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

2. WATER HELD IN CLUSTER OF PORES 5

3. TYPICAL CAPILLARY PRESSURE--SATURATION RELATIONSHIP 33

4. SATURATION--RELATIVE PERMEABILITY RELATIONS 34

5. COORDINATE SYSTEM FOR AXI-SYMMETRIC

TRIANGULAR ELEMENT 47

5. DOMAIN DISCRETIZATION ILLUSTRATION 55

7. DISCRETIZATION FOR THE TEST PROBLEM 56

8. LOCATION OF IDALOU AIR INJECTION TEST SITE 59

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

application of the chain rule

dS as,, cp^ cS cp dp - ^ = ^ ^ . - ^ = -^(^--^) (3.11)

dt Cp^ Ot cp ct ct

c ^ and

— - = nr ^ = --r^i^r- - -^zr) (3.12) dt dt cp dt ct ^ '

33

t

LU OC 3 (O (/) UJ OS

a. >

<

Q. <

SATURATION, S w

Figure 3: TYPICAL CAPILLARY PRESSURE--SATURATION RELATIONSHIP

34

< UJ

OH LU o. UJ

UJ OC

0 0

SATURATION, S w

Figure 4: SATURATION--RELATIVE PERMEABILITY RELATIONS

35

dS The coefficient w

' P. can be obtained from Equation (3.10).

Equations (3.4) and (3.5) are reduced to

p k K ^w rw

w (Vp^+P^g Vh)

cS cp dp w , a ^w Wcp^ dt ct

(3. 13)

and

p k K ^a ra (Vp^+P^g Vh)

dS dp,, t'p^ w , w _ ^a ^^a^p ^ ct t (3.14)

Even though the relative permeability k is saturation

dependent, the value of term can be assumed to be known as

saturation is known initially and can be updated as the

saturation changes.

By letting

M = w

p k K ^w rw

w

and

M = a

p k K a ra

E q u a t i o n s ( 3 . 1 3 ) and ( 3 . 1 4 ) become

M4 v2p.,+ 7,yh w 'w

dS dp^ dp w , ^ a '^w X

Wcp ' dt ct ( 3 . 1 5 )

r 2 2 1 ^^a '^^ ""Pa . a^ a a j a ^p c t tZ

( 3 . 1 5 )

35

Assumptions Made in this Study

The following assumptions were made in the derivation

of Equations (3.15) and (3.15)

1. isothermal conditions exit;

2. water and the solid medium are incompressible;

3. air behaves as a perfect gas;

4. air in the porous media is continuous and

initially at atmospheric pressure;

5. only air and water are present in pores; and,

5. hysteresis effect of wetting and drying phases are

neglected.

With the governing equations developed, the next step

was to find a solution for these equations. In spite of the

assumptions made in simplifying them, these equations cannot

be solved by a closed form solution technique because of

nonlinearity. Hence, there was a need to use a suitable

numerical technique. The numerical techniques used in this

study are discussed in detail in the following section.

Formulation of Finite Element Equations

The most commonly used numerical methods for solving

complex physical problems are finite difference and finite

element methods. Both FDM and FEM can be classified as

domain methods, as the unknowns are in the domain. The FDM

37

approximates the governing equations of the problem using a

local expansion for the variables, generally a truncated

Taylor series. On the other hand, the FEM deals with

equivalent integral equations.

Lappala (1982) compares these two methods and

situations where each method is more appropriate. The

following were the conclusions from his research:

1. The Finite Element Method is capable of tracking

steeper fronts than the Finite Difference Methods;

2. For linear problems with steeper fronts, the Finite

Element Method is much faster for an equivalent

accuracy than the Finite Difference Method. On the

other hand, for non-linear problems the Finite

Element Method is slower than the Finite Difference

Method;

3. As the front becomes steeper, the number of required

spatial nodes or grid points increases and the time

step required to minimize oscillations

correspondingly decreases;

4. The preferred method for dry soil is a Finite

Difference Method with averaged K (permeability),

because it introduces dispersion.

He pointed out that if non-linear material properties are

rising on an element rather than on a nodal basis, and if

38

simple triangles are used in the finite element

discretization so that the integrals of the basis function

can be evaluated analytically, then the disadvantages of FEM

becomes less restrictive.

The FEM is used in this research for the following

reasons:

1. The interface between the clay and sand layer can be

easily represented using the FEM,

2. The boundary conditions can be easily handled in

FEM, whereas in FDM they may require special

equations.

Among the various formulations, the Galerkin method is

the most widely used in groundwater hydrology. The Galerkin

method is a special case of a more general approach called

the method of weighted residuals (MWR), where the weighting

function is also the coordinate function.

Since Equations (3.15) and (3.16) are similar, the

discussion on the formulation is limited to the water

2 equation. V p in the r-Z coordinate system can be written

W

as

V2p = !!^.i[f (r.5 )l (3.17) ^w „2 r r cr ^ '

cZ,

where

r = radial coordinate

39

Z = vertical coordinate

Using the definition of the second order partial operator

and the r-Z coordinate system (3.17), Equation (3.15) can be

written as

dp T 3 cp

w^' .^2 r or^ cr cZ

w ,_2 r cr rr 6"Z

cS cp cp w / ^a ^w X = TIP -:; ( -^—)

'^Wcp ^ at ct ' (3.18)

Since - — is equal to the zero vector in the 'r' direction, cr

Equation (3.18) becomes

M w

2 c P 1 3 cp,. ) )+Y

ar " 'w az2

cS cp cp w , a _ ^w X ^^Wop ^ ct ct ^c

(3.19)

Galerkin Formulation

The Galerkin finite element formulation begins with the

governing differential equation, D , as

Dp (p) = 0 (3.20)

where p signifies the field variable. In this study the

field variables are water and air pressures. The next step

in the Galerkin formulation is to approximate the field

variable behavior, i.e..

P = ^N^Pi = [N][^} i

(3.21)

40

where [N] is a row vector of the shape or the interpolation

function and {p} is the column vector containing the nodal

values of the appropriate field variable.

Substituting p from Equation (3.21) into the

differential operator of Equation (3.20) gives

Dp(^) = R = 0 (3.22)

where R is the residual. If the exact solution p and the

approximate solution p are the same, then the residual of

equation (3.22) would be zero. However, the approximate

solution does not generally equal the exact solution,

resulting in the nonzero residual as indicated. Since the

governing equation cannot be satisfied pointwise throughout

the domain, Q, its satisfaction is sought in the sense of a

weighted average over the the domain, i.e.,

J W Dp(^)dn = 0 (3.23)

where W is the weighting function.

