Waterflooding under fracturing conditions

WATERFLOODING

UNDER

FRACTURING CONDITIONS

O Q

O O

E.J.L. Koning

- > l *

!■, .f

WATERFLOODING UNDER FRACTURING CONDITIONS

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Universiteit Delft, op gezag van de rector magnificus,

Prof. Drs. P.A. Schenck, in het openbaar te verdedigen voor een commissie, aangewezen door het College van Dekanen, op dinsdag 27 september 1988 te 10.00 uur

door

Eric Jan Leonardus Koning

Doctorandus in de Wiskunde en Natuurwetenschappen geboren te Haarlem

TR diss

1664

II

V

-CONTENTS P a g e

CHAPTER ONE

INTRODUCTION 1

1.1. Waterflooding 2 1.2. Waterflooding under fracturing conditions 2

1.2.1. Scope and definition 2 1.2.2. Objectives of thesis 3

1.2.3. Previous work 3 1.2.4. New elements in thesis 5

1.3. Organisation 6

References 7

CHAPTER TWO

THE PORO- AND THERMO-ELASTIC STRESS FIELD AROUND A WELLBORE 9

Summary 10 2.1. Introduction 11

2.2. Solution for the stress field 14

2.2.1. Assumptions 14 2.2.2. Notation and basic equations 14

2.2.3. Solution with Goodier's displacement potential 17 2.2.4. The particular solution in cylindrical coordinates 18

2.2.5. Complete formulation of the problem 20

2.2.6. Method of solution 22 2.2.7. Asymptotic expansion of the stress solution 23

2.2.8. Simplified solution method 27 2.2.9. Numerical method to evaluate Ao — 30

ÖT

2.3. Solution for the pressure and temperature fields 30

2.3.1. Temperature field 30 2.3.2. Pressure field 32 2.4. Analytical solution for thermo-elastic stress variations 38

2.5. Analytical solution for poro-elastic stress variations 47

2.6. Fracture initiation pressure 59 2.7. Evaluation of two field cases 64

2.8. Other applications 73

2.9. Conclusions 73 List of symbols 75 References 77 Appendix 2-A - Basic equations 78

Appendix 2-B - The particular stress solution in cylindrical coordinates 81 Appendix 2-C - The complete stress solution in cylindrical coordinates 84

Appendix 2-D - Asymptotic expansions of the stress solution 89

Appendix 2-E - Simplified solution method 93 Appendix 2-F - Numerical method to evaluate Ao — 97

CTT

Appendix 2-G - Solution for the pressure distribution 101

CHAPTER THREE

ANALYTICAL MODELLING OF FRACTURE PROPAGATION 103

Summary 105 3.1 Introduction 106

3.2 Fracture propagation in an infinite reservoir in the absence

of reservoir stress changes 109

3.2.1 Assumptions 109 3.2.2 One-dimensional leak-off 109

3.2.3 Two-dimensional leak-off - Pseudo-radial solution 114 3.2.4 Two-dimensional leak-off - Elliptical solution 116 3.3 The effect of poro- and thermo-elastic stress changes on fracture

propagation pressure 120 3.3.1 Definition of fracture propagation pressure 120

3.3.2 Analytical calculation of poro-elastic stress

changes at the fracture wall 121 3.3.3 Numerical calculation of poro-elastic stress changes

at the fracture wall 129 3.4 Fracture propagation in an infinite reservoir under the

influence of reservoir stress changes 131

3.4.1 Assumptions 131 3.4.2 Consistency checks 132

VII

-3.4.3 Two field cases 133 3.5 Fracture propagation in a pattern flood.

