United Slates
Environmental Protection
Agency
Office of Radiation Programs
Las Vegas Facility
P.O Box 15027
Las Vegas NV89114
Technical Note
ORP/LV 78 8
September 1978
Radiation
oEPA
Water Movement in
Uranium Mill
Tailings Profiles
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Technical Note
ORP/LV-78-8
WATER MOVEMENT IN
URANIUM MILL TAILINGS PROFILES
by
A. Klute and D. F. Heermann
U. S. Department of Agriculture
Science and Education Administration
Agricultural Research
Fort Collins, CO 80523
A report of research conducted
under IAG - DG - H026
between USDA - SEA and the
U.S. Environmental Protection Agency
Project Officer
Robert F. Kaufmann
Field Studies Branch
Office of Radiation Programs-Las Vegas Facility
U.S. Environmental Protection Agency
Las Vegas, Nevada 89114
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DISCLAIMER
This report has been reviewed by the Office of Radiation Programs -
Las Vegas Facility, U.S. Environmental Protection Agency, and approved for
publication. Approval does not signify that the contents necessarily reflect
the views and policies of the EPA. Mention of trade names or commercial
products does not constitute endorsement or recommendation for their use.
Neither the United States nor the EPA makes any warranty, expressed or
implied, or assume any legal liability or responsibility for any information,
apparatus, product or process disclosed, or present that its use could not
infringe on privately owned rights.
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FOREWORD
The Office of Radiation Programs of the U.S. Environmental Protection
Agency carries out a national program designed to evaluate population
exposure to ionizing and non-ionizing radiation, and promote the development
of controls to protect the public health and safety.
This report was prepared as an integral element of the overall program
to control radiation from uranium mill tailings piles situated in the
western United States. Such piles are composed of materials containing con-
centrations of radium-226 two to three orders of magnitude more than back-
ground rock and soil. Radon-222, the noble gas progeny of radium-226, is
produced within the pile and diffuses to the atmosphere as a continuing
long-term source resulting in radiation exposure of the surrounding populace.
Pathways of exposure from uranium mill tailing piles include airborne
particulates and contamination of surface or ground water. This report
assists in understanding the impact of normal precipitation cycles on
recharge of shallow ground water with contaminated leachate from tailings
piles. Knowledge of the vertical distribution of soil moisture also assists
in evaluating the magnitude of radon diffusion from a tailings pile.
This study represents what we believe is the first application of
digital modeling techniques to simulate the unsaturated or soil moisture
regime in a uranium tailings pile under average and extreme precipitation
and evaporation cycles typified by a climatologic setting in Salt Lake City,
Utah. Beginning with initial gravity drainage of the saturated pile, long-
term soil moisture variations as a function of time, depth, and tailings
lithology are simulated. The results clearly indicate that long-term
recharge to underlying ground water can occur and that the upper 10 to 20
centimeters of the profiles tend to dry out rapidly. The presence of vege-
tation would increase the rate and probably the depth of dewatering. Clear-
ly then, there is a tradeoff at least qualitatively identified between
management/stabilization schemes calling for minimal recharge to ground
water and reduction of radon diffusion by maintenance of relatively high
water content in the very near surface tailings. Future studies will be
needed for identification and analysis of the tradeoffs. This was well
beyond the scope of the present study.
iii
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Readers of this report are encouraged to inform the Office of Radiation
Programs - Las Vegas Facility of any omissions or errors. Requests for
further information or comments concerning this report are also welcome.
Donald W. Hendricks
Director, Office of
Radiation Programs, LVF
iv
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ABSTRACT
The general objective of this study was to characterize the behavior of
water in profiles of uranium mill tailings. The approach taken consisted of
(1) measurement of the water retention and transmission properties of selected
tailings materials, (2) numerical simulation of water flow in selected profiles
of tailings subjected to specific boundary conditions, and (3) analysis and
interpretation of the simulation results within the framework of unsaturated
soil water flow theory.
The sequence of flow events in a tailings profile without vegetation and
with a water table at a given depth is: (1) an initial drainage from satura-
tion, with evaporation at the surface, (2) infiltration of varying amounts of
rain at irregular intervals, (3) and periods of evaporation and drainage from
the profile, with redistribution within the profile, between infiltration
events. The water flow regime in the upper 90 cm, particularly the upper 10-
20 cm is transient and dynamic. Rainfall events produce wetting, which lasts
only a few days, in the upper portion of the profile. The wetted zone is
dried by evaporation to depths of about 10-15 cm in evaporation periods lasting
15 to 30 days. Infiltrated water, which has penetrated to depths more than
about 10 cm, continues to flow downward within the profile while evaporation
is occurring from the surface.
The lower part of the profile, below about 70-90 cm, tends to behave in
a quasi-steady downward flow condition. The average flow rate in this portion
of the profile is determined by the fraction of the rainfall trapped in the
profile and not re-evaporated during the time interval of averaging. The
water content of the profile at depths below 70-90 cm and at distances more
than 50-70 cm above the water table is that corresponding to a hydraulic
conductivity equal to the magnitude of the downward flux.
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CONTENTS
Page
FOREWORD i13-
ABSTRACT v
FIGURES vlii
TABLES • xii
SYMBOLS xiii
ACKNOWLEDGMENT xiv
1. INTRODUCTION i
2. CONCLUSIONS ..... 2
3. RECOMMENDATIONS 4
4. SOIL WATER FLOW CONCEPTS AND TERMINOLOGY 5
5. DETERMINATION OF THE HYDRAULIC PROPERTIES OF TAILINGS ... 12
Sampling and Basic Physical Characteristics 12
Water Retention Characteristics . . ...... 13
Sample Preparation 15
Method for Determination of Hysteresis 15
Hydraulic Conductivity-Water Relationships 20
6. SIMULATION OF WATER FLOW IN TAILINGS PROFILES 28
The Numerical Solution Process 28
7. ANALYSIS OF WATER FLOW IN TAILINGS PROFILES 34
The Sequence of Flow Events 34
Drainage of Initially Saturated Profiles 34
Drainage Without Evaporation 35
Drainage With Evaporation 40
Drainage of Layered Profiles with Evaporation .... 45
vi
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Page
Infiltration Events 51
Infiltration into a Uniform Coarse Profile 51
Infiltration into the M/F/M Profile 55
Evaporation Following a Rain
60
Uniform Coarse Profile 60
Evaporation from the M/F/M Profile 61
Discussion of the Flow in the Tailings Profile 70
The Development of the Quasi-Steady State Profile . . 72
REFERENCES 77
GLOSSARY 79
vii
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Number
FIGURES
Page
1 A diagram of the relationships between hydraulic head,
H, pressure head, h, and gravitational head, z ........ 6
2 Diagram of the hysteresis in the water content-pressure
head relation, 6(h) ..................... 9
3 Diagram of the pressure chamber apparatus used to determine
the water retention characteristics of the tailings
materials ..........................
4 Water retention functions for the Coarse tailings ....... 17
5 Water retention functions for the Medium tailings ....... 18
6 Water retention functions for the Fine tailings ........ 19
7 Hydraulic conductivity-water content function for Coarse
tailings .................. ......... 25
8 Hydraulic conductivity-water content function for Medium
tailings ........................... 26
9 Hydraulic conductivity-water content function for Fine
tailings ........................... 27
10 Diagram showing the division of the space-time domain into
increments and the numbering of the nodes .......... 29
11 Expanded view of the nodes about the general node n, m
showing the identification of the nodal values of
pressure head ........................ 31
12 Water content profiles at several times during the drainage,
without evaporation, of a uniform Coarse tailings profile . . 36
13 Water content profiles at several times during the drainage,
without evaporation, of a uniform Fine tailings profile ... 36
14 Water content versus time at several depths in a uniform
Coarse tailings profile draining without evaporation at
the top ........................... 37
viii
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Number Page
15 Water content versus time at several depths in a uniform
Fine tailings profile draining without evaporation at
the top ........................... 37
16 Flux across the water table (0 to 1 day) from uniform
profiles of Coarse, Medium and Fine tailings, draining
without evaporation at the upper end ....... . ..... 38
17 Flux across the water table (0 to 30 days) from uniform
profiles of Coarse, Medium and Fine tailings, draining
without evaporation at the upper end ............. 38
18 Water content profiles at several times in a uniform Coarse
tailings profile draining with a 1 cm/day potential
evaporation rate at the top ................. 41
19 Water content profiles at several times in a uniform Medium
tailings profile draining with a 1 cm/day potential
evaporation rate at the top ................. 41
20 Water content versus time at several depths in the uniform
Coarse tailings profile of Fig. 18 .............. 42
21 Water content versus time at several depths in the uniform
Medium tailings profile of Fig. 19 .............. 42
22 Flux versus depth in the upper 30 cm of the uniform Coarse
profile of Fig. 18 ...................... 44
23 Flux versus depth in the upper 30 cm of the uniform Medium
profile of Fig. 19 ...................... 44
24 Flux across the surface (actual evaporation rate) of the
uniform Coarse and Medium profiles of Fig. 18, and 19 .... 45
25 Water content profiles at various times in a two-layer,
Coarse over Medium, profile draining with a potential
evaporation rate of 1 cm/day ................. 46
26 Pressure head distributions for the profile of Fig. 25 ..... 46
27 Flux across the water table (0 to 1 day) for draining profiles
of uniform Coarse (Fig. 18), Coarse over Medium (Fig. 25),
and Coarse over Fine (Fig. 29) ................ 48
28 Same as Fig. 27, except 0 to 15 days .............. 48
29 Water content profiles at several times in a two-layer,
Coarse over Fine profile, draining with evaporation ..... 50
ix
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Number Page
30 Pressure head distributions for the profile of Fig. 29 50
31 Flux across the surface (negative of the rainfall rate) of
the uniform Coarse profile 52
32 Water content distributions in the uniform Coarse profile
subjected to a 2.3 cm rainfall 53
33 Detail of water content distributions in the upper 30 cm of
the profile of Fig. 32 54
34 Pressure head profiles corresponding to the water content
profiles of Fig. 33 54
35 Flux versus depth in the upper 30 cm of the profile of
Fig. 32 55
36 Flux across the surface (negative of the rainfall rate)
of the layered M/F/M tailings profile 57
37 Water content distributions in a layered (M/F/M) profile
at several times during a 12 cm rain 58
38 Flux versus depth at various times in the upper 90 cm of
the M/F/M profile of Fig. 37 58
39 Detail of water content distributions in the upper 90 cm
of the M/F/M profile of Fig. 37 59
40 Water content distributions in a uniform Coarse tailings
profile subjected to a potential evaporation rate of
0.5 cm/day after a 2.3 cm rain 62
41 Detail of water content distributions in the upper 30 cm
of the uniform Coarse profile of Fig. 40 62
42 Flux versus depth at various times during the evaporation from
the profile of Fig. 40 63
43 Flux across the surface during evaporation from the uniform
Coarse profile (Fig. 40), M/F/M profile, 12 cm rain
(Fig. 44), and M/F/M profile, 6 cm rain (Fig. 47) 64
44 Water content distributions in the M/F/M profile subjected
to a 0.5 cm/day potential following a 12 cm rain 65
45 Detail of water content distributions in the upper 90 cm
of the M/F/M profile of Fig. 44 65
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Number ^£
46 Flux versus depth at several times during evaporation
from the M/F/M profile of Fig. 44 67
47 Water content distribution in the M/F/M profile with a
0.5 cm/day potential evaporation rate following a
6 cm rain 69
48 Detail of the upper 90 cm of the M/F/M profile of Fig. 47
69
49 Flux versus depth in the upper 90 cm of the M/F/M
profile of Fig. 47 71
50 Water content distributions for two states of steady flow
in a uniform Medium tailings profile 74
51 Water content distributions for two states of steady flow
in a profile of Medium tailings with a Fine layer at
30 to 60 cm depth 75
52 Water content distributions for two states of steady flow
in a profile composed of alternating 30 cm layers of
Medium and Fine tailings 75
xi
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TABLES
Number Page
1 IN SITU BULK DENSITY AND PARTICLE DENSITY OF THE SELECTED
FRACTIONS OF MILL TAILINGS, VITRO MILL SITE 12
2 PARTICLE SIZE DISTRIBUTION OF THE MILL TAILINGS FRACTIONS 13
3 PARAMETERS OF THE EMPIRICAL 6(h) AND K(6) FUNCTIONS OF THE
COARSE TAILINGS 21
4 PARAMETERS OF THE EMPIRICAL 6(h) AND K(6) FUNCTIONS OF THE
MEDIUM TAILINGS 21
5 PARAMETERS OF THE EMPIRICAL FUNCTIONS OF THE FINE TAILINGS 22
6 SUMMARY OF CONDITIONS FOR SIMULATIONS OF DRAINAGE WITHOUT
EVAPORATION 39
7 SUMMARY OF CONDITIONS FOR SIMULATIONS OF DRAINAGE WITH
EVAPORATION AT THE SURFACE 40
8 SUMMARY OF CONDITIONS FOR SIMULATION OF DRAINAGE WITH
EVAPORATION OF A TWO-LAYER, COARSE OVER MEDIUM PROFILE 47
9 SUMMARY OF CONDITIONS FOR SIMULATION OF DRAINAGE WITH
EVAPORATION OF A TWO-LAYER COARSE OVER FINE PROFILE 49
10 SUMMARY OF CONDITIONS FOR SIMULATION OF THE FILTRATION INTO
A UNIFORM COARSE PROFILE 52
11 SUMMARY OF CONDITIONS FOR SIMULATION OF INFILTRATION OF 6
AND 12 CM RAINFALL INTO THE M/F/M PROFILE 56
12 SUMMARY OF CONDITIONS FOR SIMULATION OF EVAPORATION FOLLOWING
A RAIN ON A UNIFORM COARSE TAILINGS PROFILE 60
13 SUMMARY OF CONDITIONS FOR SIMULATION OF EVAPORATION FOLLOWING
A 12 CM RAIN ON THE M/F/M PROFILE 64
14 SUMMARY OF CONDITIONS FOR SIMULATION OF EVAPORATION FOLLOWING
A 6 CM RAIN ON THE M/F/M PROFILE 68
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LIST OF SYMBOLS
Symbol
A
B
Definition
Units
d
C
h
b
e
0,
K
Empirical curve fitting parameter in the hydraulic
conductivity function, K(0) = Aexp(B9), Eq. (18).
Empirical curve fitting parameter in the hydraulic
conductivity function, See A, above.
Empirical curve fitting parameter in the relation
6(h) = (h/d)a, Eq. (11).
Ditto.
Water capacity, d9/dh
Pressure head of the soil water
Empirical curve fitting parameter in the water
retention function, Eqs. (8), (9) and (10)
Ditto.
Volumetric water content
Empirical curve fitting parameter in the water reten-
tion function, Eqs. (8), (9) and (10). The water
content at zero pressure head.
Ditto. The water content limit as h ->• -°° .
Volumetric water content at the intersection of the
empirical 9(h) functions, Eq. (8) and Eq. (11).
Pressure head at the intersection of the empirical
0(h) functions, Eq. (8) and Eq. (11).