In the finite element Galerkin formulation, the

weighted integral of Equation (3.23) is summed over the

element subdomain, Q , and the element level shape

functions are utilized as the weight function, i.e.,

n

f W D^(^)di^ = I I ^i ^^{P)<^^ = 0 J" P e=l P-e ^ P

for j = 1, 2, ..., n (3.24)

41

where

n^ = number of the elements e

e th N. = j shape function for element e

n = number of shape functions

The Galerkin formulation possesses the disadvantage of

requiring inter-element continuity (i.e., first derivative

continuity along the element boundaries), and the exact

satisfaction of all boundary conditions. These

disadvantages can be relaxed or eliminated by generating a

'weak' Galerkin formulation obtained through integration by

parts and using Green's theorem. Now substituting the

approximate values for water and air pressures

Pw = 'NlfP„i

Pa = (NK^a!

The Equation (3.19) can be written as

^ ,2 2

- 1 [N]^(M^(i.^.(r.3^) + ^ ^ Y ^ ^ ) )

as ap^ ap,,

C

where f is the volume integral. Using differentiation by V

;52

a p parts, the term y- can be written as

az^

42

cz cz ,„2 cz az cZ

On r e a r r a n g i n g

[.,T.!!^ . 4(lN)-.%) - ^ . % (3.26) -ry2 cZ oZ cZ cZ ^ ' cZ

.2, Similarly, the term - - can be written as

az^

1 a T ^Pw Now the term — .-T-.([N] r-—--) can be written as

r cr ar

A. .((Nl^r^) = i(ilMlr^) . A([NlT4_( .i )) r ar cr r ar cr r cr cr

On rearranging

i([N)T^(r.^)) = l.^.dNl^r^) r ar cr r cr cr

-i(£lNlIr%) (3.28) r cr cr ^ '

S u b s t i t u t i n g Equat ion (3.26) through Equat ion (3 .28) in

Equat ion (3 .25)

-J („ (i4-([Nl\5i) - (^^'-^) J \f T cr c r cr cr V

. A(iN]^f^)-iMZi^ az ^ az ' az az

43

^ y ( A ( [N]^^) - i i N i l i ^ ) ) "w ^ cz ^ az ' cz az ^ ^

m as cp cp rxTiT w , ^ a ^w X X , - IN] TIP ^ ( _ ^ - _ ^ ) ) . d v Wcp c t c t

^ c = {0} ( 3 . 2 9 )

on r e a r r a n g i n g

J w^ dz az ar c r ' V

as ap cp ap

V ^c

m t / O UU G U

J WW cz az V

- J ( ^ i ( t « " l - - ^ ) - M w ^ ( I N l ^ . ^ ) ) .dv V

- f M Y ( A ( [ N ^ ] i ^ ) ) . d v J w'w ^ az az V

= [0} ( 3 . 3 0 )

The f o u r t h and f i f t h i n t e g r a l s i n t h e a b o v e e q u a t i o n c a n b e

t r a n s f o r m e d i n t o s u r f a c e i n t e g r a l u s i n g G r e e e n ' s t h e o r e m

- f ( ^ ^ ( [ N l ^ . r . ^ : ^ ) + M„ J - ( [ N ) ^ . ^ ) ) . d v J ^ r a r cr w cZ cZ r c r V

J (M [ N ] - ^ ^ - ^ c o s 9 + M [N] - ^ s i n 0).dr

V

44

= - J (M V [Nl^i^sin 0).dr

where J is the surface integral. Substituting these into r

E q u a t i o n ( 3 . 3 0 ) , y i e l d s

j „ ^ ( ( l i M ^ ^ ) . (ilNlZ;^), .av ^ ^ cZ cZ cr cr

+ f M Y ( i M ^ i ^ ) .dv ^ W^W^ cZ cZ ^

m as ap ap + J ([Nl^p _ « ( ^ - 4 i ) ) .dv

r. w cp c t c t V ^Pc

- j ( M ^ [ N ] ' ^ - ^ c o s 9 + M ^ [ N ] ' ^ - ^ s i n 9) . d r

- I (VwfNJ^f-i-^)-^^

= 10} ( 3 . 3 1 )

cp The t e r m —r^ c a n b e w r i t t e n a s

cZ

-^ = miip I az az ^^w^

ap cp Using similar substitutions for the terms —— , —— , and

cr cZ

P 'h — — , and noting that 4— = 1/ Equation (3.31) becomes cr cZ

V

a[N]

45

T

J W^W cZ

' J (nPw^tN]"tN](_^-_^.)) .dv

- j (M^[N]T(^cos 9 s i n 9 ) [ 1 ) . dr

- J M^V^lNl'^sin 9 .dr •J W W r

= {0} (3.32)

Repeating these operations for the air equation, it

becomes

J „ ((ItMlilNI , ^[Kl^c[N| ^ •^ a cZ cZ cr cr a

V

V

as T, af^ 1 c[^ ] V ^c

- j (M [N]^(4^cos 9 + 4|^sin 9)f6 }) . dr ^ a cr cZ a

- f M^Y [Nl' sin 9.dr a'a

r

= fO] (3,33)

Element Shape Function

The next phase in the finite element formulation is to

choose a specific finite element(s) to be used in the

45

discretization. In this study, linear triangular elements

are used. The shape function for the three noded triangular

element is expressed in terms of local coordinates

(Segerlind, 1983)

[Pl = N.^. + N.p. + N^^^

where ^i = 2X

(a. + b. r + c.Z) ^ 1 1 1 '

^j = -2A-3 (a. + b.r + c.Z)

^k = 2A^"k (a,, + bj r + Cj^Z)

where A is the area of the triangular element.

The coefficients a, b, c are defined as follows

^i=^i\-\^i- •^i^^j'^K' c . = R, - R . 1 k :

a.=R^Z.-R.Z^; b.=Z^-Z,; c . = R. - R, ] 1 k

a, = R,Z.-R.Z.; ''k'^i-Zj; c, =R . -R. k ] 1

The coordinate system is illustrated in Figure 5.

