SECONDARY RECOVERY OF GROUNDWATER BY AIR INJECTION—A FINITE ELEMENT MODEL by NEELAKANDAN SATHIYAKUMAR, B.E., M.E., M.S. in C.E. A DISSERTATION IN CIVIL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1987
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
SECONDARY RECOVERY OF GROUNDWATER BY AIR
INJECTION—A FINITE ELEMENT MODEL
by
NEELAKANDAN SATHIYAKUMAR, B.E., M.E., M.S. in C.E.
A DISSERTATION
IN
CIVIL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
December, 1987
Mr) lie i<^^'
op' ^ ACKNOWLEDGEMENTS
The author expresses his wholehearted gratitude to Dr.
Billy J. Claborn for his valuable guidance, assistance and .
encouragement throughout the course of this study. The
author wishes to express his deep appreciation to Dr. C. V.
G. Vallabhan for his assistance and guidance, especially in
the area of numerical techniques. The author appreciates
the valuable suggestions and encouragements of Dr. R. H.
Ramsey III, Dr. R. E. Zartman and Dr. K. A. Rainwater as
members of the committee.
Special thanks are due to Dr. Ernst W. Kiesling,
Chairman of the Department of Civil Engineering, Dr. Robert
M. Sweazy, the former Director of the Water Resources Center
and Dr. Lloyd V. Urban, present Director of the Water
Resources Center, for the financial assistance provided.
The author appreciates his wonderful wife, Anandhi, for
her sacrifice, patience, inspiration and unlimited support
during the entire period of study.
Finally, the author wishes to express his deep regards
to his parents and his wife's family members for their love,
encouragement and sacrifices during his education.
11
CONTENTS
ACKNOWLEDGEMENTS i i
LIST OF TABLES v
LIST OF FIGURES vi
CHAPTER
I. INTRODUCTION AND OBJECTIVES 1
Need for Secondary Recovery of Groundwater .... 1 Mechanism of Air Injection 2 Need for this Study 7 Objectives of the Study 7
II. LITERATURE REVIEW 9
Methodology 9 Comparison Between Oil and Water Recovery 10 Mathematical Modelling Aspects 11 Hydraulic Conductivity 20 Other Related Works 21 Earlier Investigations of Secondary Recovery 23
III. MODEL DEVELOPMENT AND NUMERICAL TECHNIQUES 29
Governing Equations 29 Assumptions Made in this Study 35 Formulation of Finite Element Equations 35 Galerkin Formulation 39 Element Shape Function 45 Step-By-Step Integration Method 51 Numerical Verification of Model 54
IV. RESEARCH FINDINGS 58
Idalou Air Injection Program 58 Test Details 50 Formation Pressures 58 Soil Parameters Used in this Study 70 Comparison of Results 74 Scheme I 79 Formation Pressure Comparison 81
iii
Comparison of Water Level Changes 88 Scheme II 92 Scheme III 104 Scheme IV 115 Summary 125
V. CONCLUSIONS AND RECOMMENDATIONS 129
Conclusions 129 Recommendations 131
BIBLIOGRAPHY 134
IV
LIST OF TABLES
1. OBSERVED FORMATION PRESSURE EQUATIONS 59
2. COMPARISON OF FORMATION PRESSURE EQUATIONS--SCHEME I 83
3. COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME I 93
4. COMPARISON OF FORMATION PRESSURE EQUATIONS--SCHEME II 95
5. COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME II 103
5. COMPARISON OF FORMATION PRESSURE EQUATIONS--SCHEME III 107
7. COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME III 115
8. COMPARISON OF FORMATION PRESSURE EQUATIONS--SCHEME IV 117
9. COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME IV 125
LIST OF FIGURES
1. CONCEPTUAL ILLUSTRATION OF EFFECTS OF DRAINAGE ON WATER HELD IN STORAGE 3
9. MONITOR WELLS USED IN IDALOU AIR INJECTION TEST .... 51
10. NORTH-SOUTH CROSS SECTION OF IDALOU TEST SITE 52
11. WELLS LOCATED IN NORTH-SOUTH CROSS SECTION 53
12. WELLS LOCATED IN EAST-WEST CROSS SECTION 54
13. PRE-INJECTION MOISTURE PROFILES AT NH#4 (HPUWCD #1, 1982b) 55
14. AIR INJECTION RATE VERSUS TIME, IDALOU AIR INJECTION PROGRAM (HPUWCD#1, 1982b) 55
15. AIR INJECTION PRESSURE VERSUS TIME, IDALOU AIR INJECTION PROGRAM (HPUWCD#1, 1982b) 57
15. CAPILLARY PRESSURE--SATURATION RELATIONSHIP FOR BOTANY SAND 71
17. CAPILLARY PRESSURE--SATURATION RELATIONSHIP FOR CHINO CLAY 72
18. RELATIVE PERMEABILITY--SATURATION
RELATIONSHIPS 73
19. BOUNDARY CONDITIONS USED IN THIS STUDY 75
vi
20. INITIAL DOMAIN DISCRETIZATION 77
21. DOMAIN DISCRETIZATION FOR SCHEMES I AND II 80
22. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME I 84
23. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEME I 84
24. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY--SCHEME I 85
25. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME I 85
25. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME I 85
27. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SIXTH DAY--SCHEME I 85
28. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECT I ON--SCHEME I 87
29. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME I 89
29. COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME I 89
31. COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME I 90
32. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME I 90
33. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME I 91
34. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME II 95
35. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEME II 95
VI1
35. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY--SCHEME II 97
37. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME II 97
38. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME II 98
39. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SIXTH DAY--SCHEME II 98
40. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECT I ON--SCHEME II 99
41. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME II 100
42. COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME II 100
43. COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY—SCHEME II 101
44. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME II 101
45. COMPARISON OF WATER SURFACE CHANGES
AT THE END OF FIFTH DAY--SCHEME II 102
45. DOMAIN DISCRETIZATION FOR SCHEMES III AND IV 105
47. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME III 108
48. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END THE SECOND DAY--SCHEME III 108
49. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY--SCHEME III 109
50. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME III 109
51. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME III 110
52. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SIXTH DAY--SCHEME III 110
viii
53. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECT I ON--SCHEME III Ill
54. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME III 112
55. COMPARISON OF WATER SURFACE CHANGES AT THE END OF THE SECOND DAY--SCHEME III 112
55. COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME III 113
57. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME III 113
58. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME III 114
59. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME IV 118
50. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEME IV 118
51. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY--SCHEME IV 119
62. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME IV 119
53. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME IV 120
54. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SIXTH DAY--SCHEME IV 120
55. COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECT I ON--SCHEME IV 121
55. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME IV 122
57. COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME IV 122
58. COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME IV 123
IX
59. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME IV 123
70. COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME IV 124
71. COMPARISON OF MOISTURE PROFILES AT NH#4 127
X
CHAPTER I
INTRODUCTION AND OBJECTIVES
When the water level in an unconfined aquifer declines
because of pumping, some of the water remains behind, held
in small interstices by the capillary forces. Successful
recovery of oil held in a similar manner has been practiced
by the petroleum industry since early in this century. Of
the secondary recovery methods applicable to oil, such as
air drive, surfactant/foam, thermal, and vibration, the air
drive method seems to be both economical and appropriate for
recovery of water (HPUWCD#1, 1982a).