Hydraulic conductivity
L/T, cm/day
None
None
L, cm
L-1, cm'1
L, cm
L, cm
None
None
None
None
None
cm
L/T, cm/day
xiii
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ACKNOWLEDGMENTS
The United States Department of Agriculture, Science and Education
Administration, Agricultural Research, and Colorado State University, carry
out research on problems of mutual interest under a broad form cooperative
agreement. The cooperation of the Department of Agronomy, Colorado State
University, in this investigation is gratefully acknowledged.
xiv
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SECTION 1
INTRODUCTION
The tailings materials which result from the extraction of uranium
from ore are deposited in piles formed from a slurry containing up to 80-90%
water. The piles consist of interbedded layers of materials ranging from
clay to coarse sand in texture. The radionuclides in the tailings are a
potential source of airborne, waterborne, and direct gamma radiation expo-
sure to people. Radon-222, a noble gas daughter of radium-226 is formed
within the pile and transported to the surface.!./ A primary variable affect-
ing the diffusion of radon to the surface of a pile is the distribution of
the water content of the tailings as a function of time and depth. It was
the general objective of the study reported here to develop background
information on the behavior of water in profiles of tailings materials.
The specific objectives of this project were:
1. Characterize the movement and storage of water in tailings
profiles consisting of clay, silt and sand-sized particles.
2. Estimate future moisture conditions under typical (average) and
extreme precipitation, and evaporative conditions.
A simulation approach to the objectives was taken, which involved:
(1) the determination of the hydraulic conductivity and water retention
functions for each tailings material of interest. (2) selection and mathe-
matical description of the appropriate boundary and initial conditions to
characterize specific flow problems, and (3) numerical solution of the
equation of water movement. The numerical solutions gave the water content,
and soil water flux distributions within the tailings profile subjected to
specific boundary and initial conditions. The results from the numerical
simulations were then interpreted and analyzed in the framework of the
concepts of unsaturated flow theory.
!/ Radon, in turn, decays by alpha emission to several daughters most
notably lead-210 and polonium-210, both of which are particulates.
Inhalation of radon and its daughters, therefore, results in exposure to
alpha particles and is related to an increase in cancer incidence.
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SECTION 2
CONCLUSIONS
In a profile of tailings without vegetation and with a water table at a
given depth, the sequence of water flow events may be described as:
1. An initial period of drainage from saturation, with evaporation at
the surface.
2. Infiltration of varying amounts of rain at irregular intervals.
3. Periods of evaporation and drainage from the profile, and redistri-
bution of water within the profile, between rainfall events.
The initial drainage from saturation, if not disturbed by rainfall
events, could last for several years. However, rain events will prevent the
profile from attaining a steady state condition in the upper profile. The
water flow regime in the upper profile (depths to about 90 cms) will be one
of dynamic, transient fluctuations in water content and flux. The lower
part of the profile will tend toward a quasi-steady state downward flow
condition.
Rainfall infiltration events will wet the upper profile to volumetric
water contents, 9, on the order of 0.25 to 0.4 in the coarser fractions of
tailings and to about 0.6 in the finer fractions. These levels of 6 will
be maintained for only a short time (a fraction of a day) during and just
after a rainfall. Infiltration produces a "wetting front" which moves down-
ward into the profile. Above the front, the general level of 9 will be about
as given above. Below the front the water content will not be affected
during the period of infiltration. Wetting front penetration increases with
the magnitude of the rainfall event and the initial level of water content
in the profile. An approximation of the depth of wetting, d, can be obtained
from d = r/(6u-9i), where 9U is the water content in the wetted zone, Q^ is
the initial water content of the profile, and r is the depth of rainfall
applied. This relation holds only if 9U and 9^ are constant with depth, and
normally they are not. The level of water content reached in the wetted zone
during a rain event will depend on the intensity of the rain — the higher
the intensity, the higher the water content.
In the simulations conducted in this study a rainfall of 2.3 cm pene-
trated 20 cm into coarse tailings (c.f. Fig. 33) and a rain of 5.5 cm pene-
trated about 18 cm into medium tailings. The initial water content of the
latter was lower than that of the coarse tailings. In view of the character-
istics of the rainfall at a location such as the Vitro Mill site, Salt Lake
City, Utah, where rains of greater than 2 cm occur several times a year,
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and rains of 5 cm or more very infrequently, the depth of wetting during
rainfall infiltration will usually be less than 20 cm.
Layers of finer textured material at depths greater than about 30 cm
will have no appreciable effect on the dynamic transient behavior of the
water content profile above that depth.
Evaporation at the surface of a profile following a rain event will dry
only the upper 5 to 10 cm of the profile even when the evaporation period
extends to 15 or more days. If the preceding rain has penetrated beyond
this depth the water in the wetted zone will continue to move downward and
eventually moves to the water table.
The long term (e.g. annual) average flux in the lower part of the pro-
file, which is in the quasi-steady state condition, will be downward and its
magnitude will be between zero and R/At, where R is the amount of rainfall
in the averaging time interval At.
The water content distribution in the lower part of the profile when
it is in quasi-steady state will be determined by the texture of the material
in that region, the magnitude of the average downward flux, and the proximity
to the water table. At depths below approximately 70-90 cm and at distances
more than 50-70 cm above a water table the water content of such a profile
approximates that corresponding to a hydraulic conductivity equal to the
magnitude of the downward flux. For example, for a downward average flow
rate of .05 cm/day (about 10 cm/year), the water content of coarse tailings
would be about 0.3, that of the fine tailings about 0.5 to 0.6.
Under limited rainfall conditions, such as at the Vitro Mill site in
Salt Lake City, any significant vegetative cover on the tailings pile would
use all the available precipitation, leaving little or no water to flow
below the root zone to greater depths.
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SECTION 3
RECOMMENDATIONS
Many aspects of water movement in tailings profiles could not be
adequately explored within the scope of the project. The simulations and
analyses should be extended to include a series of rainfall and evaporation
events with a fuller exploration of the effects of various rainfall intensi-
ties and magnitudes on water flow within the profile. The extraction of
water by plant root systems should be included in the simulations and its
effect on the behavior of water in the profile determined. A more extensive
and systematic examination should be made of the effect of profile layering
on the water content distribution in the upper 30 to 60 cm of the tailings
profile. Another factor that should be explored Is the effect of water table
depth on the flow in the profile, including the case of a water table at
great depth or the "no water table" case.
A logical extension of the present work on simulation of water flow in
tailings profiles would be the combination of a model for radon transport in
the profile with the water transport model. Such a combination could provide
a method of systematic examination of the effect of various profile layering
sequences on radon transport to the pile surface. Also, a combination of the
soil water flow model with a model for transport of dissolved chemical
species would seem to be a possible method of examining transport of contam-
inants to the ground water.
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SECTION 4
SOIL WATER FLOW CONCEPTS AND TERMINOLOGY
The salient features of the theory of flow of water in unsaturated
porous materials, such as soils, will be outlined in order to provide the
reader, who may be unfamiliar with this subject, with the essential concepts
and definitions of terms required for the subsequent discussion and analysis
of flow systems. For further, more detailed, background on the subject the
reader is referred to the cited literature (Childs 1969, Klute 1973, Swartz-
endruber 1969). Selected terms used in this report are defined in the
Glossary of this report.
Water movement in soil can occur in the liquid phase or the gas phase.
In each phase water may be transported by convective motion or by diffusion
processes. The convective motion of a phase is driven by mechanical forces
such as gravity and pressure gradients. Movement by diffusion within a phase
occurs in response to gradients of the thermodynamic chemical potential of
the water within the phase. In many cases diffusion may be considered to be
driven by the concentration gradients of the water.
In a water saturated medium the water transport is of necessity solely
in the liquid phase and is dominantly a process of convective transport
driven by mechanical forces. These forces are given by the negative gradient
of a potential function. A commonly used potential function is the energy
per unit weight of soil water or the hydraulic head, H. The hydraulic head
is made up of two components, the gravitational head and the pressure head.
The gravitational head of the soil water at a point in the soil is the
elevation of that point above an arbitrarily chosen reference and is a
measure of the gravitational potential energy per unit weight of the soil
water at a given point in the gravity force field of the earth. In saturated
soils the pressure head, h, is related to the pressure in the soil water by
the relation:
P -P
h = w atm (1)
d g
where d is the soil water density, g is the acceleration of gravity, P is
the pressure of the soil water and Patm is atmospheric pressure. The W
hydraulic head and its components may be measured in those cases where P >
patm with a Piezometer, which is a cased well, usually of small diameter™
terminated at the point in the soil at which the head is to be measured.
Fig. 1 shows a piezometer, and the interpretation from it, of the hydraulic
head and its components. In reference to a piezometer, the hydraulic head
is defined as the elevation of the free water surface in the piezometer tube
relative to an arbitrarily chosen reference elevation.
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SOIL-WATER SYSTEMS
Saturated
Piezometer
Unsaturated
Tensiometer
Porous cup
Reference level
Fig. 1. A diagram of the relationships between hydraulic
head, H, pressure head, h, and gravitational head, Z. The
pressure head is measured from the level of termination of
the piezometer or tensiometer in the soil to the water
level in the manometer and is negative in the unsaturated
soil.
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In unsaturated soils the pressure in the liquid phase PW is usually
less than atmospheric, and the piezometer cannot be used to measure the
hydraulic head. A. tensiometer, which consists of a porous permeable membrane
or cup connected to a manometer or other pressure measuring device (See Fig.
1), is used to extend the measurement of hydraulic head to unsaturated soils.
The pressure of the water inside the "cup" when the water in the tensiometer
is in equilibrium with the soil water is then used in equation (1) to obtain
the pressure head. The pressure that must be applied to the tensiometer
water to establish equilibrium between it and the soil water is determined
by the water content of the soil which in turn is related to factors such as
the curvature and surface tension of the air water interface within the soil
water system, and the magnitude and extent of the adsorptive forces between
the soil solid matrix and the soil water.
The equation of soil water transport, which was used in the simulations
and analyses described in this report, is developed by combining a mass bal-
ance for the soil water with a flux equation. The mass balance equation
relates the time rate of change of water content of a differential volume
element of soil to the excess of inflow over outflow from the element and to
the time rate of production of soil water within the element. If it is
assumed that the soil water is of constant density, the mass balance may be
written as a volume balance, which for one dimensional flow becomes:
£--£*« (2>
In equation (2) 9 is the volumetric water content, z is the vertical space
coordinate taken as positive upward, t is the time, Q is the soil water flux
and S is a source term or the time rate of production of soil water per unit
soil volume. The latter may be used to represent the extraction of water
from soil by plant roots in which case the production rate is generally nega-
tive. Equation (2) expresses the fact that there are two reasons for the
change in water content of a differential soil volume element at a point in
the flow system, (1) the excess of inflow to the element over outflow from
the element, which is mathematically described by - 9Q/(Dz), and (2) the
production (positive or negative) of water by any processes or reactions in
the soil within the element. In the simulations described in this report we
have not considered extraction of water by plant roots, and have set the
source function S equal to zero.
The soil water flux is assumed to be given by the Darcy equation which
is the product of the hydraulic conductivity, k, and a driving force express-
ed as the negative gradient of the hydraulic head. For a saturated soil the
hydraulic conductivity is a maximum for the given soil. As the water content
of the soil decreases from saturation the ability.of the soil to transmit
water, which is measured by the hydraulic conductivity, decreases very
strongly. The Darcy equation has been extended to unsaturated soils by con-
sidering the conductivity to be a function of the water content. The primary
reasons for the decrease of K as 9 decreases are (1) the decrease in effec-
tive cross section of the soil which is available for liquid flow, (2) the
drainage of water from the largest pores, which contribute strongly to the
conductivity, and a decrease in the average size of the liquid-filled flow
-------
channels and, (3) the increase in tortuosity (crookedness) of the liquid
flow path.
In one dimension the Darcy equation or flux equation may be written as:
3H
Q - - K(6)ff (3)
where K(6) is the hydraulic conductivity water content function. The con-
ductivity is a positive quantity, and since the hydraulic gradient may in a
given case be either positive or negative, the flux Q may be either positive
or negative. The algebraic sign of the flux indicates the direction of the
flow with respect to the chosen coordinate axis, a positive flux being in
the positive z axis direction, and vice versa for a negative flux. Since
the hydraulic head is the sum of the pressure and gravitational heads, equa-
tion (3) may be written:
Q - - K (e)<|£+ 1) (4)
where 3h/3z is the pressure head gradient. Combining equation (4) with (2)
with S = 0 yields:
+ 1) ) (5)
dt dZ dZ
Equation (5) is a partial differential equation in two dependent variables,
6 and h. In order to obtain an equation for water flow in one dependent
variable, another relation between 6 and h is required. This relationship
is the water retention function, 8(h). The schematic form of this function
is shown in Fig. 2. In general, the water content decreases as the pressure
head decreases (becomes more negative). The retention function displays a
phenomenon called hysteresis, i.e., the water content at a given pressure
head depends on the historical sequence of pressure heads which has been
imposed on the soil. If the flow process is entirely one of drainage from
saturation or of wetting from a given initial water content, the relation
between 9 and h is single valued and non-hysteretic. If, however, the flow
process involves sequences of wetting and drying the water content-pressure
head relation is multi-valued and hysteretic. There is an infinite number
of possible 6(h) relationships or scanning curves lying between the initial
drainage curve and the main wetting curve (See Fig. 2). The scanning curve
to be applied at any time and place in a flow process depends on the sequence
of reversal points that have occurred at that place. A reversal point is
described by the paired values of pressure head and water content (such as
nl» ®1 or ^2» ®2 *n ^-8' 2} at which a reversal in the time-trend of water
content change has taken place. In non-hysteretic flow simulation the func-
tion 9(h) is single valued and may easily be represented empirically. In
hysteretic flow simulation the 6(h) relation is much more complicated because
some method must be devised to keep account of the necessary reversal points
and select the appropriate scanning curve. We have devised procedures to
treat the hysteresis in the 6(h) relation that can be used in the numerical
solution process for the flow equation to be described in another section of
this report.
-------
(-) PRESSURE HEAD
Fig. 2. Diagram of the hysteresis in the water content-pressure head relation, 0(h).
IDC - initial drainage curve. MDC - main drainage curve. MWC- main wetting curve.
a and b - primary wetting scanning curves, c - primary drying scanning curve. Higher
order (e.g., secondary, etc.) are possible and reverse from lower order scanning curves.
-------
Assuming that proper mathematical description of the 6(h) relation and
its hysteresis is available, the time derivative term in equation (5) can be
transformed to
li _ !§. lh _ _3h
9t ~ dh 3t ~ 9t (6)
In equation (6) the water capacity function, C, is the rate of change of
water content with pressure head, d6/dh, and is the derivative of the appro-
priate applicable drying or wetting scanning curve of the 9(h) relationship.
The water capacity is a function of the water content. It may be seen from
the general shape of the 6(h) function (Fig. 2) that the water capacity will
have a maximum at some pressure head less than zero.