Equations (3.31) and (3.32) can be combined and written

at the element level as

[SRUfi) + [CDll^! = {F| (3.34)

where

[SR]

= 11° = 12° = 13°

•^11° - 12° 13 =21° =22° =23°

" 21° • 22° 23

=31° =32° =33 °

" 31° • 32° 33

47

Z •

^ r

Figure 5: COORDINATE SYSTEM FOR AXI-SYMMETRIC TRIANGULAR ELEMENT

[ p ]

r 'wl

^al

\2

^a2

48

[CD] = -

+ c

V3

1 1

'a3

- c

3)

11 "^^12 ^12 "^^13 " " " i s

- ^ 1 1 ^ ^ 1 1 - ^ 1 2 -^^12 - ^ 1 3 ^^13

' " ^21 - ^ 2 1 ^ ^ 2 2 - ^ 2 2 -^^23 " ^ 2 3

- ^ 2 1 ^ ^ 2 1 - ^ 2 2 "^22 - ^ 2 3 " ^ 2 3

^ ^ 3 1 - ^ 3 1 -^^32 ^32 "^^33 ^33

- ^ 1 ^ ^ 1 3 - ^ 3 2 "^32 - ^ 3 ^^33 J

^ d t ^

[ f |

• a l

•w2

• a 2

•w3

a 3

49

The formulation shown here for [CD] is called

consistent or distributed formulation. There is another

formulation called lumped formulation which can be obtained

by adding the coefficients of [CD] in a row and keeping the

sum on the main diagonal of [CD]. The [CD] matrix for the

lumped formulation is

[CD]

-^^11

- ^ 1 1 0

0

0

0

- ^ 1 1

^ ^ 1 1 0

0

0

0

0

0

•^^22

" ^ 2 2 0

0

0

0

" ^ 2 2

^ ^ 2 2

0

0

0

0

0

0

•^^33

- ^ 3 3

0

0

0

0

" ^ 3 3

^ ^ 3 3

T h e e l e m e n t s a p p e a r i n g i n [ S R ] , [ C D ] , {F} a r e d e f i n e d a s

aN. aN.. cN. cN^

' i , j J w^ dr dr dZ dZ V

) . d V

'^1,3 = I "a* V

aN^ aN.

ar ar -I-

cN^ cN .

~az az" ) . d V

as.

V c dV

1 , : J a c p ^ 1 3 dV

[N] [N] f . = f (M N . ( 4 ^ c o s 9 + H ^ s i n 9 ) {PJ ) • d r

w i J w 1^ c r cZ w

: N ^ j V w ^ i ^ i n 9 .dr - j (M^y^^) .dV

50

^^i = I (M N. ( 4 ^ c o s 9 + 4 ^ s i n 9 ) {p ] ) .df ai J a 1 cr cZ a .

+ f M Y N.sin 9 .dP - f(M Y c^).dV J a'a 1 J a'a cZ •• V

where i and j = 1 to 3, represents respective coefficients

shown in matrices. These ecjuations can be further

simplified by treating the terms M and M as 'constants'

and taking them to the right hand side. Now the elements

appearing in [SR],[CD],{F} are

aN. aN. aN. aN. s. . = f ( ^—3_ + —i:—l).dv 1,3 J ar cr az az '

V aN. aN. cN. aN.

r. . = f( ^ — i + —^-rJ-).dV 1/3 J ar cr az cZ '

V

as

^p^ V w ' c

as

. CS . . = f(Ji_.Tip _-!lN,N,).dV (3.35) 1,3 J M wcp^ 1 3

d. . = f(—TIP ^-^N,N.). 1,3 J M '^acp^ 1 3'

dV .. '"acp 1 3'

V a

f . = f (N.(ii^cos 9 + 4|^sin 9 ) {6 } ) .dP wi J 1 ar cZ w

cN. ^ J Y^N.sin 9 .dr - j(y^-^).dV

r V

f . = r (N. ( 4 ^ c o s 9 + 4 ^ s i n 9 ){pj) .dr ai J ^ 1 cr cZ a

+ J y^N^sin 0 .dr - j(y^J^).dV r V

51

The advantage of this operation is that this matrix, [SR],

and the second part of 'f' vectors need be evaluated only

once and stored in files, as they contain only properties of

the geometry.

Once the equations are assembled at the local element

level, they can be assembled at the global level. However,

due to the coupling of water and air ecjuations, the

resulting 'C' matrix (from the 'CD' matrix) is banded and

nonsymmetric. Hence, the advantage of a symmetric banded

matrix can not be utilized in this situation.

The assembled ecjuation at the global level is

[G]{P1 + [C](H} = {F1 (3.35)

where

n [G] = y [SR].

i=l ^

n e [c ] = y [CD].

i = l

Step-By-Step Integration Method

While it is possible to use a variety of schemes for

solving Equation (3.36), a relatively simple and

satisfactory procedure uses the finite difference in time

(Pinder and Gray, 1977). In this approach. Equation (3.35)

is rewritten as

52

[G](9[Pi^"'^ + (1-9){P1^) + [c](-l^)({Pl''''^-{P|'') At

= 9{F|"*'-'- + (1-9){F1^ = TFj (3.37)

When 9 is selected to be one, the scheme is a fully implicit

or backward difference approximation. When 9 is selected to

be zero, the scheme is said to be explicit or forward

difference approximation and when 9 is equal to one-half,

the scheme becomes a central difference or Crank- Nicolson

approximation. However, unconditional stability of linear

partial equations generally requires 9 to lie between one

half and one. In this study, the value of 9 is taken as

0.567 which will be corresponds to a Galerkin form of the

integration scheme. Ecjuation (3.37) can be rearranged so

that the unknown values appear on the left hand side and

known values on the right hand side

(9[G] + (^)[C]{Pl''''^)

= ((9-l)[G] + (^[C]){P}'' + TF'I) (3.38)

The final expression becomes

[A]{P]^''^ = [B]^{P}^ + m (3.39)

where

[A] = 9[G] + (^[C])

[B] = ((9-l)[G] + (;^)tC])

53

[Tl = 9[F] " - + (1-9{F1 )

The solution of (3.39) appears relatively straight­

forward, except for the highly nonlinear behavior of the

coefficients appearing in [G] and [C]. Among the different

approaches in solving (3.38), such as the Picard,

Chord-slope and Newton Raphson iterative schemes, the Picard

scheme is relatively simple to use (Neuman 1973). In matrix

notation, Ecjuation (3.39) becomes

j^jm^pjn+l,m+l ^ [B]"^{P|"^ + [F]"^ (3.40)

where 'm' is the iteration level. Computationally, one

first solves (3.40) for [P} ' using the initial

conditions to evaluate the coefficients of [A] , [B] and

T?}"^. ipjJ " !/" " ! is used to update the saturation at the

nodes. New element saturations are calculated using the

nodal saturation. Next, the calculated element saturation

is used to update [A]" '*'"'-, [B]" "*"" and {T}" " -'-. Equation (3.40)

is once again solved for [P] ' . This cycle is repeated

until the difference between successive iterations is within

a specified tolerance in terms of element saturation.

