Doctor ofPhilosophy in Petroleum Engineering New Mexico ...

LINE SOURCE SOLUTIONS OF INTERFACE CRESTING

PROBLEMS FOR HORIZONTAL WELLS

UNDER STEADY STATE FLOW CONDITIONS

by

Boyun Guo

Submitted In Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy in Petroleum Engineering

New Mexico Institute of Mining and Technology

Socorro, New Mexico

July 1992

ACKNOWLEDGMENTS

I would like to express my gratitude to Dr. Robert L. Lee, my advisor, for his

encouragement and support throughout the duration of this work. In addition, I wish

to thank the other members of my committee, Dr. Stefan Miska, Dr. Randy S. Seright,

Dr. Robert E. Bretz, and Dr. Chia S. Chen, for the guidance and help they have offered

to me, and for their time and patience in reading and evaluating this thesis.

This work was financed by the Engineering Foundation Grant, the Graduate School

of the New Mexico Institute of Mining and Technology, and the Gaz de France. I am

deeply grateful for their support.

Special thanks go to Mr. Dick Carlson for his assistance in typing this thesis.

This work is dedicated to my father, Guo Jun, and my mother, Li Suzhen, for their

sacrifices they made that made my education possible, and to my wife, Wang Huimei, for

her seemingly endless patience and help in encouraging me in fulfilling my study in the

United States of America.

- 1 -

ABSTRACT

The number of horizontal oil wells has been increasing rapidly in the past few years

because of several advantages of their application. However, further improvement in the

oil production rate of horizontal wells is limited by the encroachment of water or gas

when bottom-water or gascap exists. This encroachment of water or gas is called interface

cresting. The behavior of the cresting in horizontal wells is unknown. The critical oil

production rate (defined as the maximum water-free or gas-free production rate) and

location of the water-oil or gas-oil interface need to be determined in horizontal oil wells.

This work focuses on the theoretical analysis of the cresting behavior in line-sink-

like horizontal wells under steady state flow conditions. The principle objectives of this

investigation are (1) to develop an approach to determination of the critical oil production

rate of horizontal wells and (2) to determination of the position of the water-oil or gas-oil

interface.

The mechanism of the water encroachment process is analyzed graphically in this

study. This analysis reveals that the existence (or non-existence) of an unstable water

crest depends upon the flowing pressure gradient beneath the well bore. If this flowing

pressure gradient is less than the hydrostatic pressure gradient of water, a stable water

crest exists right below the wellbore, i.e., no unstable water crest exists.

By treating the horizontal weUbore as a line sink, conformal mapping theory was

adopted in this study to solve the interface cresting problems under steady state flow

conditions. The critical oil production rates and locations of the interfaces for two water-

oil cresting systems and two gas-oil cresting systems were determined analytically in this

investigation. The difference between the two water-oil cresting systems is the different

locations of wellbores in the two flow domains. The wellbore is located at the top of the oil

• •

- 11 -

zone in the first system and below the roof of the oil zone in the second system. The same

difference exists in two gas-oil cresting systems. In the first system, the wellbore is located

at the bottom of the oil zone; while in the second, the wellbore is located above the bottom

of the oil zone. Both the critical case and the subcritical case were studied for the four

systems. The resulting solutions show that the critical oil production rate is proportional

to the conductivity and thickness of the oil reservoir and the density contrast between oil

and the cresting fluid. This critical oil rate is also a function of the wellbore location in

the oil reservoir. Example calculations using the analytical solutions are compared with a

numerical reservoir simulator and an agreement is indicated. These solutions provide an

approximate approach to evaluating the performance of horizontal weUs.

- m -

TABLE OF CONTENTS

ABSTRACT ii

1. INTRODUCTION 1

2. LITERATURE REVIEW 3

3. MECHANISM OF INTERFACE CRESTING 10

3-1. Critical Oil Rate 10

4. STATEMENT OF THE PROBLEM 14

4-1. Assumptions 14

4-2. Governing Equation 14

4-3. Boundary Conditions 15

5. SOLUTIONS TO WATER-OIL CRESTING SYSTEMS 17

5-1. Wellbore at the Roof of the Reservoir, Critical Case 17

5-2. Wellbore at the Roof of the Reservoir, Subcritical Case 25

5-3. Wellbore below the Roof of the Reservoir, Critical Case 32

5-4. Wellbore below the Roof of the Reservoir, Subcritical Case 39

6. SOLUTIONS TO GAS-OIL CRESTING SYSTEMS 48

6-1. Wellbore at the Bottom of the Reservoir, Critical Case 48

6-2. Wellbore at the Bottom of the Reservoir, Subcritical Case 56

6-3. Wellbore above the Bottom of the Reservoir, Critical Case 63

6-4. Wellbore above the Bottom of the Reservoir, Subcritical Case 71

7. RESULTS 80

7-1. Definitions of Dimensionless Variables 80

7-2. The Critical Oil Rate 81

7-3. Location of the Crest 82

- iv-

8. COMPARISON WITH NUMERICAL SIMULATION 83

8-1. Introduction to the Numerical Model 83

8-2. Validation of the Numerical Model 83

8-3. Sensitivity Runs the Numerical Model 84

8-4. Comparison of the Analytical Solution and Simulation 85

9. DISCUSSIONS 86

9-1. Anisotropy 86

9-2. Capillarity and Relative Permeability 88

9-3. Well Bore Condition 89

9-4. The Critical Crest Height at the Cusp Point 90

9-5. Solution Justification 91

10. CONCLUSIONS 96

11. SUGGESTIONS FOR FUTURE RESEARCH 98

12. NOMENCLATURE 99

13. REFERENCES 102

APPENDIX A. METHOD OF SOLVING THE PROBLEMS 107

APPENDIX B. SOURCE CODE OF THE NUMERICAL SIMULATOR 112

APPENDIX C. DATA FILES FOR THE NUMERICAL SIMULATOR 129

— V -

LIST OF TABLES

Table 8.1 - Relative Permeabilities, Normalized 83b

Table 8.2 - Helder Field Parameters, Well A5(RD) 83b

Table 8.3 - Simulation Input Data and Results 85a

— VI —

LIST OF FIGURES

Figure 2-1 - Computed critical rates

Figure 3-1 - Water-cresting for a high pressure gradient case

Figure 3-2 - Water-cresting for a low pressure gradient case .,,.lib

Figure 4-1 - Irregular boundaries in a water cresting system , 15a

Figure 4-2 - Irregular boundaries in a gas cresting system ,,,,16a

A

Figure 5-1 - Mappings of flow Domain for a water cresting system (Case 1) 17a




Figure 6-1 - Mappings of flow Domain for a gas cresting system (Case 1) .... 48a

Figure 6-2 - Mappings of flow Domain for a gas cresting system (Case 2) 56a

Figure 6-3 - Mappings of flow Domain for a gas cresting system (Case 3) .... 63a

Figure 6-4 - Mappings of flow Domain for a gas cresting system (Case 4) ....71a

Figure 7-1 - Solution of critical rate (wellbore at the roof of the reservoir) ,,,,81a

Figure 7-2 - Solution of critical rate (wellbore below the roof of the reservoir) ,81b

Figure 7-3 - Location of critical crest (wellbore at the roof of the reservoir)

Figure 7-4 - Location of critical crest (L^ = 0.5) , , 82b

Figure 7-5 - Location of critical crest (Ly, = 0.7) 82c

Figure 7-6 - Location of critical crest {L^j = 0.9) , 82d

Figure 7-7 - Location of crest (wellbore at the roof of the reservoir) 82e

Figure 7-8 - Location of crest {L^ —1.0) 82f

Figure 7-9 - Location of crest {L^ —0.9) 82g

Figure 7-10 - Location of crest {Lro —0.8) .... 82h

Figure 7-11 - Location of crest —0.7) 82i

Figure 7-12 - Location of crest {Ly, = 0.6) 82j

Figure 7-13- Location of crest (Ly, - 0.5) .... 82k

Figure 7-14- Location of crest (Ly, —0.4) 821

Figure 8-1 - Simulation grids for a horizontal-well-penetrated reservoir 83a

- vii-

CHAPTER 1

INTRODUCTION

The production of oil through a vertical well that penetrates an oil layer underlain

by water or overlain by gascap gas causes the oil-water or gas-oil interface to deform into

a cone shape. As the production rate is increased, the height of the water cone above the

original oil-water contact or the depth of the gas cone below the original gas-oil interface

also increases until, at a certain production rate, the cone becomes unstable and the coning

fluid flows into the well. The maximum water-free or gas-free oil production rate is called

the critical oil rate.