Effect on sweep efficiency 139 3.5.1 Zero voidage in the absence of reservoir stress changes 139

3.5.2 Zero voidage with reservoir stress changes 145 3.5.3 General flooding conditions and the use of

a reservoir simulator 147

3.7 Conclusions 148 List of symbols 150 References 152

Appendix 3-A Calculation of poro-elastic stresses in elliptical

coordinates 154 Appendix 3-B A numerical method for calculating

poro-and thermo-elastic stress changes 163 Appendix 3-C Calculation of thermo-elastic stresses

and of the axes of the elliptical fluid fronts 168

CHAPTER FOUR

A PRESSURE FALL-OFF TEST FOR DETERMINING FRACTURE DIMENSIONS 171

Summary 172 4.1 Introduction 173

4.2 Calculation of the pressure fall-off with a closing fracture 175

4.2.1 Assumptions 175 4.2.2 Integral equation for the dimensionless pressure function 175

4.2.3 Solution for the dimensionless pressure function 179

4.3 Analysis of a pressure fall-off test 183 4.3.1 Four methods to determine fracture length 183

4.3.2 Discussion 191 4.4 Conclusions 193 List of symbols 194 References 196

Appendix 4-A Solution for dimensionless pressure function in Laplace

space 198 Appendix 4-B Relationship between fracture closure constant and

fracture length 202

CHAPTER FIVE

A PRACTICAL APPROACH TO WATERFLOODING UNDER FRACTURING CONDITIONS 207

Summary 208 5.1 Introduction 209

5.2 An example 209 5.3 Conditions for a successful process 210

5.4 Preliminary investigations 210

5.5 Basic data gathering 214 5.5.1 Measurements of in-situ stress, fracture orientation and

elastic moduli 214 5.5.2 Injectivity test and fall-off testing 215

5.5.3 Matching with propagation model 217 5.6 Determination of optimal reservoir pressure, injection rate

and well pattern 217 5.6.1 Determination of maximum reservoir pressure 217

5.6.2 Determination of maximum injection rate 221 5.6.3 Fractured well pattern and reduction in the number of wells 221

5.7 Full-scale implementation 230 5.8 Special applications 232

5.9 Conclusions 233 List of symbols 235 References 237

Appendix 5-A Determination of the ratio of producers to fractured

injectors and the reduction in the number of wells 238

SUMMARY 243 SAMENVATTING 247 ACKNOWLEDGEMENTS ' 251

1

-CHAPTER ONE

INTRODUCTION 1.1. Waterflooding

1.2. Waterflooding under fracturing conditions 1.2.1. Scope and definition

1.2.2. Objectives of thesis 1.2.3. Previous work

1.2.4. New elements in thesis 1.3. Organisation

INTRODUCTION

1.1 WATERFLOODING

When an oilfield is exploited by simply producing oil and gas from a number of wells, the reservoir pressure, in many cases, drops rather quickly and so does the production rate.

Therefore, water is often injected into the reservoir to maintain the reservoir pressure. The injection wells are located at carefully chosen points so that as much oil as possible is displaced by the water to the production wells before water starts to break through in the producers. The process of recovering oil by water-injection is called waterflooding. The degree to which the water is capable of sweeping the oil to the producers without bypassing it, is called the sweep efficiency.

Even in the optimal case not all the oil can be produced by

waterflooding. A certain amount of oil always remains trapped inside the pores of the rock by capillary forces. This so-called residual oil may be as much as 40% of the total pore volume of the rock.

1.2 WATERFLOODING UNDER FRACTURING CONDITIONS

1.2.1 Scope and definition

A major saving in the exploitation costs of an oilfield can be obtained by a reduction in the number of wells. To reduce the required number of injectors, the injection capacity of the well has to be increased. An

effective way of doing this is by rupturing the reservoir rock along a plane intersecting the wellbore. Such an induced fracture in the rock enlarges the surface area of injection considerably so that with the same pressure

gradient a dramatic increase in injection rate can be obtained.

Rupturing of the reservoir rock can be effected by injecting water at a pressure above the formation breakdown pressure. The fluid pressure is then high enough to overcome the rock stress and the tensile strength that keep the rock particles together.