Using equation (6) in (5) yields
C (9)f = fe(K(9)(f +1) > <7>
which is the one-dimensional soil water flow equation used as a basis for
the simulations described in this report. Particular solutions of equation
(7), subject to given initial and boundary conditions, predict and describe
the dependence of the pressure head, h, upon position and time. An initial
condition is a mathematical statement of the spatial distribution of pressure
head at the time arbitrarily designated as zero time. Boundary conditions
are mathematical statements that specify either the pressure head or the soil
water flux at the boundaries of the flow region. The functions C(0) and K(0)
are the hydraulic functions or properties of the soil in the region of flow,
and are the avenue by which the soil affects the flow behavior. A solution
of equation (7), viz, the function h(z, t), may be used to obtain the water
content position-time function, 6(z, t), by using the water retention func-
tion 9(h). The soil water flux distribution, Q(z, t), may also be obtained
by using h(z, t) in the Darcy law, equation (4). The assumptions implicit
in equation (7) have been summarized by Klute (1973). One of these assump-
tions is that the air phase is at atmospheric pressure, and has essentially
a zero pressure gradient. This assumption is sometimes not well satisfied
in practice. For various reasons it is not possible to give a complete as-
sessment of the errors introduced by this assumption. Some qualitative
statements may be made, however. Air pressure build up may occur during
rapid infiltration from ponded water into a soil profile bounded at its
lower end by a water table or a soil layer of low air permeability. The
qualitative effect in this case is to decrease the rate of infiltration and
the extent of penetration of the infiltrating water. Conversely, drainage
of a layered soil profile, when air access to the soil layers is restricted,
will not occur as rapidly as would be predicted from theory based on the
assumption of constant atmospheric pressure within the gas phase. In general,
a more correct approach to water flow in unsaturated soils would involve a
two-phase flow treatment, and a flow equation for each phase (gas and water)
would have to be solved. Such an approach is inherently more complex than
the one-phase approach used in this study. We do not feel that the errors
introduced in the analyses described herein due to neglect of the air phase
flow are extremely significant. In the infiltration cases that were studied
herein, no ponding occurred on the soil surface, hence, the situation of a
10
-------
pressure build-up in a gas phase trapped between a water table and overlying
ponded water could not occur. Simulations of drainage of an initially satu-
rated layered soil column may be significantly in error when a fine textured
layer overlies a coarser textured layer and air must enter the coarse layer
through the overlying wetter fine material, which may have a low air perme-
ability. The error would be greatest during the initial (first one or two
days) rapid drainage of the layered profile. The water content of a coarse
layer under a finer textured layer would be higher than predicted by the
method of simulation used in this study. However, after a few days, there
is an increased probability of leakage or access of air to the buried coarse
layers, and furthermore, the rate of flow and drainage would be decreasing.
Both of these factors suggest that the error due to neglect of air flow would
decrease in significance with time of drainage.
The conductivity water content relationship K(6) has been found not to
be very much hysteretic. In many cases the hysteresis in K(6) if it exists
is hidden within the uncertainty of determination of the relationship. For
the purpose of the analyses described in this report, it was assumed that
K(6) is not hysteretic. However, the K(h) function will be hysteretic be-
cause of the hysteresis in the 9(h) function.
11
-------
SECTION 5
DETERMINATION OF THE HYDRAULIC PROPERTIES OF TAILINGS
SAMPLING AND BASIC PHYSICAL CHARACTERISTICS
Upon consultation with representatives of the Office of Radiation
Programs, U. S. Environmental Protection Agency, Las Vegas, Nevada, the
tailings pile at the Vitro Mill site in Salt Lake City, Utah was selected
as a test area. A visit to the site was made in April, 1976, to become famil-
iar with the general features and characteristics of tailings materials.
Three size fractions of tailings material were selected for characterization.
These are hereafter designated as Coarse, Medium and Fine. A second trip to
the site was made in June, 1976, to collect bulk samples of these fractions.
Due to the generally single-grain structure of the tailings material in situ,
it was decided to use core samples which were packed in the laboratory from
bulk material for the determination of hydraulic properties. A limited
number of core samples was taken at the site for the purpose of establishing
the general level of in situ bulk density of the selected tailings fractions.
These results are summarized in Table 1.
TABLE 1. IN SITU BULK DENSITY AND PARTICLE DENSITY OF THE
SELECTED FRACTIONS OF MILL TAILINGS. VITRO MILL SITE
Fraction Bulk Density Particle Density
Coarse
Medium
Fine
g/cm
1.48 to 1.57
1.28 to 1.50
0.9 to 1.1
g/cnH
2.70
2.71
2.89
12
-------
The bulk samples were packed in drums, with plastic liners, and
transported to the laboratory. Particle size distributions were determined
by sieving and by the hydrometer method (Day, 1965). Table 2 lists the sum-
mation percentage for each of the fractions. The particle density of each
fraction was determined by the picnometer method (Blake, 1965). The results
are shown in Table 1.
TABLE 2. PARTICLE SIZE DISTRIBUTION OF MILL
TAILINGS FRACTIONS SUMMATION PERCENTAGES^'
I/
Particle
Diameter
1000 pm
500
250
125
63
48
25
16
9
7
3
Coarse
92 %
76
41
22
15
10
7.5
5
3.4
-
-
Tailings Fraction
Medium
98 %
95
81
55
33
24.4
4.2
2.1
1.3
-
-
Fine
-
-
-
100 %
95
92
84
80
57
42
5.9
— Percent of the fraction with particle diameters less than or equal to
the indicated diameter.
WATER RETENTION CHARACTERISTICS
The 0(h) relationship for each fraction was determined using a combina-
tion of pressure chamber apparatus or suction apparatus (Rose, 1966; Taylor
and Ashcroft, 1972). Fig. 3 illustrates the pressure apparatus. The suction
apparatus was similar, except that the chamber enclosing the samples was
maintained at atmospheric pressure, and suction was applied to the samples
by lowering the water bottle relative to the porous plate to an appropriate
distance. Generally, pressure chamber apparatus was used to obtain data in
13
-------
To Regulated Air Pressure
(Pressure Chamber
. I
Air Trap
Porous Ceramic Plate
Plastic Screen 4 4
Roller
Rubber Backing Sheet
Water-filled Tubing
Fig. 3 Diagram of the pressure chamber apparatus used to determine the water retention
characteristics of the tailings materials.
-------
the range of pressure heads less than -60 cm of water, and suction apparatus
for h > -60 cm.
Sample Preparation
It was desired to pack core samples to the field value of bulk density.
The following procedure was used for the Coarse and Medium fractions. A bulk
quantity of a given fraction was thoroughly mixed with sufficient water to
bring it to a water content of 6 - 10% by weight, and placed in a plastic
bag to prevent loss of water. Sub-samples were taken to determine the actual
weight percentage of water. The weight of moist material to be packed in a
sample ring (volume 48.3 cm3, height 2.5 cm) to obtain the desired bulk
density was calculated. A hydraulic press was used to pack the desired
weight of moist tailings material into the sample rings. Cores were packed
to bulk densities of 1.53 and 1.47 g/cm3 for the Coarse and Medium fractions,
respectively.
Cores of the Fine fraction were prepared by a filtration procedure, in
which a filter cake of the material was formed in the sample ring by suction
filtration from a slurry of the Fine material. Cores with bulk densities in
the range 0.95 to 0.99 g/cm3 were obtained by this technique.
After packing the cores, a piece of cheesecloth held in place by a
rubber band, was placed over the lower end of the core to prevent loss of
material from the core in subsequent handling.
Method for Determination of Hysteresis
In the simulation procedure the main drainage and main wetting curves
for each fraction were required. Because any rewet curve of 6(h) starting
at pressure heads less (i.e., more negative) than -7000 cm does not differ
much from the rewet curve starting at -7000 cm, it was arbitrarily decided
to determine the "main" wetting curve starting at h = -7000 cm and neglect
any hysteresis at lower pressure heads. Pressure heads of -2, -7.5, -15,
-30, -60, -120, -250, -500, -1000, -3000, and -7000 cms of water were select-
ed as the standard points of measurement on the 6(h) curves.
To determine the main wetting curve (MWC) beginning at h = -7000 cm, a
large number of core samples (30 or more) was packed and soaked in water for
at least 24 hours. They were then placed on a porous plate in a pressure
chamber at a gas phase pressure of 7000 cm of water (corresponding to h =
-7000 cm) and brought to hydraulic equilibrium. Equilibrium was determined
by removal of the cores from the porous plate and weighing them. When the
weight of the core remained constant (within approximately .05 gm, corres-
ponding to an uncertainty in 9 of about 0.001) the cores were distributed to
a series of pressure chambers at gas phase pressures of 3000, 1000, 500, 250,
and 120 cms of water and to suction plates at pressure heads of -60, -30,
-15, -7.5 and -2 cms of water, and allowed to take up water until they were
at equilibrium. Gas bubbles tend to accumulate in the water beneath the
porous plate and in the connecting tubing to the water bottle. These bubbles
may block the flow of water to the ceramic plate and the core samples.
Periodically, the gas bubbles were removed to the air trap by closing
15
-------
stopcock S~ (See Fig. 3) and circulating the water in the closed loop
consisting of the air trap, the space below the ceramic plate and the connect-
ing tubing, using a roller on one of the tubes. This modification of the
apparatus allowed us to determine the wetting 9 (h) curve.
After equilibrium was reached the water content of each core was deter-
mined by standard gravimetric procedures. The data thus obtained were used
to develop the MWC.
The main drying curve MDC was determined by preparing a large number of
cores, which were soaked for 24 hours or more, equilibrated in a pressure
apparatus at h = -7000 cm, and then on a suction apparatus at h = -2 cm.
This sequence was necessary to establish the rewet saturation at h = -2 cm
so that the main drainage curve would be properly determined. After equili-
bration at h = -2 cm, the cores were distributed to a series of suction
plates at pressure heads of -7.5, -15, -30, and -60 cms of water, and to a
series of pressure chambers at pressure heads of -120, -250, -500, -1000,
-3000, and -7000 cms of water. After equilibration the water content of
each core was obtained. The data thus obtained were used to develop the MDC.
The water content-pressure head relation for the Coarse and Medium frac-
tions was determined in the range of pressure heads less than -7000 cm water
(-7 bars) by a vapor pressure technique (Croney, et al., 1952). Samples of
the materials were equilibrated in vacuum dessicators over saturated salt
solutions with relative humidities corresponding to water potentials of -71,
-145, -290, -900, and -1,568 bars. The weight percentage water contents were
determined at equilibrium and converted to the volumetric basis by multiply-
ing by the bulk density of the cores packed for the 0(h) determination at
pressure heads greater than -7 bars. Due to experimental difficulties (e.g.
shrinkage at high suction) with the Fine fraction of tailings, it was decided
to limit this study to simulations where the Fine fraction would not be
exposed at the profile surface, and therefore, retention data at pressure
heads less than -7 bars would not be needed for this fraction.
The water retention data obtained from the measurements described above
were represented by empirical functions for use in the numerical simulation.
In the range of pressure head, -7000 cm <_ h <_ o, the following function was
used (Gillham, et al., 1976):
where
3 = (h/h )b b < o (9)
o
r = (eo - er) / (10)
This empirical function for 0(h) has four parameters, hQ, b, 9Q, and Q^,
which were selected to fit the function to the measured main wetting curve
(MWC) and main drying curve (MDC) data. These parameters were selected with
16
-------
T - 1 - 1
1 - 1 - 1
-f
1 - 1 - 1
COARSE TAILINGS
Bulk Density 1.53 g/cc
Total Porosity 0.433
1 - 1
1 - 1 — r
IDC
I
I
0.5
0.4
0.3
M
CJ
s
E9
0.2 8
0.1
-10*
-1000 -100 -50 -20 -10 -5
PRESSURE HEAD, CM
-2 -r
Fig. 4,
Water retention functions for the Coarse tailings. Each point is an average
of 3 to 15 determinations. The vertical lines represent jK 1 standard
deviation from the mean. The curves are the empirical functions used in
the simulations of water flow. The IDC was calculated from the MDC by the
theory of Mualem,
-------
00
I 1 1
I 1 1
r 1 1
MEDIUM TAILINGS
Bulk Density 1.47 g/cc
Total Porosity 0.458
-10*
-1000 -100 -50
PRESSURE HEAD, CM
-20 -10 -5
g
O
-2 -1
Fig. 5.
Water retention functions for the Medium tailings. Each point is an average
of 6 to 9 individual measurements. The vertical lines represent + 1
standard deviation from the mean. The curves are the empirical functions
used in the simulations of water flow. The IDC was calculated from the MDC
by the theory of Mualem.
-------
T
T 1 T
T
T 1 T
FINE TAILINGS
Bulk Density 0.986 g/cc
Total Porosity 0.66
IDC & MDC
t L
J L-
-10*
_L
JL
_L
_L
_L
0.7
0.6
M
O
0.4 §
0.3
-1000 -100 -50
PRESSURE HEAD, CM
-20 -10 -5
-2 -1
Fig. 6.
Water retention functions for the Fine tailings. Each point is an average
of 4 to 12 individual determinations. The vertical lines represent + 1
standard deviation from the mean. The solid lines are the empirical functions
used in the simulations of water flow.
-------
the constraint that the MDC and MWC should intersect at (&it h-^), with h±
about -7000 cm. The empirical functions for the MWC and MDC were used to
generate the scanning curves interior to the MWC and MDC. A theory developed
and tested by Mualem (1974) was used for this purpose. The initial drainage
curve (IDC) was generated by his theory from the MDC. Scanning curves interi-
or to the IDC and MWC were also obtained by his theory. Mualem's theory pro-
vides a reasonably accurate method of obtaining scanning curves, which is
suited to use in the numerical simulation process, and greatly reduces the
necessary amount of experimental measurement required.
At pressure heads more negative than about -7000 cm the 9(h) data were
represented by the empirical form:
0(h) = &a (ID
d
where a and d are parameters which were selected to fit the data, and with
the constraint that the function of Eq. (11) should pass through the inter-
section point (Gj, hj) of the MDC and MWC (see above).
Figs. 4, 5 and 6 show the empirical 9(h) functions obtained for the
Coarse, Medium, and Fine fractions, respectively. The solid curves in these
figures are plots of Eq. (8) subject to Eqs. (9) and (10). The dashed line
in Figs. 4 and 5 is a plot of Eq. (11). The parameters of the functions are
given in Tables 3, 4 and 5.
HYDRAULIC CONDUCTIVITY - WATER CONTENT RELATIONSHIPS
The measurement of the K(9) relation may be accomplished by a number of
methods (Klute, 1972). All of the available methods require a substantial
amount of time and in some cases the methods are rather difficult. Because
of these and other difficulties encountered in the direct measurement of the
K(6) function there has been considerable interest in schemes for the calcu-
lation of hydraulic conductivity from pore size distribution data (Jackson,
et al., 1965; Kunze, et al., 1968). The latter may be obtained from the
water content - pressure head relation.
Lalibertie, et al. (1968) described a method of using water content-
pressure head data to calculate the conductivity-water content function.
These authors developed an integral equation for the conductivity, which may
be written as:
9 - 9 .
r
/
J
In Eq. (12), p is the density of the soil water, n is its viscosity, a is its
surface tension, g is the acceleration of gravity, and 9g and 9 are param-
eters of the 9(h) function. To evaluate the integral, and obtain K(9), a
functional relation between h and 9 is needed.