One of the reasons for drilling horizontal wells is to enhance the oil production rate

in those areas where the water or gas breakthrough is a severe problem in vertical wells.

However, the encroachment of the water or gas is still a limitation for further improving

oil production rate in horizontal wells, especially for some cases where the thickness of

the oil reservoir is small. Determination of the critical oil rate is a very practical problem

for horizontal well evaluation. It is also of vital importance to know the location of the

crest-shaped interface for evaluating the oil recoveryfrom horizontal wells and determining

the crest subsidence time. Since horizontal wells have been drilled commercially in the

oil industry only in recent years, very few studies have been reported about the behavior

of the unwanted fluid encroachment, and conflicting results have been given by different

investigators. Therefore, this problem needs to be reinvestigated.

The encroachment of the unwanted fluid is referred to as interface cresting through

out this thesis. In some figures, the cross section of the water or gas crest is referred as

- 1 -

- 2 -

cone since it looks like a cone. In this investigation, the mechanism of the water cresting

process is analyzed graphically. This analysis shows that a stable water crest exists right

below the wellbore if the pressure gradient beneath the wellbore is lower than the hydro

static pressure gradient of water. If the pressure gradient beneath the wellbore is higher

than the hydrostatic pressure gradient of water, an unstable water crest exists.

A line-source horizontal wellbore is assumed and conformal mapping theory is em

ployed to solve the interface cresting problems under steady state flow conditions in this

study. The critical oil production rates and interface locations for two water-oil cresting

systems and two gas-oil cresting systems are determined analytically. The difference be

tween the two water-oil cresting systems is the different positions of wellbores in the oil

reservoirs. The wellbore is placed at the roof of the oil reservoir in the first system and

below the roof of the oil reservoir in the second system. This same difference exists in

the two gas-oil crest systems. However, in the first gas-oil cresting system, the wellbore is

placed at the bottom of the oil reservoir; while in the second, the wellbore is placed above

the bottom of the oil reservoir. Both the critical case and the subcritical case were solved

for the four systems. The solutions are presented in terms of dimensionless variables.

CHAPTER 2

LITERATURE REVIEW

The interface coning problem in vertical wells was first investigated in 1930's.

Muskat and Wyckoff [1935] presented an analysis of water-coning more than fifty years

ago. Their results indicate the existence of a critical cone height which, for the particular

system represented by their electrical conduction models, is in the vicinity of 76 % per

cent of the total distance between the bottom of the well and the normal water level.

Muskat concluded in his work (Muskat [1946]) that this critical cone height is about 60 %

to 75 % of that distance. Because of the difficulty of the simultaneous determination of

the pressure distribution and the shape of the cone, they did not give an explicit solution,

instead, their method consisted of a tedious graphic procedure which yielded the critical

oil rate. Their theory gives a critical oil rate which is too high (Schols [1972]) because they

assumed that the pressure distribution for the water-coning system is effectively the same

as that for the case where there is no cone and the oil is flowing in a sand between two

parallel impermeable boundaries and flowing into a well partially penetrating the sand.

Meyer and Garder [1954] came up with an approximate equation assuming radial

flow. They derived a simple formula giving the maximum theoretical flow rate into a well

under the condition that no water would be produced, with consideration of the existence

of water-cone. This maximum theoretical flow rate is referred to as the production rate

at which stable cone reaches the bottom of the well. This rate can be taken as the critical

rate only for cases where the conductivities of the sands are very high. Moreover, since

they assumed that radial flow prevails, their results are too low in value (Schols [1972]).

- 3 -

-4-

Chaney et al. [1956] developed a set of curves by which maximum water free pro

duction rates can be determined for wells having a gas-oil contact, water-oil contact, or

both, which are based on Muskat's mathematical analysis and potentiometric analyzer

study.

Karp et al. [1962] analyzed the horizontal barriers for controlling water-coning

experimentally and theoretically based on Muskat's theory and Meyer's theory.

Chierici [1964] presented an approach to determination of the maximum permissible

oil rate without water or gas production starting from Muskat's theory and using the

potentiometric model technique.

Welge and Weber [1964] presented a 2-D method for calculating water coning be

havior. Their computer output matched coning behavior of the laboratory sandpacked

model and of several producing wells experiencing water or free gas production by coning.

Sobocinski and Cornelius [1965] presented a correlation for predicting the behavior

of water cone as it builds from the static water oil contact to breakthrough conditions.

The correlation was based on a limited amount of experimental data from a laboratory

sandpacked model and on results from a computer program for a 2-D incompressible

system.

Wheatley [1985] presented a 3-D approximate theory of water-coning. He considered

the effect of existence of cone on the pressure distribution in such a way that a uniform

pressure distribution in a sand for a pure radial flow case is modified by superposition of

line and point source flow strengths. The critical flow rate can be obtained, according to

his theory, by trial and error (iteration) technique. His approximate approach is good for

values of dimensionless drainage radius in the range of 2 to 10.

Hoyland et al. [1986] presented two methods for predicting critical oil rate in

anisotropic formations with well completed from the top of the formation. One is an

analytical solution, and the other is a numerical simulation. Their analytical solution is

similar to that of Muskat's but the assumption of uniform flux at the wellbore was re

placed by that of an infinitely conductive wellbore. The pressure distribution function

- 5 -

they used was given by Papatzacos [1987]. However, they still assumed that the potential

distribution in the oil zone is unperturbed by the water cone.

Dagan and Bear [1968] studied the salt-fresh water interface coning problem. They

concluded that the critical cone height is 1/3 to 1/2 of the distance between the bottom

of the well and the normal interface.

In all the above mentioned investigations, the effect of limited entry of the wellbore

on the oil productivity of the well was not taken into account. Therefore, according to

their theories, the maximum water-free production rate would be obtained when wellbore

penetration is zero, which is physically impossible.

Abass and Bass [1988] studied the performance of water coning under different

boundary conditions analytically, numerically and experimentally. In their study a fully

implicit, strongly coupled mathematical model was used to treat rapid pressure-saturation

change problems. On the other hand, a plexiglass model was constructed to obtain qual

itative and quantitative descriptions of water coning phenomenon. According to their

experimental study, there is no critical oil rate for the real case. However, they derived an

analytical solution for calculating the water free oil rate for steady state and pseudosteady

state flow conditions in 2-D radial flow systems using average pressure concept. Although

the adoption of their 2-D radial flow assumption and average pressure concept are not

very apropos for water coning systems, they were the first group of investigators who took

into account the effect of limited entry of wellbore on the maximum available water-free

oil rate. According to their solution, the optimum fractional wellbore penetration interval

(completed from the top of the formation) should be 0.5 in oil-water coning systems.

Yang and Wattenbarger [1991] present two correlation methods for water coning

calculations based on numerical simulation.

Although the critical oil rate was shown to exist analytically (Muskat [1935], Wheat-

ley [1985], Hoyland [1986]) and wasobservedexperimentally by someinvestigators (Henley

[1961], Sobocinski [1965], Khan [1972], Dagan [1968]), it was not observed in certain sit

uations (Abass [1988]).

- 6 -

Guo and Lee [1992] presented an approach to optimization of the completion in

terval in oil-water coning systems. Their graphical analysis explained why sometimes the

unstable water cone and the critical rate defined from it do not exist. According to their

theory, the maximum water-free production rate can be obtained for an optimum wellbore

penetration which is less than one-third of the pay zone thickness but greater than zero.

In summary, the interface coning in vertical wells is a problem whose analytical

solution is known to be difficult because the shape of the cone is not known a priori and

is part of the required solution. A number of investigators (Muskat [1935], Meyer [1954],

Chaney [1956], Karp [1962], Rechardson [1971], Bournalzel [1971], Wheatley [1985], Hoy-

land [1986], Piper [1987], Abass [1988]) have studied this coning behavior analytically.

Some authors (Welge [1964], Blair [1969], Letkman [1970], Woods [1975], Chappelear

[1976], Trimble [1976], Abass [1988]) studied the water-coning problem by numerical sim

ulation, while others (Henley [1961], Karp [1962], Sobocinski [1965], Khan [1970], Bor-

rnalzel [1971]) studied this problem through experiments. Although many researchers

have studied this coning behavior while determining the critical oil rate, their results are

frequently conflicting. Fig. 2-1 is from Guo and Lee [1992]. It shows the comparison of

critical flow rates given by six investigators. Since the eifect of limited entry of wellbore on

the oil productivity of the well was not taken into account in the critical rate calculations

in the first four investigations, all four curves in the figure show that the maximum water-

free production rate would be obtained when the wellbore penetration is zero, which is

physically impossible. The differences between the calculated critical flow rates given by

the four methods are quite large because they made different assumptions to simplify the

problem.