Need for Secondary Recovery of Groundwater
For most of the Ogallala aquifer, which underlies the
High Plains of Texas, water levels have declined more than
50 feet due to pumping which began about 50 years ago. From
studies conducted by the High Plains Underground Water
Conservation District No. 1 (HPUWCD#1) in 1974, the total
water stored in the saturated zone at that time was 340
million acre feet, and the annual pumping rate was 8.1
million acre feet of water. By the year 2030, the
corresponding storage will decrease to 135 million acre feet
and the pumping rate will decrease to 2.21 million acre feet
of water per year. Recovery of 355 million acre feet of
water may be possible in the High Plains of Texas; this
represents more that has been withdrawn to date (Claborn,
1983). The mechanism by which the secondary recovery of
water is possible is discussed in the next section.
Mechanism of Air Injection
Figure 1 shows a typical portion of porous media before
and after drainage of water. The capillary fringe, which
can be defined as the water which is continuous with the
water table, moves up and down as the water table
fluctuates. After the water level is lowered due to
pumping, some of the voids that were once completely filled
with water contain both water and air, as shown in Figure
lb. A significant volume of water is retained in the pores
by capillary forces. Smaller amounts are retained on the
surface of the soil particles (the hydroscopic moisture).
The water is acted on by air pressure, gravity, soil
pressure and surface tension forces. As the water table
continues to fall, the water at the top of the capillary
fringe is so high above the water table that the surface
tension forces can no longer hold the water up and
separation occurs. Some of the water in the fringe then
WATER TABLE
a. BEFORE DRAINAGE b. AFTER DRAINAGE c. ISOLATED WATER
Figure 1: CONCEPTUAL ILLUSTRATION OF EFFECTS OF DRAINAGE ON WATER HELD IN STORAGE
drains and some becomes isolated and suspended above the
water table, as shown in Figure Ic. Drainage ceases when
these forces attain equilibrium.
When air is injected from a well bore beneath some
layer with capability to prevent or seriously retard the
upward motion of air, the movement of the air will be
radially outward from the injection well. Figure 2 shows a
typical cluster of pores when air injection begins. The air
pressure at A exceeds the pressure at B by some amount, A p,
when the injected air flows past the cluster. In response
to the unbalanced force on the water created by this
pressure differential, A p, both menisci will be displaced
to the right. If the unbalanced force is sufficiently
large, water will leave the cluster at B (or at a larger
menisci) until a smaller menisci is formed at A, which
produces a force to the left to balance the force caused by
A p. As the injected air passes the cluster and forces the
water out, this water at first reduces the pore space
available for air flow in pores at a lower elevation.
Reduced pore space means increased pressure difference, and
more water will be obtained. However, gravity is moving the
water downward and the soil eventually becomes drier than
the drainage equilibrium value. The air pressure drop will
become much less since there will be greater pore space
DIRECTION OF
AIR FLOW
SOLID MATERIAL
WATER
Figure 2: WATER HELD IN CLUSTER OF PORES
available for the flow of air. This means that, with the
same pressure difference, drainage will occur at one
moisture content, but not at lower water content. The
difference in pressure must be increased to cause additional
drainage.
There is an another mechanism by which the water may
move in the air injection process. If the unsaturated zone
is thin compared to the saturated zone, then the water table
in the vicinity of the air injection well is subjected to an
increased downward pressure. Assuming that the water is
incompressible in the pressure ranges of the air injection
operation, the water table drops (dewatering) in the area
beneath the well and rises at some distance away from the
injection well, resulting in an outward-moving wave. When
this water wave combines with the water draining from the
pores from the unsaturated zone a wall of water may form
filling the unsaturated region. This creates a trap for the
injected air, resulting in a quasi-pressure vessel. As more
and more air is injected, this wall is pushed farther and
farther away from the injection well. The soil becomes
saturated as the wave moves by, and water drains by gravity
after the wave passes. This capillary water is subjected to
the air pressure gradient and drains towards the water table
as described in the previous mechanism.
Need for this Study
Three field tests for the secondary recovery of
groundwater by air injection have been conducted by the
HPUWCD#1, one each at Slaton, Idalou and Wolfforth, all near
Lubbock, Texas. These tests indicated that the results of
air injection for secondary recovery of groundwater are
unpredictable. Had a model been available, that model could
have been used to predict the results at these test sites.
Some modelling work has been done in saturated-unsaturated
simultaneous air-water flow in porous media using finite
differences and finite element methods. These models are
discussed in detail in the Chapter II. However, a search of
the literature revealed no reported work in axisymmetric
simultaneous air-water flow using the finite element method.
There is a need for a predictive model for design and
operation of secondary recovery efforts.
Objectives of the Study
The basic objective of this study is to develop a model
to predict the secondary recovery of groundwater by air
injection. The specific objectives are:
1. To formulate a suitable mathematical model to
predict the secondary recovery of groundwater by
air injection.
8
2. To solve the mathematical equations by the finite
element method,
3. To calibrate the model using the results obtained
at the Idalou test site, and
4. If possible, use the calibrated model to predict
the results at the other two sites.