20
-------
TABLE 3. PARAMETERS OF THE EMPIRICAL HYDRAULIC
FUNCTIONS OF THE COARSE TAILINGS
Function and
Parameter
Equation No. Symbol
Water Retention
Eq. (10)
Eq. (9)
Eq. (11)
Intersection Point
Hydraulic Conductivity
Eq. (18)
IDC
6 0.433
6r 0.076
h0, cm - 131
b -0.468
d, cm
a
h.£, cm
ei
A, cm/ day
B
Value
MDC
0.385
0.0763
- 131
-0.468
-1.71 x 10~2
-0.1964
-7320
0.07842
5.40 x 10~8
54.29
MWC
0.385
0.0776
- 67.4
-0.504
TABLE 4. PARAMETERS OF THE EMPIRICAL HYDRAULIC
FUNCTIONS OF THE MEDIUM TAILINGS
Function and
Parameter
Equation No. Symbol
Water Retention
Eq. (10)
Eq. (9)
Eq. (11)
Intersection Point
Hydraulic Conductivity
Eq. (18)
IDC
60 0.458
er 0.090
h0> cm - 281
b -0.451
d, cm
a
h.£ , cm
ei
A, cm/ day
B
Value
MDC
0.401
0.0959
- 281
-0.451
-1.102 x ID'2
-0.1749
-7121
-.09589
1.99 x 10~8
50.0
MWC
0.401
0.0939
- 166
-0.503
21
-------
TABLE 5. PARAMETERS OF THE EMPIRICAL HYDRAULIC
FUNCTIONS FOR THE FINE TAILINGS
Function and
Equation No.
Water Retention
Eq. (10)
Eq. (9)
Eq. (11)
Symbol
9o
e
r
h , cm
b
d, cm
a
Parameter
Value
IDC & MDC
0.66
0.31
-2500
-0.28
—
^^
MWC
0.66
0.35
-400
-0.28
Intersection Point
Hydraulic Conductivity
Eq. .(18)
cm
9i
-6982
0.3727
A, cm/day
B
5.38 x 10
45.1
-12
In application of most of the theories which have been developed to
calculate K(9) from 9(h), it has been found that the calculated K(9) function
can be brought into reasonable agreement with measured K(6) data by the use
of a matching factor. The matching factor M is defined as:
-ft]
(13)
where KM is a measured conductivity at some known water content, Qit and KC
is the calculated conductivity at that water content. If Kc(9) is the calcu-
lated conductivity function, the matched, calculated function ^(9) is
obtained from
Kmc(9)
M K (9)
c
(14)
Nielsen, et al. (1973) conclude that the procedure of calculating K(6) from
9(h), with a matching factor, is a satisfactory method of evaluating the
hydraulic conductivity of a field soil, especially when one considers the
spatial variability of the hydraulic conductivity.
To avoid the time consuming procedure of determining K(9) by one of the
available methods of direct measurement (Klute, 1972), and in view of
Nielsen's conclusion cited above, as well as the experience of others, I
decided to calculate K(9) from Eq. (12) and use a matching factor to adjust
the calculated function as described above. For this purpose, the h(6)
function obtained by solving equation (8) (with Eqs. 9 and 10) for h was
used in the integral of Eq. (12). The result is:
22
-------
c
K_(9) = TE7TT—o l«—z-) I (In (v + (v-lV) d9 (15)
where
e + e (e - e )
v = r (_2 ) r = _o r'
7 VV
-------
used in the 6(h) determinations. Evaporation from the upper end was allowed
to proceed. The bottom end of the column was closed. The water content as
a function of depth in each column was measured at a series of times over a
15 to 30 day period, by gamma attenuation (Gardner, 1965). The soil water
flux Q as a function of depth and time was calculated from the continuity
equation for soil water (Eq. (2) ). Integration of the continuity equation,
with S = 0, yields:
z
o
Q(Z) = Q(Z ) - [(e(z,t2) - e(z,tl)]dz (16)
where Q(z) is the average flux at depth z for the time interval At = t£ - ti,
Q(z0) is the same for depth zo, and 6(z,t2) and 0(z,t1) are the measured
water content profiles in the drying column at times t£ and t-^. The flux at
a depth z may be calculated by Eq. (16) if the flux at some depth z? is known.
Since the bottom of the drying column was closed, the flux there is zero,
and was chosen as the value of Q(zQ). Repeated application of Eq. (16) was
used to develop numerical values of Q(z,t), the flux-depth-time function,
for the drying column.
The water content gradient 30/(3z) = G(z,t) as a function of depth and
time was also calculated from the measured water content profile data.
The soil water diffusivity D(0) (Klute, 1972) was then calculated from
the ratio of soil water flux to water content gradient:
DUO - - an
Eq. (17) is derived from Darcy's law, Eq. (3), by assuming a single valued
functional relation between 0 and h exists (as would be the case in the dry-
ing column), and neglecting the gravitational driving force for the flow.
The latter is a reasonable approximation at the low water contents involved
in the drying column. The diffusivity function D(z,t) obtained from Eq. (17)
was converted to a diffusivity water content function, D(0), by using the
measured 9(z,t) data.
The conductivity-water content relation was calculated from the soil
water diffusivity using the relation K(0) = D(6)C(0). The water capacity
function C(6) = d6/(dh), in the range of water contents found in the drying
columns, was calculated from the measurements of 0(h) as described earlier
in this section.
Figs. 7, 8, and 9 show the hydraulic conductivity-water content relation
for each of the tailings fractions. The data were represented by an empiri-
cal function of the form
K = AeB6 (18)
where A and B are parameters. The solid lines in these figures are plots of
the empirical function for each material. The parameter values are also
given in Tables 3, 4 and 5.
24
-------
10
I I I I
Matching Point
COARSE TAILINGS
K - Ae
A = 5.4 x 10'" cm/day
B » 54.3
Matched, calculated conductivity
A Conductivity by drying method.
10
Fig. 7.
0.2 0.3 0.4
VOLUMETRIC WATER CONTENT
Hydraulic conductivity - water content function for
Coarse tailings.
25
-------
A « 1.99 x 10 cm/day
-|- Matched, calculated conductivity.
A Conductivity by drying method.
1
0.1 0.2 0.3 0.4
VOLUMETRIC WATER CONTENT
0.5
0.6
Fig. 8. Hydraulic conductivity - water content function for
Medium tailings.
26
-------
103
10
I ID'1
-2
io
iio-3
o
o
I"-4
10"
10
,-6
FINE TAILINGS
Matching Point
K - Ae
BO
A - 5.38 x 10~12 cm/day
B - 45.1
I
I
I
I
I
0.2
0.6
0.7
0.3 0.4 0.5
VOLUMETRIC WATER CONTENT
Fig. 9, Hydraulic conductivity - water content function for
the Fine tailings.
27
-------
SECTION 6
SIMULATION OF WATER FLOW IN TAILINGS PROFILES
THE NUMERICAL SOLUTION PROCESS
The flow of water in vertical profiles of tailings is in principle
described by solutions of the equation of flow Eq. (7). This equation
is a partial differential equation and has an infinite number of solutions.
A particular solution which describes the flow in a specific profile is
obtained by imposing boundary conditions and an initial condition. In addi-
tion, the hydraulic functions K (8) and C (6), which characterize a given
tailing material, must be specified. If the profile is non-uniform, e.g.,
is made up of layers, the hydraulic functions appropriate to each layer must
be specified. In the case of vertical one-dimensional flow, two boundary
conditions are needed, one at each end of the flow region. These conditions
are mathematical statements of the time dependence of either the pressure
head h, or the flux Q at the ends or boundaries of the flow region. The
initial condition is a statement of the spatial distribution of pressure
head at the time selected as time zero.
The equation of flow is non-linear due to the dependence of K and C
upon 6 and hence in turn upon h. Analytic solutions are not available except
for idealized situations, such as constant initial pressure head, constant
boundary pressure heads, semi infinite regions of flow, non-hysteretic flow,
or specific types of conductivity and water retention functions. These
idealized conditions are not descriptive of the cases to be examined in this
study. We want to deal with non-uniform profiles (layered), time dependent
boundary conditions, and hysteretic flow regimes. In such cases the solu-
tions to equation (7) must be obtained by numerical methods, utilizing a
computer.
The partial differential equation of flow is solved numerically by
replacing it with a set of algebraic equations. The space and time deriva-
tives are approximated by finite difference equations. We desire a solution
to equation (7), which is a function h(z, t) for the space-time domain
described by - L < z < o and t > o . To obtain finite difference approxima-
tions to the derivatives in equation (7) the space-time domain of the solu-
tion is divided into increments At and Az by imposing a grid on the region
(See Fig. 10). The profile of length L is divided into a set of intervals
Az which are, in general, of non-uniform size. The time domain t > o is
divided into increments At, which are also of non-uniform size. The inter-
section points of the grid or nodes are identified by indices n and m where
n is the space index and m is the time index. The pressure head and other
28
-------
z=0
z=-l_
m=0
n=N
N<4 ,
V
V,
n+i '
n-1 '
X,
X.
2,
1
1
n o
x x,
x N,
X X,
X X,
/ / n
\\
\\ 1
//
X X,
X X,
// 1
n
t t
) f
"x. V
X Xx
; ;
I f ( 1
1 > >
I ITU
k. x,
X X,
•x. x.
X "^
1-1
x x^
•v x*
-x V,
X. V
^-
»—
x.
X.
X
X
Fig. 10. Diagram showing the division of the space-time
domain into increments, and the numbering of the nodes
for the identification of variables in the numerical
solution process.
29
-------
quantities at each node are identified by these indices. For example the
pressure head at node n, m is denoted as ^ The initial condition'for a
given solution is specified by the set of nodal values of pressure head, h
i.e., m=oando_ o . In the Crank-Nicolson finite
difference scheme used in this study (von Rosenberg, 1969), six nodal values
of pressure head are required. The general case in the interior of the flow
region is shown in Fig. 11. The approximation to the time derivative in
Eq. (7) is:
C — « C n. m + 1 " n. m
9t n, m + % At
Where Cn, m + h ' (Cn, m + Cn, m + l)/2. A function F is defined as;
rill
L9z
F = K(J + 1) so that K ( + 1) ) = and
_ 3h 9F
C 3t = 3t (20)
Then the r.h. side of equation (7), which is 3F/9z, is approximated by:
F
-21 * n + *• m + % ' n m -r ^ n - *g. m + * (21)
0 O
-2* (r + r2)
In Eq. (21):
r = Azu/Az£ (22)
F - v i /3h\
n + Jj, m + Jj~Kn + %, m + J ' (hn, m +1 + h
n, m
(26)
30
-------
At
m+1
n+1
•n+l,nr
n-1 ,m
n-U.nrt-r
n ,nrf 1"
Fig. a Expanded view of the nodes about the general
node n,m, showing the identification of the nodal
values of pressure head. Other variables are
subscripted in the same way.
31
-------
(—)
az'm-Js.m+Jg
To develop the algebraic equations, which replace the partial differential
equation (7), Eqs. (26), 27) and (28) are substituted into Eqs. (23) (24) and
(25) respectively. The results are then substituted into Eq. (21), and then
used to replace the right hand side of Eq. (20). Eq. (19) is used to replace
the left hand side of Eq. (20). The terms in the resulting equation are then
regrouped so that all pressure heads at the m+1 time level appear on the left
side of the equation. The result of all these manipulations is a set of
algebraic equations of the form:
- An hn+l,m+l + Bn hn,m+l ' Cn hn-l,m+l = Dn <29>
for o <_ n £ N, 0 _< m. The unknowns in Eq. (29) are the pressure heads at
the m+1 time level. The coefficients An, B , C and Dn are functions of the
«oa8™". ?8 at th^ m tlme leve1' and the average conductivities and aver-
and Cn m+W' These average
(Kn,m + Kn,m+l)
In these equations, the conductivities and water capacities at the mfl time
level are functions of the pressure heads at the m+1 level, which are the
unknowns in Eq. (29). Consequently, Eq. (29) represents a set of non-linear
algebraic equations in the unknowns hn ^ for o £ n < N. The solution pro-
cess uses the initial condition, h 'to solve for the set tu , , i.e., the
pressure head at the end of the first time step. The set h_ T 'is then used
in turn to solve for hn 2< The solution process thus "marches forward" in
time by increments At. 'At each time step the set of non-linear algebraic
equations represented by Eq. (19) is solved by an iterative process. The
solution to the original equation of flow (Eq. (7) ) is thus approximated by
a set of nodal values of pressure head, h^ for o < n < N, and o < m.
The necessary values of conductivity and water capacity were evaluated
from the empirical functions that were established for each tailings material
fraction (See Section 5). The water content at each node 8 was obtained
from hn>m using the empirical water retention function, 9(h)! The flux at
32
-------
each of the nodes, Q , was calculated from a finite difference approximation
of the Darcy equation;1^. (4), in which hfl m and 6n>m was used.
The computer program for the solution process contains several signifi-
cant features:
(a) The size of the time step At is under program control and is
determined by the change in water content across the time step.
(b) The space grid Az may be of variable size and distributed in an
arbitrary manner. This allows the use of a fine grid in those portions of
the flow region known to contain steep gradients.
(c) Provision is made to sense the presence of hysteresis (i.e.,
reversals in the sign of 96/3t) and to utilize the appropriate scanning
curve in the 0(h) relationship to determine the water content from the
pressure head and to determine the water capacity. All scanning curves are
calculated from the main drainage and main wetting curves of the 9(h) rela-
tion, by a theory developed by Y. Mualem (1974). This process also generates
scanning curves between the main drying curve and the initial drying curve.
(d) The boundary conditions on the flow may be time dependent, and may
be switched from potential to flux type and vice versa during the coarse of
the solution process.
(e) The hydraulic properties assigned to each node are arbitrary
allowing the representation of non-uniform media.
33
-------
SECTION 7
ANALYSIS OF WATER FLOW IN TAILINGS PROFILES
THE SEQUENCE OF FLOW EVENTS
Two general types of tailings profile situations may be identified,
one with a water table at relatively shallow depth - perhaps less than 4
or 5 meters - and located within the tailings materials, and a second with a
water table at great depth and located below the tailings material. The
Vitro mill site is an example of the first situation, and in this study
we have focused on it. A water table depth of 3 meters was chosen for
the simulations.
In a profile of tailings without vegetation, the general sequence
of water flow events may be described as follows:
1. An initial period of drainage from complete saturation, with
evaporation at the surface.
2. Infiltration of varying amounts of rain at irregular intervals.
3. Periods of evaporation and drainage from the profile and
redistribution of water within the profile between rainfall
events.
In view of the above sequence, numerical simulations of flow were
made (1) for drainage to a water table, with and without evaporation at
the surface, (2) for infiltration of rainfall, and (3) for evaporation,
drainage and distribution following infiltration events.