Continued injection at this pressure causes the fracture to propagate into the reservoir. Water injection in this manner is called waterflooding under fracturing conditions.

3

-1.2.2 Objectives of thesis

Although an increase in injectivity is favourable, excessive lateral growth of the fracture may adversely affect the sweep efficiency of water

injection and result in premature water breakthrough. Moreover, if the fracture is vertical, excessive vertical growth may establish communication with other reservoirs resulting in loss of injection water or even worse, loss of oil.

It is the purpose of this thesis to:

a) investigate the mechanism of fracture initiation and fracture propagation under the influence of continued water injection,

b) evaluate the effect of fracture growth on sweep efficiency, c) improve the methods for determining fracture dimensions,

d) give rules for designing the process of waterflooding under fracturing conditions in such a way that sweep efficiency is unimpaired and vertical fracture growth is limited.

1.2.3 Previous work

Fracturing of production wells to stimulate productivity is a well-established technique. Here, fractures are created with a special injection fluid that contains proppant particles (e.g. sand) to keep the fracture open when the well is producing. These fractures thus retain a fixed length after the fracture has been created. Furthermore, since their length is generally small with respect to the interwell spacing, sweep efficiency is hardly affected.

Extensive literature exists on the subject of fracturing for production stimulation .

For waterflooding under fracturing conditions the effect of continuous fracture growth on sweep efficiency is of paramount importance. Moreover, this fracture growth is strongly dependent on changes in reservoir rock stress and fluid leak-off resulting from changes in reservoir pressure and temperature. Therefore, the theory developed for the stimulation of

producers is not directly applicable.

Relatively little is known in the petroleum-engineering literature on the mechanism of fracture propagation during waterflooding and its effect on sweep efficiency.

In the early literature on waterflooding, studies appeared on the effect of fractures with a fixed length on sweep efficiency using physical

. 2'3 model experiments

4 A major step forward was the construction by Hagoort et al. of a numerical model that could simulate the growth of a vertical fracture of constant height in a simple, vertically homogeneous reservoir. They studied fracture propagation as a function of reservoir and injection/production conditions. One of the important conclusions of this study was that the leak-off from the fracture into the reservoir should essentially be modelled as two-dimensional in the plane of the reservoir. Therefore previously

developed analytical models with a one-dimensional description of leak-off are generally inadequate for modelling waterflood-induced fractures.

Later, Hagoort presented a broad and innovative study on the subject in his thesis "Waterflood-induced Hydraulic Fracturing" . Apart from the

numerical simulation model, analytical calculations of sweep efficiency for a 5-spot containing a fractured injector with a fixed fracture length were presented. The calculations were also extended to stratified reservoirs. The effect of reservoir pressure on rock stress and fracture propagation

pressure were discussed using two-dimensional poro-elastic stress calculations. Finally, a first step towards the. monitoring of fracture length by pressure fall-off testing was presented. The technique consists of recording the downhole fluid pressure during an interruption of injection. The declining fluid pressure with time can be analysed to get an indication of fracture length.

Recently, the effect of a change in temperature on reservoir rock stress and on fracture propagation was analysed by Perkins and Gonzalez . They showed that injection of large amounts of cold water into reservoirs with a good permeability can eventually result in thermal fracturing of the rock due to a decrease in rock stress by cooling. With a simple

semi-analytical model they showed that thermal fractures can reach a considerable length. They also showed that the common practice of calculating pressure and temperature-induced changes in rock stress using two-dimensional rather than three-dimensional poro- and thermo-elasticity, greatly underestimates the stress changes for large lateral penetration depths of pressure or temperature.

5

-1.2.4 New elements in thesis

The thesis presented here takes the work of Hagoort and Perkins & Gonzalez as a starting point. The main new elements are:

- A complete analytical description of the stress field surrounding an unfractured wellbore. The theory of three-dimensional poro- and thermo-elasticity is used to calculate the effect of pressure and temperature changes on reservoir rock stress.

