches
= subsurface inflow into segment j during time step i, inches
Note that segments are numbered from top to bottom and therefore the drainage into
segment j + 1 from above equals the drainage out of segment j through its lower
boundary. This routing is applied to all segments except liners and the segment directly
above liners. Drainage into the top segment of the top subprofile is equal to the
infiltration from the surface; drainage into the top segment of other subprofiles is equal
to the leakage through the liner directly above the subprofile. Water is routed for a
whole day in a subprofile before routing water in the next subprofile. The leakage from
a subprofile for each time step during the day is totalled and then uniformly distributed
throughout the day as inflow into the next subprofile.
The only unknown terms in the water routing equation are SM^) and DRj(j +1); all
other terms have been computed previously or assigned during input. Subsurface inflow
is specified during input. Infiltration and evapotranspiration are computed for' the day
before performing subsurface water routing for the day. leachate recirculation is known
from the calculated lateral drainage for the previous day. The drainage into a segment
is known from the calculation of drainage out of the previous segment or from the
computation of leakage from the previous subprofile. The two unknowns are solved
simultaneously using the continuity and unsaturated drainage equations.
The number of time steps in a day can vary from subprofile to subprofile. A
minimum of 4 time steps per day and a maximum of 48 time steps per day can be used.
The number of time steps for each subprofile is computed as a function of the design of
69
-------
the lateral drainage layer in the subprofile and the potential impingement into the lateral
drainage layer. The time step is sized to insure that the lateral drainage layer, when
initially wetted to field capacity, cannot be saturated in a single time step even in the
absence of drainage from the layer.
= T(k) [POR(k) - FC(k)]
(126)
N =
int\
I
Ar
48
for A t > 0.25 days
for 0.021 days <; A t <. 0.25 days
for A t < 0.021 days
(127)
where
Ar = maximum size of time step, days
T(k) = thickness of lateral drainage layer k, inches
POR(k) = porosity of layer k, vol/vol
FC(k) = field capacity of layer k, vol/vol
IRma* = maximum impingement rate into lateral drainage layer, inches/day
N = number of time steps per day, dimensionless
The maximum impingement rate is the lowest saturated hydraulic conductivity of the
subprofile layers above the liner, but not greater than maximum daily infiltration
(estimated to be about 10 inches/day).
Drainage out of the bottom segment above the liner of a subprofile is the sum of
lateral drainage, if a lateral drainage layer, and the leakage or percolation from the
segment or the liner of the subprofile. Drainage from this segment is also a function of
the soil moisture content of the segment. The soil moisture content, lateral drainage and
leakage are solved simultaneously using continuity, lateral drainage and
percolation/leakage equations.
The water routing routine does not consider the storage capacity of the lower
segments when computing the drainage out of a segment. Therefore, the routine may
route more water down than the lower segments can hold and drain. Any water in
excess of the storage capacity of a segment (porosity) is routed back up the profile into
the segment above the saturated segment. In this way, the water contents of segments
are corrected by backing water up from the bottom to the top, saturating segments as the
corrections are made. If the entire top subprofile becomes saturated or if water is routed
70
-------
back to the surface, the excess water is added to the runoff for the day. If runoff is
restricted, the excess water is ponded on the surface and subjected to evaporation and
infiltration during the next time step.
4.14 VERTICAL DRAINAGE
The rate at which water moves through a porous medium as a saturated flow
governed by gravity forces is given by Darcy's law:
q = K i = K (128)
dl
where
q = rate of flow (discharge per unit time per unit area normal to the
direction of flow), inches/day
K = hydraulic conductivity, inches/day
/ = hydraulic head gradient, dimensionless
h = piezometric head (elevation plus pressure head), inches
/ = length in the direction of flow, inches
This equation is also applicable to unsaturated conditions, provided that the hydraulic
conductivity is considered a function of soil moisture and that the piezometric head
includes suction head.
The HELP program assumes pressure head (including suction) to be constant within
each segment of vertical percolation and lateral drainage layers. This assumption is
reasonable at moisture contents above field capacity (moisture contents where drainage
principally occurs). In circumstances where a layer restricts vertical drainage and head
builds up on top of the surface of the layer, as with barrier soil liners and some low
permeability vertical percolation layers, the program assumes the pressure head is
uniformly dissipated in the low permeability segment. For a given time step these
assumptions yield a constant head gradient throughout the thickness of the segment. For
vertical percolation layers with constant pressure, the piezometric head gradient in the
direction of flow is unity, and the rate of flow equals the hydraulic conductivity:
i = = 1 (129)
dl
q = K (130)
71
-------
For low permeability vertical percolation layers and soil liners, the hydraulic head
gradient is
i = ML = h* *l (i3i)
dl I
where
hw = pressure head on top of layer, inches
The unsaturated hydraulic conductivity is estimated by Campbell's equation
(Equation 5). Multiplying the water content terms (6, Or and 4>) in Equation 5 by the
segment thickness yields an equivalent equation with the water content terms expressed
in terms of length:
K = K SM-RS (132)
* ( UL - RS )
Here, SM, RS, and UL represent the soil water content (0), residual soil water content
(0r), and saturated soil water content (<£) of the segment, each expressed as a depth of
water in inches. The HELP program uses Equation 132 to compute unsaturated
hydraulic conductivity.
Based on Equations 130 and 132, the drainage from segment] during the time step
i, DRi(j+l), is as follows:
DT
3
2
*(/> (133)
SMt(j) - RS(j)
UL(j) - RS(j)
where
K,(j) = saturated hydraulic conductivity of segment j, inches/day
DT = the time step size, days
= 1 / N
Rearranging Equation 133 to solve for SMfi) and substituting it into Equation 124
for SMfd) yields the following non-linear equation for the remaining unknown, DRjQ+1):
72
-------
= -2[UL(j)
(134)
The HELP program solves this equation for DRfl+l) iteratively using D/fw (/ +7J as its
initial guess in the right hand side of Equation 134. If the computed value of DRfl+l)
is within 0.3 percent of the guess or 0.1 percent of the storage capacity of segment j, the
computed value is accepted; else, a new guess is made and the process is repeated until
the convergence criteria are satisfied. After DR^'+l) is computed, the program
computes SM^) using Equation 124. Constraints are placed on the solution of DRi(j+l)
and SMj(j) so as to maintain these parameters within their physical ranges; 0 to Ks • DT
for DR,(j+l), and WP(j) to VL(j) for SMt(j).
4.15 SOIL LINER PERCOLATION
The rate of percolation through soil liners depend on the thickness of the saturated
material directly above it. The depth of this saturated zone is termed the hydraulic
(pressure) head on the soil liner. The average head on the liner is a function of the
thicknesses of all segments that are saturated directly above the liner and the moisture
content of first unsaturated segment above the liner. The average head on the entire
surface of the liner is computed using the following equation:
7S(m)
SM,(m) - FC(ro)
• - l—- - - — +
UL(m) - FC(m)
for SMt(m) > FC(ro) else
<135)
where
hw(k)t = average hydraulic head on liner k during time step i, inches
TS(j) = thickness of segment], inches
m = number of the lowest unsaturated segment in subprofile k
n = number of the segment directly above the soil liner in subprofile k
73
-------
The percolation rate through a liner soil layer is computed using Darcy's law, as
given in Equation 128. As presented in Equation 131, the vertical hydraulic gradient
through the soil liner, segment n+ 1, is
LlL (136)
dl
The HELP program assumes that soil liner remains saturated at all times.
Percolation is predicted to occur only when there is a positive hydraulic head on top of
the liner; therefore, the percolation rate through a soil liner is
0 for hw(k\ = 0
-------
hole in the geomembrane. As such, the two components of a composite liner
complement each other, geomembrane and soil liners also have complimentary physical
and chemical endurance properties.
Giroud and Bonaparte (1989) provided a detailed summary of procedures for calcu-
lating leakage through composite liners. Methods described in this section were derived
from Giroud and Bonaparte's work, summarizing also the work of Brown et al. (1987),
Jayawickrama et al. (1988) and Fukuoka (1985, 1986). In these procedures Giroud and
Bonaparte (1989) assumed that the hydraulic head acting on the landfill liners and the
depth of liquid on these liners are equivalent since the effects of velocity head are
relatively small for landfill liners.
4.16.1 Vapor Diffusion Through Intact geomembranes
Intact geomembranes are liners or sections of liners without any manufacturing or
installation defects, that is without any holes. Since the voids between the molecular
chains of geomembrane polymers are extremely narrow, leakage through intact
geomembranes is likely only at the molecular level, regardless of whether transport is
caused by liquid or vapor pressure differences. Therefore, transport of liquids through
intact sections of composite liners is controlled by the rate of water vapor transport
through the geomembrane. The hydraulic conductivities of the adjacent soil layers are
much higher than the permeability of the geomembrane and therefore do not affect the
leakage through intact sections of geomembranes.