DRAINAGE OF INITIALLY SATURATED PROFILES
Although uniform profiles of tailings do not occur, it is desirable
to simulate drainage of such profiles for background purposes. The
drainage of uniform profiles of each of the tailings fractions, with a
water table at 300 cm depth, and with and without evaporation at upper
end, was simulated. A water table at the lower end of the profile is
represented by a zero pressure head at that boundary.
The surface condition of no evaporation is simulated by an imposed
hydraulic gradient of zero at the soil surface. Evaporation at the soil
surface was simulated by imposing a constant positive flux upper boundary
condition. The constant rate of evaporation was imposed as long as the
pressure head at the soil surface was greater than - 15,000 cm (15 bars
suction). When the soil surface pressure head decreased to - 15,000 cm,
it was subsequently held at that value. This procedure was followed in
order to simulate the "energy-limited" and "soil-limited" phases of
34
-------
evaporation. The evaporation rate from a very wet soil is determined
primarily by the energy available for vaporization of the water - the
"energy-limited" evaporation regime. As the soil surface dries, a point is
reached when the soil cannot supply water to the surface sufficiently fast
to maintain the energy-limited rate, and a "soil-limited" rate of evaporation
occurs. The latter is less than the energy-limited rate. Sometimes these
phases of evaporation are called the constant rate and falling rate periods.
Use of these terms assumes that the energy for evaporation is supplied at an
essentially constant rate. Due to the diurnal fluctuations of energy
exchange at the soil surface the energy available for evaporation of water
undergoes corresponding diurnal fluctuations. Because of the time-span
of the evaporation events (about 15 days) we decided not to simulate the
diurnal fluctuations, and a constant rate was imposed.
The choice of -15,000 cm as the pressure head for switching from a
flux to a potential type upper boundary condition is arbitrary. It is
felt that the value chosen is reasonable and that another, perhaps more
negative choice, would not change the results significantly.
The initial condition chosen for the drainage simulations was that
corresponding to saturated downward flow with barely ponded water at the
soil surface. In this case, the average hydraulic gradient initially
applied to the profile is approximately unity. In a uniform profile of
constant conductivity the pressure head would be essentially zero through-
out the profile.
Drainage Without Evaporation.
A summary of conditions for the simulations of drainage without
evaporation at the surface is given in Table 6. The water content profiles
at various times for draining uniform 300 cm profiles of the Coarse and Fine
tailings fractions, without evaporation at the top t are shown in Figures
12 and 13. The long time limit (t •*• °°) of these profiles is that correspond-
ing to static equilibrium, with flux Q equal to zero and the pressure head
at a given depth equal to the negative of the elevation of that depth above
the water table. Water content profiles during drainage of a uniform Medium
profile were qualitatively similar to those for the Coarse profile and are
not shown. Figures 14 and 15 show the time dependence of water content, 6,
at the depths indicated. The equilibrium limits of water content for each
depth are shown by the short dashes at the right of the graph.
The water flux across the water table versus time is shown in Figures
16 and 17. Figure 16 shows the flux during the early period of drainage
(0-1 day) and Figure 17 during the later period (up to 30 days) of drainage.
Also included is the flux versus time for a draining column of Medium
tailings. The flux across the water table is downward and since it is in
the negative z axis direction, it is negative. At the onset of drainage
35
-------
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5
100
200
300
Fig. 12. Water content distributions at
several times during the drainage
to a water table at 300 cm of a
uniform Coarse tailings profile
without evaporation at the surface.
The numbers on the curves are the
time of drainage in days.
VOL. WATER CONTENT
0.3
0.4
100
200
00
300
Fig. 13,
0.0
Water content distributions at
several times during the drainage
to a water table at 300 cm of a
uniform Fine tailings profile
without evaporation at the surface.
The numbers on the curves are the
time of drainage in days.
-------
Water table at 300 cm.
OJ
0
15
5 10
TIME, DAYS
Fig. 14. Water content versus time at the depths
Indicated In a uniform Coarse profile drain-
ing without evaporation at the top.
Water table at 300 cm.
0 5 10
TIME,DAYS
Fig. 15. Same as Fig. 14, except for a uniform
Fine tailings profile.
37
-------
TIME, DAYS
0.5
Water table at 300 cm
1.0
-300
Fig. 16. Flux across the water table (0-1 day) from
uniform profiles of Coarse, Medium and Fine tailings
draining without evaporation at the surface.
TIME, DAYS
-12
-16
Fig. 17. Same as Fig. 16, except 0 to 30 days,
38
-------
TABLE 6
SUMMARY OF CONDITIONS FOR SIMULATIONS OF
DRAINAGE WITHOUT EVAPORATION
Profile: Uniform, either Coarse,
Medium or Fine tailings.
Upper Boundary Condition: No evaporation, zero flux for
t > 0.
Lower Boundary Condition: Water table, h = 0 for t > 0
at a depth of 300 cm.
Initial Condition: Saturated, downward flow with
barely ponded water at surface,
unit hydraulic gradient; zero
pressure head gradient.
the magnitude of the flux is equal to the saturated conductivity of the
profile. Thus in the case of the draining Coarse profile, the initial flux
was -874 cm/day, in the Medium profile it was -175 cm/day, and in the
Fine it was -45.5 cm/day. Within a short time (1 day or less) after the
initiation of drainage the flux decreases in magnitude and approaches zero
asymptotically.
The simulations of drainage of uniform 300 cm columns of each of the
tailings materials without evaporation were continued until the static
equilibrium condition was reached. From the results, the time of drainage
required to reach equilibrium was estimated. These times were 100-150 days,
1000-2000 days, and 10000 days or more for the Fine, Medium and Coarse
fractions, respectively. The times are indefinite because of the asymptotic
approach toward equilibrium. The order of time to drainage to equilibrium
may seem contrary to what might be expected. However, these results are
consistent with experimental observations on draining soil columns. The
Coarse fraction does drain very rapidly at first. After a very rapid period
of initial drainage the water content in the profile is reduced and con-
sequently the hydraulic conductivity. The latter part of the drainage of
the Coarse fraction then must occur at a low level of conductivity, and
the drainage rate is very low. Observation of the water content in the
profile of Coarse material during this period might lead one to conclude
that the profile was at equilibrium due to the slow rate of change. However,
in terms of the potential of the water in the profile, equilibrium has not
been reached, and will only slowly be approached.
In the Fine profile the initial drainage rate is lower than in the
Medium or Coarse profiles. The water content of the Fine material decreases
relatively less than it does in the Coarse profile. This means that the
decrease in conductivity during drainage of the Fine profile will be less
39
-------
than the conductivity decrease in the Coarse material. For example, the
upper part of the Fine column profile drains to a volumetric water content
of about 0.52, which corresponds to a conductivity of about 6 x 10~2 cm/day
(See Fig. 9). The water content upper part of the Coarse profile decreases
to about 0.12, corresponding to a conductivity of about 2 x 10"5 cm/day.
Thus the latter part of the drainage of the Coarse profile functions at a
much lower level of conductivity than is found in the drainage Fine profile,
and consequently, the Coarse profile is slow to reach equilibrium.
Drainage with Evaporation.
A summary of conditions for the simulations is given in Table 7.
Figs. 18 and 19 show the water content profiles at various times during
the drainage of uniform profiles at Coarse and Medium tailings when a
potential evaporative flux of 1 cm/day was imposed at the soil surface. The
most notable difference between these profiles and those for drainage without
evaporation is the very pronounced drying in the surface 5 to 10 cm.
(Compare Figs. 12 and 18 for the Coarse tailings). The water content
versus time at selected depths in these profiles is shown in Figs. 20
and 21. The surface (depth 0 cm) dries quickly until the pressure head
at the surface reached the chosen value of -15,000 cm for switching from
a constant flux upper boundary condition to a constant potential upper
boundary condition. Examination of the water content-depth curves in
Figs. 12 and 18, and the water content-time curves in Figs. 14 and 20
shows that the influence of the evaporation at the surface on the 8-profile
in the Coarse material, does not extend beyond 90 cm depth and perhaps not
that far. The same was true of the Medium profiles. Simulations of
TABLE 7
SUMMARY OF CONDITIONS FOR SIMULATIONS OF
DRAINAGE WITH EVAPORATION
Profile: Uniform, either Coarse, or
Medium tailings.
Upper Boundary Condition: Potential evaporation rate of
1 cm/day until pressure head
at the surface decreased to
-15,000 cm. Thereafter, a
constant pressure head of
-15,000 cm.
Lower Boundary Condition: Water table, h = 0 at a depth
of 300 cm for t > 0.
Initial Condition: Saturated, downward flow, with
barely ponded water at the sur-
face; zero pressure head gradi-
ent.
40
-------
0
100
w
200
300
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5
. Water table at 300 en
t I t
Fig. 18. Water content distributions at several
times in a uniform Coarse tailings
profile draining to a water table
with a 1 cm/day potential evaporation
rate at the surface. Numbers on the
curves are the time in days. The profile
labeled oo is the steady state limit of
such a flow.
0
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5
100
3
w
200
300
0.026-
Water table at 300 cm.
l i i
Fig. 19. Same as Fig. 18, except for a
uniform Medium tailings profile.
-------
Water table at 300 cm.
5 10 15
TIME, DAYS
Fig. 20. Water content versus tirae at the indicated depths in
the uniform Coarse tailings profile of Fig. 18.
Water table at 300 cm.
132 cm
5 10 15
TIME, DAYS
Fig. 21. Water content versus time at the indicated depths in
the uniform Medium tailings profile of Fig. 19.
42
-------
drainage with evaporation at the surface for the Fine material were not
done due to the limitations of time for this study. The water content
profiles in the Fine material draining with evaporation at the surface
would be quite similar to those shown in Fig. 13 below a depth of approx-
imately 70 to 90 cms. At shallower depths the evaporation at the surface
would cause a rapid decrease of water content, which would be qualitatively
similar to that displayed in the upper portion of the Coarse and Medium
profiles. The water content at the surface in the Fine material would
become equal to that corresponding to a pressure head of -15,000 cm, which
is about 0.34.
If the upper boundary pressure head condition of -15,000 cm were
to be imposed for a sufficiently long time (and with a water table at the
bottom of the profile), the long-time limit of the flow in these profiles
would be a condition of steady state with an upward flow from the water
table. The water content profile labled t -*• » in Figs. 18 and 19 are
those corresponding to this steady flow. In this condition the flux Q
would be constant and in the upward direction. The values of Q shown in
Figs. 18 and 19 are these steady state upward flow values. These water
content profiles at t -»• °° represent a lower bound on the water content
distribution that might be found in the profile. Steady state upward
flow will not be attained, because of the rainfall events that occur at
irregular intervals. These infiltration events will keep the upper part
of the profile wetter than that labeled t •*• °°.
The flux versus depth in the Coarse and Medium profiles draining with
evaporation at the surface is shown in Figs. 22 and 23. The plane of zero
flux, above which the flow is upward (flux positive) and below which the
flow is downward (flux negative), moves downward into the profile as time
increases. The position of the plane of zero flux is another measure of the
depth of influence of the evaporative condition at the surface. At about
5 days the zero flux plane had moved to a depth of about 10-15 cm, which
illustrates that the influence of the evaporation at the surface on the
water behavior in the profile is confined to depths less than perhaps
30 cm at drainage times less than 15 to 30 days.
The linearity of flux versus depth over most of the profile shows
that the time rate of change of water content is essentially the same at
all depths. This follows from the continuity equation for the water, viz.:
li - _ 12
3t 3z
The flux across the upper end of the profiles is shown in Fig. 24.
The imposed 1 cm/day evaporation is sustained for only a short time -
about 0.3 day for the Coarse profile and 0.6 day for the Medium profile.
After this time the flux decreases with time asymptotically toward the very
low flux value for a steady upward flow with a low pressure head (e.g.,
-15 000cm) at the soil surface. This flux is about 0.00039 cm/day for the
Coarse and 0.0059 cm/day for the Medium tailings.
43
-------
Fig. 22. Flux versus depth in the upper 30 cm of the
uniform Coarse profile of Fig. 18.
Fig. 23. Flux versus depth in the upper 30 cm of the
uniform Medium tailings profile of Fig. 19.
44
-------
1.0
0.8 -
Potential Evaporation Rate
0.2 .
TIME, DAYS
Fig. 24. Flux across the surface (actual evaporation
rate) of the uniform Coarse and Medium profiles of
Figs. 18 and 19, draining to a water table with a
potential evaporation of 1 cm/day imposed at the surface.
During the 15 day drainage and evaporation period the evaporative
losses were 1.6 and 2.1 cm for the Coarse and Medium profiles, respectively.
During this same period the drainage losses were 51.6 and 37.3 cms for
these two profiles. The small evaporative loss compared to the drainage
loss indicates further that the evaporation has little effect on the
behavior of the water content profiles except at very shallow depths.
Drainage of Layered Profiles with Evaporation.
The drainage of two 300 cm profiles each with two layers of tailings
material was simulated. The first, designated C/M, had a 150 cm Coarse layer
over a 150 cm Medium layer. The second, designated C/F, had a 150 cm Coarse
layer over a 150 cm Fine layer. Each was subjected to a potential evaporation
rate of 1 cm/day at the surface.
C/M. Conditions for this simulation are summarized in Table 8. The
water content profiles, and pressure head profiles at several times during
the drainage are shown in Figs. 25 and 26. Within the Coarse layer above
110 cm depth the water content profile at a given time is the same as that
obtained for a uniform Coarse profile, i.e., the Medium layer below 150
cm depth did not significantly affect the drainage of the upper part of
the Coarse layer. For example, the profiles at 5.1 days above 100 cm
depth in the uniform Coarse profile (Fig. 18) and in the C/M profile
45
-------
0
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5
0.091 days
0.32
5.1
15.1
-180
PRESSURE HEAD, CM
-120 -60
300
Fig. 25. Water content distributions at various
times in a two-layer, Coarse over
Medium, profile draining to a water
table at 300 cm and with a 1 cm/day
potential evaporation rate at the
surface.
0.091 days
0.32
5.1
15.1
-100
8
I - 200
300
Fig. 26. Pressure head distributions
for the profile of Fig. 25.
-------
TABLE 8
SUMMARY OF CONDITIONS FOR SIMULATION
OF DRAINAGE OF A TWO-LAYER, COARSE OVER MEDIUM
(C/M) PROFILE
Profile: Coarse tailings, 0 to 150 cm
depth. Medium tailings, 150 to
300 cm depth.
Upper Boundary Condition: Potential evaporation rate of
1 cm/day until pressure head at
the surface decreased to -15,000
cm. Thereafter, a constant pressure
head of -15,000 cm.
Lower Boundary Condition: Water table, h=0 at a depth of
300 cm, t > 0.
Initial Condition: Saturated. Downward flow with
barely ponded water at the surface.
(Fig. 25) are essentially identical. There was, however, a zone in the
Coarse layer between 115 and 150 cm depth just above the Medium material,
which drained more than the material above it. This occurs because of the
development of pressure heads in this zone which are more negative than
those found at somewhat shallower depths in the Coarse material (See Fig.
26).
It is required by the physical laws of soil water flow that the pressure
head be continuous across the boundary between layers of differing texture.
Since the layers have different water retention properties (different
values of 9 at a given h) there will be a discontinuity in the water content
profile at the boundary between the layers. This is displayed in Fig. 25.