The calculation of the stress field is used to evaluate the pressure at which fractures can be initiated.

- An analytical model of fracture propagation with a complete two-dimensional description of fluid leak-off into the reservoir.

- A focusing on the mode of propagation for which the fluid flow around the fracture is pseudo-radial. This means that the pressure transients are travelling radially into the reservoir.

It is shown that this mode of fracture propagation does not influence the sweep efficiency.

- An extension of the pseudo-radial model to include a three-dimensional analytical calculation of the effect of pressure- and temperature-induced stress changes on fracture propagation.

- The development of a three-dimensional numerical method to calculate poro-and thermo-elastic changes in reservoir stress that can be incorporated in the numerical fracture simulator as developed by Hagoort et al.

- An extension of Hagoort's method to determine fracture length from a • pressure fall-off test by including: a description of fracture closure,

two-phase elliptical fluid flow and different fracture geometries. It is shown that the pressure fall-off with time is characterised by four distinct periods, each of which gives independent information on the fracture size.

- A practical programme for designing the application of waterflooding under fracturing conditions in a given field.

1.3 ORGANISATION

This thesis is written in five self-contained chapters that can be read independently of the others.

Chapter 2 deals with the calculation of the stress field around an unfractured wellbore. The numerical method presented here was published earlier in Ref. 7. The analytical methods will be published in Ref. 8.

Chapter 3 describes the modelling of fracture propagation and its effect on sweep efficiency. Most of this chapter was published in Ref. 9.

Chapter 4 describes the method of determining fracture length from a fall-off test. This chapter was published in Ref. 10. Recently, an extension of the method together with a field application was published in Ref. 11.

Finally, Chapter 5 concludes this thesis with practical design rules for field implementation.

7

-1. Howard, G.G. & Fast, C.R., Hydraulic Fracturing. SPE Monograph Volume 2, 1970.

2. Crawford, P.B. & Collins, R.E., Estimated effect of vertical fractures on secondary recovery.

Trans. AIME (1954), 201, p. 192.

3. Dyes, A.B., Kemp, C.E. & Caudle, B.H., Effect of fracture on sweep-out pattern.

Trans. AIME (1958), 213, p. 245.

4. Hagoort, J., Weatherill, B.D. & Settari, A., Modelling the propagation of waterflood-induced fractures.

SPEJ (Aug. 1980), p. 293.

5. Hagoort, J., Waterflood-induced hydraulic fracturing. PhD. Thesis, Delft Technical University, 1981.

6. Perkins, T.K. & Gonzalez, J.A., The effect of thermo-elastic stresses on injection well fracturing.

SPEJ (Feb. 1985), p. 78.

7. Koning, E.J.L. & Niko, H., The effect of reservoir pressure and temperature variations on fracturing conditions during a waterflood operation.

Proceedings of the 3rd European Meeting on Improved Oil Recovery (April 1985), Rome, p. 219.

8. Koning, E.J.L., The poro- and thermo-elastic stress field around a wellbore.

SPE paper in preparation.

9. Koning, E.J.L., Fractured water-injection wells. Analytical modelling of fracture propagation.

SPE 14684, 1985.

10. Koning, E.J.L. & Niko, H., Fractured water-injection wells. A pressure fall-off test for determining fracture dimensions.

SPE 14458, 1985.