A combination of Pick's and Darcy's laws results in a relationship between geomem-
brane water vapor diffusion coefficient, obtained from water vapor transmission tests,
and "equivalent geomembrane hydraulic conductivity". Giroud and Bonaparte (1989)
recommended using the term "equivalent hydraulic conductivity" since water transport
through intact geomembranes is not described truly by Darcy's law for transport through
porous media. The following equation for water transport through intact geomembranes
was developed by substituting this relationship into Pick's law:
WVT = diffusmty • Ap = Permeability • A p Fick/s ^ (13g)
Darcy 's law (139)
where
WVT = water vapor transmission, g/cm2-sec
A/? = vapor pressure difference, mm Hg
75
-------
Tg = thickness of geomembrane, cm
qL - geomembrane leakage rate, cm/sec
p = density of water, g/cm3
Kg = equivalent saturated hydraulic conductivity of geomembrane, cm/sec
AA = hydraulic head difference, cm H20
Expressing Ap in terms of hydraulic head, Ah, diffusivity (also known as permeance or
coefficient of diffusion) and hydraulic conductivity are related as follows:
diffusivity . Tg Q
P
Equation 139 applies to the diffusion of water through the geomembrane induced by
hydraulic head or vapor pressure differences. The program applies Darcy's law to
geomembrane liners in the same manner as for soil liners (Equation 137). Diffusivity
is expressed in the program as equivalent hydraulic conductivity. Table 8 provides
default "equivalent hydraulic conductivities" for geomembranes of various polymer types.
Leakage through intact sections of geomembranes is computed as follows:
*!,<*>, • '
for hg(k)t = 0
(141)
where
QL, fc)i = geomembrane leakage rate by diffusion during time step i,
inches/day
Kg(k) = equivalent saturated hydraulic conductivity of geomembrane in
subprofile k, inches/day
h,(k)i = average hydraulic head on geomembrane liner in subprofile k
during time step i, inches
Tg(k) = thickness of geomembrane in subprofile k, inches
4.16.2 Leakage Through Holes in geomembranes
Properly designed and constructed geomembrane liners are seldom installed
completely free of flaws as evident from leakage flows and post installation leak tests.
76
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TABLE 8. GEOMEMBRANE DIFFUSIVITY PROPERTIES"
Geomembrane Material
Butyl Rubber
Chlorinated Polyethylene (CPE)
Chlorosulfonated Polyethylene
(CSPE) or Hypalon
Epichlorohydrin Rubber (CO)
Elasticized Polyolefin
Ethylene-Propylene Diene
Monomer (EPDM)
Neoprene
Nitrile Rubber
Polybutylene
Polyester Elastomer
Low-Density Polyethylene
(LDPE)
High-Density Polyethylene
(HOPE)
Polyvinyl Chloride (PVC)
Saran Film
Coefficient of
Migration, cmVsec
2x10-"
6xian
SxlO"11
3xlO-9
ixia11
2xia11
4xian
5x10-'°
7xia12
2x10-'°
5xia12
3xia12
2xlO-10
9xlO-13
Equivalent Hydraulic
Conductivity, cm/ sec
IxlO'12
4xlO-12
3xlO-12
2xlO-10
8xlO-13
2xlO-12
3xlO-12
3xlO-u
5xlO'13
2x10-"
4xlO-13
2xlO-13
2xlO-u
6xlO'14
* From Giroud and Bonaparte (1985)
geomembrane flaws can range in size from pinholes that are generally a result of
manufacturing flaws such as polymerization deficiencies to larger defects resulting from
seaming errors, abrasion, and punctures occurring during installation. Giroud and
Bonaparte (1989) defines pinhole-sized flaws to be smaller than the thickness of the
geomembrane. Since geomembrane liner thicknesses are typically 40 roils or greater, the
HELP program assigns the diameter of pinholes to be 40 roils or 0.001 m (defect area
= 7.84xlO'7m2). Giroud and Bonaparte (1989) indicates that pinhole flaws are more
commonly associated with the original, less sophisticated, geomembrane manufacturing
77
-------
techniques. Current manufacturing and polymerization techniques have made pinhole
flaws less common. Giroud and Bonaparte (1989) defined installation defect flaws to be
of a size equal to or larger than the thickness of the geomembrane. Based on 6 case
studies that produced consistent results, Giroud and Bonaparte (1989) recommended using
a defect area of 1 cm2 (20 x 5 mm) for conservatively high predictions of liner leakage
on projects with intensive quality assurance/quality control monitoring during liner
construction. Therefore, the HELP program uses a defect area of 1 cm2. Finally, the
HELP program user must define the flaw density or frequency (pinholes or defects/acre)
for each geomembrane liner. Giroud and Bonaparte (1989) recommended using a flaw
density of 1 flaw/acre for intensively monitored projects. A flaw density of 10
flaws/acre or more is possible when quality assurance is limited to spot checks or when
environmental difficulties are encountered during construction. Greater frequency of
defects may also result from poor selection of materials, poor foundation preparation and
inappropriate equipment as well as other design flaws and poor construction practices.
Geosyntec (1993) indicated that geomembranes may undergo deterioration due to
aging or external elements such as chemicals, oxygen, micro-organisms, temperature,
high-energy radiation, and mechanical action (i.e., foundation settlement, slope failure,
etc.). Although geomembrane deterioration can create geomembrane flaws or increase
the size of existing flaws, the HELP program does not account for this time-dependent
deterioration in the liner.
The liquid that passes through a geomembrane hole will flow laterally between the
geomembrane and the flow limiting (controlling) layer of material adjacent to the
geomembrane, unless there is perfect contact between the geomembrane and the
controlling soil or free flow from the hole. The space between the geomembrane and the
soil is assumed to be uniform. The size of this space depends on the roughness of the
soil surface, the soil particle size, the rugosity and stiffness of the geomembrane, and the
magnitude of the normal stress (overburden pressure) that tends to press the
geomembrane against the soil, The HELP program ranks the contact between a
geomembrane and soil as perfect, excellent, good, poor, and worst case (free flow). The
HELP program also permits designs where a geomembrane is separated from a low
permeability soil by a geotextile. The leakage is controlled by the hydraulic
transmissivity of the gap or geotextile between the geomembrane and the soil. This
interracial flow between the geomembrane and soil layer covers an area called the wetted
area. The hole in the geomembrane is assumed to be circular and the interracial flow
is assumed to be radial; therefore, the wetted area is circular. Giroud and Bonaparte
(1989); Bonaparte et al. (1989); and Giroud et al. (1992) examined steady-state leakage
through a geomembrane liner for all of these qualitative levels of contact and provided
either theoretical or empirical solutions for the leakage rate and the radius of interracial
flow. Leakage and wetted area are dependent on the static hydraulic head on the liner;
the hydraulic conductivity and thickness of the surrounding soil, waste, or geotextile
layers; the size of the flaw; and the contact (interface thickness) between the
geomembrane and the controlling soil layer.
78
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The HELP program designates the controlling soil layer as either high, medium or
low permeability. High is a saturated hydraulic conductivity greater than or equal to
IxlO'1 cm/see; medium is greater than or equal to IxlO'4 cm/sec and less than IxlO"1
cm/sec; and low is less than Ix 10"4 cm/sec. The low permeability layers are assumed
to remain saturated in the wetted area throughout the simulation. As mentioned earlier,
geomembranes are geosynthetics with a very low cross-plane hydraulic conductivity (See
Table 8). On the other hand, geotextiles are geosynthetics with a high cross-plane
hydraulic conductivity and high in-plane transmissivity (See Table 9). The fabrics can
help minimize damage to the membrane by the surrounding soil or waste layers. The in-
plane transmissivity of geotextiles used as geomembrane cushions isused to compute the
radius of interracial flow and leakage through a geomembrane separated from the
controlling soil by a geotextile.
Worst Case (Free Flow) Leakage
Giroud and Bonaparte (1989) theoretically examined free flow through a
geomembrane surrounded by infinitely pervious media such as air or high permeability
soil or waste layers (Brown et al., 1987). However, Giroud and Bonaparte (1989)
cautioned that if the leachate head on the geomembrane liner is very small, surface
tensions in the surrounding high permeability layers could prevent free flow through the
geomembrane flaw. With time, leachate-entrained, fine-grained particles can clog the
high permeability layers, greatly reducing the permeability of these layers and possibly
preventing free flow. Free flow is assumed whenever the layers above and below the
geomembrane have high permeability.