Segerlind (1984) states that the lumped formulation has

a full operating range for the step-by-step integration

parameter (9). It is claimed that larger time steps can be

54

used, when the lumped formulation technicjue is employed. He

also points out that the violation of physical reality and

numerical oscillations cannot be avoided with the consistent

formulation along with the forward difference or central

difference methods. The discretization of the domain to be

employed in this study is illustrated in Figure 6.

Numerical Verification of Model

The model developed in this study should be verified by

some means before using it on a real problem. This can be

achieved by application to a problem with an analytical

solution. Search for a standard problem with analytical

solution failed; other investigators used a physical model

to verify their numerical model. Hence, the model was

verified for numerical stability. This was carried out as

follows:

A small domain eighteen feet by ten feet was

considered. The discretization of this domain is shown in

Figure 7. No air injection was considered so that the

original water content does not change with time. The upper

and lower boundaries were considered as no-flow boundaries.

Soil parameters (discussed in the following chapter) used in

this test problem were that of the actual problem. The

simulation was carried for 100 days at one-day time

55

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Figure 6: DOMAIN DISCRETIZATION ILLUSTRATION

56

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57

intervals. As anticipated, the water content and water

pressures remained constant with respect to time. The nodal

air pressures were not ecjual to zero as expected. Instead,

very small positive air pressures ( 1 to 100 times 10""^, ft)

were observed at most of the nodes. These errors in air

pressure are very small, so that the model is assumed to be

stable.

In addition to checking for stability of the test

problem, the test problem was used to evaluate the mass

balance. For this purpose, only the water equation was

considered and initially the entire domain was considered to

be fully saturated. The top and bottom horizontal

boundaries were considered as no-flow boundaries. Known

water pressures were prescribed along the entire left

vertical boundary for the simulation. The inflow and

outflow were calculated based on the pressure gradients and

the permeability across the elements at the boundaries. The

inflow and outflow ratio were in the range of 0.94 to 0.98

and hence, it was judged that the model performed reasonably

well.

The application of this model in this study are

discussed along with the comparison of results are presented

in the following Chapter.

CHAPTER IV

RESEARCH FINDINGS

To compare the results obtained from the model with the

field results obtained from the Idalou air injection test,

it is necessary to describe the Idalou air injection

program. The first part of this chapter is devoted to a

description of the Idalou air injection program while the

second is devoted to the comparison and discussion of

research findings.

Idalou Air Injection Program

An air injection test was performed during the period

June 17, 1982 through June 23, 1982 on Mr. Clifford Hilber's

farm, near Idalou, Texas. The location is shown in Figure

8. The Idalou air injection program had one air injection

well and sixteen monitoring wells/holes. The wells

installed at the Idalou site were as follows:

1) an air injection well,

2) seven vadose zone air pressure monitor holes,

3) five neutron monitor holes, and

4) four water level observation wells ecjuipped with

continuous recorders and air pressure monitors.

58

59

Scale 1:28800

Figure 8: LOCATION OF IDALOU AIR INJECTION TEST (HPUWCD#1, 1982b)

60

The air pressure monitor holes are denoted as 'AM'

followed by the well number; the neutron monitor holes as

'NH' followed by the hole number; and the water level

observation wells by 'WELL' followed by the well number. An

additional eight wells in the surrounding area were used to

monitor the regional water levels. Figure 9 shows the

location of all the wells and holes. Figure 10 shows the

north-south cross section of the Idalou test site. Figure

11 shows the wells located in a north-south cross section

while Figure 12 shows the wells located in an east-west

cross section.

Pre-injection data collection consisted of atmospheric

pressure, formation pressure, water levels, formation soil

moisture, and pressure plate analysis. The pre-injection

moisture profile from NH#4 is shown in Figure 13.

Test Details

The Idalou air injection test started at 1248 hours on

June 17, 1982. The test lasted until 1138 hours on June 23,

1982, slightly less than six days, for a total elapsed time

of 142.8 hours. There were five stages of air injection and

between each stage the air injection rate was increased.

The plot of air injection rate (flow rate) versus time is

shown in Figure 14. The plot of air injection pressure

versus the time is shown in Figure 15.

51

Scale 1:1600

(See text for well type designation)

Figure 9: MONITOR WELLS USED IN IDALOU AIR INJECTION TEST (HPUWCD#1, 1982b)

52

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APPROXIMATE CLAY LAYER

•PRE* MOISTURE PROFILE ( P R E - I N J E C T I O N « DAY AVERAGE)

% MOISTURE

Figure 13: PRE-INJECTION MOISTURE PROFILE AT NH#4 (HPUWCD#1, 1982b)

66

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Formation Pressures

In the Idalou test program, regression ecjuations were

fitted to the measured formation pressures as a function of

radial distance. Different ecjuations were developed for the

depth intervals below the land surface of 115 feet to 135

feet and for 155 feet to 180 feet. The general form of the

regression equations is

P = a + bx log r

where

P=formation pressure

a,b= constants

r= radial distance from the air injection well

These equations employed the following assumptions:

1) the formation pressure one foot radially from the

center of the air injection well ecjualed the

pressure within the well; and

2) pressure decayed as a function of the logarithm

of the radial distance.

Ecjuations were developed for each day for each depth

interval. The depth interval 115 to 135 feet is above the

clay layer (see Figures 10 and 13), and hence this portion

was not considered in this simulation. The ecjuations for

the depth interval 155 to 180 feet are shown in Table 1.

59

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70

The results from this study will be compared to the

data collected from the Idalou air injection test program.

Specifically, the water level, the formation pressure and

the moisture variation at 290 feet away from the injection

well (NH#4) are used.

Soil Parameters Used in this Study

The soil parameters recjuired in this model are: 1) the

saturation-capillary pressure relationship; and 2) the

saturation-relative permeability relation. These soil

parameters vary depending upon the type of soil. In this

study, it is assumed that only sand and clay layers are

present in the study profile. The saturation-capillary

pressure relationship for sand is shown in Figure 15 and for

clay in Figure 17. The saturation-relative permeability

relationships were assumed to be independent of the soil

type, as shown in Figure 18. Soil parameter tests conducted

on the core samples, taken at the sand and clay layers at

the the Idalou test site, could not be used in this study

because of no clear-cut difference in properties between

them. Hence, there was a need for such soil properties.