11. Koning, E.J.L. & Niko, H., Application of a special fall-off test in a fractured North-Sea injector.

CHAPTER TWO

THE PORO- AND THERMO-ELASTIC STRESS FIELD AROUND A WELLBORE

Summary

2.1. Introduction

2.2. Solution for the stress field 2.2.1. Assumptions

2.2.2. Notation and basic equations

2.2.3. Solution with Goodier's displacement potential 2.2.4. The particular solution in cylindrical coordinates 2.2.5. Complete formulation of the problem

2.2.6. Method of solution

2.2.7. Asymptotic expansion of the stress solution 2.2.8. Simplified solution method

2.2.9. Numerical method to evaluate Ao — 0T

2.3. Solution for the pressure and temperature fields 2.3.1. Temperature field

2.3.2. Pressure field

2.4. Analytical solution for thermo-elastic stress variations 2.5. Analytical solution for poro-elastic stress variations 2.6. Fracture initiation pressure

2.7. Evaluation of two field cases 2.8. Other applications

2.9. Conclusions List of symbols References

Appendix 2-A - Basic equations

Appendix 2-B - The particular stress solution in cylindrical coordinates Appendix 2-C - The complete stress solution in cylindrical coordinates Appendix 2-D - Asymptotic expansions of the stress solution

Appendix 2-E - Simplified solution method

Appendix 2-F - Numerical method to evaluate Ac — Appendix 2-G - Solution for the pressure distribution

10

-SUMMARY

Changes in reservoir pressure or temperature can change the stress field in the reservoir rock. Such changes in the stress field influence the conditions under which fracturing of the reservoir rock can occur. They also influence the geometry and direction of propagation of induced fractures.

This chapter presents new analytical and numerical methods for calculating the stress field around a single vertical well in an infinite reservoir. Poro- and thermo-elastic variations in the stress field resulting from axisymmetric changes in reservoir pressure and temperature are

incorporated. The methods are general in the sense that no use is made of the assumption of plane strain. The formulae presented allow application of arbitrary axisymmetric pressure and temperature fields.

Simple analytical formulae for thermo-elastic stress variations are presented for temperature distributions with a step profile. Simple

analytical formulae for poro-elastic stress variations are presented for quasi, steady-state pressure profiles including discontinuities in fluid mobility.

Analytical expressions for the fracture initiation pressure including poro- and thermo-elastic effects are presented.

The numerical and analytical methods are used to evaluate the change in fracturing conditions for two realistic field cases. The first is a high-permeability reservoir in which a large thermo-elastic reduction in rock stress occurs following the injection of cold water. Thermal fractures may be induced under such conditions. These fractures are likely to remain contained within the cooled reservoir region.

The second case is a low-permeability reservoir for which the poro-elastic increase in rock stress due to the rise in pore-pressure is dominant with respect to the cooling effect. The fracture initiation

pressure of the reservoir has therefore increased. The stresses in cap and base rock have decreased because of cooling. The induced stress gradients may force created fractures to grow vertically into cap and base rock.

THE PORO- AND THERMO-ELASTIC STRESS FIELD AROUND A WELLBORE

2.1 INTRODUCTION

In this chapter an analysis is made of the factors that influence the stress field in the reservoir rock surrounding a wellbore. In principle, knowledge of this stress field allows one to determine the downhole fluid

injection pressure at which tensile failure or fracturing of the rock occurs.

This so-called fracture initiation pressure is an important parameter in selecting a suitable downhole pressure during injection. The injection pressure should be lower if fracturing is to be prevented, for instance to avoid communication with other reservoir zones. Or, it should be higher if fracturing is desired, for instance to increase the injection capacity of the well.

Since the concept of stress is generally considered to be a difficult one, we start out with some introductory remarks on the definition of

stress. For a more complete introduction see, for instance, Ref. 1.

Consider a slightly deformed material body in equilibrium. Owing to the deformation internal forces have been generated. Imagine an arbitrary point inside the body blown up to form an infinitesimal cube. The sides of this cube are subject to forces which hold it in equilibrium. A cube can be found with an orientation such that the sides of the cube only experience forces that are normal to its six surfaces. Since the cube is in

equilibrium, there are three independent forces, one for each pair of opposite surfaces.