The time dependence of the flux at the surface in the C/M profile was
the same as that for the uniform Coarse profile (See Fig. 24). This is
to be expected in view of the fact that the water content distributions in
the upper part (depth less than 100 cm) for the C/M and uniform Coarse
profiles are the same. The cumulative evaporation in 15 days was consequently
the same for both columns (1.6 cm).
The flux across the water table in the C/M profile is shown in Figs. 27
and 28. The initial drainage rate of the profile is determined by the
conductivity of the lower layer - in this case 175.5 cm/day (See Fig. 27).
After a short time the drainage rate from the C/M profile becomes essentially
the same as that from a uniform Coarse profile.
47
-------
TIME, DAYS
Water table at 300 cm.
1 cm/day evaporation (potential)
rate at the surface.
-300
Fig. 27.
Flux across the water table (0-1 day) for
the draining profiles of uniform Coarse tailings (Fie 18)
Coarse over Medium (Fig. 24), and Coarse over Fine
tailings (Fig. 29).
TIME, DAYS
I
o-
§
Fig. 28. Same as Fig. 27, except 0 to 15 day
48
-------
C/F. Conditions for the simulation are summarized In Table 9. Water
content profiles and pressure head profiles for the drainage, with 1 cm/day
evaporation at the surface of this profile, are shown in Figs. 29 and 30.
As in the C/M profile, the water content profiles above a depth of about
110 cm in the Coarse layer are the same as in a uniform Coarse profile. A
zone just above the Fine layer is initially wetter than the same depths in
the uniform Coarse profile but at later times drains faster. The discon-
tinuity in the 6-profiles is more pronounced in the C/F profile than in the
case of the C/M profile. This is because of the greater disparity between
the water retention properties of the Coarse and Fine layers.
The time-dependence of the evaporation rate at the surface was the
same as that for the uniform Coarse profile (See Fig. 24), hence the total
evaporation in the 15-day period will be the same, viz, about 1.6 cm.
TABLE 9
SUMMARY OF CONDITIONS FOR SIMULATION
OF DRAINAGE OF A TWO-LAYER
COARSE OVER FINE PROFILE
Profile: Coarse tailings 0 to 150 cm depth
Fine tailings 150 to 300 cm depth
Upper Boundary Condition: Potential evaporation rate of 1 cm/day
until the pressure head at the surface
decreases to -15,000 cm. Thereafter,
a pressure head of -15,000 cm.
Lower Boundary Condition: Water table, h=0, at 300 cm depth.
Initial Condition: Saturated. Downward flow.
The flux at the water table, shown in Figs. 27 and 28, is initially
equal to the negative of the saturated conductivity of the lower Fine
layer (- 47 cm/day). After a short time (about 0.2 day) the drainage rate
becomes the same as that for a uniform Coarse profile. After about 1 day
the drainage rate is somewhat less than that for either the uniform Coarse
or C/M profile (Fig. 28).
49
-------
0
VOL. WATER CONTENT
Q 0.1 0.2 0.3 0.4 0.5 0.6 0.7
100 -
Ln
O
200 -
300
Fig. 29,
PRESSURE HEAD,CM
- 100
200
300
Water content distributions at
various times in a two-layer
Coarse over Fine profile draining
to a water table at 300 cm and with
a potential evaporation rate of 1
cm/day at the surface.
Fig. 30. Pressure head distributions for
the profile of Fig. 29.
-------
INFILTRATION EVENTS
The characteristics (duration, intensity, frequency, etc.) of the rains,
to which a particular tailings pile is subjected, will depend on its geo-
graphic location. With particular reference to the Vitro mill site, hourly
rainfall data from the Salt Lake City, Utah weather station were examined
in order to develop some idea of the nature of the rainfall in that area,
and to provide guidance in the selection of the upper boundary condition
for simulation of rainfall infiltration. Examination of the data for the
years 1966 through 1970 showed the following characteristics for the rain-
fall: There were 456 rainfall events in the 5 year period. Thus the average
spacing of events was about 4 days. The spacing between events was most
often less than two days (273 out of 456 cases). In 10 cases out of the
456 the spacing was 15 or more days. Rains of 2 cm or more occurred 14
times. Most of the rains (322) delivered less than 0.5 cm. The average
annual rainfall at the Salt Lake station is about 20 cm. In the 5 year
period there was one rainfall that delivered about 6 cm of rain.
Rainfall infiltration events were simulated on two 3 meter profiles,
a uniform Coarse profile, and a layered profile, with 30 cm of Medium
tailings on 30 cm of Fine tailings which was in turn over Medium tailings
from 60 to 300 in depth. This profile will be designated hereafter by
M/F/M.
Infiltration into a Uniform Coarse Profile.
The initial condition for this simulation was the 6 and h profile
resulting from 15 days of drainage with 1 cm/day evaporation at the surface.
In the simulation of rainfall infiltration, a specified flux is imposed
as the upper boundary condition. This flux is downward, in the negative
z-axis direction, and hence is negative, and equal in magnitude to the
rainfall rate. The flux may be time dependent. As long as the pressure
head at the surface of the profile is less than zero, the flux condition
is maintained. If the pressure head at the surface becomes equal to or
greater than zero, the upper boundary condition is switched to a potential
condition with a pressure head equal to or greater than zero. This procedure
simulates the ponding of water on the soil surface that may occur under
rainfall. Conditions for the simulation of infiltration into the uniform
Coarse tailings profile are summarized in Table 10.
The time dependence of flux at the profile surface, which is the
negative of the rainfall rate, is shown in Figure 31. The total amount of
rain applied in the event was 2.28 cm with a peak intensity of 11.3 cm/day,
and an average intensity of 6.4 cm/day. The intensity during the last half
of the event was 7.5 to 8 cm/day.
Figure 32 shows the water content profiles at various times during
the rainfall infiltration event. The total depth of penetration of the
rain into the tailings profile was less than 20 cm as indicated by the
water content profiles. Fig. 33 shows the water content distributions in
the top 30 cm. Corresponding to these 9 profiles is the set of pressure
head profiles shown in Fig. 34. These profiles display the characteristic
51
-------
1
8
,15.0
-4
-8
-121
15.1
TIME, DAYS
15.2
15.3
15.4
Fig. 31. Flux across the surface (negative of the
rainfall rate ) of the uniform Coarse profile of Fig. 32.
TABLE 10
SUMMARY OF CONDITIONS FOR SIMULATION
OF INFILTRATION INTO A UNIFORM COARSE PROFILE
Profile:
Upper Boundary Condition:
Lower Boundary Condition:
Initial Condition:
Uniform coarse tailings.
Time dependent rainfall rate
simulated by a time dependent
negative flux at the surface
(shown in Fig. 31). Total of
2.28 cm of rain applied.
Water table h = 0 at 300 cm depth.
Water content and pressure head
profiles resulting from 15 days
of drainage with a potential
evaporation rate of 1 cm/day at
the surface. (See Table 7, and
Fig. 18).
52
-------
WATER CONTENT
) o.i 0.2 0.3 Q.A n.s
100
u
200
300
Fig. 32. Water content distributions
in the uniform Coarse profile subjected
to the upper flux shown in Fig. 31
(2.3 cm rain). Times shown are from
the start of the initial drainage of
the profile.
infiltration wetting front. The water content and pressure head at the
surface approach values corresponding to a hydraulic conductivity equal to
the rainfall rate, or the negative of the surface flux. The pressure head
and water content decrease slowly with depth until the wetting front is
reached.
Figure 35 shows the soil water flux versus depth at various times,
some of which correspond to the water content profiles of Fig. 33. The flux
profile at 15.1 days corresponds closely to flie peak in the rainfall rate
(See Fig. 31). Since the lower part of the profile is still draining, below
the position of the wetting front the flux is downward, and hence negative,
and essentially zero in magnitude. As the rainfall rate decreased from its
peak value to the general level of 7.5 cm/day the flux at the surface
shifted accordingly.
53
-------
10
I
20
VOL. WATER CONTENT
0.1 0.2 0.3
0.4
Initial Condition
15.346 days
30
Fig. 33. Detail of water content distributions
in the upper 30 cm of the uniform
Coarse tailings profile of Fig. 32.
-180
PRESSURE HEAD, CM
-120 -60
Initial Condition
Fig. 34. Pressure head distributions
corresponding to the water
content distributions of
Fig. 33.
-------
FLUX, Q, CM/DAY
I
30
Fig. 35. Flux versus depth in the upper
30 cm of the profile of Fig. 32 at several
times during the 2.3 cm rainfall event.
Infiltration into the M/F/M Profile.
The initial condition for this simulation was the steady state pressure
head profile corresponding to upward flow from a water table at a depth of
300 cm and a surface pressure head of - 15,000 cm. Two rainfall-infiltration
events were considered, one in which 6 cm of rain was applied and another
in which 12 cm was applied. The conditions for these simulations are
summarized in Table 11.
The flux conditions imposed at the surface for these events are shown
in Fig. 36. Both rain events had the same intensity-time characteristic,
with the 6 cm rain terminating sooner than the 12 cm rain. The peak
rainfall intensity was about 47 cm/day, at 0.09 day after the initiation of
the rain. In the latter part of each rain the intensity was approximately
13-14 cm/day. These rains represent extreme events when considered in
relation to the Salt Lake precipitation data. In the rainfall data for
Salt Lake City there were no single events in which as much as 12 cm of
rain was obtained. There was one event in which about 6 cm was obtained.
55
-------
TABLE 11
SUMMARY OF CONDITIONS FOR SIMULATION OF
INFILTRATION OF 6 AND 12 CM RAINFALL
INTO THE M/F/M PROFILE
Profile:
Upper Boundary Condition:
Lower Boundary Condition:
Initial Condition:
Medium tailings, 0 to 30 cm
depth. Fine tailings, 30
to 60 cm depth. Medium tail-
ings, 60 to 300 cm depth.
Time-dependent rainfall rate
simulated by a time-dependent
flux at the surface (shown in
Fig. 35). Two rainfall
events, a 6 cm rain, and a 12
cm rain.
Water table, h
depth.
0, at 300 cm
The steady state water content
and pressure head profiles
corresponding to upward flow
from a water table at 300 cm
depth and a surface pressure
head of -15,000 cm.
Water content profiles at a sequence of times during the infiltration
are shown in Fig. 37. The detail of the top 90 cm of the profile is shown
in Fig. 39. The wetting front advances through the 0-30 cm Medium layer,
until at some point between 0.497 and 0.606 days the front enters the Fine
layer. Note the discontinuity in the 0-profiles due to the layering. The
front penetrated to slightly more than 50 cm by the time the rain was ended
at 0.803 days. In Fig. 39 the number in parenthesis on each profile is
the depth in cm of infiltrated water at the time corresponding to that
profile. Thus, e.g., at 0.301 days, 5.5 cm of water had infiltrated and
the penetration into the profile was about 20 cm. Flux profiles corre-
sponding to the water content profiles for this case are shown in Fig. 38.
These flux profiles are qualitatively similar to those shown in Fig. 35
for infiltration into a uniform Coarse profile.
56
-------
TIME, DAYS
1
g
h4
Pn
-20
Fig. 36. Flux across the surface (negative of the
rainfall rate) of the layered (M/F/M) profile.
The depth of penetration of a given rainfall infiltration event will
depend on the amount of water infiltrated and the initial water content
in the profile. For a given amount of rain the greater the initial profile
water content, the greater the depth of penetration. The initial condition
for the infiltration simulation on the uniform Coarse profile is probably
somewhat wetter than would on the average be found at a long time (>2-3 hrs.)
after the initial drainage of the profile. In that simulation about 2.3
cm of rain penetrated 20 cm. If the initial water content profile had been
drier, the depth of penetration would be correspondingly less. The initial
condition for the rainfall infiltration into the M/F/M profile is probably
somewhat drier than the average initial condition that would be found in
situ. In the M/F/M case, 2.75 cm of water penetrated about 10 cm (See
Fig. 39) compared to the 20 cm of penetration for 2.3 cm of water in the
uniform Coarse profile with its higher initial level of water content.
The depth of wetting front penetration, d, of a given amount of rain,
r, may be estimated if we assume that the initial water content 6j_ is
constant with depth, and that the water content above the wetting front
9U is also constant with depth. Then the depth of the wetting front is
given by:
d = r / (6U - 6±)
57
-------
Ul
00
0
100
g 200
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5 0.6
M
Initial Cond]
0.146 days -
0.396
0.606
0.802
1 I
300
Fig. 37. Water content distributions in a
layered (M/F/M) profile at several
times during a 12 cm rain.
-24
FLUX, Q, CM/DAY
-18 -12 -6
0
M
Fig. 38. Flux versus depth at various
times in the upper 90 cm of
the M/F/M profile of Fig. 37.
-------
0
30
13
60
90
.1 0.
WATER CONTENT
0.3 0 4
0.097 da
ys
M
0.7
12.0)
M
Fig. 39. Detail of water content
profiles in the upper 90 cm of the
M/F/M profile of Fig. 37. Numbers in
parentheses are the depths of infiltrated
water at the times corresponding to each
profile.
From the characteristics of the rainfall events at Salt Lake City,
it appears that most of the rain events would not penetrate more than
5 to 10 cm, and only infrequently to depths of 15 to 20 cm. Penetration
in this context is measured in terms of the water content profile. Layers
of Fine material, if present below these depths, would not affect the
penetration of the rain during the event. However they may have an influence
on the redistribution within the profile following the rain.
59
-------
EVAPORATION FOLLOWING A RAIN
Three simulations of an evaporation period after a rain event were
made:
(1) 0.5 cm/day potential evaporation rate imposed for 15 days
on the uniform Coarse railings profile following the 2.28
cm rain event.
(2) 0.5 cm/day potential evaporation rate imposed for 15 days follow-
ing the 12 cm rain on the M/F/M profile.
(3) 0.5 cm/day potential evaporation rate imposed for 60 days
following a 6 cm rain on the M/F/M profile.
Uniform Coarse Profile.
The 9 and h-profiles at the end of the 2.28 cm rain event (t =
15.357 days) were the initial condition for the subsequent evaporation.
The approximate nature of these profiles may be seen in Figs. 33 and 34,
by observing the profiles at 15.346 days. Conditions for the simulation
of evaporation from the uniform Coarse profile are summarized in Table 12.
TABLE 12
SUMMARY OF CONDITIONS FOR SIMULATION
OF EVAPORATION FOLLOWING A RAIN
ON A UNIFORM COARSE TAILINGS PROFILE
Profile:
Upper Boundary Condition:
Lower Boundary Condition:
Initial Condition:
Uniform coarse tailings, 0 to 300 cm
depth.
Potential evaporation rate of 0.5
cm/day simulated by a flux boundary
condition, initial pressure head
the surface decreased to -15,000 cm.
Thereafter, a constant pressure head
of -15,000 cm. Total period of
evaporation, 15 days.
Water table, h=0, at 300 cm depth.
Water content and pressure head
distributions resulting from the
infiltration of 2.28 cm of rain
(See Figs. 33 and 34).