The idea of using horizontal wells to increase the area of penetrated reservoir dates

back to the early 1940's (Ranney [1941]). Considerable effort was directed toward horizon

tal drilling during the 1950's and 1960's, mostly in the USSR. Until early of 1980's, very

few horizontal weUs had been drilled in western countries. The current focus on horizontal

well production resulted from the combined effects of improvements in directional-drilling

and the need for new oil. Since Stramp [1980] reported that two horizontal wells were

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- 7-

drilled in the Empire Abo Unit in New Mexico in order to reduce gas coning tendencies,

the number of horizontal wells has increased rapidly every year. For the general use of

horizontal wells in industry, Sherrard et al. [1987] reported drilling techniques and testing

results from several horizontal wells.

Production behavior of some horizontal weUs were reported by Reiss [1987], Bro-

man [1990], and Murphy [1990]. Pressure Transient Analysis and inflow performance for

horizontal wells were investigated by Goode [1987] and Kuchuk et al. [1988].

Chaperon [1986] studied the behavior of coning toward horizontal weUs in an

anisotropic formation assuming constant interface elevation at a finite distance. Her ap

proach is identical to the well known approach used by Muskat, equilibrium conditions

are stated first. This approach requires expression of the viscous flow potential pertinent

to the geometry of the flow created by a horizontal well. However, as in Muskat's ap

proach, the presence of the immobile crest which reduces the cross section was not taken

into consideration in Chaperon's solution. Since she neglected the flow restriction due to

the immobile water in the cone, her theory should give an overly optimistic evaluation of

critical rate.

Assuming the horizontal well is situated near the roof of the reservoir for the lateral

edge drive and the bottom water drive cases, Giger [1986] presented analytical 2-D models

of water cresting before breakthrough for horizontal wells. Since he used the free surface

boundary condition (velocity directed toward the surface at a given distance) and assumed

that the free surface is at a large distance, the oil height in the models is difficult to choose.

In fact, he showed that the interface is tilted with an asymptotic parabolic shape, which

means that nowhere does it tend to its initial shape, but instead, at some distance from

the well, it drops lower than initial water-oil-contact, as low as noiinus infinity. Although

these mathematical solutions were modified by the author, he still suggested that these

solutions not be used for small values of dimensionless drainage radius. In fact, an ideal

solution similar to Giger's solution was also given by Efros [1963] in the early sixties. But

again, this cannot be used in practice.

- 8 -

Kuchuk et al [1988] analyzed the pressure transient behavior of horizontal weUs

with and without a gas cap or aquifer. They considered the irregular two-phase boundary

(cone-shaped interface) as a constant pressure boundary and assumed that the gravity

effect is negligible and the viscosity of fluid is constant throughout the medium. According

to their solution, the existence of a gas or water crest does not aifect the pressure response.

Ozkan and Raghavan [1988] investigated the time-dependent performance of hori

zontal wells subjected to bottom water drive. They assumed the reservoir boundary at

the top of the formation and the boundaries at the lateral extent of the formation to be

impermeable, and that an active aquifer at the bottom of the reservoir would yield an

effect identical to that of a constant pressure boundary located at the original water-oil

contact. The most unacceptable assumption they made is that the mobility of water in

the flooded portion of the oil zone (water in water-crest) is the same as the mobility of the

oil. Furthermore, they assumed the density difference between the oil and water is negli

gible. These two assumptions are equivalent to an assumption that the moving water-oil

interface does not exist.

Joshi [1988] investigated the well productivity of slant and horizontal weUs using

Giger's theory. Joshi [1990] also compared the results given by the above theories and

found out that they are conflicting and different by a factor as high as twenty.

Papatzacos et al. [1988, 1989] solved the two-phase cresting problem using the

moving boundary method assuming gravity equilibrium in the cones. The results in their

papers are based on new semi-analytical solutions for time development of a gets or water

cone and simultaneous gas and water cones in an anisotropic, infinite reservoir with a

horizontal well placed in the oil column. The essential assumption they introduced con

cerning gas and water is that they are, at each time, in static equilibrium. In other words,

they participate in the movement of the interface boundaries by expanding when pressure

falls, yet their flow is neglected. This assumption of compressible flow contradicts their as

sumption of incompressible flow. Their solution should be valid only in the infinite-acting

period at low rates. In their semi-analytical solution, the total value of numerical errors

(introduced by the timestep procedure, the convergence acceleration and roundoff) was

estimated to be 5 —10 percent.

CHAPTER 3

MECHANISM OF INTERFACE CRESTING

In reality, the water-oil or gas-oil interface in a porous medium is a transition zone

due to capillarity and relative permeability effects. The thickness of the transition zone

depends on the magnitude of the capillary pressure and relative permeabilities. In this

study, these effects are assumed to be negligible and an abrupt contact between water

and oil zones or between gas and oil zones are assumed to exist. A discussion is made in

Chapter 9 on this assumption.

Water-oil-interface cresting (water cresting) is the process by which water in an

underlying aquifer is sucked up against the forces of gravity into production perforations

in the oil zone. Similarly, with an overlying gas cap one may experience gas-oil-interface

cresting (gas cresting). For convenience, only water cresting is treated in this chapter. All

results can be translated to cover gas cresting.

3-1. Critical Rate

In a steady state flow condition, a constant production rate causes a constant pres

sure drawdown at every point within the constant potential boundaries, which results in

a stable potential distribution in a reservoir. When the oil production rate and pressure

at the outer boundary are fixed, the spatial variation (distribution) of pressure in the

reservoir is a function of the conductivity, which is defined as the permeability of reservoir

rock divided by the viscosity of the flowing fluid in the reservoir. In this section, we study

- 10 -

-li

the change in height of the crest between two steady state flow conditions. The transient

flow between the two steady state flow conditions is not dealt with in this investigation.

In water cresting systems, the pressure drawdown causes water at the bottom of the

oil zone to rise to a certain height at which the upward force is balanced by the weight of

water beneath this point. As the lateral distance from the wellbore increases, the pressure

drawdown decreases, so, the height of the balance-point decreases. Therefore, the locus

of the balance-point is a stable crest-shaped water oil interface when the production rate

is less than a certain value. Oil flows above the water-oil interface, while water remains

stationary below the interface.

Figures 3-1 and 3-2 are schematic diagrams used for interpreting the mechanism

of water cresting. Figure 3-1 illustrates the high pressure gradient case, and the low

pressure gradient case is shown by Figure 3-2. In both of the figures, the stabilized

pressure distribution along the vertical direction beneath the wellbore is plotted on the

right. The line A—B represents the pressure distribution in the oil zone when the oil flow

rate is zero (stationary oil pressure), and the line B—C represents the pressure distribution

in the water crest (stationary water).

In a system represented in Fig. 3-1, if the oil flow rate is increased from zero to

Qoi, the flnal stabilized oil pressure distribution curve corresponding to the flow rate Qoi

is represented by the curve marked "Qoi". This curve intersects the line B-C at one

point. The height of this point from the original water-oil-contact is the height of the

stable crest 1. If the oil flow rate is increased from Qoi to Qo2i the final stabilized oil

pressure distribution curve corresponding to the flow rate Q02 is represented by the curve

marked "^02"* curve intersects the line B-C at two points. The height of the lower

point from the original water-oil-contact is the height of the stable crest 2. If the oil

flow rate is increased from Q02 to the final stabilized oil pressure distribution curve

corresponding to the flow rate Q03 is represented by the curve marked "Q03". This curve

intersects the line B-C at point D. The height of point D from the original water-oil-

contact is the height of the stable crest 3. Now the flowing pressure gradient above point

D is every where higher than the hydrostatic water pressure gradient represented by line

unstable /cone

cone 3

cone 2

cone 1

Qo1 < Qo2 < Qo3 = Qocr

Fig. 3-1 - Water-coning for a higli pressure gradient case

- lla -

Qo3 > Qo2 > Qo1

cone 3

cone 2

cone 1

Fig. 3-2 - Water-coning for a low pressure gradient case

- lib -

- 12 -

B-C. If the oil flow rate is further increzised, no more stable crest can be established. Even

a slight increase in oil rate above Qo^ can cause the stable crest 3 to become unstable and

water will flow into the wellbore. The maximum water-free production rate {Qoz) is called

critical rate which is denoted as Qocr in Fig. 3-1.