These forces taken per unit area are called the three principal

stresses in the point. If a cube with a different orientation is chosen, for instance, such that its sides line up with the axes of a given Cartesian coordinate system, the three principal stresses can be decomposed into their components parallel to the coordinate axes. This then generally results in nine nonvanishing components, three for each pair of opposite surfaces. These nine components are called the elements of the stress tensor in the point under consideration. Since the cube is prevented from rotating, there can only be six independent components.

12

-The relation giving the stress resulting from an applied deformation or the deformation resulting from an applied stress is called the

stress/strain relation. In this work we consider only relations of a linear form and therefore our analysis falls within the framework of the theory of linear elasticity.

When fluid is injected from a wellbore into a reservoir a certain pressure is required to squeeze the fluid into the pores of the reservoir rock. This fluid pressure inside the wellbore exerts a radially outward force (or equivalently a radial stress or radial load) on the surrounding reservoir rock. For simplicity, we consider here an open hole without a casing. As a result of the radial load the rock is deformed (expanding radially outward) and stresses are generated in the rock.

Normally the rock is already in a state of stress before the well is drilled. These so-called tectonic stresses result from the weight of the overburden and from tectonic movement of the earth's crust. The radial

expansion of the borehole now results in a decrease in the horizontal stress that is tangential to the borehole circumference. If the fluid pressure is high enough, this so-called tangential stress may change from a compressive stress into a tensile stress. If this tensile stress exceeds the tensile strength of the reservoir rock formation fracturing will occur.

During fluid injection the pressure of the fluid inside the pores of the rock surrounding the well increases. As a result of this increase in pore pressure the rock frame or rock matrix tends to expand. Since there is little room for such a poro-elastic expansion, internal compressive stresses are generated. An increase in tangential stress results which works against the decrease induced by the radial loading of the wellbore.

A change in temperature will also affect the stress state of the reservoir rock. If, for instance, injection of cold fluid cools the reservoir rock, the rock grains and the rock matrix tend to contract,

resulting in a thermo-elastic decrease in the rock stress. The corresponding lowering of the tangential stress will facilitate formation fracturing. Similarly, injection of a hot fluid will tend to impede fracturing.

Since this work is based on the linear theory of elasticity, the results for the stress field apply only to reservoir rocks that have a

should be small with respect to the in-situ tectonic stresses so that non­ linear effects on material constants such as Young's modulus and Poisson's ratio may be neglected if the latter are taken at their in-situ values.

The linearity of the theory enables us to calculate the stresses

induced by poro-elastic effects, thermo-elastic effects and wellbore loading separately. The combined effect is then obtained by simply adding the

respective stress components.

In the past poro-elastic effects have been included in the calculation 2-4

of the stress field around a wellbore . However, these works relied on the assumption that the induced poro-elastic deformations occur only in the horizontal plane of the reservoir. This so-called assumption of plane strain allows the vertical variation of the stresses to be neglected and results in a considerable simplification of the differential equations involved.

Although in most cases the assumption of plane strain is acceptable when calculating the stresses induced by the loading of the wellbore

(Section 2.2.7 of this work), this assumption is generally not valid when poro- or thermo-elastic stresses are considered.

This was first demonstrated by Perkins and Gonzalez . They considered combined poro- and thermo-elastic stresses for disc-shaped regions of

uniform change in pressure and temperature. It was concluded that as long as the penetration depth of the pressure or temperature front is small compared to the reservoir height plane strain conditions would prevail. However, as the fronts advance during injection the state of rock deformation changes from horizontal strain initially, to vertical strain when the penetration depths have become large with respect to the reservoir height. The

horizontal stress may differ by as much as 100% between the case of horizontal and vertical strain even though the pressure or temperature inside the disc remained constant.

In their work Perkins and Gonzalez did not provide expressions for the stresses induced by a more realistic pressure profile. Moreover, the effect of a wellbore and an analysis of the fracture Initiation pressure were not included.

This chapter attempts to address these remaining problems. In Section 2.2 new analytical expressions for the tangential stress are presented for an arbitrary axisymmetric pressure or temperature profile. No assumption of plane strain has been made. In Section 2.3 simple but realistic expressions for the pressure and temperature field are derived.