60
-------
Water content profiles during the evaporation are shown in Figs. 40
and 41. While the surface was being dried by evaporation the wetting front
continued to penetrate the column. Eventually the wetted zone at the top
of the profile was removed by a combination of upward flow to the surface
and downward flow toward the water table. Over the 15 day period there was
a further slight drainage of the lower part of the column at depths to
about 270 cm. (See Fig. 40). The profile at 20.8 days shows that the
water content at depths between 35 and 70 cm increased slightly above that
found in this region at the end of the rain (15.357 days), and then decreased
again at later times.
Flux profiles for the 0-30 cm depth are shown in Fig. 42. Immediately
at the end of the rain the plane of zero flux, above which flow is upward
and below which flow is downward, is at the soil surface. At later times
there is a negative peak of flux (downward flow) in the range 15-20 cm,
within the zone wetted by infiltration and below the plane of zero flux.
This peak becomes smaller with the passage of time as the infiltrated water
moves downward.
The flux at the surface as a function of time from the end of the
rain is shown in Fig. 43. The 0.5 cm/day potential evaporation rate was
maintained for only a short time - a little more than a day - at which time
the surface dried to a pressure head of -15,000 cm and the boundary condi-
tion was switched from a flux condition of 0.5 cm/day to a potential condition
of h = -15,000 cm. Subsequent to the switch of boundary condition the flux
at the surface decreases. This behavior of the surface flux is a display of
the classical constant rate and falling rate periods. The total evapora-
tion during the 15.8 day period following the rain was 1.14 cm. In the
same time period the change in storage in the profile calculated from the
water content profiles above a depth of 26.8 cm was 2.41 cm. Since the
change in storage is the sum of the evaporation loss and the total flow
across the depth of 26.8 cm, the latter was 1.27 cm. Since the profile
at 31.1 days shows continued drainage at all depths, all of the 1.30 cm
of water - and more - has flowed down through the profile to the water
table. The general level of water content in this drainage profile at
times of 15 to 30 days after the initiation of drainage from saturation is
still high enough so that the conductivity will allow a sufficient flow
rate to dissipate more than half of the infiltrated water to the water
table. The flow of this fraction of the infiltrated water to the water
table took place with only a very minor increase in water content at depths
of 30 to 60 cms (e.g., see the 9-profile in Fig. 40 at 20.8 days). Below
100 cm depth the water content always decreased with time, while the
infiltrated water was flowing to the water table.
Evaporation from the M/F/M Profile.
Two simulations were made with a 0.5 cm/day potential evaporation
rate imposed on the layered prof ile M/F/M. In the first, the 9 and fa-
profiles at the end of the 12 cm rain event (t = 0.809 days) were the
initial condition for the subsequent evaporation. In the second, a 0.5
cm/day potential evaporation rate was imposed on theprofHe after it had
61
-------
0
100
I
w
200
300
VOL. WATER CONTENT
0.1 0.2 0.3 0.4
0,5
Fig. 40. Water content distributions in a
uniform Coarse tailings profile
subjected to a potential evaporation
rate of 0.5 cm/day after a 2.3 cm
rain.
VOL. WATER CONTENT
0.2 0.3
Fig. 41. Detail of water content distributions
in the upper 30 cm of the uniform
Coarse profile of Fig. 40.
-------
FLUX, Q, CM/DAY
zi.
-2
15.402 days
.10
,20.81 I
.20
.30
Fig. 42. Flux versus depth at various times during
the evaporation from the profile of Fig. 40.
received 6 cm of infiltrated water. The 0 and h profiles at that time
(t = 0.337 days) were used as the initial condition for this evaporation
event.
Evaporation following a 12 cm Rain on the M/F/M Profile. The
conditions for this simulation are summarized in Table 13, The water
content profiles during a 15 day period of evaporation following a 12 cm
rain are shown in Figs. 44 and 45. The profile at 0.82 days is just
shortly after the end of the rain. During the course of the evaporation
there was water movement to greater depths, i.e., redistribution occurred.
There was penetration of the wetting front into the Medium material under-
lying the Fine layer, with a general decrease of water content throughout
the upper Medium layer. The evaporation caused the water content to be
63
-------
0.5
UNIFORM COARSE 2.3 CM RAIN
M/F/M 12 CM RAIN
M/F/M 6 CM RAIN
0.1 -
TIME, DAYS
Fig. 43. Flux across the surface during the evaporation
from the uniform Coarse profile (Fig. 40), M/F/M profile
12 cm rain (Fig. 44), amd M/F/M, 6 cm rain (Fig. 47).
TABLE 13
SUMMARY OF CONDITIONS FOR SIMULATION
OF EVAPORATION FOLLOWING A 12 CM RAIN
ON THE M/F/M PROFILE
Profile:
Upper Boundary Condition
Lower Boundary Condition:
Initial Condition:
Medium tailings, 0-30 cm depth
Fine tailings, 30-60 cm depth
Medium tailings, 60-300 cm depth
Potential evaporation rate of 0.5 cm/
day until pressure head at surface
decreased to -15,000 cm. Thereafter,
a constant pressure head of -15,000 cm.
Total evaporation period, 15 days.
Water table, h=0, at 300 cm depth.
Water content and pressure head
profiles at the end of a 12 cm
rainfall (see Fig. 39).
64
-------
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5 0.6
100
200
300
Fig. 44. Water content distributions in the
1I/F/M profile subjected to a 0.5
cm/day potential evaporation rate
after a 12 cm rain.
VOL. WATER CONTENT
I Oil^ I °l5 l °l6
i y i i i i
Detail of water content distributions
in the upper 90 cm of the M/F/M
profile of Fig. 44, showing the
redistribution of water during the
evaporation period.
-------
reduced most in the upper 3-4 cm. Also, the zone in the Medium layer a
few centimeters dee? just above the Fine layer tended to dry faster than
depths in the middle of the Medium layer, because at these depths just
above the Fine layer, the pressure head is more negative than it is in the
mid-part of the Medium layer. Within the Fine layer, the water content
decreased in the upper 9-10 cm of the layer. Due to the downward redistribu-
tion there was an increase in the water content in the lower 2/3 of the Fine
layer. There was also an increase of water content in the 60-75 cm zone
of the underlying Medium layer.
Flux profiles are shown in Fig. 46 at times corresponding to some
of the 6 profiles of Fig. 45. At early times during the evaporation, there
is a negative "peak" of flux in the general zone of the profile wetted by
the previous rain. This peak is gradually dissipated by downward flow,
and the plane of zero flux moves downward into the profile from the surface.
The net change in storage in the depth interval 0 to 80 cm, as
evaluated from the water content profiles at the end of the rain (t =
.809 days) and at the end of the evaporation period (t = 15.22 days), was
-1.54cm i.e., there was a net decrease of water in this part of the profile.
During this same time period the total evaporation calculated from the time
integral of the evaporation rate was 1.53 cm. Thus the evaporation accounts
for all the decrease in storage and no flow of significance is occurring
across the 80 cm depth.
The flux at a depth of 82 cm, calculated from the pressure head
profiles at various times during the 15 day evaporation and redistribution
period, were all in the range 0.00057 cm/day to 0.00062 cm/day i.e., a small
upward flow was indicated. This flow is probably a "residual" effect from
the initial condition for the rain event which was a steady state profile
with a constant upward flux and with a pressure head of -15,000 cm at the
top of the profile. Assuming that there is an average upward flux of 6 x
10~4 cm/day for 15 days, only 9 x 10~3 cm of water would have flowed upward
across the 82 cm depth. This amount is too small to be reliably detected
by a mass balance on the profile above that depth. The reason these flow
rates are small is due to the low level of water content and consequent low
conductivity at these depths. These low levels of Q are due to the initial
condition chosen for the infiltration, viz. a steady state upward flow
profile with h = -15,000 cm at the top of the profile. As was pointed out
earlier, this initial condition is drier than would probably be found in
profiles in situ.
Within the 0-30 cm upper Medium layer there was a net loss of 4.69 cm
of water. Since the evaporation amounted to 1.53 cm there was a total
downward flow of 3.16 cm in the 15 day period across the upper surface of
the Fine layer. In the upper 10.5 cm of the Fine layer there was a loss of
0.38 cm of water. The lower part of the Fine layer gained 2.04 cm for a
net gain in the entire Fine layer of 1.66 cm. The lower Medium layer in
the zone 60 to 80 cm depth gained 1.49 cm of water.
66
-------
-12
FLUX, CM/DAY
-6
PM
W
Q
M
90
Fig. 46. Flux versus depth at several times
during the evaporation and redistribution within
the M/F/M profile of Fig. 44.
Since 12.1 cm of water entered the profile during the preceeding
rain event, and only 1.53 cm have been lost by evaporation, there has been
a net gain of 10.06 cm of water within the upper 90 cm of the profile. If
the evaporative condition were maintained for a longer time the evaporative
loss would occur at a rate which is about .035 cm/day at 15 days and will
decrease asymptotically toward essentially zero as time increased. (See
Fig. 43). Another 15 days of evaporation would probably account for about
0.5 cm of evaporation. This amount is small compared to the 10.6 cm net
gain of water by the profile at 15 days. It is most likely then that the
10 cm of water will remain within the profile and move slowly toward the
water table. Within a time period of up to 30 days after a rain event
it is fairly likely that there would be another rain event of sufficient
magnitude to cause another net gain of water in the upper part of the
67
-------
profile, thus contributing further to the tendency toward a downward
flow.
Evaporation from the M/F/M Profile after a 6 cm rain. A potential
evaporation rate of 0.5 cm/day was imposed on the layered profile after
approximately 6 cm of rain had infiltrated (t = 0.337 days). The evaporation
lasted about 60 days. Conditions for the simulation are summarized in
Table 14.
TABLE 14
SUMMARY OF CONDITIONS FOR THE SIMULATION
OF EVAPORATION FOLLOWING A
6 CM RAIN ON THE M/F/M PROFILE
Profile:
Upper Boundary Condition:
Lower Boundary Condition:
Initial Condition:
Medium tailings, 0-30 cm depth
Fine tailings, 30-60 cm depth
Medium tailings, 60-300 cm depth
Potential evaporation rate of
0.5 cm/day until pressure head
at surface decreased to -15,000 cm.
Thereafter, a constant pressure head
of -15,000 cm. Total evaporation
period, 60 days.
Water table, h=0, at 300 cm depth.
Water content and pressure head
distributions resulting from
infiltration of 6 cm of rain
on the M/F/M profile.
Water content profiles at selected times during the evaporation
period are shown in Figs. 47 and 48. At the end of the proceeding rain
6.06 cm of water had infiltrated. Fifteen days after the rain, 1.66 cm
of water had been lost by evaporation. There was no significant change of
water content in the Medium layer at 60 cm depth. The 0-30 cm Medium
layer had a net loss of 2.22 cm and the Fine layer a net gain of 0.56 cm,
resulting in a net loss for the profile of 1.66 cm, which matches the
evaporative loss. At this time (15 days) 4.4 cm of the 6.06 cm of
infiltrated water are still within the profile.
68
-------
0
100
200
300
VOL. WATER CONTENT
JK1 0.2 0.3 0.4
J
!•«••.• crLrTI Medium
_JL«t JJl..:
0.377 days
15.1
60.6
Medium
Fig. A7. Water content distributions in the
M/F/M profile with a 0.5 cm/day
potential evaporation rate after
a 6 cm rain.
VOLUMETRIC WATER CONTENT
P 0.1 0.2 0.3 0.4 0.5
0.377 days
0.707
1.46
Fig. 48. Detail of the water content
distributions in the upper
90 era of the M/F/M profile of
Fig. 47.
-------
In the period 15 to 60 days an additional 0.87 of water was lost
by evaporation, for a total evaporative loss (0-60 days) of 2.53 cm. At
60 days there was a slight increase in water content in the 60 to 80 cm
zone. At 60 days, 3.53 cm of the 6.06 cm of infiltrated water were still
in the profile. The 0-30 cm Medium layer contained 2.97 cm of the 3.53
cm. At 82 cm depth the flux was about 0.00069 cm/day (upward). The effect
of this flux on storage above 82 cm in the 60 day period is negligible
(less than .04 cm). It is unlikely that evaporation uninterrupted by a
rain would occur for as long as 60 days in the Salt Lake City environment.
Thus 3.5 to 4 cm of the 6 cm of infiltrated water would continue to flow
downward to the water table.
Flux profiles at times corresponding to some of the 6-profiles of
Fig. 47 are shown in Fig. 49. Immediately after the end of the infiltration
the flux in the upper profile decreases in magnitude and an upward flow
(the evaporation) begins from a region close to the surface. A negative
peak of flux occurs at a depth near the bottom of the zone that was wetted
by infiltration. This peak gradually becomes smaller with increasing time.
The plane of zero flux is above a depth of 20 cm even at 60 days, showing
that the evaporation loss that occurs during this period comes from the
upper 20 cm of the profile.
In the two evaporation simulations from the M/F/M profile, there was
no downward flow of any part of the infiltrated water to the water table.
This is because of the low level of water content at depths of about 90 -
100 cm which was consistent with the initial condition for the 6 and 12 cm
rains. At the low water contents found at these depths the conductivity
is low, and also the hydraulic gradient is acting in a direction to cause a
small upward flow. Consequently there is no drainage to the water table
and the net gain of water to the profile is "trapped" in the upper part of
the profile. Additional rain events will each cause an additional net gain
of water to the profile which will slowly force the water deeper into the
profile.
DISCUSSION OF THE FLOW IN THE TAILINGS PROFILE
After the initial drainage of the saturated pile, the general nature
of the water content behavior in the profile would seem to be as follows:
Each rain event causes an increase in water content in the upper part of
the profile. The level of water content at the surface reached during the
rain depend on the intensity and duration of the rain. If the rain is
very short in duration and of low intensity there will be only a small
increase in water content at the surface and only a small depth of penetra-
tion of the wetted zone. If the rain lasts for more than perhaps 30 minutes
or an hour the water content at the surface tends toward the value of 6
associated with a hydraulic conductivity equal to the rainfall rate. This
is because the hydraulic gradient at the surface tends toward unity, and
thus according to Darcy's law the flow rate at the surface is numerically
equal to the hydraulic conductivity. Examples of this condition are shown
in Figs. 32 and 33 for infiltration into the uniform Coarse profile and
70
-------
FLUX, Q, CM/DAY
-1.0
MEDIUM
FINE
MEDIUM
90
Fig. 49. Flux versus depth in the upper 90 cm of the M/F/M
profile of Fig. 47.
Fig. 37 for the infiltration into the M/F/M profile. The higher the
rainfall rate, the higher the water content at the surface of the profile.
If the rainfall rate exceeds the conductivity at the rewet water content V
of the tailings at the surface, ponded water will develop on the surface.
This will accumulate until runoff occurs. In view of the high hydraulic
conductivities at rewet water content and zero pressure head of the Coarse
\J See definition in Glossary.
71
-------
and Medium fractions (874 and 175 cm/day, respectively) it is unlikely that
ponding will occur if either of these materials is on the surface in depths
greater than 20-25 cm. If the Fine fraction were at or near the surface
(at depths of less than 10-15 cm) it is possible that ponding could occur.