TABLE 9. NEEDLE-PUNCHED, NON-WOVEN GEOTEXTILE PROPERTIES*
Applied
Compressive
Stress, kPa
1 to 8
100
200
Resulting
Geotextile
Thickness,
cm
0.41
0.19
0.17
In-Plane Flow
Geotextile
Transmissivity,
cnWsec "
0.3
0.04
0.02
Horizontal
Hydraulic
Conductivity,
cm/ sec
0.7
0.2
0.1
Cross-Plane
Flow
Vertical
Hydraulic
Conductivity,
cm/sec
0.4
—
—
* Geotechnical Fabrics Report—1992 Specifiers Guide
International, 1991), and Giroud and Bonaparte (1985).
** Transmissivity = horizontal hydraulic conductivity x thickness.
(Industrial Fabrics Association
79
-------
Pinholes. Giroud and Bonaparte (1989) concluded that percolation through pinholes
surrounded by high permeability layers can be considered as flow through a pipe and
recommended using Poiseuille's equation. Therefore, the HELP program uses the
following form of Poiseuille's equation to predict free flow leakage through geomembrane
pinholes:
= (86,400) TC n2(k) p15 g hg(k\ d} (142)
= (4,046.9) (128) r,15
1.775 x 1Q-* nz(*) Ag(*),
where
qL2(k)i = leakage rate through pinholes in subprofile k during time step i,
inches/day
86,400 = units conversion, 86,400 seconds per day
n2(k) = pinhole density in subprofile k, #/acre
pi5 = density of water at 15°C = 999 kg/m3
g = gravitational constant, 386.1 inches/sec2
d2 = diameter of a pinhole, 0.001 meters
4,046.9 = units conversion, 4,046.9 m2/acre
T|15 = dynamic viscosity of water at 15°C = 0.00114 kg/m sec
Tg(k) = thickness of geomembrane in subprofile k, inches
1.775 x 10"4 = constant, 1.775 x 10"4 inches acre/day
Installation Defects. Giroud and Bonaparte (1989) also concluded that leakage
through defects in geomembranes surrounded by high permeability layers can be
considered as flow through an orifice and recommended using Bernoulli's equation.
Therefore, the HELP program uses the following form of Bernoulli's equation to predict
free flow leakage through geomembrane defects:
86,400
(144)
4046.9
80
-------
\ = 0.0356 «3(*) fKffi (145)
where
Qi3(k)i = leakage rate through defects in subprofile k during time step i,
inches/day
CB = head loss coefficient for sharp edged orifices, 0.6
n3(k) = installation defect density for subprofile k, #/acre
a3 = defect area, 0.0001 m2
hg(k)i = average hydraulic head on geomembrane liner in subprofile k
during time step i, inches
0.0356 = constant, 0.0356 inches05 acre/day
Perfect Liner Contact
Perfect geomembrane liner contact means that there is no gap or interface between
the geomembrane liner and controlling soil or waste layer. Perfect contact is not
common but can be achieved if the geomembrane is sprayed directly onto a compacted,
fine-grained soil or waste layer or if the geomembrane and controlling layers are
manufactured together. Problems associated with the installation of spray-on liners (e.g.
application, polymerization, etc.) has limited their use. Perfect liner contact results in
only vertical percolation through the controlling layer below the liner flaw; however,
both vertical and horizontal flow can occur in the layer opposite the controlling soil or
waste layer.
Giroud and Bonaparte (1989) indicated that a lower bound estimate of leakage for
perfect contact conditions can be estimated using Darcy 's law assuming vertical flow
through the controlling layer only in the area below the hole. An upper bound prediction
can be obtained by assuming radial flow in the controlling layer and integrating Darcy's
law in spherical coordinates to obtain the following equation:
w Ks hff d
O = * g
^" 0.5 d (146)
1 -
where
Qh = leakage rate through pinholes and installation defects, nWsec
K, = saturated hydraulic conductivity of soil layer, m/sec
81
-------
hg = hydraulic head on geomembrane, m
d0 = diameter of the geomembrane flaw, m
Ts = thickness of soil layer, m
A geomembrane flaw diameter of 0.1 cm is used for pinhole flaws. Considering the
density of pinholes, converting units and assuming d/Ts« 0, the leakage rate for
pinholes in geomembrane with perfect contact is
_ n n2(k) Ks(k) hg(k)t 0.04 (14?)
6,272,640
where
Qi^^i = leakage rate through pinholes in subprofile k during time step i,
inches/day
n2(k) = pinhole density in subprofile k, #/acre
Ks(k) = saturated hydraulic conductivity of soil layer at the base of
subprofile k, inches/day
hg(k)i = average hydraulic head on liner in subprofile k during time step i,
inches
0.04 = diameter of a pinhole, 0.04 inches
6,272,640 = units conversion, 6,272,640 square inches per acre
Since the area of defect flaws was identified to be 1 cm2, an equivalent defect
diameter was calculated to be 1.13 cm. Considering the density of installation defects
and converting units, the leakage rate for installation defects in geomembrane with
perfect contact is
qL(k\ = * "3(*) *.(*)
6,272,640
(0.5) (0.445)
(148)
T .(*)
where
QL3(k)i = leakage rate through installation defects in subprofile k during time
step i, inches/day
n3(k) = installation defect density in subprofile k, #/acre
0.445 = diameter of an installation defect, 0.445 inches
82
-------
Ts(k) = thickness of soil layer at base of subprofile k, inches
Interfacial Flow
Problems associated with the installation of geomembrane liners typically causes an
interface or gap to develop between the installed geomembrane liner and the adjacent
materials. Even with a large overburden pressure on the geomembrane liner, gaps exist
due to geomembrane wrinkles from installation; clods, large particle size and
irregularities in the subsoil; and the stiffness of the geomembrane preventing the liner
from filling the small voids between soil particles. However, the thickness of the
interface is dependent on the effective stress on the liner. Percolation through
geomembrane flaws typically involves radial flow through the interface and vertical flow
through the controlling layer (See Figure 9). This flow also occurs in reverse when the
controlling layer is placed over the geomembrane (See Figure 10). Layer erosion and
consolidation can increase the interface thickness over time; however, such increases are
not considered in the HELP program.
The head acting on the geomembrane liner decreases from a maximum at the edge
of the geomembrane flaw to zero at the edge of the wetted area. Flow through the
interface and controlling layer completely dissipates the leachate head or, as with intact
liners, the total head on the liner. The leachate is assumed to flow radially until this
head is dissipated; this radial distance is called the wetted area.
Giroud and Bonaparte (1989) indicated that the interracial flow is dependent on the
hydraulic transmissivity (thickness) of the air or geotextile cushion occupying the
interface, the hydraulic head on the geomembrane, the hydraulic conductivity of the
controlling soil layer, and the size of the geomembrane flaw. Vertical flow through the
controlling layer is dependent on the hydraulic conductivity of the layer, the hydraulic
gradient on the layer at various locations in the wetted area, and the area of the wetted
area.
Giroud and Bonaparte (1989) and Giroud et al. (1992) used Darcy's law for flow
through a porous media, considering both radial and interracial flow, and developed the
following equation, modified for flow per unit area and temperature corrected, for
estimating leakage through circular flaws in geomembranes with interracial flow.
-H20.J (149)
where
qh - interracial flow leakage rate through flawed geomembrane, m/sec
Ks = saturated hydraulic conductivity of controlling soil layer, m/sec
83
-------
:P0jme:al>Jl:ity:5Qif:br:W«isie:
in-
L
****
* i*
to*
or Waste fK^>
Figure 9. Leakage with Interfacial Flow Below Flawed geomembrane
;Higb;:P.ermeabiliiy;Sbil>C3:r: Waste;
T
ii
Figure 10. Leakage with Interracial Flow Above Flawed geomembrane
84
-------
/av, = average hydraulic gradient on wetted area of controlling soil
layer, dimensionless
n = density of flaws, # per m2
R = radius of wetted area or interracial flow around a flaw, m
Tj20 = absolute viscosity of water at 20°C, 0.00100 kg/m-sec
rj,5 = absolute viscosity of water at 15°C, 0.00114 kg/m-sec
Since the U.S. Geological Survey defined hydraulic conductivity, in Meinzer units,
as the number of gallons per day of water passing through 1 ft2 of medium under a unit
hydraulic gradient (1 ft/1 ft) at a temperature of 60°F (15 °C) (Viessman et al., 1977;
Linsley et al,, 1982), Equation 149 was corrected to reflect an absolute water viscosity
at 15 °C (Giroud and Bonaparte, 1989).