In a system shown in Fig. 3-2, if the oil flow rate is increased from zero to Qoi,

the final stabilized oil pressure distribution curve corresponding to the flow rate Qoi is

represented by the curve marked "Qoi"- This curve intersects the line B-C at one point.

The height of this point from the original water-oil-contact is the height of the stable

crest 1. If the oil flow rate is increased from Qoi to Q02, the final stabilized oil pressure

distribution curve corresponding to the flow rate Q02 is represented by the curve marked

"Qo2"- This curve still intersects the line B-C at one point. The height of this point

from the original water-oil-contact is the height of the stable crest 2. If the oil flow rate is

increased from Q02 to the final stabilized oil pressure distribution curve corresponding

to the flow rate Q03 is represented by the curve marked "Q03". This curve intersects the

line B-C at bottom of the wellbore. The height of the stable crest 3 equals the thickness

of the oil zone. In this system, no unstable water crest exists. The maximum water-free

production rate, or critical rate, is Qoa-

In summary, if the reservoir conductivity is high, the flow resistance of rock to

flowing oil is low and the wellbore pressure cannot be dropped below the hydrostatic water

pressure of the system before water breakthrough into the well occurs. Therefore, the final

stabilized oil pressure distribution curve intersects the water pressure distribution line at

only one point. The height of the intersection point is the height of stable water crest if

capillary pressure is negligible. When the wellbore pressure is dropped to the hydrostatic

water pressure of the system due to the increased flow rate, the stable crest contacts the

bottom of the wellbore. However, if the reservoir conductivity is low, the resistance of the

rock to flowing oil is high and the wellbore pressure can be dropped below the hydrostatic

water pressure of the system before water breakthrough into the well occurs. In this case

the flnal stabilized oil pressure curve can intersect the water pressure distribution line at

two points. The height of the lower intersection point is the height of stable water crest.

- 13 -

As production rate is increased, the pressure drawdown increases everywhere in the oil

zone. Therefore, the lower intersection point shifts upward and the upper intersection

point shifts downward until they meet at one point (point D in Fig. 3-1). The height

of this point is the maximum height of stable water crest. Since the oil pressure gradient

above this point is everywhere greater than hydrostatic water pressure gradient, a further

increase in flow rate causes the water crest to become unstable.

An example of the high conductivity case is the experiment conducted by Abass and

Bass [1988]. In their physical model, they used grooves cut on the surface of a piece of

plexiglass to simulate a porous medium. Since the conductivity of this "porous medium"

is very high (resulting in a very low pressure gradient), it was impossible for them to

observe the unstable water crest. However, for the general case in the oil reservoirs, where

the conductivities of formations are low as compared to the conductivities of man-made

porous media in laboratories, unstable crests exist at high oil production rates.

CHAPTER 4

STATEMENT OF THE PROBLEM

4-1. Assumptions

The following assumptions are made in this investigation:

1. Oil reservoir is homogeneous and isotropic,

2. Steady state flow condition prevails,

3. Capillary pressure is negligible, an abrupt interface exists,

4. The horizontal wellbore is long enough, so that a two-dimensional flow dominates,

5. The horizontal wellbore behaves like a line sink in the reservoir,

6. Fluids are of constant and known properties.

Discussions on these assumptions will be given in Chapter 9 in detail.

4-2. Governing Equation

Oil is a slightly compressible fluid. Oil flow through a homogeneous porous medium

under a steady state flow condition is governed by Laplace's Equation. Laplace's equation

is expressed as

vv = 0 (4-1)

- 14-

- 15 -

where the Laplacian operator in Cartesian system is

dx

d<i>

V' =^ +^. (4-2)dx^ dy^ '

The velocity potential, in Eq. (4-1) is defined as

<i>=~{po9y + p) (4-3)H-

where

X= lateral distance, ft [M]

y = vertical distance, ft [M]

k = isotropic permeability of oil reservoir, md

y. = viscosity of oil, cp [mPa-s]

po = density of oil, Ibm/ft^ [gram/cm^]

g = 980 cm/s^ = 32/</5^, gravitational acceleration

p = pressure, psi [Pa].

4-3. Boundary Conditions

It should be noted that although the boundary conditions are stated in this section

in terms of velocity potential, they are not applied in this form when conformal mapping

theory is employed. In fact, instead of velocity potential, only the location of the oil-water

and oil-gas interfaces are solved in the investigation.

Figure 4-1 illustrates a cross section of an oil reservoir underlain by a water zone.

The wellbore is treated as a line sink and assumed at point E. The boundary conditions

of oil flow in the water-oil cresting system are described as below:

d<f> =00, for X = 0, y = ys, (4 - 4)

d(f>= 0, for x = 0, (4-5)

= 0, for y = yAF, (4 - 6)dy

y//////////////////////

////

)^^

////

///////

Vx

=q

/2d

AB

CD

Flo

win

gO

il

ui

Sta

tio

nary

Wate

r

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

77

7

Fig

.4

-1~

Irre

gu

lar

bo

un

dari

es

ina

wate

rcre

stin

gsy

stem

- 16 -

Tx =h = (4-7)where

q = flow rate per unit length of wellbore, bbl/day/ft [M^/s/M]

d = thickness of oil reservoir, ft [M].

The difficulty in treating the interface boundary A-B-C-D is that its shape is flow rate

dependent and unknown. However, the velocity potential along the water-oil interface

may be expressed as

k<i> = --{po9y + P), for y = vabcD' (4 - 8)

Since Fig. 4-1 shows that p = —pw9y for y = yABCDt Eq. (4-8) becomes

kg^p . /. /vN(l> = y, for y = yABCD, (4 - 9)

where Ap = py,—po is the density difference between water and oil in reservoir conditions.

The cross section of an oil reservoir overlain by a gascap gas zone is shown in Fig.

4-2. The wellbore is again treated as a line sink and assumed at point E. The boundary

conditions of oil flow in the gas-oil cresting system are also given by Eq. (4-4) through Eq.

(4-7) except for the gas-oil interface boundary. However, the corresponding parameters

are shown in Fig. 4-2. The velocity potential along the gas-oil interface may be expressed

as

k<!> = —{PoQy + p), for y = yABCD' (4 - 10)

Since Fig. 4-2 shows that p = -pgQy for y = yABCD-, Eq. (4-10) becomes

kgAp -(f>= for y = yABCD, (4-11)

where A/? = p^ —pg is the density difference between oil and gas in reservoir conditions.

////////////////////

X

AB

CD

sta

tio

nary

Gas

Vx

=q

/2d

Flo

win

gO

il

Fig

.4

-2~

Irre

gu

lar

bo

un

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es

ina

gas

cre

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gsy

stem

o>

(U

CHAPTER 5

SOLUTIONS TO WATER-OIL CRESTING SYSTEMS

In this chapter, four cases in two water-oil cresting systems are studied, and the

critical oil rates and locations of the water-oil interfaces in the two systems are determined.

Case 1: Wellbore is located at the roof of an oil reservoir: critical case as shown in

Fig. 5-l(a).

Case 2: Wellbore is located at the roof of an oil reservoir: subcritical case as shown

in Fig. 5-2(a).

Case 3: Wellbore is located below the roof of an oil reservoir: critical case as shown

in Fig. 5-3(a).

Case 4'' WeUbore is located below the roof of an oil reservoir: subcritical case as shown

in Fig. 5-4(a).

The conformal mapping method is employed in solving the water-oil-interface cresting

problem. This method is discussed in Appendix A.

5-1. Wellbore at the Roof of an Oil Reservoir, Critical Case

5-1-1. Flow Domain in Physical Plane z = x + iy

Since the flow region in physical plane is symmetric, only half of the flow region,

as shown in Fig. 5-1(a), is studied. Point D is a. sink which represents the cross section

- 17-

- 18 -

of horizontal wellbore. The wellbore is treated as a line-sink. Other boundary conditions

were discussed in Chapter 3.

5-1-2. Flow Domain in Hodograph Plane i/ = Vx-\- iVy

Mapping of the flow domain in the hodograph plane is based on the mapping of the

domain in physical plane.