The results from the Idalou test site will be used to
calibrate the model because more data is available from this
test than from the test at either Slaton or Wolfforth,
Previous work in this area is reviewed in Chapter II. The
development of the model and the numerical technique to be
used in this study are discussed in Chapter III. The
discussion of the results of this study is presented in
Chapter IV. The conclusions and some thoughts on the future
direction of continued research are presented in Chapter V.
CHAPTER II
LITERATURE REVIEW
The review of previous work related to this research
can be classified into two major areas: first, secondary
recovery (methodology), and, second, the mathematical
modelling of secondary recovery of groundwater. Whetstone
(1982) has done an extensive literature search in this area.
His study indicates that most work has been on methodology
rather than the modelling. In this section, the methodology
aspects of secondary recovery are discussed. The
mathematical aspects of secondary recovery will be discussed
in the next section.
Methodology
Whetstone (1982) could find no work reported in the
literature with the primary purpose of recovering water in
the vadose zone by air injection. However, he summarized
works reported in the literature which are closely related
to secondary recovery of groundwater by air injection. In
this section, some of the studies reviewed by Whetstone are
summarized.
10
Water was driven away from the well, not produced in
the experiments conducted by E.R. Cozzen in 1935. In a
study conducted by Evan 'Ev and M.F. Karimn in 193 5 to
examine water displacement efficiencies produced by the
injection of various gases, ammonia gas was found to be the
most efficient in water displacement. Studies conducted by
C D Robert in 1957, he concluded that the injection of air
can be used to retard or to accelerate the movement of
groundwater.
Whetstone (1982) also conducted a literature review in
the following areas.
1. horizontal wells,
2. hydrologic papers of possible applicablity, and
3. secondary recovery of petroleum.
He reported more than 150 references in the hydrologic area
alone. For the secondary recovery of petroleum. Whetstone
identified more than 100 references. The first serious
study of air injection for the recovery of petroleum was
conducted by James 0. Lewis in 1917.
Comparison Between Oil and Water Recovery
The petroleum industry has practiced the secondary
recovery of oil for more than sixty years. It was from this
practice that the concept of water recovery by a similar
11
mechanism was evolved. Reddell et al. (1985) compared oil
reservoirs and water aquifers. The similarity between oil
reservoirs and groundwater aquifers are many and include: 1)
liquid occupies some of the pore space; 2) gases may also
occupy some of the pore space; 3) at times both the liquid
and gases are simultaneously present in the pore spaces; and
4) the liquids and gases move through pores of the medium
according to Darcy's law. The dissimilarities between
aquifer and petroleum reservoirs are: 1) liquids in the two
systems possess different properties; 2) petroleum
reservoirs tend to be deeper, under more pressure and have
lower permeabilities than groundwater aquifers; 3) the
wetability of the liquids in the two systems is vastly
different; and 4) solubility of gases in the two liquids is
significantly different.
Mathematical Modelling Aspects
Many articles in the literature, relate to saturated-
unsaturated flow of one, two and three phase fluid flow.
The only available literature related to the secondary
recovery of groundwater was of the work conducted by the
researchers who were directly involved in this activity.
Even though many of the other reported works were not
related to this research, the concepts from these projects
12
are very valuable. In this section the review of such
literature is discussed.
Blanford (1984), summarizing the saturated-unsaturated
models in the literature, reported the early work in
numerical analysis of two-dimensional saturated-unsaturated
porous media flow [Rubin (1968), Freeze (1971, 1972) and
Green (1970)] were performed by using the finite difference
methods (FDM).
Neuman (1973) was one of the first investigators to use
the finite element method (FEM) for the analysis of
saturated-unsaturated porous media flow. Neuman used a
Galerkin's spatial finite element formulation with linear
triangular elements and an under relaxation scheme in time
for the saturated-unsaturated seepage flow. He also
correctly pointed out that triangular and quadrilateral
elements for the two dimensional flow can be extended for
the analysis of axi-symmetric subsurface flow problems.
Fedds et al. (1975) compared the predicted finite
element solution of Neuman et al. (1973) with field data on
one and two dimensional problems. Reeves and Duguid (1975)
used a spatial FEM with bilinear quadrilateral elements and
a weighted scheme in time for the analysis of two
dimensional saturated-unsaturated problems. They also
included the pressure dependent boundary condition.
13
Narasimhan and Witherspoon (1982) gave an overview of
development of the finite element models citing the pioneers
in this area. They pointed out that Neuman was among the
earliest researchers to apply the Finite Element Method to
analyze the fluid flow in saturated-unsaturated porous
media. They also discussed the situations wherein the
higher order interpolation functions can be used. Another
issue addressed in this overview was that of whether to
distribute or to lump the capacity matrix arising from the
time derivative. They recommended lumping the capacity
matrix because of its consistence with the physics rather
than distributing the matrix. This concept is discussed in
detail in the next chapter.
Green (1970) proposed a two-dimensional finite
difference model describing isothermal, two-phase fluid flow
in porous media. He considered a linear relationship
between the saturation and capillary pressure. The
hysteresis effects were neglected in his research. He
tested the validity of his model using experimental
infiltration data. Even though his model is too simplified,
he showed the approach in the formulation of two phase,
saturated-unsaturated fluid flow problems.
Narasimhan et al. (1975) developed an integrated finite
difference method (IDEM). This method combines the
14
advantages of an integral formulation with the simplicity of
finite difference gradient. The IDEM and FEM are
conceptually similar and differ mainly in the procedure
adopted for measuring spatial gradients. They considered
the following single phase (water) equation
k grad 0 + g = C _£0_ ct
(2.1)
where
K = permeability
grad = partial differential operator
9 = moisture content (volumetric)
g = acceleration due to gravity
C = slope of water retention curve
t = time
They integrated this equation after neglecting the spatial
variation of permeability. They used the divergence theorem
to convert the first volume integral to a surface integral.
Many aspects of their formulation were similar to the finite
element formulation.