Soil parameters used in this study were obtained from data

compiled by Maulem (1976). The Botany sand-fraction and the

Chino clay soil are two of many soils for which Maulem has

71

220

MOISTURE CONTENT

Figure 15 CAPILLARY PRESSURE--SATURATION RELATIONSHIP FOR BOTANY SAND (Mualem, 1975)

72

0

Figure 17:

iO 20 30 40 50

MOISTURE CONTENT

CAPILLARY PRESSURE--SATURATION RELATIONSHIP FOR CHINO CLAY (Mualem, 1976)

73

SATURATION, S;

1.00

0 .00 40 60

^ SATURATION, S

30 10 0

W

Figure 18 RELATIVE PERMEABILITY--SATURATION RELATIONSHIPS (Mualem, 1976)

74

reported data. These two soils were selected in this

research because it is believed that their properties are

well matched with the Ogallala acjuifer soil properties.

These relationship were smoothed for inflection points. In

particular, the saturation- capillary pressure curves were

smoothed to have continuity in pressure between the

saturated and unsaturated region.

Comparison of Results

From the data collected during the Idalou air injection

test, it was possible to compare model results in terms of

formation pressure, water level changes, amount of water

gained above the original water table and moisture

variations in the unsaturated zone. The boundary conditions

that were used in this study are shown in Figure 19. The

bottom of the acjuifer was considered as a no-flow boundary.

The top of the domain consisted of a clay layer. Air is

permitted to pass through this clay layer and air pressure

was maintained at atmospheric level just above it. Water

was not permitted to pass through this layer. The vertical

boundary condition at the right side was maintained at

initial conditions, as it was assumed that this vertical

boundary was so far away from the air injection well that no

influence was felt from air injection. The portions of

INITIAL CONDITIONS

75

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75

vertical boundary at the left side, containing the injection

well casing and the line of symmetry beneath the injection

well, were considered as a no-flow boundary. In the

screened portion, either the pressure or flow can be

prescribed. Though the program developed in this study is

capable of handling the prescribed pressure or flow boundary

conditions, only the pressure boundary conditions were

prescribed in the simulation.

An eight-foot clay layer at the top of the zone was

considered in the simulation as the plot of pre-injection

moisture profile at NH#4 (located at 290 feet from the air

injection well) indicted that there was only one clay layer

present in this section. In addition to this single clay

layer, a 15 foot unsaturated zone (target vadose zone) and

100 foot of saturated zone were also considered. The

overall dimensions of the domain were 124 feet in the

vertical direction and 4430 feet in the radial direction.

The domain was discretized using triangular elements as

shown in Figure 20. The discretization contained 584

elements and 380 nodes. The nodal numbering was done in

such a way that the band width of the resulting ecjuation

was a minimum. The first column of elements had a radial

thickness of one foot with each succeeding elements being

150 percent larger than the preceding one. Settari and Aziz

77

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78

(1974) successfully used this concept of increasing the cell

widths radially to reduce the number of unknowns.

Nodes 9 through 14 were prescribed at the known

injection air pressures from the time versus air pressure

curve (Figure 15). The soil parameters from Figures 16

through 18 were also used. The initial moisture content at

all the nodes were set according to the pre-injection

moisture values (Figure 13). The air and water pressures at

nodes 352 through 380 (outer vertical boundary) were

maintained at their initial values. The other boundary

conditions were maintained as described earlier (Figure 19).

The computer program was used to predict the moisture and

water level changes for a three day period. At the end of

the three days, it was observed that the air pressure

values, the water levels and the moisture levels were not

changed as expected. The extent of dewatering of the

saturated zone was not calculated as had occurred in the

field. The possible reasons for not achieving the expected

results may include: 1) the discretization near the water

table was too coarse; and 2) the relative permeability of

air in the saturated zone should not be ecjual to zero, as

100 percent saturation seldom occurs in nature.

Although this first simulation did not give the

expected results, it led to the following important

modifications that were to be incorporated in the model:

79

1) a finer discretization near the air injection zone

was needed (near the water table), and

2) instead of altering the relative permeability

curve, it was decided that the minimum relative air

premeability to be set at 0.075 of absolute

permeability.

Scheme X

In this Scheme, the aforementioned modifications were

implemented. A saturated thickeness of 40 feet, sixteen

feet of unsaturated thickness and eight feet of clay were

considered. The total number of elements and nodes were the

same as that of the initial problem. The domain

discretization used in Scheme I-is shown in Figure 21.

Maintaining the initial values at the far right vertical

boundary did not seem to be appropriate, and hence the

boundary conditions (from the fourth and fifth surface

integrals of ecjuations 3.32 and 3.33) were included. Thus,

the flow of air and water across this boundary was permitted

in accordance with the pressure gradient and permeablity.

The formation pressures, the water level changes and the

volume of net water gained for this Scheme are discussed in

the following paragraphs.

r .7- '.9 . : ds . 01

81

Formation Pressure Comparison

Comparison of formation pressures from the field

observation and the model prediction must be carried out for

the same location in the field and in the model. An

automatic nodal/element generation technicjue was utilized in

the model and hence few of the nodes were at the location

where the field observation were available. Even though

this problem could have been overcomed by a finite element

interpolation technicjue (using the shape function), another

limitation was encountered. The ecjuations presented in

Table 1 were from various times during the air injection

program. In order to store the pressure data (results from

the simulation) at time periods corresponding to the field

observation times, an enormous volume of data would have to

be stored. Moreover, this would have increased the computer

use time. To overcome these drawbacks, a decision was made

to compare the formation pressure ecjuations rather than the

formation pressures. As the results from the simulation

were stored on a daily (simulation time, not the real time)

basis, another decision was made to fit the formation

pressure ecjuations based on the predicted and observed

formation pressures at uniform time intervals. This step

was carried out for convenience, not based on a necessity.

The observed formation pressure data from wells AM#1, AM#2,

82

AM#3, AM#4, AM#5, AM#7, AM#8, NH#1 and NH#3 were used for

this purpose (HPUWCD#1, 1982b). A one-day time interval was

chosen for this purpose. Though the air injection was

carried out for less than six days, the valve at the air

injection well was not vented to the atmosphere (Figure 15)

for another day after the injection process terminated.

Because the air injection pressure data was available for

5.8-days, the simulation was carried out for this entire

period. The duration of air injection referred in the

remainder of this text means a period of 6.8-days instead of

the actual 5.8-days of air injection. The formation

pressure ecjuations from the observed pressures of the field

test and model results are compared in Table 2.

The similarities between the observed (field) and

predicted (model) values of the formation pressure ecjuations

are shown graphically in Figure 22 through Figure 28. The

predicted values are considerably lower than the observed

values. This difference may be explained as follows.