14

-In Sections 2.4 and 2.5 these expressions are used in the general formulae of Section 2.2 to derive closed form analytical expressions for the poro- and thermo-elastic change in tangential stress. In Section 2.6 an analysis of poro- and thermo-elastic effects on fracture initiation pressure is presented. Both open and cased hole are considered. In Section 2.7 the change in tangential stress is calculated for two realistic field cases. In Section 2.8 other applications of the solution for the stress field are discussed. Finally, in Section 2.9 the conclusions are presented.

2.2 SOLUTION FOR THE STRESS FIELD

2.2.1 Assumptions

In a reservoir of axial symmetry the fracture initiation pressure for vertical fractures depends on the tangential stress field of the reservoir rock. The following assumptions have been made in calculating this stress field.

1. The reservoir has the shape of a horizontal disc of which the axis

coincides with that of the injection well (Fig. 2.1). The disc has finite height and infinite radius and consists of linearly elastic, isotropic and homogeneous, permeable bulk reservoir rock.

2. The reservoir is bounded at top and bottom by an infinite, linearly elastic, isotropic and homogeneous, impermeable cap and base rock with the same elastic constants as the bulk reservoir rock.

3. Coupling between elastic behaviour and fluid or heat flow is neglected.

2.2.2 Notation and basic equations

The linear stress/strain relations for combined poro- and thermo-elastic deformations are :

e. . = - [(1+u) Aa. . - vAo&. .] - a ApS.. - amAT8.. (2.1)

13 E l] ij p r 13 T 13

i,j=l,2,3 where:

Ap = change in pore pressure AT = change in temperature

GEOMETRY OF A DISC-SHAPED RESERVOIR SURROUNDED BY AN INFINITE MEDIUM.

The reservoir is considered to have infinite radial extent.

16

-Aa. . = change in total stress tensor with respect to the stress state at Ap = AT = 0.

3

Ao = trace of stress tensor = Z A a. 1-1 X 1

c.. = strain tensor

5.. = Kronecker delta (1 if i=j, 0 if i/j)

i ] J J

E = Young's modulus of bulk reservoir rock v = Poisson's ratio of bulk reservoir rock a = linear thermal expansion coefficient a = linear poro-elastic expansion coefficient

The linear poro-elastic expansion coefficient is defined as:

%

=

-3

( c

b - v

=

i r

( 1

- ^

} (2

-

2)

where c, and c are the compressibilities of the bulk reservoir rock and the rock grains, respectively.

The total stress is the stress exerted by both the rock matrix and the fluid in the pores of the rock. The formulation in terms of total stress offers certain advantages over a formulation in terms of effective stress,

eff

which is defined as: Ao. . = Ao. . - Ap6... One of the advantages is

i] ij r i]

continuity of stress across a boundary between permeable and impermeable media. For further discussion, see Ref. 5.

From (2.1) it is clear that the mathematical treatment of poro- and thermo-elastic effects is completely equivalent. For compactness, we introduce the following generic notation for a combined poro- and

* thermo-elastic deformation :

* The notation for the generic linear expansion coefficient is chosen as a, in line with the usual notation for the linear thermal expansion

coefficient. It should not be confused with a as used in M.A. Biot's classic paper: "General Theory of Three-dimensional Consolidation". J. Appl. Phys., 1941, Vol. 12, p. 155. Biot's a is related to our a

l-2u P

according to: a = —— a.

oAT =•a Ap + a AT (2.3) p r T

so that Eq. (2.1) becomes:

e.. = è [(1+u) Aa.. - uAoS..] - aAÏ 6.. (2.4) 13 E 13 13 13

The sign convention in Eq. (2.1) and (2.4) takes compressive stresses and strains as positive.