Faust's (1978) model, developed for three phase fluid
flow in terms of water, a non-aqueous phase, and air is
k K rx [i
(Vp^-Pj^gVD) X
+ q X ct (2.2)
where
X = fluid in consideration
15
k = relative permeability
K = absolute permeability
P = pressure
D = depth
q = flow
S = saturation
H = absolute viscosity of the fluid
p = density of the fluid
<t> = porosity of the medium
He assumed that: the air is always at atmospheric pressure,
the densities and viscosities are pressure independent, and
summation of saturations equal to one. Faust compared his
simulated results of fluid transport (non-aqueous fluid)
with the experimental results.
Faust also proposed a method of obtaining the relative
permeability of a nonaqueous phase fluid in terms of water
and air permeabilities. He also proposed the relationship
between the porosity of the medium with the formation
pressure
O = <D° [l + C^(p-p") (2.3)
where
p = pressure
0
p = reference pressure
<t> = porosity
15
0
'P = porosity at the reference pressure
C = aquifer compressibility
Lin (1987) developed a two phase flow model in porous
media. The fluids considered were water and
tricholoroethylene (TCE). He injected the TCE and simulated
the water and TCE movement. The hysteresis effects were not
considered. Lin states that the stability of the model
cannot be mathematically analyzed due to the complexity of
the numerical method as well as the mathematical model
itself.
Yortsos and Grgavalas (1981) developed an analytical
model for oil recovery by steam injection. Three phases
(water, oil and steam) were considered. The water and oil
phases were governed by mass balance criteria. The steam
phase was governed by the thermal energy balance. They also
considered condensation and heat conduction in their model.
Narasimhan et al. (1978) developed an explicit-
implicit scheme for the FEM in subsurface hydrology. The
governing equation considered in their formulation is
V (K Vh) - q = C (| ) (2.4)
where
h = total head
C = specific capacity
17
Using the Galerkin FEM, the resulting equation was a system
of first order linear or quasi linear differential equations
of the form
[A]h+ [D] h = Q (2.5)
where
A = conductance matrix
D = capacity matrix
Pi = temporal variation of head
Q = flow
They recommended lumping the capacity matrix to avoid the
numerical difficulties. Another advantage of lumping the
capacity matrix is that a larger time interval is
permissible. They also discussed in detail how to designate
the implicit and explicit nodes and how to incorporate the
automatic determination of time intervals.
Pinder and Huyakorn (1982) identified three different
nodal categories of saturated-unsaturated flow:
1) nodal points that remain unsaturated during the
time interval, 5t
2) nodal points that remains saturated; and
3) nodal points that undergo a change from a state of
saturated to a state of unsaturated during the
time interval, 5t, and vice versa.
18
They suggested that for the nodal points of category 1 the
central difference scheme (implicit) should be employed.
For the nodal points of category 2, they recommended
employing a backward difference (explicit) scheme, because
cS the governing equations becomes elliptic (—^ = 0). For the
ct
nodal points of category 3 they advised to use the central
difference scheme. But they also remarked that the
variation of capillary potential with time exists only for
the unsaturated region. For the saturated region, the
variation of capillary potential with time will be zero (the
contribution to the capacity matrix will be zero). This is
similar to the suggestion given by Narasimhan (1978).
Cooley (1983) proposed the sub-domain method as a new
procedure for the numerical solution of variably saturated
flow problems. He also recommended lumping the time
derivative terms in formulating the capacity matrix. He
verified his model using various simple subsurface flow
problems with known solutions (flow to a well, drainage from
square embankment, and one-dimensional infiltration).
Javandel and Witherspoon (1958) studied the
applicability of Finite Element Methods to transient flow in
porous media. Unlike the conventional Galerkin method, they
used the variational principles in formulation. They used
19
triangular elements and considered the axisymmetric flow of
a water phase. The central difference scheme was employed
for time.
Huyakorn et al. (1984) discussed the techniques for
making the Finite Element Methods competitive with the
Finite Difference Methods in modelling flow in variably
saturated porous media. They proposed a modified Picard
method. The conventional Picard method requires knowledge
of the tangent of the capillary pressure versus the
saturation curve at any given saturation. The proposed
chord slope method (instead of tangent) is as follows
__w ^ ^v w (2.6) c\u r+1 r
where
r+l,r = current and previous iteration levels
y = capillary pressure
They compared this modified Picard scheme with the Newton-
Raphson scheme. Their findings were: 1) the Picard scheme
requires less computer time than Newton-Raphson scheme; and
2) the Newton-Raphson scheme normally requires a fewer
number of iterations (particularly true in steady state
simulations).
20
Hydraulic Conductivity
Many investigations related to hydraulic conductivity
were reported in the literature. Even though these are not
directly related to the secondary recovery of groundwater,
values of hydraulic conductivity are required for modelling
movement of fluids in a porous media.
Campbell (1974) proposed a simple method of determining
the relative hydraulic conductivity as a function of the
degree of saturation from the soil water retention curve.
He cautioned that the proposed method is valid only if there
is an exponential relationship between the potential and
moisture content, i.e., the water retention function plots
as a straight line on logarithm scales. Since that
relationship breaks down near saturation, Clapp and
Hornberger (1978) used a short parabolic section in this
region to represent a gradual air entry.
Shani et al. (1987) proposed a field method for
estimating hydraulic conductivity and the matrix potential
versus water content relations. This method is based on the
observation that water when applied at a constant rate to a
point on the soil surface creates a ponded zone in a short
time interval with a constant area. Thus, steady state
solutions of the two dimensional flow equation can be
applied to find hydraulic conductivity and matrix
potentials.
21
Mantoglou and Gelhar (1987) proposed a method to
evaluate the effective hydraulic conductivity of transient
unsaturated flow in stratified soils based on a three
dimensional stochastic approach.
Other Related Works
The effects of compressibility of air and hysteresis,
are considered in this section. Hoa (1977) considered the
influence of the hysteresis effect on transient flows in
saturated-unsaturated porous media. An analytical
expression for primary and secondary scanning curves such as
0-Oo T
(2.7) Og-Go l + a(v,/ -y)P
was proposed,
where
a, p = constants
9 = water content
9o = irreducible water content
y = potential
y ^ = minimum potential
Brustsaert and El-Kadi (1984) stated that
compressibility (expansion or compression) of air, water and
the solid matrix should be considered in any rigorous
formulation of flow. They identified four different and
22
distinct types of formulations. The first of these is an
upper zone partially saturated with water where the flow of
air is neglected. This is called the diffused upper zone
and air compressibility is considered. In the second zone,
the water and solid matrix are incompressible. The Richards
equation is valid for this zone. A sharp interface exists
between air and water in the third zone and compressibility
effects are considered. In the fourth zone, the upper
boundary of groundwater is assumed to be a true free surface
and solid material and water are considered incompressible.