At point Ay Vx = qc/2d and Vy = 0, so point A is located at point (qc/2d,0) in the

hodograph plane.

At point ByVy = 0. According to Eq. (A-8) Vx = 0. Hence, point jB is a stagnation

point, and it should be located at origin in hodograph plane.

The mapping of boundary B - C is a, semicircle according to Eq. (A-8).

Since point D represents a cross section of line-sink, the velocity at the point is

assumed to be infinite. If we denote Db and Di as two points approaching point D from

below and from the left of point D, respectively, we may obtain that = 0, Vy = +oo at

point Db and Vjp = oo, Vj, = 0 at point Di.

Straight boundaries in physical plane are mapped onto the hodograph plane as

straight lines.

Based on the above analysis, the flow domain in hodograph plane is mapped in

Fig. 5-l(b). Comparison of Fig. 5-1(a) and Fig. 5-l(b) shows that the mapping from

physical plane to the hodograph plane is not conformal, but isogonal.

5-1-3. Flow Domain in a Modified Hodograph Plane e =

Because of the inconformity of mapping from the physical plane to the hodograph

plane, we need to modify the hodograph plane to use conformal mapping theory. Among

many transformations, the following one may be utilized to obtain the conformity:

. -a£ = ot-\- ip =^1 +

- 19 -

= (5-1-0)Vx - tVy

This transformation is chosen because it simply relates to the flow domains represented

in the z—plane and in the ft-plane through Eq. (A-5) in Appendix A. In fact, combining

it with Eq. (A-5) givesdO. i

from which one obtains

dz = —ied^. (5-1-1)

Eq. (5-1-1) is the fundamental equation for the solution of our problem.

Using Eq. (5-1-0), the flow domain in hodpgraph plane is mapped onto the modified

hodograph plane (e-plane) as shown in Fig. 5-l(c).

5-1-4. Flow Domain in Complex Potential Plane ft = ^ -f-

Since the value of velocity potential <j> is relative, it does not matter where we map

each point of flow boundary on the complex potential plane, as long as these points are

arranged in such an order as to follow one direction on a stream line. Choosing the

stream function as zero along the lower boundary of the flow domain in physical domain

arbitrarily and assuming the velocity potential at point B is zero, we map the flow domain

in the complex potential plane as shown in Fig. 5-1(d).

5-1-5. Flow Domain in a Intermediate Plane §? = i -|- is

Since it is difficult to directly relate the flow domain represented in the modified

hodograph plane and that in the complex potential plane, we may indirectly relate them

through a third plane. We can first map the flow domains represented in both of the

two planes to the upper half of a intermediate plane by two transformations and then

combine the two transformations. The flow domain represented by the upper half of the

intermediate plane, ^ = t + is^ is shown in Fig. 5-l(e). In the figure, = -l^ts = 0

and tc = oo are chosen arbitrarily.

20 -

5-1-6. Transformation Between e-Plane and Plane

The transformation between £—plane and plane may be obtained by employing

the Transformation of Schwarz and Christoffel:

e=Aj(n- +B. (5-1-2)Since we have chosen to ——= 0 and tc = oo, Eq. (5-1-2) may be simplified as

'•"Jim*'which can be integrated along the real axis as

£ = Alnl 1 +B. (5-1-4)/ \/t + 1 - 1\

x\Zr+T-t-1/

To determine the constant A, we examine the change in e at point B:

For t <tB = 0, one has

Vi+T-lVvrri+iy

For t>tB = Oi

=lnV%/<TT+iy VF+T+I

Therefore the change in e at point according to Eq. (5-1-4), is

Vt+T 1

-v/T+T—1

Ae = i4(i0 —27r)

= —iwA.

Yet the mapping on the £—plane shows

hence,

or

A£ = -kgAp'

kgAp

A = -

= —iirA,

tfi

irkgAp'

-|- iT.

-I- iO.

(5-1-5)

- 21 -

Substitution of Eq. (5-1-5) into Eq. (5-1-4) gives

ifie = -

irkgAp \\/r+T-i-1/

To determine the constant B, apply Eq. (5-1-6) at point D:


e =

0 = -ifi

irkgAp

kgAp

B = -kgAp'

Substituting Eq. (5-1-7) into Eq. (5-1-6) gives

\yjt -f 1 -1- 1/

(5-1-6)

(5-1-7)

(5-1-8)

5-1-7. Transformation Between ft-Plane and 3I?-Plane

The transformation between ft-plane and K-plane may be again obtained by em

ploying the Trsinsformation ofSchwarz and Cbristoffeh

fl =AJiU - + (5-1-9)Since we have chosen = —1,^5 = 0 and tc = oo, Eq. (5-1-9) may be simplified as

- (5-1-1.)

ft =+

which can be integrated along the real axis as

A

+ B

1^(—)1 vt+iy+ B.

To determine the constant A, we examine the change in ft at point A:

For t < tA,

For t > tAy

t-tA

^ + 1

t-tA

t-\-l

4- ITT.

-1- iO.

(5-1-11)

- 22 -

Therefore the change in Q at point A, according to Eq. (5-1-11), is

A^ i4 , . . AAft = -—-^(lO —ir) = —ZTT

hence

or

t>l + l

But the mapping on the ft plane shows that

Aft = -2 '

2 <^ + 1

9c

^>1 + 1

U +1 27r


ft = —In -H B.27r V <+ 1 y

To determine the constant 5, apply Eq. (5-1-13) at point Bi

Qc0 =



Sl=^]n2ff L«^(i + l)J

(5 - 1 - 12)

(5 - 1 - 13)

(5 - 1 - 14)

(5 - 1 - 15)

5-1-8. Solution to Water-Oil Interface, Critical Oil Rate

The solution of water-oil interface can be obtained solving Eq. (5-1-1). Differenti

ating Eq. (5-1-15) gives

= T-^ dt. (5 - 1- 16)2ir\t-tA t + lj ^ '

Substitutions of Eq. (5-1-8) and Eq. (5-1-16) into Eq. (5-1-1) gives

dz = —2TkgAp

-In '̂̂ '̂ ZI—) - i] (— ) dt..TT vvm+1/ . \t-tA t+ij (5-1-17)

- 23 -

To solve the water-oil interface, we need to separate the imaginary part from this complex

equation.

Between points B and C, < > 0, so

VVi+T+l/

The imaginary part of Eq. (5-1-17) is then

dy = (— dt. (5 - 1- 18)^ 2Trkgl^^p\t-tA ^ ^

The height of water crest (he) can be obtained by integration of Eq. (5-1-18):

y

hc =y-yB =j dyVB

= / Mc (I l_\.J 2irkgAp\t-tA t+l)

(5 - 1 - 19)

ta

t-tAIn

2TrkgAp

At the cusp point C, t approaches infinity. The height of the water crest {He) is

To determine the constant tA, one can apply Eq. (5-1-8) at point A:

(5 - 1 - 21)

L-«A(t+l).

.2d _ ngc kgb.p

Solving Eq. (5-1-21) for Ia gives

ri /ym-n itir \^/i7TT+lJ

where

c„, = exp . (5-1-23)

Assuming the dimensionless critical-crest-height is HcDi the real critical-crest-height

JIc is then Hcd^- Therefore, the critical oil production rate for unit length of wellbore,

according to Eq. (5-1-20), is2TllaDdkgAp

Mln(=b) •

- 24-


-2TrHDdkgAp9c =

/^In ~(i+fe)(5-1-25)

However, since C^i in Eq. (5-1-25) is still a function of gc? we must solve for Qc

using a numerical method (see Chapter 7).

The real part of Eq. (5-1-17) is

(5-1-26)27r2A;flrAp \<-H 1 t-tAj \y/tTT-\-lJ

which can be integrated as

Now, Eq. (5-1-27) can now be integrated numerically.

The height of water crest, given by Eq. (5-1-19) can be plotted versus lateral

distance, x, given by Eq. (5-1-27) with parameter t. The plot gives the location of the

water-oil-interface (see Chapter 7).

Eq. (5-1-24) indicates that the critical rate is a function of critical crest height at

the cusp point which is an unknown variable. This limits the application of the theory.

Theoretically, this can be avoided. In fact, Eq. (5-1-19) shows that at the outer boundary

where the crest is null, the parameter t takes a value of zero. Substituting t = 0 into the

upper boundary of the integral in Eq. (5-1-27) yields

Theoretically, the critical rate (gc) can be numerically solved by coupling Eq. (5-1-28)

and Eq. (5-1-23). However, it is found that the integral converges extremely slowly. It

may be impossible to numerically solve these two equations using the currently available

computers. Therefore, Eq. (5-1-24) has to be used for determining the critical rate.