A rainfall rate greater than 45 cm/day would theoretically produce ponding
on the Fine material.
The increase of water content in the upper part of the profile
produced by the rain is transient, and does not last more than a day or so
after the end of the rain. Evaporation removes water from the upper part
of the profile, perhaps the top 10-15 cm and over a period of 15-30 days
following a rain. Downward flow will occur below this depth and reduce the
level of water content in the wetted zone. Examples of this kind of behavior
are shown in Fig. 40 for evaporation from a uniform Coarse profile, in
Fig. 44 for evaporation from the M/F/M profile following a rain.
The Development of the Quasi-Steady State Profile
Some fraction of each of the rainfall amounts applied to the tailings
profile is retained in the profile and does not evaporate during the
evaporation period following the rain. If the amount of applied rain is
very small (a few tenths of a centimeter or less) probably all of it is
soon returned to the atmosphere by evaporation. Larger amounts of rain will
have a larger fraction of the water retained in the profile, which is then
available to move slowly downward to the water table. For example, of the
2.3 cm of infiltrated water on the uniform Coarse profile (See Figs. 32, 33,
34, and 35), about 1.1 cm evaporated in a 15 day evaporation period following
the rain. The rest drained to lower depths in the profile. In the case of
the 12 cm rain on the M/F/M profile, about 10 cm of water were still in
the profile after a 15 day period of evaporation. This water is available
to move slowly downward in the profile.
Each rain and evaporation event produces fluctuations in the water
content and flux in the upper part of the profile. The amplitude of
these fluctuations decreases with depth (Klute and Heermann, 1974). At a
sufficiently great depth - perhaps 1 meter - the flow condition approaches
a quasi-steady state downward flow. The flow rate associated with this
condition averaged over time periods longer than a few days - perhaps over
a month or a year - could be estimated if we knew the fraction of each rain
event that was retained in the profile during the time period taken for
averaging. If the amount of rain in a given period At is R, and if f is the
fraction of this amount retained in the profile, the magnitude of the average
downward flux |Qd j in the lower part of the quasi-steady state profile
would be
M -
72
-------
Since the wetting front associated with most rain infiltration events
does not penetrate below 20-30 cms, layering in the profile below this depth
would not seem to have any effect on the fraction of the rains "trapped"
in the profile. Consequently, the value of |Q
-------
VOL, WATER CONTENT
10Q.
200-
300
Fig. 50. Water content distributions
for two states of steady flow in a
uniform Medium tailings profile.
layers is much higher than that of the Medium layers due to the water
retention capability of the Fine fraction, and the presence of these Fine
layers will cause the average level of water content in the profile to be
increased.
The effect of water extraction by vegetation on the profile distribu-
tion of water was not simulated in this study due to the lack of sufficient
time to do so. However, several comments about the probable effect of
extraction of water by roots of vegetation may be made. In a low rainfall
environment such as the Vitro site, the vegetation will use all the water
that falls as precipitation. This means that transpiration losses added
to soil surfaceevaporation losses will usually return all the infiltrated
water to the atmosphere, and little water will flow to the water table.
74
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Ui
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5 0.6
100
200
U=__^—-M_—•-———w-i————•
I I I II I I I
I J Medium
to • ^*«^- «k.V^ 4Bfc Mb. «. 4» * «b 4»*h* 4» *
Fine I \
Q - 0
Medium
Q - -0.54 cm/day
300 ~
Fig. 51,
Water content distributions for
two states of steady flow in a
profile of Medium tailings with
a Fine layer at 30 to 60 cm depth,
0
VOL. WATER CONTENT
0.1 0.2 0.3 0.4 0.5 0.6
100
200
I ii n i i i
H I _ ijm _ f _ _ __ •
a
~r —. • — — .«,». ^_X.
_tl
- M Q = 0
M
- F
-_ IL
\f Q =-0.054 cm/day
\:"".
300
Fig. 52.
Water content distributions for
two states of steady flow in a
profile composed of alternating
30 cm thick layers of Medium
and Fine tailings.
-------
Some rain events may be of sufficient magnitude to allow a portion of the
rain to flow to the water table simply because evapotranspiration at its
prevailing rate cannot intercept all the infiltrated water before some
of its gets below the root zone to a water table. It appears that,
generally, vegetation would tend to keep the upper part of the profile
at a lower water content than would be found without vegetation.
Within the time available for the present study, the simulations
and analysis conducted did not show the full development of the quasi-
steady condition in the lower profile. Neither were we able to fully
examine the effect of layering and water extraction by vegetation on the
water regime in the tailings profiles.
76
-------
REFERENCES
Blake, G. R. 1965. Particle density. In Methods of Soil Analysis, Part I.,
Monograph No. 9, Am. Soc. of Agron., C. A. Black, et al. (Eds.),
Ch. 43, pp. 371-373, Madison, Wisconsin.
Childs, E. C. 1969. The Physical Basis of Soil Water Phenomena. John
Wiley and Sons., New York, N. Y.
Croney, D., Coleman, J. D. and Bridge, P. M. 1952. The suction of moisture
held in soil and other porous materials. Road Research Technical
Paper No. 24, Road Research Laboratory, Dept. of Sci. and Industrial
Research, London.
Day, P. R. 1965. Particle fractionation and particle-size analysis.
In Methods of Soil Analysis, Part I., Monograph No. 9, Am. Soc. of
Agron., C. A. Black, et al. (Eds.), Ch. 43, pp. 545-566, Madison,
Wisconsin.
Gardner, W. H. 1965. Water content. In Methods of Soil Analysis, Part I,
Monograph No. 9, Ch. 7, pp. 82-125, C. A. Black, et al. (Eds.),
Am. Soc. of Agron., Madison, Wisconsin.
Gillham, R. W., Klute, A., and Heennann, D. F. 1976. Hydraulic properties
of a porous medium: measurement and empirical representation.
Soil Sci. Soc. Am. J. 40:203-207.
Jackson, R. D., Reginato, R. J. and von Bavel, C. H. M. 1965. Comparison
of measured and calculated hydraulic conductivities of unsaturated
soils. Water Resources Res. 1:375-380.
Klute, A. 1973. Soil water flow theory and its application in field
situations. In Field Soil Water Regime, R. R. Bruce et al. (Eds.),
Special Publication No. 5, Soil Sci. Soc. Am., Madison, Wisconsin.
Klute, A. 1972. The determination of the hydraulic conductivity and
diffusivity of unsaturated soils. Soil Sci. 113:264-276.
Klute, A. and Heermann, D. F. 1974. Soil water profile development under
a periodic boundary condition. Soil Sci. 117:265-271.
Kunze, R. J., Uehara, G. and Graham, K. 1968. Factors important in the
calculation of hydraulic conductivity. Soil Sci. Soc. Am. Proc.
32:760-765.
77
-------
Lallberte, G. E., Brooks, Royal H., and Corey, Arthur T. 1968. Perme-
ability calculated from desaturation data. J. Irrig. and Drainage
Div., ASCE 94:57-71.
Mualem, Y. 1974. A conceptual model of hysteresis. Water Resources
Research 10:514-420.
Nielsen, D. R., Biggar, J. W., and Erh, K. T. 1973. Spatial variability
of field-measured soil-water properties. Hilgardia 42:215-260.
Rose, C. W. 1966. Agricultural Physics. Pergammon Press, N. Y., N. Y.
Swartzendruber, Dale. 1969. The Flow of Water in Unsaturated Soils in
Flow in Porous Media. Roger J. M. DeWiest, (Ed.), Academic Press
New York.
Taylor, S. A. and Ashcroft, G. L. 1972. Physical Edaphology. W. H.
Freeman, San Francisco.
Von Rosenberg, Dale V. 1969. Methods for the Numerical Solution of
Partial Differential Equations. American Elsevier Publication Co
N. Y., N. Y.
78
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GLOSSARY
Evaporative flux, potential: The water flux across the soil surface under
conditions such that the rate of evaporation is determined by the
energy available to evaporate water.
Flux: The volumetric flux density: The volume of soil water transported
through unit cross sectional area of soil per unit time.
Symbol: Q
Dimensions: Volume of Soil water 1,3 L
Area soil x time or I2f~ or I
Gravitational head: The energy per unit weight of the soil water due to
its elevation above a reference level. A component of the
hydraulic head. Symbol: Z
Dimens ions: Length,
Hydraulic conductivity: The ratio between the soil water flux and the
negative of the hydraulic gradient. A measure of the ability of
soil or porous medium to transmit water by mechanically driven
flow. A maxmium for a given soil at saturation of that soil.
In unsaturated soil is a function of the water content. Decreases
strongly with decreasing water content. Symbol: K
Dimensions: same as those for flux, 1^ .
?
Hydraulic gradient: The rate of increase of hydraulic head with position.
In one dimension, the vertical, the gradient is 3H/9Z where H is
the hydraulic head. Dimensionless. A measure of the driving force
for Darcy flow.
Hydraulic head: The energy per unit weight of the soil water due to the
combined effects of gravity and pressure. The elevation of the
free water surface in the open arm of a manometer connected to the
point in the soil at which the hydraulic head is to be measured
(see Figure 1).
Symbol: H. Dimensions: Length.
Hydraulic properties: The water retention function, 9(h) and the
hydraulic conductivity function, either K(6) or K(h).
79
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Hysteresis: In the context of soil water, the term refers to the fact that
the hydraulic functions, in particular 0(h),are not single valued, i.e.,
the relation between 6 and h depends on the past sequence of pressure
heads which have prevailed, or whether the soil is wetting or drying.
See Figure 2.
Mass balance: A mathematical statement of the conservation of mass principle.
Considering a volume element of a medium in which mass transport
of a substance is occurring, the partial time derivative of the con-
centration increases because of (1) an excess of flow into the
element over flow out of the element, and/or (2) the production of
the substance by some reaction or process within the element. For
the soil water and one dimensional flow in the Z direction, a
mass balance may be written as:
IfiL . _ 9(dL Q) + G
9t §Z~
where PL - the mass of water per unit bulk volume of soil
G - the mass of water produced per unit soil volume and
per unit time by reactions
dL - the density of the soil water.
Since pL = dL6, where 9 is the volumetric water content, and
assuming that dL is constant, the density may be factored out of
the time and space derivative terms to yield a volume balance
IT = " 3Z + S
where S = G/d^ in the volume of soil water produced by reactions or
processes per unit soil volume per unit time.
Pressure head: The energy per unit weight of the soil water due to the com-
bined effects of pressure and solid-liquid interactive forces. A
component of the hydraulic head. Symbol: h.
Dimensions: Length of fluid column, quite commonly that of a
water column. See also Figure 1 for the evaluation of pressure head
from piezometer and tensiometer measurements. The pressure head
is negative in unsaturated soil.
Profile: (1) The vertical distribution of a substance in a soil column
or soil profile, e.g., the "water content profile."
(2) Also used to refer to the vertical array of layers the
matrix of a soil, as, e.g., a "soil profile."
80
-------
Rewet water content: The water content attained by wetting a relatively
dry soil to zero pressure head. Due to entrapment of air the soil
will not be completely saturated, i.e., the rewet volumetic water
content is less than the total porosity. The water content attained
by wetting the soil along the M W C to zero pressure head. See
Figure 2.
Source term: A term in a mass or volume balance equation for a substance
which describes the time rate of production of the substance per
unit volume of medium. See, e. g., S in Equation (2). Production
may be positive or negative, i.e., consumption is negative production.
Steady state: Refers to a state of flow, e.g., of water, which is time-
invariant, i.e., all pertinent physical variables such as water
content, flux, hydraulic head, etc., are constant with time. In the
volume balance equation for the soil water, Eq.(2), steady state
flow is described by 90/8t = 0.
Suction: Soil water suction. The negative of the soil water pressure
head. A measure of the combined effect of soil water pressure and
solid-liquid interactive forces on the energy per unit weight of
the soil water.
Water capacity: The rate of increase of volumetric water content with
soil water pressure head, d0/dh. Symbol: C. Dimensions: IT1.
The slope or derivative of the water retention function, 6(h)
(See Figure 2).
Water content, volumetric: The volume of soil water per unit bulk volume
of soil. Symbol: 8. Dimensions: None, but on occasion it maybe
useful to identify it as the ratio of soil water volume to bulk
soil volume.
Water content, weight basis: The ratio of the mass of soil water to the
over-dry mass of soil solids.
Water retention function: The relation between water content and soil
water pressure head. In this report, the function 6(h) (See Figure
2) .
Water table: The locus of points in the soil water system at which the
soil water pressure is atmospheric, or the pressure head is zero.
Wetting front: A region of relatively steep rate of change of water
content with distance which is characteristically found when water
is imbibed or infiltrated into a dry soil.
"U.S. GOVERNMENT PRINTING OFFICE: 1978--684-041/2035
81
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TECHNICAL REPORT DATA
(Please read instructions on the reverse before completing)
1. REPORT NO.
ORP/LV 78-8
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
Water Movement in Uranium Mill Tailings Profiles
5. REPORT DATE
o
Sprit
.PERFC
FORMING ORGANIZATION CODE
7. AUTHOR(S)
A. Klute, D. F. Heermann
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
U.S. Department of Agriculture
Science and Education Administration
Agricultural Research
Fort Collins, CO 80523
10. PROGRAM ELEMENT NO.
Interagency Agreement
IAG - DG - H026
12. SPONSORING AGENCY NAME AND ADDRESS
Office of Radiation Programs
U.S. Environmental Protection Agency
P. 0. Box 15027
Las Vegas, NV 89114
13. TYPE OF REPORT AND PERIOD COVERED
Final April 1976-Sept. 1978
14. SPONSORING AGENCY CODE
U.S. EPA
IB. SUPPLEMENTARY NOTES
16. ABSTRACT
This study characterizes the behavior of water in profiles of uranium
mill tailings through (1) measurement of the water retention and transmission
properties of selected tailings materials, (2) numerical simulation of water
flow in selected profiles of tailings subjected to specific boundary conditions,
and (3) analysis and interpretation of the simulation results within the
framework of unsaturated soil water flow theory. The sequence of flow events
in a tailings profile without vegetation and with a water table at a given
depth is: (1) an initial drainage from saturation, with evaporation at the
surface, (2) infiltration of varying amounts of rain at irregular intervals,
(3) and periods of evaporation and drainage from the profile, with redistribu-
tion within the profile, between infiltration events. The water flow regime
in the upper 90 cm, particularly the upper 10-20 cm is transient and dynamic.
Infiltrated water, which has penetrated to depths more than about 10 cm,
continues to flow downward within the profile. The lower part of the profile,
below about 70-90 cm, tends to have a quasi-steady downward flow condition.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lOENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
soil physics, soil profiles, soil proper-
ties, soil water, water pollution,
simulation, tailings, radioactive wastes,
ground water recharge
Uranium mill tailings
Salt Lake City
0808
0813
1201
18. DISTRIBUTION STATEMENT
Release unlimited; available on request
at address shown in Block 12
19. SECURITY CLASS (ThisReport)
unclassified
21. NO. OF PAGES
82
20. SECURITY CLASS (This page)
unclassified
22. PRICE
No charge
EPA Form 2220-1
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