If the material is uniform in the fourth zone, the equation
is a Laplace equation governs.
Vachaud et al. (1973) studied the effect of air
pressure on a stratified vertical column. They considered a
constant flux infiltration and gravity drainage. The local
soil air pressure was found to differ significantly from
external atmospheric pressure. When a simulated rain was
applied with an intensity of 3 cm/hour, the air pressure was
+50 mB (milliBar) and the air pressure was -15 mB in the
case of gravity drainage. From these results they concluded
that air pressure must be considered and the governing flow
equation must be written in terms of two-phase immiscible
fluid flow.
23
Gray and Pinder (1974) proposed a Galerkin
approximation for the time derivatives. They claimed that
the Galerkin approximation permits a high order
approximation in time as well as in space. The chief
limitation of this proposed method, depending upon the order
of the approximation, is that the formulation requires
solution for more time intervals simultaneously resulting in
an increase in computer time. They also pointed out when
the conventional finite difference approximation with o is
set to 0.57 equals the linear Galerkin approximation.
Dakshnamurty and Lend (1981) proposed a mathematical
model for predicting moisture flow in an unsaturated soil
under a set of hydraulic and temperature gradients. They
used the well-known Darcy's law as the governing equation
for the water phase and Pick's law for the air phase. They
used the FDM with an "explicit" solution scheme.
Parker et al. (1987) developed a model to describe the
relative permeability, as well as saturation-fluid pressure
functional relationships, in a two or three fluid phase
porous media system subject to monotonic saturation paths.
Earlier Investigations of Secondary Recovery
Two reports were submitted to the Texas Department of
Water Resources by previous investigators directly involved
24
with this study. One concerned the physics of flow and the
other dealt with mathematical modelling aspects of secondary
recovery of water. Claborn (1985) reported on mathematical
modelling aspects and Redell (1985) reported on the physics
of flow.
Claborn (1985) used a finite difference model to
predict the secondary recovery of groundwater. He proposed
the following modification to the Darcy's equation
q=-ii-l-/P +YZX - — ^ (2.8) ^ 1 oL ( w ' H cL
where
q = fluid flow flux
P = water pressure w
P = air pressure a '^
L = length along the flow path
7 = specific weight of water
Z = vertical distance from the datum
[i = water viscosity
k = intrinsic permeability of water
k = pseudo-permeability
The contribution due to the term that includes k is a
modification of the conventional Darcy's law. Claborn
(1985) discusses the justifications for adding this term.
They are:
25
...there is an unbalanced force on pores containing capillary water... The extent of this force is proportional to the drop in air pressure across the pore (the air pressure gradient); there is a drag force exerted on the water film by the air passing through pores. If the flow of air is assumed to be laminar, with parabolic velocity distribution within the pore, the drag will be proportional to the air pressure gradient. These two cases are mutually exclusive within a pore, e.g., capillary water excludes the presence of film water. They are, however, continuous; when the capillary cell breaks, it leaves a film subject to the drag force. The pseudo permeability in the first case is probably largely related to the size of the pore, while in the second case, the thickness of the film controls the permeability.
The finite difference operator for the first-order partial
difference equation consisted of four nodal values in
Claborn's model. The conventional method is to consider
only two nodal values. Claborn used increasing cell widths
in the radial direction, as the distance from the well
increased. The amount of air injection was calculated in
the cells adjacent to the well based on the air pressure
gradient and air permeability value. The solution technique
used was called the 'Strongly Implicit Procedure' (SIP)
which was developed by Stone (1958). Claborn proposed three
different solution schemes in his work.
In scheme I, the air equation was solved for the
assumed water pressure values, until the air pressures
values converged. These 'converged' air pressure values in
the water equation, were used to solve the water equation
26
until it converged. Using results from this water equation,
the convergence was checked for the air equation. These two
equations were solved until both of them converged
simultaneously. Numerical stability was checked by
operating the model for several time steps without including
any air injection. When this model was used with air
injection, the equations did not converge, even at low rates
of injection. It was observed that the air equations
converged readily while the water equations diverged.
In solution scheme II, The solution logic was modified
to alternate between the air equations and the water
equations within each iteration. As in scheme I, the
numerical stability was obtained, but the convergence could
not be obtained.
In scheme III, the entire set of equations were to be
solved at one time (air and water equations simultaneously),
was proposed. Schemes II and III resulted from discussions
with Dr. Donald Redell, Department of Agricultural
Engineering, Texas A & M University. One of Dr. Redell's
student tried Scheme III for a small system and realized
some success.
Even though Claborn did not have success in modelling
the secondary recovery of groundwater by air injection, he
developed the approach to be taken in such modelling.
27
Claborn discussed the probable reasons why his model
failed to simulate the field observations. One of issue
discussed was the validity of using Darcy's law to model the
secondary recovery of groundwater by air injection. As
justification for the validity of using Darcy's law, he
referred Abriola and Pinder (1985) who had stated that
Darcy's law has been used extensively in the soil science
and petroleum literature to model multiphase flow in porous
medium since many experimental investigations in two and
three phase fluid system had shown its applicability.
Reddell et al. (1985) reviewed the principles of
similitude methodology for their applicability in the design
of a physical model to predict the air injection process.
Difficulties were encountered in meeting some of the
similitude criteria, and hence, a decision was made to
design a sand tank model to verify the numerical model to
simulate the air injection process. A numerical model of
two-phase flow from the petroleum industry was adapted to
the flow of water and air. These results from this
numerical model were then used to design the sand tank.
They pointed out the importance of considering the
hysteresis effects of relative permeabilities versus
saturation for both water and air. They also conducted
experiments on the cores collected from the Idalou test site
28
to determine the capillary pressure versus saturation
relationships, and the relative permeabilities versus
saturation relationships.
From this literature search, it is apparent that, no
work has been reported with the primary objective of
recovering water in the vadose zone by air injection.
However, the concepts from these works are very valuable.