Giroud et al. (1992) developed the following equation to describe the average
hydraulic gradient on the geomembrane; a description of the development is presented
in the following paragraphs.
i = 1
'avg
2T, In
(150)
where
hg = total hydraulic head on geomembrane, m
Ts = thickness of controlling soil layer, m
r0 = radius of a geomembrane flaw, m
Methods for calculating the wetted area radius for various liner contact conditions and
design cases are presented in the following sections.
The HELP program applies Equations 149 and 150 as follows:
0.00100 \
0.00114 j
6,276,640
(151)
85
-------
u,.(*)l •
*,<*)«
2 T,(*) In
(152)
where
= leakage rate through pinholes (2) or installation defects (3) with
interracial flow in subprofile k during time step i, inches/day
= saturated hydraulic conductivity of controlling soil layer in
subprofile k, inches/day
= average hydraulic gradient on wetted area of controlling soil
layer from pinholes (2) or installation defects (3) in subprofile k
during time step i, dimensionless
n2j(k) = density of pinholes (2) or defects (3) in subprofile k, #/acre
R2j(k)i ~ radius of wetted area or interracial flow around a pinhole (2) or an
installation defect (3) in subprofile k during time step i, inches
6,272,640 = units conversion, 6,272,640 square inches per acre
= radius of flaw; pinhole r0 = 0.02 inches; defect r0 = 0.22 inches
= average hydraulic head on liner in subprofile k during time step i,
°2,3
ht(k)t
inches
Ts(k) = thickness of soil layer at base of subprofile k, inches
Geotextile Interface
Giroud and Bonaparte (1989) assumed a unit hydraulic gradient for vertical flow
through the controlling layer (i.e., Equation 149 without the i,vg term) and applied the
principle of conservation of mass to the radial and vertical flow through the
geomembrane. They integrated the resulting equation and developed the following
equation for estimating the radius of the wetted area:
R =
4 *, e<»
(153)
86
-------
where
e,
int
= hydraulic transmissivity of the interface or geotextile, nWsec
However, assuming a unit hydraulic gradient indicates that the depth of saturated material
on the geomembrane is substantially smaller than the thickness of the controlling layer.
This was a limitation of Giroud and Bonaparte's (1989) method for estimating
geomembrane liner leakage. However, Giroud et al. (1992) used a simplified and
conservative form of Equation 153, the principle of conservation of mass for flow
through the two layers, and integrated the resulting equation to obtain an equation similar
in form to Darcy's law (Q = Kia). The hydraulic gradient term in the resulting equation
was identified as the average hydraulic gradient on the geomembrane liner and is
provided in Equation 150.
Equation 153 is solved iteratively by using (hg + r0) as the initial guess and then
substituting the computed R into the right hand side until it converges. Equation 151 is
also limited by the fact that the thickness between the installed geomembrane liner and
the controlling layer is not easily determined, especially for multiple design cases.
However, Giroud and Bonaparte (1989) provided information on the hydraulic
transmissivity (loaded thickness times in-plane hydraulic conductivity) of geotextile
cushion under a variety of effective stresses (See Table 9). Therefore, the HELP
program only uses Equation 153 to estimate the leakage through flaws in geomembrane
liners installed with geotextile cushions. The cushion is assumed to completely fill the
interface between the liner and controlling layer. For other liner design cases, Giroud
and Bonaparte (1989); Bonaparte et al. (1989); and Giroud et al. (1992) used laboratory
and field data and theoretically based equations to develop semi-empirical and empirical
equations for estimating the wetted area radius for excellent, good, and poor contact
between the geomembrane liner and controlling layer.
Pinholes. The HELP program applies Equation 153 for computing the radius of the
wetted area of leakage from pinholes and through a geotextile interface and a controlling
soil layer as given in Equation 154 for each time step and each subprofile. The radius
is then used in Equation 152 to compute the average hydraulic gradient. The radius and
average hydraulic gradient is then used in Equation 151 to compute the leakage rate for
pinholes.
e,.tt,(*)
2 In
*2(*),
0.02
0.02
*a(*),
(154)
where
hydraulic in-plane transmissivity of the geotextile in subprofile k,
inchesVday
87
-------
Installation Defects. The HELP program applies Equation 153 for computing radius
of leakage from installation defects and through a geotextile interface and a controlling
soil layer as follows:
*,<*), =
4 *,(*), 6,n,(*)
*,(*)
2 In
0.22
°'22
-
(155)
This radius is then used in Equation 152 to compute the average hydraulic gradient. The
radius and average hydraulic gradient is then used in Equation 151 to compute the
leakage rate for installation defects.
Excellent Liner Contact
Excellent liner contact is achieved under three circumstances. Medium permeability
soils and materials are typically cohesionless and therefore generally are able to conform
to the geomembrane, providing excellent contact. The second circumstance is for very
well prepared low permeability soil layer with exceptional geomembrane placement
typically achievable in the laboratory, small lysimeters or small test plots. The third
circumstance is by the use of a geosynthetic clay liner (GCL) adjacent to the
geomembrane with a good foundation. The GCL, upon wetting, will swell to fill the gap
between the geomembrane and the foundation, providing excellent contact.
Medium Permeability Controlling Soil. Giroud and Bonaparte (1989) indicated that
if a geomembrane liner is installed with a medium permeability material (saturated
hydraulic conductivity greater than or equal to 1x10^ cm/sec and less than IxlO"1 cm/sec)
above the geomembrane as the controlling soil layer, the flow to the geomembrane flaw
will be impeded by the medium permeability layer and the leakage through the flaw will
be less than free flow leakage. Similarly, if medium permeability material below the
geomembrane acts as the controlling soil layer, the flow from the flaw will be impeded
by the medium permeability layer and leakage will also be less than free flow.
Whenever a medium permeability soil acts as the controlling soil layer, the contact is
modeled as excellent. However, even with excellent contact, there will be some level
of flow between the geomembrane and medium permeability layer. Bonaparte et al.
(1989) used a theoretical examination of the flow in the interface between the medium
permeability soil and the geomembrane liner to develop several empirical approaches that
averaged the logarithms of the perfect contact leakage and free flow leakage predictions
to obtain the following equation for the radius of convergence of leakage to a flaw in a
geomembrane placed on high permeability material and overlain by medium permeability
material:
-------
R = 0.97 a.0'38 A,038 JC;0"25 (156)
where
R = radius of interfacial flow around a geomembrane flaw, m
a0 = geomembrane flaw area, 7.84xlO"7 m2 for pinholes and 0.0001 m2
for installation defects
hg = total hydraulic head on geomembrane, m
K, = saturated hydraulic conductivity of controlling soil layer, m/sec
This equation is also used to calculate the wetted radius of interracial flow for
geomembranes overlain by high permeability soil and underlain by medium permeability
soil. This radius, as it is actually computed below in Equations 157 or 155, is then used
in Equation 152 to compute the average hydraulic gradient. The radius and average
hydraulic gradient are then used in Equation 151 to compute the leakage rate for
geomembrane flaws.
Pinholes. By inserting the pinhole area, converting units and simplifying, Equation
156 is converted to the following equation for radius of interracial flow from pinholes
in geomembranes with medium permeability controlling soil layers.
R2(k)f = 0.0494 hg(k)^* Ks(k) -025 (157)
where
R2^)i = radius of wetted area or interracial flow around a pinhole in
subprofile k during time step i, inches
Ks(k) = saturated hydraulic conductivity of controlling soil layer in
subprofile k, inches/day
hg(k)f = average hydraulic head on liner in subprofile k during time step i,
inches
Installation Defects. By inserting the installation defect area, converting units and
simplifying, Equation 156 is converted to the following equation for radius of interracial
flow from installation defects in geomembranes with medium permeability controlling soil
layers.
= 0.312 hg
89
-------
where
= radius of wetted area or interracial flow around a pinhole in
subprofile k during time step i, inches
Low Permeability Controlling Soil. Giroud and Bonaparte (1989) indicated that,
when a geomembrane liner is installed on or under a low-permeability soil or waste
layer, excellent geomembrane liner contact can be obtained if the liner is flexible and
without wrinkles and the controlling layer is well compacted, flat, and smooth; has not
been deformed by rutting due to construction equipment; and has no clods or cracks.