- 25 -

5-2. Wellbore at the Roof of an Oil Reservoir, Subcritical Case

5-2-1. Flow Domain in Physical Plane z x-\-iy

Since the flow region in physical plane is symmetric, only half of the flow region as

shown in Fig. 5-2(a), is studied. Point is a sink which represents the cross section


were discussed in Chapter 3,

5-2-2. Flow Domain in Hodograph Plane i/ = Vx-\- iVy

Mapping of the flow domain in hodograph plane is based on the mapping of the


At point A, Vx = g/2rf and Vy = 0, so point A is located at point (g/2d,0) in the

hodograph plane.

At points jB, Vy = 0. According to Eq. (A-8) in Appendix A v® = 0. Hence, point

is a stagnation point, and it should be located at the origin in the hodograph plane. So

is point D.

The mapping of boundary B - C - D is part of a circle according to Eq. (A-8).

Since point E represents a cross section of a line-sink, the velocity at the point is

assumed to be infinite. If we denote Eb and Ei as two points approaching point E from

below and from the left of point E^ respectively, we obtain that Vx = 0, = -}-oo at point

Eb and Vx = -|-oo, Vy = 0 at point Ei.

Another fact for hodograph mapping is that straight boundaries in physical plane

be mapped onto the hodograph plane as straight lines.


Fig. 5-2(b). Comparison of Fig. 5-2(a) and Fig. 5-2(b) shows that the mapping from


Wkgl^P

- 26 -

5-2-3. Flow Domain in a Modified Hodograph Plane £ = a +1/?

Because of the inconformity of mapping from physical plane to hodograph plane,

we need to modify the hodograph plane to use conformal mapping theory. Among many

transformations, the following one may be utilized to obtain the conformity:

+ vj

Vfc - tv.(5-2-0)

jx — n'y


in 2—plane and in ft—plane through Eq. (A-5) in Appendix A. In fact, combining it with

Eq. (A-5) gives:dil _ idz ~


dz = —iedn. (5-2-1)

Eq. (5-2-1) is the fundamental equation for solving our problem.

Using Eq. (5-2-0), the flow domain in hodograph plane is mapped onto the modified

hodograph plane (e-plane) as shown in Fig. 5-2(c).

5-2-4. Flow Domain in Complex Potential Plane Q = +

Since the value of velocity potential <l> is relative, it does not matter where we map





in complex potential plane as shown in Fig. 5-2(d).

-27-

5-2-5. Flow Domain in a Intermediate Plane U = t + is






intermediate plane, ^ = t-\- is, is shown in Fig. 5-2(e). In the figure, <£? = = 0

and to = oo are chosen arbitrarily.

5-2-6. Transformation Between £-Plane and Plane

The transformation between £-plane and 3fi-plane may be obtained by employing

the Transformation of Schwarz and Christoffek

e=Ajiit- +B. (5-2-2)Since we have chosen ts = -l,tB = 0 and to —oo, Eq. (5-2-2) may be simplified as

(S-<c)


e = A\Vt+T+iJ.

+ B.


For t <tB = 0,

ViTT-i

For t>tB = 0,

+1 +1/

Vv^r+T-i-iy

1 "I" 1

v^+T-1

Vi+T-H

Therefore the change in s at point B, according to Eq. (5-2-4), is

+ ZTT.

+ iO.

Ae ——Atc{i^ —i^) = iirAtc-

(5-2-3)

(5-2-4)

- 28 -

But the mapping on the £-plane shows

hence,

or

ifie =

irkgAptc

To determine the constant B, apply Eq. (5-2-6) at point E:


A£ = -kgAp'

kgAp= iirAtc,

tilA =

irkgAptc'


2Vi+T -tchiXx/i+T 1/.

+ B.

0 =tfj,

irkgAptc[—tc ln(—1)] -f-

B =kgAp'


'2y/iTl 1, /\/r+T-l"e =

kgAp iirtc \ + 1 + 1 /

(5-2-5)

(5-2-6)

(5-2-7)

(5-2-8)

5-2-7. Transformation Between Jl-Plane and K-PIane

The transformation between H-plane and 32—plane may be again obtained by em

ploying the Trans{orma.tiott ofScJiwarz and Christoffel

(1 =AJ(U- )"/""'(» - +B. (5-2-9)Since we have chosen ts = -l^ts = 0 and <£> = oo, Eq. (5-2-9) may be simplified as

t^)(Ji+l)+ B (5 - 2 - 10)

- 29 -


AQ =

+

To determine the constant A, we may examine the change in ft at point A:

For -l<t< tAj

For i > tA,

Km)- t-tA

t+1

t-tA

t + 1

+ ITT.

+ tO.

Therefore the change in ft at point A, according to Eq. (5-2-11), is

Aft = ^ (fO —tTr) =-tTT—tA + V ^ tA + 1

But the mapping onto the ft-plane shows

hence

or

An =-f,

iq . A""2 " "'''iTTT'

tyl -H 1 27r


27r \t + l J

To determine the constant apply Eq. (5-2-13) at point B:

0=^lnH4) +B,



fi = -;iln2w

'-tA{t+iy(t-tA) 1

(5-2-11)

(5-2-12)

(5 - 2 - 13)

(5-2-14)

(5-2-15)

- 30 -

5-2-8. Solution to Water-Oil Interface

The solution of water-oil interface can be obtained solving Eq. (5-2-1).

Differentiating Eq. (5-2-15) gives

dSl =-^(— dt.2t: \t-tA <+ 1/


dz =fiq

2irkgAp

To solve the water-oil interface, we need to separate the imaginary part from this complex

equation:

Between points B and D^t > 0, so

(5 - 2 - 16)

t--^VT+t)] (5-2-17). * V V«TT+1 tc ). \t-tA t + lj ^

\y/^ + 1 + 1/


dy= M (_L27rkgAp \t-tA t + lj

The height of water crest (h) can be obtained by integration of Eq. (5-2-18):

(5 - 2 - 18)

y

=y-yB =jVb

t

-ItB

dy

2irkgApyt-tA t-j-l)m1— ]

t-tA2'KkgAp

In—tA{t -i-1)

dt

(5 - 2 - 19)

At the cusp point Z?, t approaches infinity. The height of the water crest (^) is

q kgAp

fiqH = VD - VB - 2'KkgAp

To determine the constant one can apply Eq. (5-2-8) at point A:

1 + —IT V + 1 + 1/ tc

(5 - 2 - 20)

(5-2-21)

which can be rearranged as

2d'jrkgAp 2fiq

= —VtA + 1tc

- 31 -

/1_^XVl+ v/ilTTy

(5 - 2 - 22)

The constant tA can be solved numerically from Eq. (5-2-22) if the constant ic is

known.

To determine the constant one can apply Eq. (5-2-8) at point C

kgAp kgAp

Since ic > 0, Eq. (5-2-23) can be rearranged as

/? = -irkgAp

In Eq. (5-2-24), /? is a parameter. When the oil production rate approaches the critical

rate, /3 approaches 0. For the critical case, Eq. (5-2-24) becomes

1+i [in f Vte+r"TT _ \y/^C + 1 + 1/ ic

from which tc can now be solved numerically.


fiq

(5 - 2 - 23)

(5 - 2 - 24)

(5 - 2 - 25)

dx =2'ir'̂ kgAp

which can be integrated as

^ vT+t)] f— (5-2-27)Jo2ir^kgApl\ VT+T+l tc /Jv' + l i-iAj

Now, Eq. (5-2-27) can be integrated numerically.

The height of water crest, h, given by Eq. (5-2-19) can be plotted versus lateral



—i-—vm) (5-2-26).\ y/t + 1+1 tc /. + 1 t —tA/

- 32 -

5-3. Wellbore Below the Roof of an Oil Reservoir, Critical Case

5-3-1. Flow Domain in Physical Plane z = x-\-iy


as shown in Fig. 5-3(a), is studied. Point D is sink which represents the cross section



5-3-2. Flow Domain in Hodograph Plane i/ = Vx + ivy



At point A, Vx = qd^d and Vy = 0, hence, point A is located at point (gc/2rf,0) in

the hodograph plane.

At point J3,Vy = 0, Va; = 0 according to Eq. (A-8) in Appendix A. Hence, point B

is a stagnation point, and it should be located at origin in hodograph plane.