Most of the researchers have utilized the conventional
Darcy's law describing the movement of fluids in the porous
media, in thier models. As a starting point, the Darcy's
law for each of the fluid flux along with the mass
conservation principles were used in this study. A
description of model development and the numerical
techniques is presented in the next chapter.
CHAPTER III
MODEL DEVELOPMENT AND NUMERICAL
TECHNIQUES
To solve any physical problem by numerical methods, the
problem must be represented by a set of equations. An
accurate representation of a physical system will generally
require a complex system of equations. By making use of
some assumptions, one may be able to simplify the model to a
considerable extent. The assumptions which were made for
this study are discussed in appropriate places and
summarized at the end of the next section.
Governing Equations
For simultaneous air-water flow (multiphase flow), the
governing equations have been published in standard texts
(Bear 1982, Pinder and Gray 1975, Pinder and Huyakorn 1983).
These equations are based on conservation of mass and the
use of Darcy's equation for the fluid flux under
consideration.
The continuity equation is
Mass. - Mass . = Change in Mass in out
which can be written as
29
30
ct + V .(p^q^) = 0 (3.1)
and Darcy's equation is
*x ^ ^ ( V p ^ + P^gVh) ^X
(3.2)
where
P^ = density of fluid x
r| = porosity of soil medium
[i^ = viscosity of fluid x
q„ = flux of fluid x
S^ = saturation of fluid phase x
k = relative permeability of fluid x rx
K =
g =
h =
absolute permeability of porous medium
acceleration due to gravity
distance from datum in vertical direction
p^ = pressure of fluid x
V = partial operator
Substituting (3.2) into (3.1) gives
p k K ^x rx (Vp^+P^g Vh) = n-c(S p ) ^ x^x'
ct (3.3)
Equation (3.3) can be written for both the water and
the air phases. The resulting equations are
31
V . Pw^rw^
w
(Vp^+P^g Vh) d{S p ) ^ w^w^
dt (3.4)
V . p k K ^a ra
(Vp^+p^gVh) = ^ •
^tVa) ct
(3.5)
where the subscripts a and w refer to the air and water
phases respectively.
Equations (3.4) and (3.5) contains ten variables,
namely p , p , k , k , S , S , p , p , u , and p . The • w ^a' rw' ra' w a' ^w' ^a' ^w ^a
viscosities of water and air can be taken as constants if
isothermal conditions are assumed. The density of water can
be taken as constant because water is incompressible for the
range of pressures expected during air injection. If the
air pressure is known, the density of air can be calculated
using the ideal gas law. Six unknowns would then remain in
the two equations. Therefore, four auxiliary equations are
needed. By stipulating that only air and water are present
in the pores, thus
S + S w a 1.0 (3.5)
A second equation follows from the definition of
capillary pressure which states that in a partially
saturated porous medium, there is a difference between the
pressures of the water and the air. This is described by
^c ^a ^w (3.7)
32
where p^ is capillary pressure. Although equation (3.7)
relates water and air pressures, it introduces a new
variable, p^, into the problem. Therefore, only three
additional equations are required. These are empirically
determined and relate the relative permeablities and the
capillary pressure to saturation
\ w = fl(S„) (3.8)
A typical saturation-capillary curve is shown in Figure
3. Typical saturation-relative permeablity curves are shown
in Figure 4. Using equations (3.5) to (3.10), equations
(3.4) and (3.5) can be expressed in terms of
p , p , S and S The time derivative of the saturations * w ^a' w a.
S and S can be expressed in terms of p.. and p through the w a ^ w " a
Fioure 64- COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SIXTH DAY--SCHEME IV
100
90 -
80 -
121
« a >-^ u OC cn (/) UJ OC Q .
70
60
50
40
30
20
10
F i e l d
Model
T 1 1 1 1 1 1 1 1 r 0.2 0.4 0.6 0.8 1
— T 1 1 1 1 1 1 r—
1.2 1.4 1.6 1.8 2 (Thousands)
RADIAL DISTANCE ( feet )
Figure 65: COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECTION--SCHEME IV
122
« «
u o z
(n a
OC
70
60 -
50 -
40 -
30
20 -
10 - O
Original
Model
-Field
F i g u r e 55
1 2 3 4 CThousands)
RADIAL DISTANCE (feet)
COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME IV
70
« e
UJ
u
i o t-oc UJ >
60 -
50 -
40 -
30
20 -
10 -
.a-
F i e l d Model
O r i g i n a l
F i g u r e 67
1 1 1 1 1 1 \— 1 2 3 4
O^ousands) RADIAL DISTANCE (feet)
COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME IV
123
o
UJ (J
o
OC
F i g u r e 68
1 2 3 4 O^ousands)
RADIAL DISTANCE (feet)
COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME IV
u o z
o
OC
T 2 ^ 3
CThousands) RADIAL DISTANCE (feet)
Figure 69: COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME IV
124
70
60 -
50 -
UJ
u
a
OC
40 -
30
20 -
10 -
T 1 1 r 2 3
(Thousands) RADIAL DISTANCE (feet)
"7-4
Figure 70: COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME IV
125
Day
1
2
3
4
5
TABLE 9
COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME IV
Estimated Net Water Gained (acre-
Field Model
62.82
110.27
127.34
152.43
215.75
18.12
45.45
130.82
244.09
285.40
•feet)
* Approximately at one day time interval, since the start of air injection
126
pressure ecjuations and water surface change comparisons, the
results from this scheme were closer to the field
observations than the previous schemes. However, the daily
simulated net volumes of water recovered from this scheme
were lower in the first two days. When the model parameters
were adjusted to obtain agreement between the formation
pressure ecjuations, more water recovery was simulated.
The moisture content changes were also reported in
HPUWCD#1 (1982b) for NH#4, at the end of air injection. The
predicted moisture content changes at 290 feet from the
injection well are compared to the reported values
graphically in Figure 71. From this figure, one can
conclude that the model indicates more dewatering of the
unsaturated zone. This may be due to the assumption of
axial symmetry of the formation.