Excellent contact is also possible when using a GCL with a good foundation as the low
permeability soil layer. Using the theoretical techniques previously mentioned and
laboratory data, Brown et al. (1987) developed charts for estimating the leakage rate
through circular flaws in geomembrane liners for what Giroud and Bonaparte (1989)
defined to be excellent liner contact. Leakage rates predicted using these charts are
dependent on the flaw surface area, the saturated hydraulic conductivity of the controlling
soil or waste layer, and the total leachate head on the geomembrane. Giroud and
Bonaparte (1989) summarized and extrapolated or interpolated the data in these charts
and developed the following equation for the wetted area radius with excellent liner
contact with low permeability soil (saturated hydraulic conductivity less than Ix 10"*
cm/sec); units are as in Equation 156:
R = 0.5
This radius, as it is actually computed below in Equations 160 or 161, is then used
in Equation 152 to compute the average hydraulic gradient. The radius and average
hydraulic gradient are then used in Equation 151 to compute the leakage rate for
geomembrane flaws.
Pinholes. By inserting the pinhole area, converting units and simplifying, Equation
159 is converted to the following equation for radius of leakage from pinholes in
geomembranes with excellent contact with low permeability controlling soil-layers.
= 0.0973 h(k)°5 K(k) -°06
Installation Defects. By inserting the installation defect area, converting units and
simplifying, Equation 159 is converted to the following equation for radius of leakage
from installation defects in geomembranes with excellent contact with low permeability
controlling soil layers.
90
-------
= 0.124
Good Liner Contact
Using the equations for perfect and excellent liner contact and free-flow percolation
through geomembrane liners, Giroud and Bonaparte ( 1989) developed leakage rate curves
for a variety of conditions (i.e., leachate head, saturated hydraulic conductivity, etc.).
The worst ease field leakage was arbitrarily defined to be midway between free-flow and
excellent contact leakage estimates. The area between worst case field leakage and
excellent contact leakage was arbitrarily divided into thirds and defined as good and poor
field leakage. However, due to the lengthy calculations required to estimate good and
poor liner leakage, Giroud and Bonaparte (1989) developed empirical equations to predict
leakage through geomembrane liners under good and poor field conditions. These
equations are discussed in the following paragraphs.
Giroud and Bonaparte (1989) indicated that good geomembrane liner contact can be
defined as a geomembrane, installed with as few wrinkles as possible, on an adequately
compacted, low-permeability layer with a smooth surface. Similar to Equations 156 and
159, Giroud and Bonaparte (1989) observed families of approximately parallel linear
curves when plotting the leakage rate as a function of total head on the geomembrane
liner, geomembrane flaw area, and saturated hydraulic conductivity of the controlling soil
or waste layer. Giroud and Bonaparte (1989) concluded that the leakage rate through
damaged geomembranes is approximately proportional to equations of the form a/ hgx
K,z. Therefore, Giroud and Bonaparte (1989) proposed the following equation for
determining the wetted area radius for good liner contact:
R = 0.26 a,005 *f°'45 */°13 <162)
This radius, as it is actually computed below in Equations 163 or 164, is then used
in Equation 152 to compute the average hydraulic gradient. The radius and average
hydraulic gradient are then used in Equation 151 to compute the leakage rate for
geomembrane flaws. Similar to Equation 159, Equation 162 has the limitation that the
saturated hydraulic conductivity of the controlling soil layer must be less than Ix 104
cm/sec. Equation 162 is valid only in units of meters and seconds.
Pinholes. Inserting pinhole area, performing units conversion and simplifying,
Equation 162 is converted for radius of leakage from pinholes in geomembranes with
good contact with low permeability controlling soil layers as follows:
91
-------
= 0.174 hg(kft45
Installation Defects. By inserting the installation defect area, converting units and
simplifying, Equation 162 is converted to the following equation for radius of leakage
from installation defects in geomembranes with good contact with low permeability
controlling soil layers.
= 0.222 *(*)?* *(*)-*» (164)
Poor Liner Contact
Giroud and Bonaparte (1989) indicated that poor geomembrane liner contact can be
defined as a geomembrane, installed with a certain number of wrinkles, on a poorly
compacted, low-permeability soil or waste layer, with a surface that does not appear
smooth. Similar to Equation 162, Giroud and Bonaparte (1989) proposed the following
equation for determining the radius of leakage through a geomembrane for poor contact
with a low permeability controlling soil layer:
R = o.6i
-------
from installation defects in geomembranes with poor contact with low permeability
controlling soil layers.
= 0.521 /»(*)°45 tf(*)-°13 (167)
4.17 GEOMEMBRANE AND SOIL LINER DESIGN CASES
As previously mentioned, the HELP program simulates leakage through both the
intact and damaged portions of geomembrane liners. Leakage through geomembrane
flaws (pinholes and defects) is modeled for various liner contact conditions. The
minimum level of leakage will occur through an intact geomembrane liner. The total
leakage is the sum of leakage through (1) intact geomembrane sections and (2) pinhole-
size and (3) defect-size geomembrane flaws.
The HELP program insures that the total leakage through the geomembrane and
controlling layers is not greater than the volume of drainable water. The program also
checks to insure that the leakage rate is not greater than the product of the hydraulic
gradient and the saturated hydraulic conductivity of the controlling layer,
Giroud and Bonaparte (1989) and Giroud et al. (1992) developed their equations for
intact geomembranes, geomembranes surrounded by highly-pervious materials, and
composite liners; defined as a geomembrane installed over a low-permeable soil liner and
covered by a drainage layer. However, various other liner design cases are possible and,
although the equations were not specifically designed to address these designs, similar
physical conditions indicated that these equations would be applicable to other liner
design cases. However, the applicability of these equations to other liner designs has not
been fully verified.
geomembrane liners are frequently installed with various defects that increase as
design and installation monitoring efforts decrease. Therefore, the HELP program user
must identify the liner contact condition (perfect, excellent, good, poor, or worst case)
for each damaged geomembrane liner. The user must also identify the hydraulic
conductivity of the controlling layer and the geomembrane flaw type (pinhole or defect)
and density. The user must also identify the thickness and equivalent hydraulic
conductivity of the geomembrane for the intact portions of the liner. In some cases, the
user will have to identify the geotextile cushion thickness and in-plane hydraulic
transmissivity.
Based on the design of the geomembrane liner system (layer type, saturated hydraulic
conductivity, and location of controlling soil layer), the HELP program can compute
93
-------
leakage for 6 different geomembrane liner design cases (See Figures 11 through 16).
These design cases are discussed in the following sections.
Design Case 1. Geomembrane liner Design Case 1 consists of a geomembrane installed
between two highly permeable soil or waste layers (See Figure 11). The program uses
the free flow equations (Equation 143 for pinholes and Equation 145 for installation
defects) to calculate the leakage rate through flaws, The vapor diffusion equation
(Equation 141) is used to calculate the leakage rate through the intact portion of the
geomembrane liner. The damaged and intact leakage estimates are then added together
to predict the total leakage through the geomembrane liner.
Design Case 2. Geomembrane liner Design Case 2 consists of three design scenarios:
1) a geomembrane liner installed on top of a highly permeable layer and overlaid by a
medium permeability layer; 2) a geomembrane liner installed on top of a medium
permeability layer and overlaid by a highly permeable layer; and 3) a geomembrane liner
installed between two medium permeability layers (See Figure 12). Three levels of
contact (perfect, excellent, or worst case) between the geomembrane and medium
permeability layer are allowed for this design case. The program uses Equations 147 and
148 to calculate the perfect contact leakage rate through pinholes and installation defects,
respectively. Equations 151, 152, 157 (for pinholes) and 158 (for installation defects)
are used to calculate the excellent contact leakage rate. As in Design Case 1, free flow
equations (Equation 143 for pinholes and Equation 145 for installation defects) are used
to calculate the worst case contact leakage rate. Finally, the vapor diffusion equation
(Equation 141) is used to calculate the leakage rate through the intact portion of the
geomembrane liner for all three scenarios and levels of contact. The damaged and intact
leakage estimates are subsequently added together to predict the total leakage through the
geomembrane liner.
Design Case 3. Geomembrane liner Design Case 3 consists of a geomembrane overlying
a low permeability layer (either a soil liner or vertical percolation layer), which is the
controlling soil layer (See Figure 13). The geomembrane may be covered with either a
LEGEND
. HIGH PERMEABILITY
SOIL OH WASTE
GEOMEMBRANE
Figure 11. Geomembrane Liner Design Case 1
94
-------
(a)
LEGEND
HQH PERMEABILITY
SOL OR WASTE
MEDIUM PERMEABILITY
SO LOB WASTE
I— GEOMEMOHANE
(b)
(c)
Figure 12. Geomembrane Liner Design Case 2
(a)
LEGEND
UGH PERMEABILITY
SOIL OH WA3TE
MEDIUM PERMEABILITY
SOL Ofl WASTE
|_ LOW PERMEABILITY
SOIL OR WASTE
I— GEOMEMBRANE
LAYER TYPE 1 01 2
(b)
Figure 13. Geomembrane Liner Design Case 3
95
-------
high permeability, medium permeability, or low permeability soil or waste layer
designated as either a vertical percolation or lateral drainage layer. The level of contact
between the geomembrane and low permeability controlling soil layer maybe defined as
perfect, excellent, good, poor, or worst case. The program uses Equations 147 and 148
to calculate the perfect contact leakage rate through pinholes and installation defects,
respectively, Equations 151 and 152 are used to calculate the interracial flow leakage
rate and hydraulic head gradient for excellent, good and poor levels of contact.