Homogeneous radial-flow was assumed in the model.

However, the air pressures at the same distance from the

injection well were not the same in different directions in

the field test. For example, pressure readings from AM#4

(located at 330 feet north of air injection well) showed 28

psi on June 20 while the readings from AM#8 (located 312

83

CN

U •J QQ < H

pci D Ul Ul u Oi 04 •-•

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0} U

c • H (0

84

a

UJ Q:

(/) V) UJ Q: 0.

100

9 0 -

Figure 22:

0.6 0.8 (Thousancjs)

RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME I

100

UJ

(/) tn UJ

Q.

2 0 0 4 0 0 6 0 0 8 0 0

Figure 23 RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEME I

85

Figure 24

400 200

UJ (£

l/> in UJ i£ a.

Figure 25

400 2 0 0

PAD«AL DISTANCE ( f « « 0

X, OP FORMATION PRESSURE

FOURTH DAY—SCHEML

86

(X ^^ UJ QC _)

UJ

OL

110 -

100 -

90 -

80 -70-J

60 -

Figure 25

4 0 0 6 0 0

RADIAL DISTANCE (feet) «nTrccTTT?F

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY—SCHEME I

n Q.

* « • UJ OC

</) in U l OC a.

(Thousands)

„ COMP^rSOrrFO;;;lTION PRESSURE Figure 27: COMPARISO ^^ ^ ^ ^ ^ ^^ ^^^

SIXTH DAY--SCHEME I

87

100

« CL

UJ OC

<n UJ OC Q.

T 1 1 1 1 I I I I T

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (Thousands)

RADIAL DISTANCE (feet)

Figure 28: COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECTION--SCHEME I

88

feet west of the air injection well) showed 4.7 psi at the

same time. This indicated that there was a greater

influence from the air injection north of the injection well

than to the west of the well. Even along a radial line,

lower formation pressure was observed in a well located

nearer the injection well than one farther away. For

example, NH#3 (located 110 feet from the injection well)

showed a pressure of 20.3 psi while the pressure at AM#3

(located 140 feet from the injection well) was 42.3 psi at

1900 hours on June 19 (HPUWCD#1, 1982b). These wells are

located along a northly radial from the air injection well.

These field observations clearly indicated the inhomogeneity

of the medium, whereas the model assumes radial homogeneity.

The magnitude of the formation pressure (in the model) can

be increased by increasing the air permeability, but the

data measured at each observation hole cannot be reproduced

by a radial flow model.

Comparison of Water Level Changes

Water level comparisons are shown graphically in Figure

29 through Figure 33 for the first five days of air

injection. In the model, a maximum radial distance is 4430

feet whereas the measurement most distant from the field air

injection well was only 2800 feet. As some of the water

60

50 -

89

£ ^ Ui

u

o

g ^

30 -

20 -

1 0 -

Figure 29:

Field

Model Original

2 3 (Thousands)

RADIAL DISTANCE (feet)

T~ 4

COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME I

UJ

o

I o

Figure 30

2 3 (Thousands)

RADIAL DISTANCE (feat)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME I

60

90

50 -

£ ^ O

(A

a

OC

30 -

20 -

1 0 -

Q-\

Figure 31 :

1 2 3 4 (Thousands)

RADIAL DISTANCE (feet)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME I

Ul

u

o

OC

Figure 32:

(Thousands) RADIAL DISTANCE (feet)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME I

91

«

Ui o

(A O

OC

Figure 33

(Thousands) RADIAL DISTANCE ( feet )

COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME I

92

level observation wells became inoperative during the air

injection process, only a few water level records were

available from the field. Hence, instead of joining the

locations of these field observed water level changes, they

are plotted as discrete points. Once again, a number of

similarities can be noticed from the comparison of observed

and predicted curves. The dewatered zone in the field was

considerably larger when compared with the dewatered zone

from the model. This may be due to the lower predicted

formation pressures near the well. A wave-like water

surface was noticeable from the predicted results.

The volume of net water gained was also calculated from

the predicted values and compared with field observations.

The net volume changes were summed and are presented in

Table 3. From Table 3, one can observe that the volume of

water gained from the simulation was considerably higher

than the observed value. Based on this observation, a

decision was made to reduce the minimum relative air

permeability.

Scheme II

Scheme II is identical in all aspects to Scheme I,

except for the value of the minimum relative air

permeability. The minimum relative air permeability was

92

level observation wells became inoperative during the air

injection process, only a few water level records were

available from the field. Hence, instead of joining the

locations of these field observed water level changes, they

are plotted as discrete points. Once again, a number of

similarities can be noticed from the comparison of observed

and predicted curves. The dewatered zone in the field was

considerably larger when compared with the dewatered zone

from the model. This may be due to the lower predicted

formation pressures near the well. A wave-like water

surface was noticeable from the predicted results.

The volume of net water gained was also calculated from

the predicted values and compared with field observations.

The net volume changes were summed and are presented in

Table 3. From Table 3, one can observe that the volume of

water gained from the simulation was considerably higher

than the observed value. Based on this observation, a

decision was made to reduce the minimum relative air

permeability.

Scheme 11

Scheme II is identical in all aspects to Scheme I,

except for the value of the minimum relative air

permeability. The minimum relative air permeability was

93

Day

1

2

3

4

5

TABLE 3

COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME I

Estimated Net Water Gained (acre-

Field Model

52.82

110.27

127.34

152.43

216.75

449.77

617.77

632.10

652.57

876.05

•feet)

* Approximately at one day time interval, since the start of air injection

94

reduced to 0.005 of the absolute permeability. The

simulation was carried out for 6.8 days (from the start to

the end of air injection) as in Scheme I. The formation

pressure regression ecjuations were also fitted to the

predicted formation pressures and compared with the

ecjuations for the observed formation pressures. Table 4

shows these equations on a daily basis. Figure 34 through

Figure 40 shows graphically the comparison of the formation

pressure ecjuations. The radial distances to the point where

the formation pressure, P^, ecjuals zero, were also

considerably shorter when compared to the Scheme I, as

expected.

The water level changes were also compared and are

shown in Figure 41 through Figure 45. The water level

changes, predicted by the model, are closer to the field

observations than results from Scheme I. The volume of net

water gain was again calculated and compared with the field

observations. The magnitudes of the water level changes are

summarized in Table 5. From this table, one can observe the

improvement from Scheme I in predicting the net volume of

water recovered.