The mapping of boundary 5 - C is a semicircle according to Eq. (A-8).

Since point D represents a cross section of line-sink, the velocity at the point is

assumed to be infinite. If we denote Db and Da as two points approaching point D from

below and from above of point JD, respectively, we may obtain that = 0, Vy = -|-oo at

point Df, and v® = 0, Uy = -oo at point Da-


axe mapped onto the hodograph plane as straight lines.




- 33 -

5-3-3. Flow Domain in a Modified Hodograph Plane e = a +




ii/£ = Q + =

Vx - tv.(5-3-0)

This transformation is chosen because it simply relates to the flow domains repre

sented in 2-plane and in ft-plane through Eq. (A-5) in Appendix A. In fact, combining

it with Eq. (A-5) givesdQ i


dz = -iedQ. (5-3-1)

Eq. (5-3-1) is the fundamental equation of solving our problem.


hodograph plane (e-plane) as shown in Fig. 5-3(c).

6-3-4. Flow Domain in Complex Potential Plane fl = +

Since the value of velocity potential 0 is relative, it does not matter where we map


arranged in such an order as to foUow one direction on a stream line. Arbitrarily choosing

the stream function as zero along the lower boundary of the flow domain in physical

domain and assuming the velocity potential at point B is zero, we map the flow domain


-34-

5-3-5. Flow Domain in a Intermediate Plane U = t + is






intermediate plane, 3? = t + is, is shown in Fig. 5-3(e). In the figure, tA = = 0

and ts = oo are chosen arbitrarily.

5-3-6. Transformation Between e-Plane and 5?—Plane

The transformation between e-plane and 9f?-plane may be obtained by employing

the Transformation of Schwarz and Christoffel.

s=Aj(n- +B. (5-3-2)

Since we have chosen = —1,^5 = 0 and ts = oo, Eq. (5-3-2) may be simplified as

(^+1)- tc


d^ + B

e = 2A + B.


For t <tB = 0,

VFor t>tB = 0,

-y/W 2H—.

TT

-v^1+V^

(5-3-3)

(5-3-4)

-35-

Therefore the change in e at point 5, according to Eq. (5-3-4), is

2A /—TT'Af =

Vic-ttA

w

(f)

But the mapping on the e plane shows

hence,

or

e =irkgAp

To determine the constant B, apply Eq. (5-3-6) at point C:

p, 2py/tc

Ae = -kgAp'

fi —ttA

kgAp '

A=irkgAp'


kgAp wkgAp(o+^tan-'o)+B


e =kgAp

B = -kgAp'


(5-3-5)

(5-3-7)

(5-3-8)

5-3-7. Transformation Between ft—Plane and Plane

The transformation between ft-plane and 9fJ—plane may be again obtained by em

ploying the Transformsition of Schwarz and Cbristoffeh

a = A +B. (5-3-9)

- 36 -

Since we have chosen tA = = 0 and ts = oo, Eq. (5-3-9) may be simplified as

"-"Iwhich can be integrated along the real axis as

+ B

To determine the constant A, we may examine the change in H at point A:

For t <tA = -1 <0 < to,

t -f 1

For tn > 0 > t > tA = -1,

"m-"

t —tj)

t+l

-\-iO

t-tD+ iir

Therefore the change in ft at point Ay according to Eq. (5-3-11), is

Aft =- ^ (zV) = —.1 + 1 +

But the mapping on the ft-plane shows

hence

2 l-hto'

or

qc

l + to 27r'


2t yt-tDj

To determine the constant By apply Eq. (5-3-13) at point B:

0= ) +B,2)r X-to)

(5 - 3 - 10)

(5-3-11)

(5 - 3 - 12)

(5 - 3 - 13)


- 37 -

B= ).2t


fl = -|s.ln27r

~^d(^ +1)t-tD J

5-3-8. Solution to Water-Oil Interface, Critical Oil Rate.

(5-3-14)

(5 - 3 - 15)

The solution of water-oil interface can be obtained solving Eq. (5-3-1): Differenti

ating Eq. (5-3-15) gives

da = —)dt. (5-3-16)27r Vi 1 t —


dz = —ir^kgAp

Vtc(t - to) +tan-> - I] -7^) (5-3-tc 2

To obtain the water-oil interface, we need to separate the imaginary part from this

complex equation.

Between points B and C, tc > ^ > Oj so

17)

tan ^ =4lnto 2 1 +


dy=-tS^(-L L.)dt.^ 2nkQAp\t + l t-tnJ2nkgAp\t + l t-to

The height of water crest (he) can be obtained by integration of Eq. (5-3-18):

y

hc =y-yB =JVb

dy

t

J 2TrkgAp\t+l t —ti))t-B

2'KkgAp \ (t—tn) ^

(5 - 3 - 18)

dt

(5 - 3 - 19)

-38-

At the cusp point C,t = tc' The height of the water crest (He) is

= yc -VB = V (tc-to) )•2TrkgAp

To determine the constant toy one can apply Eq. (5-3-8) at point D:

0 =kgAp

^y/tcV̂^D —ic +

y/ic "V ic

Therefore, the constant to can be solved from Eq. (5-3-21) if constant tc is known.

To determine the constant tc, one can apply Eq. (5-3-8) at point A:

i2d /z

tan"

Qc kgAp

Since tc > 0, one hcis

tan"

Hence, Eq. (5-3-22) becomes

2'jrdkgAp

= ;r:ln2z

= 2V<c(l + <c) - In

V^+1,

\/^+V

IT

+ X-

(5 - 3 - 20)

(5-3-21)

(5 - 3 - 22)

(5 - 3 - 23)

from which tc can be solved numerically.

Assuming the dimensionless critical-crest-height is HcDi the real critical-crest-height

Hoc is then HcdC- Therefore, the critical oil production rate for unit length of wellbore,

according to Eq. (5-3-20), is

qc =2TrHcDckgAp

filn -tpitc+l){tc-to)

(5 - 3 - 24)

The critical rate, qc, can be solved by a numerical method from Eqs. (5-3-21), (5-

3-23), and (5-3-24) (see Chapter 7). It can be shown that Eq. (5-3-24) is reduced to Eq.

(5-1-25) when tn approaches infinity.


dx = -ir^kgAp

-\/<c(<C'-0 +iln-i\

/Jv+^/^ Vt + l t-tojdt,

(5 - 3 - 25)

which can be integrated:

t

- 39 -

X -Itc

ir^kgAp -y/^c{tc -1) +2^^li +V^y'J \t + l t-tDj

(5 - 3 - 26)


The height of the water crest, given by Eq. (5-3-19) can be plotted versus lateral

distance, x, given by Eq.(5-3-26) with parameter t. This plot gives the location of the


Eq. (5-3-24) indicates that the critical rate is a function of critical crest height at

the cusp point which is an unknown variable. This limits the application of the theory.

Theoretically, this can be avoided. In fact, Eq. (5-3-19) shows that at the outer boundary

where the crest is null, the parameter t takes a value of zero. Substituting t = Qinto the

upper boundary of the integral in Eq. (5-3-26) yields

xa

u

-/tc

ir^kgAp -y/^dtc -1)+2^ dt.\t + l t-toj

(5 - 3 - 27)

Theoretically, the critical rate (qc) can be numerically solved by combining Eq. (5-3-27),

Eq. (5-3-23) and Eq. (5-3-21). However, it is found that the integral In Eq. (5-3-27)

converges extremely slowly. It may be impossible to numerically solve these equations

using the currently available computers. Therefore, Eq. (5-3-24) has to be used for

determining the critical rate.

5-4. Wellbore Below the Roof of an Oil Reservoir, Subcritical Case

5-4-1. Flow Domain in Physical Plane z = x + iy


as shown in Fig. 5-4(a), is studied. Point E is a sink which represents the cross section



LueisAsSujuooj9;bmbjo^ujBLUopmo|^SDUjddBi/M-f-9"Bjd

aD8V

- 40 -

5-4-2. Flow Domain in Hodograph Plane u = Vx + iVy



At point A, Vx = ql2d and Vy = 0, hence, point A is located at point (g/2d,0) in

the hodograph plane.

At point 5, Vy = 0, Va: = 0 according to Eq. (A-8) in Appendix A. Hence, point

B is a stagnation point, and so it should be located at origin in hodograph plane. So is

point D.

The mapping of boundary B —C —D is part of a circle according to Eq. (A-8).