Equations 160, 163 and 166 are used to calculate the radius of interracial flow from
pinholes respectively for excellent, good and poor levels of contact. Equations 161, 164
and 167 are used to calculate the radius of interracial flow from installation defects
respectively for excellent, good and poor levels of contact. As in Design Case 1, free
flow equations (Equation 143 for pinholes and Equation 145 for installation defects) are
used to calculate the worst case contact leakage rate. Finally, the vapor diffusion
equation (Equation 141) is used to calculate the leakage rate through the intact portion
of the geomembrane liner for all levels of contact. The damaged and intact leakage
estimates are subsequently added together to predict the total leakage through the
geomembrane liner.
Design Case 4. Geomembrane liner Design Case 4 is simply the inverse of Design Case
3 (See Figure 14); the low permeability controlling soil layer overlies the geomembrane.
The same soil and layer types and levels of contacts may be used. The same equations
LAYER TYPE 3
LEGEND
HGH PERMEABILITY
SOIL OR WASTE
MEDIUM PERMEABILITY
SO LOR WASTE
I_ LOW PERMEABILITY
SOIL OR WASTE
I— GEOMEMBRANE
LAYER TYPE 3
LAYER TYPE 3
Figure 14. Geomembrane Liner Design Case 4
96
-------
as described for Design Case 3 are used to calculate leakage for the various contacts and
flaw sizes. This geomembrane liner design case is the exact inverse of that considered
by Giroud and Bonaparte (1989). However, since the total head loss for leakage through
damaged geomembrane liners is assumed to occur through the interface and controlling
layer, the equations proposed by Giroud and Bonaparte (1989) should apply as well for
the inverted case. However, the total head on the geomembrane for this design case and
Design Case 6 is equal to the sum of the leachate depth in the layer above the liner
system and the thickness of saturated soil liner above the geomembrane as shown in
Figure 10; the hydraulic head is the total thickness of continuously saturated soil or waste
above the geomembrane. In Design Cases 1, 2, 3 and 5, the total head is just the depth
of saturated material above the liner system as shown in Figure 9.
Design Case 5. As shown in Figure 15, geomembrane liner Design Case 5 consists of
eight scenarios that have a geotextile cushion placed between the geomembrane liner and
the controlling soil layer. The controlling soil layer may be composed of medium or low
permeability soil. The controlling soil layer may be above or below the geomembrane,
but, if above, the controlling soil layer cannot be a soil liner. The geotextile is not
connected to the leachate collect system, which would cause them to act as a drainage
layer. The geotextile functions solely as a liner cushion and defines the interracial flow
between the geomembrane and controlling soil layer. Assuming the cushion completely
fills the interface, Equations 151, 152, 154 (for pinholes) and 155 (for installation
defects) are used to estimate the leakage rate as a function of the hydraulic in-plane
transmissivity of the geotextile. Table 9 provides hydraulic transmissivity values, at
several compressive stresses, for needle-punched, non-woven geotextiles. Recall that the
hydraulic transmissivity of geotextiles is greatly affected by the applied compressive
stress and the degree of clogging.
Design Case 6. Geomembrane liner Design Case 6 consists of a geomembrane liner
installed on a high, medium, or low permeability soil or waste layer with a geotextile
cushion separating the geomembrane and a overlying soil liner (layer type 3). (See
Figure 16). Similar to Design Case 5, the program uses Equations 151, 152, 154 (for
pinholes) and 155 (for installation defects) to estimate the leakage rate as a function of
the hydraulic in-plane transmissivity of the geotextile. However, as in Design Case 4,
the total head on the geomembrane is equal to the sum of the continuously saturated
material above the liner system and the thickness of the soil liner above the
geomembrane.
Flow through the geotextile cushion in either Design Case 5 or 6 can increase the
geomembrane liner leakage due to an increase in the wetted area and possibly creating
a connection between the geomembrane flaw and controlling layer macropores. On the
other hand, laboratory tests have shown that a needlepunched, nonwoven geotextile
cushion installed between a geomembrane liner and controlling layer can decrease
leakage if the effective stress on the liner or controlling layer is adequate to push the
97
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(a)
(b)
LEGEND
HIGH PERMEABILITY
SOIL on WASTE
. UEOUU PEBHEABUJTY
SOIL OH WASTE
LOW PERMEABILITY
SOU OH WASTE
• GEOUEU8RANC
- GEOTEXTIU
LAYER TYPE 1 OB 2
(c)
(f)
LAYER TYPE 1 OH 2
(d)
LAYER TYPE 1 OR 2
(e)
Figure 15. Geomembrane Liner Design Case 5
geotextile into irregularities in the controlling layer (worst case and possibly poor contact
cases). This prevents free lateral flow between the liner and controlling layer.
However, the beneficial effects of geotextile cushions may be limited to cases of poor
design and installation.
4.18 LATERAL DRAINAGE
Unconfmed lateral drainage from porous media is modeled by the Boussinesq
98
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LAYER TYPE J
LEGEND
. HIGH PERMEABILITY
SOil OR WASTE
• IIEOIUU PERMEABILITY
SOIL Ofl WASTE
• LOW PERMEABILITY
SOIL OR WASTE
• GEOMEUBRANE
• OEOTEXTILE
LAYER TYPE S
(b)
LAYER TYPE 3
Figure 16. Geomembrane Liner Design Case 6
equation (Darcy's law coupled with the continuity equation), employing the Dupuit-
Forcheimer (D-F) assumptions. The D-F assumptions are that, for gravity flow to a
shallow sink, the flow is parallel to the liner and that the velocity is in proportion to the
slope of the water table surface and independent of depth of flow (Forcheimer, 1930).
These assumptions imply the head loss due to flow normal to the liner is negligible,
which is valid for drain layers with high hydraulic conductivity and for shallow depths
of flow, depths much shorter than the length of the drainage path. The Boussinesq
equation may be written as follows (See Figure 17 for definition sketch):
3h
dt
(h - I sin a )
dh
dl
R
(169)
where
/ = drainable porosity (porosity minus field capacity), dimensionless
h = elevation of phreatic surface above liner at edge of drain, cm
t = time, sec
KD = saturated hydraulic conductivity of drain layer, cm/sec
/ = distance along liner surface in the direction of drainage, cm
99
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Figure 17. Lateral Drainage Definition Sketch
= inclination angle of liner surface
= net recharge (impingement minus leakage), cm/sec
a
R
Where the saturated zone directly above the liner extends into more than one
modeling segment, the saturated hydraulic conductivity, KD, is assigned the
weight-averaged saturated hydraulic conductivity of the saturated zone.
where
m
n
y
X
E *,(/) • <*(/)
where y =
(170)
saturated hydraulic conductivity of segment j, cm/sec
thickness of saturated soil in segment], cm
number of the lowest unsaturated segment in subprofile
number of the segment directly above the liner in subprofile
depth of saturated lateral drainage (h - x tan a), cm
horizontal distance from drain, cm
100
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The lateral drainage submodel assumes that the relationship between lateral drainage
rate and average saturated depth for steady flow approximates the overall relationship for
an unsteady drainage event. For steady flow, the lateral drainage rate is equal to the net
recharge.
v M\ v r\ ~ «^ Lt M n
3r dx D* HD L
where
Q0 = lateral drainage rate per unit width of drain at any x, cmVsec
QD = lateral drainage rate into collector pipe at drain, x = 0, (flow rate
per unit length of collector), cm2/sec
L = length of the horizontal projection of the liner surface (maximum
drainage distance) cm
qD = lateral drainage rate at drain in flow per unit area of landfill,
cm/sec
Translating the axis from / (parallel to the liner) to x (horizontal) and substituting for
R, the steady lateral drainage equation is described as follows:
(172)
L
After expressing h in terms of y and expanding, Equation 171 can be rewritten as
follows:
. + (tana) - -- °— (173)
dx2 \dx) dx Kcos2a
Nondimensionally, it can be rewritten as follows:
dx* cos2a
101
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where
JC* = jc / L, nondimensional horizontal distance
y* = y IL, nondimensional depth of saturation above liner
9o = QD I KD > nondimensional lateral drainage rate
Assuming a unit hydraulic gradient in the direction of flow at the drain, the boundary
conditions for Equation 174 are
- tana at x* = 0 (175)
cos a
dx*
tana =0 at x' = I (176)
An alternative boundary condition is used for shallow saturated depths and small lateral
drainage rate [qD* < (sin2 a)/4]. For values of qD' > (sin2or)/4, the depth of saturated
drainage media at the upper end of the liner is greater than 0.