95

^

u u CQ < H

U Oi D en Ul u Oi t-t 0 4 »-•

2 u 0 S HH w EH K < U S cn Oi 1 0 1 bu Ul

2 1*4 0 0 HH

H 2 < 0 D cn cx HH w C3ci < 04

s 0

u

C -P 0 c: •H (U

(0 u r-A H

0) <4H ^ <4H ^ 0) 0 0 u u

c 0 •H 4J (0 3 cr u

He

Day

^ 0) TJ 0 S

TJ rH (U •H [14

rH 0) TJ 0 s

TJ rH (U •H [Z4

t 00 r-

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( CO o

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CO r-t

1

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in

t CO

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t-i

f-i fi

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(r> vO

r-{

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r-i VO (7

0 1

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r-CN

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t-t r-i

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o • H A-> U

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o Tl c

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<

0) u c

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(0

95

Figure 34

400 200

^ ^ o.sr«.cE (.. PRESSURE COMPARISON OF EORMATI ^^^

u OC Z3 in V) UJ 0^ CL

0.2 0.4 fThousonds) RADIAL DISTANCE ( f t )

Figure 35

RADIAL UiJ'""^ ^

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE IECOND DAY-SCHEME II

97

100

n

u OC D V) (/) u OC (L

2 0 0 4 0 0

F i g u r e 35 RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY--SCHEME II

n a.

UJ OC

(O V) UJ OC Q .

6 0 0

F i g u r e 37:

T r

200 '•^O

RADIAL DISTANCE ( feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME II

9 8

n a H ^

UJ K 3 V) V) UJ OC Q.

Figure 38

RADIAL DISTANCE (feet) COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME II

100

UJ OC

(/) u OC (L

_ -. , i ^ 1 1 ^ : ^ ^ ^ ^ ^ 7^ 1 ^ ' ^ ' / ' «R o a 1 1-2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 O-^^^o^lands) RADIAL DISTANCE (feet)

.Q. COMPARISON OF FORMATION PRESSURE Figure 39. ^^^^^^^j^g ^T THE END OF THE

SIXTH DAY--SCHEME II

99

(0

a. UJ OC ^ (n (/) UJ OC Q.

F i g u r e 40

T 1 1 1 1 1 1 1 \ \ 1 r 0.8 1 1.2 1.4 1.6 1.8

(Thousands) RADIAL DISTANCE ( feet )

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECTION--SCHEME II

100

UJ

u

o

a:

T 1 T 2 3

(Thousands) RADIAL DISTANCE (feet)

Figure 41: COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME II

9 «

UJ

o

o 3 0 -

20 -

10 -

Fie ld Model O r i g i n a l

F i g u r e 42:

1 2 3 4 (Thousands)

RADIAL DISTANCE (feet)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME II

101

I 5

Figure 43

1 2 3 (Thousands)

RADIAL DISTANCE (fset)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME II

u

^

OC

60

50 -

40

30 -

20 -

10 -

-BT a

1 1 1 r 2 3

(Thousands) RADIAL DISTANCE (feet)

-r 4

Figure 44: COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME II

102

« «

o

(n a

&

60

50 -

40

30 -

20 -

1 0 -

"0 -

Model

- Original

- Field

o

2 3 (Thousands)

RADIAL DISTANCE (feet)

Figure 45 COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME II

103

Day

1

2

3

4

5

TABLE 5

COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME II

Estimated Net Water Gained (acre-

Field Model

52.82

110.27

127.34

152.43

216.75

14.21

44.58

140.72

184.05

247.22

•feet)

* Approximately at one day time interval, since the start of air injection

104

Scheme III

Figure 10 indicates the presence of more than one clay

layer. The presence of multiple clay layers undoubtedly

contributed to the differences between the predicted and the

observed values. Hence, the domain was discretized with two

clay layers as shown in Figure 46. This discretization

contained 722 elements and 400 nodes. The top layer

consisted of a clay layer eight feet thick. The lower clay

layer was also eight feet thick but extented radially only

875 feet (compared to 4430 feet for the top layer) based on

the information shown in Figure 10. A saturated thickness

of 30 feet was considered in this discretization. A minimum

relative air premeability of 0.0075 of absolute permeability

was used. This value is slightly higher than the value used

in Scheme II and lower than the value used in Scheme I.

The simulation was carried out for 5.8 days (from the

start to the end of injection) as in Schemes I and II. The

variations of the permeability were also considered in this

Scheme. In the earlier two schemes, the derivatives of the

permeabilities with respect to time were not considered;

however, the values of the relative permeabilities (water

and air) were updated whenever the element saturation

changed, even within the iterations. In the current scheme,

the integrals containing the relative permeabilities were

105

i D

.37

m

. i n r^

<

o Q

- vO ^

0) LI

P ?T

Du

. 0 1

105

modified to consider such variations of permeabilities.

Rewritting the Ecjuation (3.35) with the substitution of the

term, M w

V w rw ^c

In light of assumptions made in Chapter III and by the

application of the chain rule, Ecjuation (4.1) becomes

The formation pressure regression equations were also fitted

to the predicted formation pressures and compared to the

observed field values. Table 5 shows these ecjuations on a

daily basis.

Figure 47 through Figure 53 shows graphically the

comparisons of the formation pressure ecjuations. The

predicted formation pressures were close to the observed

formation pressures. The similarities were more pronounced

in days 3 to 5 (Figure 49 through Figure 51) than the first

two schemes. The water level changes were again compared

and are shown in Figure 54 through Figure 58. The water

level changes are also closer to the field observations.

The magnitudes of net water gain are compared with the field

observations in Table 7. As occured in Scheme I, the volume

107

vO

•J

<

U Oi D Ul Ul

u Oi HH

o EH < u

Oi CO O I [X4

{ZJ

o

I

cn z o

2

o Ul

<

D Oi [z] < 04

o u

o c •H 0) 4J -H (0 U rH -H 0) ^ ^ (4H LI 0) O O O U

c o •H

0]

o

0) •H [X4

d) T3 O

T3

0) •H

>i

Q

CN lO

o I

IT)

o I

CO vO

o I

vO vO

CN

o I

OD ID

O I

CO vO C7>

vO vO C7> cr>

00

vo o I o

I

o I

o I

Oi

o

O

00 in

oi

o rH

O

CO

oi

O

rH

O

00

m

oi

o

O

CN

Oi

o t-i D O

00

Oi

o

o

00 00

vO

in ir>

CN

I

IT)

ID CN

vO CO

C^ CN

CO vO ID

O CO o

in

CO 00 '-*

Oi

o rH

O

Oi

O

o

CN

oi

o

o

in

Oi

o ft

O

CN CO

Oi

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vO CN

Oi

o

O

CN vO

C^ vO

CN

ci

CO

CO

in

in

vo

00

cr>

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CN O vO

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CO CO

Oi

o t-i

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CN

if) CO

1

a> vO

o CO

1

«* CN

CO CO

1

CO CN

o ^

1

0> vO

00 ^

1

KD

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r-\ r-i

1

(J» CO

vO

1

00 (T>

CN

c o •H •p

u 0)

> 1

OJ T)

4 : u (B 0)

»4- l

0

T3 C 4)

<U .C 4-»

-M <

LI • H

fO

V4H 0

-P Li nj 4-> (0

a; (-•

-p

0) u c

• H

01

108

100

n Q.