Since point E represents a cross section of line-sink, the velocity at the point is

assumed to be infinite. If we denote Et, and Ea as two points approaching infinitely to point

E from below and above of point E, respectively, we may obtain that Vx = 0,Vy = -Hoc

at point Eh and Vx = 0, Vy = -oo at point Ea.


be mapped onto the hodograph plane as straight lines.




5-4-3. Flow Domain in a Modified Hodograph Plane e = a-\-ip




£ = a-hi/?=

(5-4-0)- iv„

-41 -


in z—plane and in ft—plane through Eq. (A-5) in Appendix A. In fact, combining it with

Eq. (A-5) givesdO, i

rfj" ?


dz = —iedQ. (5 —4—1)

Eq. (5-4-1) is the fundamental equation of solving our problem.


hodograph plane (e—plane) as shown in Fig. 5-4(c).

5-4-4. Flow Domain in Complex Potential Plane Q = <!> + iip.

Since the value of velocity potential <f> is relative, it does not matter where we map






5-4-5. Flow Domain in a Intermediate Plane ^ = t +is






intermediate plane, U = t + is,is shown in Fig. 5-4(e). In the figure, ts = 1, = 0 and

= oo are chosen arbitrarily.

-42 -

5-4-5. Transformation Between £—Plane and Plane

The transformation between e—plane and 9li—plane may be obtained by employing

the Transformation of Schwarz and ChristoffeL

s=Aj{n-(» - + B. (5-4-2)

Since we have chosen ts = l,tp = 0 and to —oo, Eq. (5-4-2) may be simplified as

(»-i^)(«-ic)(» -1)^^


d^ + B (5-4-3)

e = 2A -I- (1 - - tc)V< + - <0 +1)In I + B. (5-4-4)

To determine the constant A, we may examine the change in e at point B:

For t>tB = h

a/7-1+ iO.

For t = 1,

In4-^ =ln

In

y/i + 1

y/t- 1

\/t + l= In

v^+1

y/l-l

\/t+l+ iTT.

Therefore the change in e at point according to Eq. (5-4-4), is

Ae = A(iAtc -^A-tc 1)(20- ztt)

= -i'KA{tAtc -tA-tc + l)

But the mapping on the £-plane shows

A£ = -kgAp'

hence,

A=-TrkgAp{tAtc - ^>1 - + 1)

(5-4-5)

-43-


-tfi

irkgApri

tAtc — —ic + ^+ {I - tA - tc)y/i

+ lny/i- 1y/i -{- 1

+ B,

To determine the constant B, apply Eq. (5-4-6) at point E:

0 =irkgAp Ja^c -tA-tc + '̂

ly/ip+(i-tA-tc)Vt^ + ln


(5-4-6)

y/is- 1ViE+l.

+ B,

B =ifi

irkgAp tAtc — + 1

y/tE- 1

(1-tA- tc)y/iE

2/i

irkgAp

+ ln\/^E + 1


(5-4-7)

tAtc -tA-tc + 1j("n/^ - v^) +(1 - - tc)iViE - \/^)

(5-4-8)y/tE + 1 y/i+1

5-4-6. Transformation Between (2—Plane and 3U-Plane

The transformation between 0—plane and 3?-plane may be again obtained by em

ploying the Transformation of Schwarz and Christoffel.

n = A +B. (5-4-9)

Eq. (5-4-9) may be simplified as

dU

(K-t^)(»-tE)


+ B

SI =—-—In (-—+B.tA-tE Kt-tEj

(5-4-10)

(5 - 4 - 11)

-44-

To determine the constant A, we may apply Eq. (5-4-11) to point F:

2 tA-tE

which gives

(5 - 4 - 12)

Eq. (5-4-12) can also be obtained by examining the change in ft at point A.


2ir \t^tEj

To determine the constant B, apply Eq. (5-4-13) at point B:

27r xtB-tE/


27r \l-tEj


(5 - 4 - 13)

(5 - 4 - 14)

n = f27r

(5 - 4 - 15)

5-4-7. Solution to Water-Oil Interface

The solution of water-oil interface can be obtained by solving Eq. (5-4-1): Differentiating

Eq. (5-4-15) gives

d{l =^(— i—) dt. (5-4-16)2Tr \t-tA t-tEj


dz =qy.

2Tr^kgAp tAic —tA —tc + ^

+ Vi)] +1. (^) -1.(^)(-\t-tA t-tEj

(5-4-17)

-45 -

To obtain the water-oil interface, we need to separate the imaginary part from this

complex equation: Since is < 0, y/ts = iy/-tE» Therefore,

y/iE-\ _ tE + l _y/tE + 1 —1 —1 '

and

Vv^E + l/ "I" 1

\ ^£ + 1 /

Between points B and D, t > 1, so

(^) =Therefore the imaginary part of Eq. (5-4-17) is then

dy =g/i

2Tr'̂ kgAp tAtc - - <c + 1

\t-tA t-tsJ(5-4-18)

The height of water crest (h) can be obtained by integration of Eq. (5-4-18):

1. e.,

y

h=y-yB= JVb

dy

qy.I

-I 2Tr'̂ kgAp tA^c — —i^c + 1[ix/^

h =

+ (l-tA- to)\^

dt,\t-tA t-tEj '

qfj,

-f tan'

2Tr'̂ kgAp

+ (l-tA- tcW-iE

In

tAtc -tA-tc + l

re - tE)'u< - 'b)(1 -tA)\

.1 (-2V=iE\\ tE + l J

(5 - 4 - 19)

-46 -

At the cusp point D, t approaches to = oo. The height of the water crest {H) is

H = yD-yBqH 2

2w^kgAp tA^C — + 1

+ (1 - - tcW-tE + tan".1 (-2V=iE\

\ tE + i J

InL(i-<a)J*

The constants tAytc-, and are determined in the following:

Applying Eq. (5-4-8) at point A gives

.2d ill0 + t— = , .

q irkgAp iA^c — + 1lis/i?- y/t?)

+ (1 - -^c)(^/^i-Vu)] +In (^qrr)

which has a real part equation of

0irkgAp i-Atc —tA —ic + ^

-{1-tA- tc)V^ -tan ^̂ ^and an imaginary part equation of

2d p, 2

q irkgAp tAtc —tA - tc + 'ii

Applying Eq. (5-4-8) at point C gives

/»

kgC^p

+ (!-'/!- tc)(\/iE - Vic) + In

\\/ic -hi/

(5 - 4 - 20)

(5 - 4 - 21)

(5-4-22)

(5 - 4 - 23)

(5-4-24)

-47-

The real part of Eq. (5-4-24) is the same as Eq. (5-4-22). The imaginary part of Eg.

(5-4-24) is

2P = irkgAp iA^c - tA - tc + I

-^Vt(

(5 - 4 - 25)

Therefore, constants tAitc and ts may be solved simultaneously from equations (5-4-22),

(5-4-23) and (5-4-25) with parameter /? which is 0 for critical condition.


dx =2T^kgAp tAtc -tA-tc + 'i- 3

which has an integral form of

Joo 2Tr^kgAp ^A^C — —^£7 + 1 4^

U\t-tA t-tEj

(5 - 4 - 26)

(5 - 4 - 27)


The height of water crest, given by Eq. (5-4-19) can be plotted versus lateral



CHAPTER 6

SOLUTIONS TO GAS-OIL CRESTING SYSTEMS

In this chapter, four cases in two gas-oil cresting systems are studied, and the critical

oil rates and the location of gas-oil interfaces in the two systems are determined.

Case 1: Wellbore is located at the bottom of an oil reservoir: critical case as shown

in Fig. 6-l(a).

Case 2: Wellbore is located at the bottom of an oil reservoir: subcritical case as shown

in Fig. 6-2(a).

Case 3: Wellbore is located above the bottom of an oil reservoir: critical case as shown

in Fig. 6-3(a).

Case 4- Wellbore is located above the bottom of an oil reservoir: subcritical case as

shown in Fig. 6-4(a).

Conformal mapping method is utilized to solve the gas-oil-interface cresting problem. This

method is discussed in Appendix A.

6-1. Wellbore at the Bottom of an Oil Reservoir, Critical Case

6-1-1. Flow Domain in the Physical Plane z = x + iy

Since the flow region in physical plane is symmetric, only half of the flow region, as

shown in Fig. 6-l(a), is studied. Point D is a sink which represents the cross section of

-48 -

Doctor ofPhilosophy in Petroleum Engineering New Mexico ...

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