v* = q° (177)
cos2 a
Equation 177 can be solved analytically for the two limiting cases by simplifying,
employing the boundary conditions and integrating from / = 0 to x = 1. For small
drainage rates or shallow saturated depths, such that qD' < (sin2 a)/4 or y" < 0.2 tan a
(y* = average depth of saturation above the entire liner),
y* = far q£ < 0.4 sin2 a
2 (sina) (cosa)
(178)
or
q£ = 2 (sina)(cosa) y* for y* < 0.2 tana
For large drainage rates, such that qD* > (sin2 a)/4 or y* > 0.2 tan a,
102
-------
_ n , »
y = —1 for qD > 0.4 sirr a
4 cos a
(179)
for y* > 0.2 tana
Equation 174 was solved numerically for a wide range of values of the parameters,
qD* and a. The nondimensional average depth of saturation on top of the liner (y*) was
computed numerically for each solution. Analysis of these solutions showed that the
relationship among qD, y, L, K, and a is closely approximated by the following equation
which converges to the analytical solutions for small drainage rates (Equation 178) and
large drainage rates (Equation 179).
0.4 sin*
This two-part function (Equations 179 and 180) is continuous and smooth and
matches the closed-form, asymptotic solutions for the cases where Y < tan a and j* >
tana. The estimate of qD' given by Equations 178 and 180 is within one percent of the
value obtained by solving Equation 174 numerically. Equations 178 and 180 are used
to compute qD in the lateral drainage sub model. The equations are applied iteratively
along with the liner leakage or percolation equations and storage equation to solve
concurrently for the average depth of saturation, the liner leakage or percolation and the
average depth of saturation above the liner during each time period. The process is
repeated for each subprofile with a lateral drainage layer for every time step.
4.19 LATERAL DRAINAGE RECIRCULATION
The lateral drainage from any subprofile may be collected or recirculated. If
collected, that fraction of the drainage is removed from the landfill and the quantity is
reported as a volume collected. If recirculated, that fraction of the drainage from the
subprofile is stored during the day and then uniformly distributed the next day throughout
the specified layer. The recirculation is then applied in the vertical water routing
procedure using Equations 124 and 134. Recirculation can be distributed to any layer
that is not a liner.
103
-------
^ FRC(kJ) qD(k\.ltli (lgl)
where
= recirculation into segment j during a timestep on day i, inches
= number of timesteps in a day for subprofile k containing segment j
k — number of the subprofile
nk = total number of subprofiles in the landfill
n = number of the timestep in day i-1
N(k) = total number of timesteps in a day for subprofile k, day"1
= fraction of the lateral drainage from subprofile k that is recirculated
to segment j
= lateral drainage rate from subprofile k during timestep n on day i-1,
inches/day
4.20 SUBSURFACE INFLOW
Subsurface inflow is treated as steady, uniform seepage into a layer. Inflow maybe
specified for any layer. If the inflow for a liner is specified, the inflow is added to the
inflow into the next lower layer that is not a liner. If inflow is specified for a liner
system that is on the bottom of the landfill profile, the inflow is added to the inflow of
the first layer above the liner system. The subsurface inflow is then applied in the
vertical water routing procedure using Equations 124 and 134. The inflow is specified
for each layer in the input. The inflow is specified as the volume per year per unit area,
which is then simply converted by the program to a volume per time step based on a unit
area. Volume per unit area is used throughout the program for storage and flows.
4.21 LINKAGE OF SUBSURFACE FLOW PROCESSES
The drainage rate out of a subprofile must equal the sum of lateral drainage rate and
the leakage rate through the liner system. The subsurface water routing, liner leakage
and lateral drainage calculations are linked as follows:
1. Water is routed through the subprofile from top to bottom by unsaturated vertical
drainage using Equation 134.
104
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2. The total drainage rate out of the segment directly above the liner system is initially
assumed to be the same as in the previous time step. Excess water is backed up
through the subprofile as necessary.
3. The average depth of saturation above the liner system and the effective lateral
hydraulic conductivity of the saturated zone are computed using Equation 135. The
depth or head is only an estimate since it is based on estimated drainage out of the
subprofile.
4. Lateral drainage and liner leakage or percolation are computed using Equation 178
or 180 for lateral drainage, Equation 137 for soil liner percolation, and Equations
141, 143, 145, 147, 148, 151, 152, 154, 155, 157, 158, 160, 161, 163, 164, 166,
167 or 168 for geomembrane liner leakage. The estimated saturated depth is used
in these computations.
5. A new estimate of the average depth of saturation is generated by updating the water
storage using the computed lateral drainage and percolation/leakage. If the new
estimate is within the larger of 5 percent or 0.01 inches of the original estimate, then
the updated water storage and the computed rates are accepted. If not, the original
and new estimate are averaged to generate a new estimate and steps 4 and 5 are
repeated until the convergence criterion is met. If the estimated and computed total
drainage are greater than the available gravity water (storage in excess of field
capacity), then the total drainage is assigned the value of the gravity water. Then,
the leakage and lateral drainage volumes are proportional to the relative rates, and
the depth of saturation is computed by Equation 178 using the assigned lateral
drainage rate.
6. The procedure is repeated for each time step in a day, and the lateral drainage
volumes are summed as are the liner leakage/percolation volumes for the subprofile
before beginning computations for the next subprofile. The daily lateral drainage is
then partitioned to removal and recirculation as specified in the input. The liner
leakage/percolation is assigned as drainage into the next subprofile or out of the
landfill. The depths of saturation for time steps during the day are averaged and
reported as average daily head.
105
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SECTION 5
ASSUMPTIONS AND LIMITATIONS
5.1 METHODS OF SOLUTION
The modeling procedures documented in the previous section are necessarily based
on many simplifying assumptions. Most of these are stated in the sections documenting
the individual procedures. Generally, these assumptions are reasonable and consistent
with the objectives of the program when applied to standard landfill designs. However,
some of these assumptions may not be reasonable for unusual designs. The major
assumptions and limitations of the program are summarized below.
Precipitation on days when the mean air temperature is below freezing is assumed
to occur as snow, Snowmelt is assumed to be a function of energy from air temperature,
solar radiation and rainfall. Solar radiation effects are included in an empirical melt
factor. In addition, groundmelt is assumed to occur at a constant rate of 0.5 mm/day as
long as the ground is not frozen. Snow and snowmelt are subject to evaporation prior
to runoff and infiltration. The program does not consider the effects of aspect angle or
drifting in its accounting of snow behavior.
Prediction of frozen soil conditions is a simple, empirical routine based on antecedent
air temperatures. Thaws are based on air temperatures and climate data. Soils while
frozen are assumed to be sufficiently wet so as to impede infiltration and to promote
runoff. Similarly, no evapotranspiration and drainage are permitted from the evaporative
zone while frozen.
Runoff is computed using the SCS method based on daily amounts of rainfall and
snowmelt. The program assumes that areas adjacent to the landfill do not drain onto the
landfill. The time distribution of rainfall intensity is not considered. The program
cannot be expected to give accurate estimates of runoff volumes for individual storm
events on the basis of daily rainfall data. However, because the SCS rainfall-runoff
relationship is based on considerable daily field data, long-term estimates of runoff
should be reasonable. One would expect the SCS method to underestimate runoff from
short duration, high intensity storms; larger curve numbers could be used to compensate
if most of the precipitation is from short duration, high intensity storms. The SCS
method does not explicitly consider the length and slope of the surface over which
overland flow occurs; however, a routine based on a kinematic wave model was
developed to account for surface slope and length.
Potential evapotranspiration is modeled by an energy-based Penman method. As
applied, the program uses average quarterly relative humidity and average annual wind
speed. It is assumed that these data yield representative monthly results. Similarly, the
program assumes that the relative humidity is 100% on days when precipitation occurs.
106
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The program uses an albedo of 0.23 for soils and vegetation and 0.60 for snow. The
actual evapotranspiration is a function of other data, also. The solar radiation and
temperature data are often synthetically generated. The vegetation data is generated by
a vegetative growth model. The evaporative zone depth is assumed to be constant
throughout the simulation period. However, outside of the growing season, the actual
depth of evapotranspiration is limited to the maximum depth of evaporation of soil water,
which is a function of the soil saturated hydraulic conductivity.