UJ OC

V) V) UJ OC (L

Figure 47

2 0 0 4 0 0 6 0 0

RADIAL DISTANCE ( feet )

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIRST DAY--SCHEME III

100

01

UJ OC D </) V) UJ OC

(Thousands) RADIAL DISTANCE

Figure 48: COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEME III

109

100

o.

UJ OC D </) in UJ cc Q .

Figure 49: COS^^VSSTSVTOUTION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY—SCHEME HI

UJ OC 3 </» in UJ OC Q.

Figure 50

••00 2 0 0

RADIAL DISTANCE ( fe«t )

COMPARISON OF FORMATION ^R|SSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME III

110

150

m a

UJ OC

«/) UJ OC

a.

6 0 0

F i g u r e 51

2 0 0 +00

RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY--SCHEME I I I

100

n o.

v ^ UJ OC

(/) V) UJ OC Q .

90 -

80 -

— I r 1.8

. , _ _ , , 1 i T 1 1 1 ' ' ' ' ^

, 0*2 ' 0 '4 • 0 6 0 .8 1 ^ 1 - 2 1 - ^ ^-^ 5 0 .2 u.^ w.o (Thousands)

RADIAL DISTANCE ( feet ) Fioure 52- COMPARISON OF FORMATION PRESSURE Figure 52. ^^^^^^^^^3 ^ HE END OF THE

SIXTH DAY--SCHEME III

I l l

100

90 -

80 -

n Q.

UJ OC D m V) UJ OC

a.

70

60

50

40

30

20

10

F i a l d

-Model

T 1 1 1 — 0.2 0.4

—I 1 1 1 1 1 1 1 1— 0.6 0.8 1 1.2 1.4

(Thousands)

— I 1 1 r

1.6 1.8 0 0.2 0.4 0.6 o.H 1 1.:^ i .-* i o i .o 2

RADIAL brSTANCE' (feet)

Figure 53: COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF AIR INJECTION--SCHEME III

112

0

70

80 -

50 -

40 -

30

20 -

10 - •

Figure 54:

Model

Original

Field

1 2 3 4 (Thousands)

RADIAL DISTANCE (feet)

COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIRST DAY--SCHEME III

70

«

UJ

o

(n a

i OC

60 -

50 -

2 3 (Thousands)

RADIAL DISTANCE (feet)

Figure 55 COMPARISON OF WATER SURFACE CHANGES AT THE END OF SECOND DAY--SCHEME III

113

«

UJ

u

(n o

F= OC

0 - t 0

Figure 56:

T 1 r 2 3

(Thousands ) RADIAL DISTANCE ( feet )

COMPARISON OF WATER SURFACE CHANGES AT THE END OF THIRD DAY--SCHEME III

« •

UJ

o

V)

o <! (J »-OC

T

2 3 (Thousands )

RADIAL DISTANCE ( f e e t )

Figure 57: COMPARISON OF WATER SURFACE CHANGES AT THE END OF FOURTH DAY--SCHEME III

70

60 -

114

« «

u

i o

g ^

50 -

40 -

30

20 -

10 -

a

T 1 1 r 2 3

CThousands) RADIAL DISTANCE (feet)

Figure 58 COMPARISON OF WATER SURFACE CHANGES AT THE END OF FIFTH DAY--SCHEME III

115

Day

1

2

3

4

5

TABLE 7

COMPARISON OF ESTIMATED NET WATER GAINED--SCHEME III

Estimated Net Water Gained (acre-

Field Model

52.82

110.27

127.34

152.43

215.75

42.48

133.75

298.50

478.00

552.73

•feet)

* Approximately at one day time interval, since the start of air injection

116

of water gained was considerably greater than that found

through the field observations.

Scheme IV

Scheme IV is identical in all aspects to Scheme III

except the minimum relative air permeability was set to

0.005 of absolute permeability. The simulation was carried

out for 6.8 days (from the start to the end of air

injection) as in all other schemes. The formation pressure

regression equations were fitted to the predicted formation

pressures and compared with the ecjuations derived from the

observed field values. Table 8 shows this comparisons on a

daily basis. Figure 59 through Figure 65 shows graphically

the comparison of the formation pressure ecjuations. The

radial distances to the point where the formation pressure,

PA/ ecjuals zero, were also shorter when compared to the

Scheme III.

The water level changes were also compared and are

shown in Figure 66 through Figure 70. The predicted water

level changes are closer to the field observed water level

changes when compared to the previous schemes. The volumes

of net water gain were also calculated and compared with the

field observations. The volume of water level changes are

summarized in Table 9. Based on comparisons of formation

117

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Figure 59

200 RADIAL DISTANCE (fee*) „^ccTTPF

COMPARISON OE^ORMA^-,^, ™ f

a

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Figure 60

0-* ° V o u s 2 n d s ) RADIAL DISTANCE (fe«0

CO^^ARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE SECOND DAY--SCHEMt. iv

119

100

a

UJ OC D V) V) UJ OC

Figure 61

200 *oo RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE THIRD DAY—SCHEME IV

a UJ OC

V) (n UJ OC

a.

Figure 62

200 *oo RADIAL DISTANCE (feet)

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FOURTH DAY--SCHEME IV

120

UJ OC

v> (/) UJ OC Q.

Figure 63

4 0 0

RADIAL DISTANCE ( feet ) ^ ^ ^ T T T ^ T T

COMPARISON OF FORMATION PRESSURE EQUATIONS AT THE END OF THE FIFTH DAY—SCHEME IV

100

« o.

>.^ UJ OC I/) Ui OC

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0 0 .2 CJ. v.*." (Thousands) RADIAL DISTANCE ( feet )

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

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— 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

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60 -

50 -

40 -

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Original

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-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

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50 -

40 -

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F i e l d Model

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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

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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.

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