Summary
Based on the comparisons of formation pressure, water
level changes and the magnitude of water gained, the trends
predicted by the model are similar to those observed in the
field. The minimum air permeability plays an important role
in the simulation. Consideration of more than one clay
layer has increased the similarity. The model has the
capability of handling layered soils. The chief limitation
127
134-1
w u < Cm Oi D cn S o Oi CM
X EH
a
1 3 3 -
1 4 2 -
1 4 6 -
150'
1 5 4 -
1 5 8 -
1 6 2 -
166
" T " " " APPROXIMATE CLAY LAYER
TARGETED VADOSE ZONE
•PRE' MOISTURE PROFILE
•POST' MOISTURE PROFILE (FIELD OBSERVATIONS)
'POST' MOISTURE PROFILE (MODEL PREDICTIONS)
J APPROXIMATE CAPILLARY FRINGE
J.
1 r T 1 1 r-0 10 20 30 40 50 60 70 80 90
MOISTURE Figure 71: COMPARISON OF MOSITURE PROFILES
AT NH#4
128
of the model is its axially symmetric formulation and hence
the each and every field observation could not be reproduced
in the model.
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
A finite element model has been developed to predict
the behavior of water and air in both saturated and
unsaturated zones due to air injection. The need for this
model and the objectives of this research were identified in
Chapter I. The conclusions drawn from the research based on
the specific objectives are presented in the following
section.
Conclusions
1. As noted in Chapter II, the literature contains no
modelling work whose primary objective is predicting the
recovery of groundwater by air injection other than the work
of earlier investigators directly involved in this research.
A two-phase (water and air) saturated-unsaturated
axisymmetric flow model was developed by utilizing the
concepts of mass conservation and Darcy's ecjuation. These
two ecjuations contained ten variables which were reduced to
two by making use of some assumptions and empirical
relationships of the soil parameters (Chapter III). The
model development with governing ecjuations were discussed in
Chapter III.
129
130
2. The mathematical ecjuations were solved by making use
of triangular axisymmetric elements. The advantages of
using the Finite Element Method over the Finite Difference
Methods were presented in Chapter III. The model was
verified for its validity in terms of stability and mass
balance.
3. The results predicted by this model were compared
with the Idalou air injection test observations (field) with
respect to the formation pressure, changes in water surface
elevations, magnitudes of water recovered and the moisture
variations. To calibrate the model, four different schemes
were used. The first two utilized a single clay layer at
the top. In the latter two schemes, two clay layers were
considered. In Schemes I and II, the permeabilities of air
and water were considered to be constant within the
iterations and in Schemes III and IV these permeabilities
were treated as variables (temporal variation) and expressed
in terms of dependent variables. Consideration of temporal
variations of the permeability had some influence on the
results. The predicted results had a number of similarities
when compared with the field observations. Consideration of
two clay layers increased the similarity. Based on these
comparisons, the trends predicted by the model are similar
to those observed in the field. Each field observations
131
could not be reproduced by the model because of the
assumption of axial symmetry. Closer representation of the
field soil properties in the model would have increased the
similarity. Based on these similarities it is safe to
conclude that, given the soil parameters, the model is
capable of predicting the effects on the groundwater caused
by air injection.
4. This model was not tested for the other two sites
due to the time constraints.
Though one can observe a number of similarities between
the predicted and observed results, the approach should be
towards three-dimensional modelling to obtain realistic
results in all directions. Three-dimensional models
generally recjuire more input data (soil properties) and
increase the complexity. The technology of obtaining the
more accurate soil properties and 'super computers' will
ease the handicaps of the three-dimensional models.
Recommendations
Based on the comparisons of predicted and observed
results, the following recommendations are made to improve
the predictive capability of the model.
132
1) Rectangular elements should be used instead of
triangular elements to reduce computer time.
Another advantage of rectangular elements is that a
longer time interval in the simulation can be used
(Segerlind, 1984).
2) The effects of pseudo permeability, k , (Claborn,
1985) should be included in the model, as this
inclusion may lead to a more realistic simulation.
3) The anisotropy of the medium should be considered,
so that the variations of permeability in the
principle axes can be studied.
4) The model should be verified with a sand tank model
similar to one suggested by Redell (1985) as the
degree of representation of the sand tank model in
the numerical model can be increased.
5) The solution technicjue should be replaced with one
capable of considering only the non-zero vectors so
that a a finer discretization with many unknowns can
be solved. This will also aid in reducing the
errors introduced by the simplifying assumptions.
6) A 110 percent increment in radial cell widths should
be tried rather than the 150 percent increment
currently used in the model so that the error
introduced by the coarse discretization can be
minimized.
133
7) Hysteresis of wetting and drying phases (capillary
pressure, and relative permeability versus
saturation relationships) should be included in the
numerical model.
8) Modelling efforts should be geared toward the
three-dimensional modelling than the axi-symmetric,
radial flow models to obtain more realistic results.
BIBLIOGRAPHY
1. Abriola, L. M., and Pinder, G. F., 1985, "A Multiphase Approach to Modelling of Porous Media Contamination by Organic Compounds 1. Ecjuation Development," Water Resources Research, Vol.21, No.l, pp 11-18.
2. Bear, J., 1979, "Hydraulics of Groundwater," McGraw-Hill, New York.
3. Blanford, G. E., 1984, "Finite Element Simulation of Saturated-unsaturated Subsurface Flow," Research Report No.155, Water Resources Research Institute, University of Kentucky, Lexington, Kentucky.
4. Brustsaert, W., and El-Kadi, A. I., 1984, "The Relative Importance of Compressibility and Partial Saturation in Unconfined Groundwater Flow," Water Resources Research, Vol.20, No.3, pp 400-408.
5. Campbell, G. S., 1974, "A Simple Method for Determining Unsaturated Conductivity from Moisture Retention Data," Soil Science, Vol.117 pp 311-314.
6. Claborn, B. J., 1983, "Secondary Recovery of Water," Journal of Irrigation and Drainage Engineering, ASCE, Vol.101, No.4, pp 357-365.
7. Claborn, B. J., 1985, "Mathematical Modelling of Secondary Recovery of Water from the Vadose Zone by Air Injection," Final Report 5 of 5, A Report Submitted to Texas Department of Water Resources by Texas Tech University, in Association with HPUWCD#1, and Texas Agricultural Experiment Station.
8. Clapp, R. B., and Hornberger, G. M., 1978, "Empirical Ecjuations for Some Hydraulic Properties," Water Resources Research, Vol.14, pp 501-604.