Vegetative growth is based on a crop growth model. Growth is assumed to occur
during the first 75% of the growing season based on heating units. Recommendations
for the growing season are based primarily for summer grasses and assume that the
growing season is that portion of the year when the temperature is above 50 to 55 'F.
However, the user may specify a more appropriate growing season for different
vegetation. The optimal growth temperature and the base temperature are based on a
mixture of winter and summer perennial grasses. It is assumed that other vegetation
have similar growth constraints and conditions. It is further assumed that the vegetation
is not harvested.
The HELP program assumes Darcian flow for vertical drainage through
homogeneous, temporally uniform soil and waste layers. It does not consider preferential
flow through channels such as cracks, root holes or animal burrows. As such, the
program will tend to overestimate the storage of water during the early part of the
simulation and overestimate the time required for leachate to be generated. The effects
of these limitations can be minimized by specifying a larger effective saturated hydraulic
conductivity and a smaller field capacity. The program does increase the effective
saturated hydraulic conductivity of default soils for vegetation effects.
Vertical drainage is assumed to be driven by gravity alone and is limited only by the
saturated hydraulic conductivity and available storage of lower segments. If unrestricted,
the vertical drainage rate out of a segment is assumed to equal the unsaturated hydraulic
conductivity of the segment corresponding to its moisture content, provided that moisture
content is greater than the field capacity or the soil suction of the segment is less than
the suction of the segment directly below. The unsaturated hydraulic conductivity is
computed by Campbell hydraulic equation using Brooks-Corey parameters. It is assumed
that all materials conducting unsaturated vertical drainage have moisture retention
characteristics that can be well, represented by Brooks-Corey parameters and the
Campbell equation. The pressure or soil suction gradient is ignored when applying the
Campbell equation; therefore, the unsaturated drainage and velocity of the wetting front
may be underestimated. This is more limiting for dry conditions in the lower portion of
the landfill; the effects of this limitation can be reduced by specifying a larger saturated
hydraulic conductivity. For steady-state conditions, this limitation has little or no effect.
The vertical drainage routine does not permit capillary rise of water from below the
evaporative zone depth. Evapotranspiration is not modeled as capillary rise, but rather
as a distributed extraction that emulates capillary rise. This is limiting for dry conditions
107
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where the storage of water to satisfy evaporative demand is critical and for designs where
the depth to the liner is shallow. This limitation can be reduced by increasing the field
capacity in the evaporative zone and the evaporative zone depth,
Percolation through soil liners is modeled by Darcy's law, assuming free drainage
from the bottom of the liner. The liners are assumed to be saturated at all times, but
leakage occurs only when the soil moisture of the layer above the liner is greater than
the field capacity. The program assumes that an average hydraulic head can be computed
from the soil moisture and that this head is applied over the entire surface of the liner.
As such, when the liner is leaking, the entire liner is leaking at the same rate. The liners
are assumed to be homogeneous and temporally uniform.
Leakage through geomembrane is modeled by a family of theoretical and empirical
equations. In all cases, leakage is a function of hydraulic head. The program assumes
that holes in the geomembrane are dispersed uniformly and that the average hydraulic
head is representative of the head at the holes. The program further assumes that the
holes are predominantly circular and consist of two sizes. Pinholes are assumed to be
1 mm in diameter while installation defects are assumed to have an cross-sectional area
of 1 cm2. It is assumed that holes of other shapes and sizes could be represented as some
quantity of these characteristic defects. Leakage through holes in geomembranes is often
restricted by an adjacent layer or soil or material termed the controlling soil layer.
Materials having a saturated hydraulic conductivity greater than or equal to Ix 10"' cm/sec
are considered to be a high permeability material; materials having a saturated hydraulic
conductivity greater than or equal to Ix 10"4 cm/sec but less than Ix 10"1 cm/sec are
considered to be a medium permeability material; and materials having a saturated
hydraulic conductivity less than IxlO"4 cm/sec are considered to be a low permeability
material. The program assumes that no aging of the liner occurs during a simulation.
The lateral drainage model is based on the assumption that the lateral drainage rate
and average saturated depth relationship that exists for steady-state drainage also holds
for unsteady drainage. This assumption is reasonable for leachate collection, particularly
for closed landfills where drainage conditions should be fairly steady. Where drainage
conditions are more variable, such as in the cover drainage system, the lateral drainage
rate is underestimated when the saturated depth is building and overestimated when the
depth is falling. Overall, this assumption causes the maximum depth to be slightly
overestimated and the maximum drainage rate to be slightly underestimated. The long-
term effect on the magnitude of the water balance components should be small. As with
leakage or percolation through liners, the average saturated depth is computed from the
gravity water and moisture retention properties of the drain layer and other layers when
the drain layer is saturated. The program assumes that horizontal and vertical saturated
hydraulic conductivity to be of similar magnitude and that the horizontal value is
specified for lateral drainage layer.
Subsurface inflow is assumed to occur at a constant rate and to be uniformly
distributed spatially throughout the layer, despite entering the side. This assumption
108
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causes a delay in its appearance in the leachate collection and more rapid achievement
of steady-state moisture conditions. This limitation can be minimized by dividing the
landfill into sections where inflow occurs and sections without inflow.
leachate recirculation is assumed to be uniformly distributed throughout the layer
by a manifold or distribution system, leachate collected on one day for recirculation is
distributed steadily throughout the following day.
5.2 LIMITS OF APPLICATION
The model can simulate water routing through or storage in up to twenty layers of
soil, waste, geosynthetics or other materials for a period of 1 to 100 years. As many as
five liner systems, either barrier soil, geomembrane or composite liners, can be used.
The model has limits on the order that layers can be arranged in the landfill profile.
Each layer must be described as being one of four operational types: vertical percolation,
lateral drainage, barrier soil liner or geomembrane liner. The model does not permit a
vertical percolation layer to be placed directly below a lateral drainage layer. A barrier
soil liner may not be placed directly below another barrier soil liner. A geomembrane
liner may not be placed directly below another geomembrane liner. Three or more
liners, barrier soil or geomembrane, cannot be placed adjacent to each other. The top
layer may not be a barrier soil or geomembrane liner. If a liner is not placed directly
below the lowest lateral drainage layer, the lateral drainage layers in the lowest
subprofile are treated by the model as vertical percolation layers. If a geomembrane
liner is specified as the bottom layer, the soil or material above the liner is assumed to
be the controlling soil layer. No other restrictions are placed on the order of the layers.
The lateral drainage equation was developed and tested for the expected range of
hazardous waste landfill design specifications. The ranges examined for slope and maxi-
mum drainage length of the drainage layer were O or 30 percent and 25 to 2000 feet;
however, the formulation of the equations indicates that the range of the slope could be
extended readily to 50 percent and the length could be extended indefinitely.
Several relations must exist between the moisture retention properties of a material.
The porosity, field capacity and wilting point can theoretically range from 0 to 1 in units
of volume per volume, but the porosity must be greater than the field capacity, and the
field capacity must be greater than the wilting point. The general relation between soil
texture class and moisture retention properties is shown in Figure 2.
The initial soil moisture content cannot be greater than the porosity or less than the
wilting point. If the initial moisture contents are initialized by the program, the moisture
contents are set near the stead y-state values. However, the moisture contents of layers
below the top liner system or cover system are specified too high for arid and semi-arid
locations and too low for very wet locations, particularly when thick profiles are being
modeled.
109
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Values for the maximum leaf area index may range from O for bare ground to 5.0
for an excellent stand of grass. Greater leaf area indices may be used but have little
impact on the results. Detailed recommendations for leaf area indices and evaporative
depths are given in the program. For numerical stability, the minimum evaporative zone
depth should beat least 3 inches.
The program computes the evaporation coefficient for the cover soils based on their
soil properties. The default values for the evaporation coefficient are based on experi-
mental results reported by Ritchie (1972) and others. The model imposes upper and
lower limits of 5.50 and 3.30 for the evaporation coefficient so as not to exceed the
range of experimental data.
The program performs water balance analysis for a minimum period of one year.
All simulations start on the January 1 and end on December 31. The condition of the
landfill, soil properties, thicknesses, geomembrane hole density, maximum level of
vegetation, etc., are assumed to be constant throughout the simulation period. The
program cannot simulate the actual filling operation of an active landfill. Active landfills
are modeled a year at a time, adding a yearly lift of material and updating the initial
moisture of each layer for each year of simulation.
110
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