9. Cooley, R. L., 1983, "Some New Procedures for Numerical Solution of Variably Saturated Flow Problem," Water Resources Research, Vol.19, No.5, pp 1271-1285.
134
135
10. Dakshanamurthy, V., and Fredlund, D. G., 1981, "A mathematical Model for Predicting Moisture Flow in an Unsaturated Soil Under Hydraulic and Temperature Gradients," Water Resources Research, Vol.17, No.3, pp 714-722.
11. Faust, C. R., 1974, "Transport of Immiscible Fluids within and Below the Unsaturated Zone : A Numerical Model," Water Resources Research Vol 21, No.4, pp 587-596.
12. Fedds, R. A., et al., 1975, "Finite Element Analysis of Two-Dimensional Flow in Soil Considering Water Uptake by Roots: II Field Applications," Soil Science American Proceedings, Vol.39, pp 231-237.
13. Gray, W. G., and Pinder, G. F., 1974, "Galerkin Approximation of the Time Derivative in Finite Element Analysis of Groundwater Flow," Water Resources Research, Vol.10, No.4, pp 821-828.
14. Green, D. W., 1970, "Numerical Modelling of Unsaturated Groundwater flow and Comparison of the Model to a Field Experiments," Water Resources Research, Vol pp 862-874.
15. Hillel, D., 1982, "Introduction to Soil Physics," Academic Press, New York.
15. Hoa, N. T., et al., 1977, "Influence of the Hysteresis Effect on Transient Flows in Saturated-Unsaturated Porous Media," Water Resources Research, Vol.13, No.6, pp 992-996.
17. HPUWCD#1., 1982, "Investigation of Secondary Recovery of Ground Water from Ogallala Formation, High Plains of Texas - Summary of Study," A Report Submitted to Texas Department of Water Resources by HPUWCD#1, in Association with Texas Tech University.
18. HPUWCD#1., 1982b, "Idalou Air Injection Test - Data Compilation," Draft Report 6 of 7, A Report Submitted to Texas Department of Water Resources by HPUWCD#1, in Association with Texas Tech University.
19. Huyakorn, P. S., et al., 1984, "Techniques for Making Finite Element Competitive in Modelling Flow in Variably Saturated Porous Media," Water Resources Research, Vol.20, No.8, pp 1099-1115.
135
20. Javandel, I., and Witherspoon, P. A., 1968, "Application of the Finite Element Method to Transient Flow in Porous Media," Society of Petroleum Engineers Journal, pp 241-251.
21. Lappala, G. E., 1982, "Recent Developments in Modeling Variably Saturated Flow and Transport," Proceeding of the Symposium on Unsaturated Flow and Transport Modeling, Seattle, Washington, pp 3-30.
22. Lin, C , 1987, "Modelling the Flow of Immiscible Fluids in Soils," Soil Science, Vol.143, pp 293-300.
23. Mantoglou, A., and Gelhair, L. W., 1987, "Effective Hydraulic Conductivity and Matric Potential-Water Content Relationships," Water Resources Research, Vol.23, No.l, pp 57-57.
24. Narasimhan, T. N., et al., 1975, "An Integrated Finite Difference Method for Analysis of Fluid Flow in Porous Media," Water Resources Research, Vol.12, No.l, pp 57-251.
25. Narasimhan, T. N., et al., 1978, "Finite Element Methods for Subsurface Hydrology Using a Mixed Explicit-Implicit Scheme," Water Resources Research, Vol.14, No.5, pp 863-873
25. Narasimhan, T. N., and Witherspoon P. A., 1982, " Overview of the Finite Element Methods in Groundwater Hydrology," Finite Elements in Water Resources, Proceedings of the Fourth International Conference, Hannover, Germany, pp 1-29.
27. Neuman, S. P., 1973, "Saturated-Unsaturated Seepage by Finite Elements," Journal of Hydrology Division, ASCE, Vol.99, pp 2233-2250.
28. Parker, J. C., et al., 1987, "A Parametric Model for Constitutive Properties Governing Multiphase Flow in Porous Media," Water Resources Research, Vol.23, No.4, pp 518-524.
29. Pinder, G. F., and Gray, W. G., 1977, "Finite Element Simulation in Surface and Subsurface Hydrology," Academic Press, New York.
30. Pinder, G., and Huyakorn, P. S., 1983, "Computational Methods in Subsurface Flow ," Academic Press, New York.
137
31. Reddy, R. N., 1986, " An Introduction to Finite Element Methods," McGraw Hill, New York.
32. Redell, D. L., et al., 1985, "Recovery of Water from the Unsaturated Zone of the Ogallala Acjuifer by Air Injection, Physics of Flow," Final Report 4 of 5, A Report Submitted to Texas Department of Water Resources by Texas Agricultural Experiment Station in Association with HPUWCD#1, and Texas Tech University.
33. Segerlind L. J., 1984, "Applied Finite Element Analysis," John Wiely and Sons, New York.
34. Settari, A., and Aziz K., 1974, "Use of Irregular Grid in Cylinderical Coordinates," Vol.257, Society of Petroleum Engineers Journal, Vol.257, pp 395-412.
35. Shani, U., et al., 1987, "Field Methods for Estimating Hydraulic Conductivity and Matric Potential-Water Content Relationships," Soil Science Society Journal, Vol.51, pp 298-302.
35. Stone, H. L., 1958, "Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Ecjuations," SIAM, Journal of Numerical Analysis, pp 530-558.
37. Vauchad, G., et al., 1973, "Effects of Air Pressures on Water Flow in an Unsaturated Stratified Vertical Column of Sand," Water Resources Research, Vol.9, No.l, pp 150-173.
38. Whetstone, G, A., 1982, "Investigation of Secondary Recovery of Ground Water from, the Ogallala Formation, High Plains of Texas, Literature Review," Report 3 of 7, Submitted to Texas Department of Water Resources, by HPUWCD#1, in Association with Texas Tech University.
39. Yortsos, Y. C., and Grgavalas, 1981, "Analytical Modelling of Oil Recovery by Steam Injection," American Society of Petroleum Engineers Journal, pp 152-192.
40. Zienkiewiez, O. C., 1977, "Finite Element Methods," McGraw Hill, New York.