Eq. (2.4) can be inverted to express the stresses in terms of the strains. If the strains are then expressed in terms of a displacement vector and the stresses are substituted into the equations of equilibrium, three differential equations in the three components of the displacement vector result. For given Ap and AT and for given boundary conditions for the

stresses the solution for the stress field is determined. The development of the differential equations is given in Appendix 2-A.

2.2.3 Solution with Goodier's displacement potential

It is shown in Appendix 2-A that a particular solution of the differential equations can be obtained by the introduction of Goodier's displacement potential. This potential is the solution to Poisson's equation and is given by:

# = 7- 7 ^ IS! dx'dy'dz' a AT (x',y',z') J (2.5)

with

2 2 2 1/2

R = [(x-x') + (y-y') + (z-z•) ] ' (2.6)

The particular solution for the stress tensor denoted as Aa. - is obtained

r 13T from (2.5) by: E a2 Aa. - = — - T —z — 0 + AAT 6. . (2.7) 13T 1+u 9x.9x. 13 1 3

where the stress consists of poro- and thermo-elastic components

Aa. - = Aa. . + Aa. .m (2.8)

18

-and the following generic notation has been introduced:

AAT = A Ap + A AT (2.9) p T

A and A are called the poro- and thermo-elastic constants, respectively. They are defined as:

Eft 1 2 c A = T-2 = T2- ^ (1 a) (2.10) p 1-u l-v c ' b and E aT Am = ^ (2.11) T 1-ü

2.2.4 The particular solution in cylindrical coordinates

In the rest of this chapter it is assumed that the pressure and temperature fields possess axial symmetry,

AT = AT(r,z) (2.12)

Poisson's equation for the displacement potential becomes in cylindrical coordinates:

320 + - 3 0 + 3 2 0 = - 7 ^ a AT (2.13)

r r r z l-v

Because of symmetry the displacement vector has only a radial and a vertical component, given by, respectively:

u = - 3 0 ; w = - 3 0 (2.14) r z

From Appendix 2-B the particular solution for the stresses is given by: Ao - = 3 0 + AAT ; Ao - = -~ 3 0 + AAT

rT 1+u r v zT 1+u z

(2.15) Aon- = 77- - 3 0 + AAT ; Aa - = r^- 3 3 0

Two limiting cases can now be considered. In the first case we assume that the vertical variation of AT inside the reservoir zone is small.

If furthermore the penetration depth of the pressure and temperature variations are small with respect to the reservoir height, the only displacement will be in the horizontal plane and:

w = - 3 0 = 0 (2.16) z

Then (2.13) can be readily integrated, giving

r

u = - 9 <t> = T ^ a - ƒ dr'r' AT(r' ,z) (2.17)

r l-i) r r

w

and for the stresses from (2.15):

A r A ar T = + ~2 ' d r'r' A T<r' 'z> r r w A r Ao - = - — ƒ dr'r' AT(r',z) + AAT(r,z) r r w Ao = AAT(r,z) Ao - = 0 rzT

The expressions in (2.18) are the well-known plane strain results for the 1

stresses .

In the second limiting case we assume that the radial variation of AT is small. When furthermore the penetration depths become very large with respect to the reservoir height, there will be displacement in the vertical direction only and therefore:

(2.18)

u = - 9 0 = 0 (2.19) r

20 -Ac - = Aa.- = AAT(r,z) rT 0T ' (2.20) Aa - = Aa - = 0 . zT rzT

We call (2.19) the condition of vertical strain.

From the above it is clear that during fluid injection into a

reservoir the conditions for poro- and thermo-elastic stresses can change from plane strain to vertical strain. In Sections 2.4 and 2.5 of this chapter this will be demonstrated with explicit examples.

In Appendix 2-B the full (r,z) dependence of the particular solution is obtained from (2.13) and (2.15) with the help of modified Bessel fuctions and Fourier integral transforms.

The particular solution does not generally satisfy all prescribed boundary conditions on the stresses. By making use of the linearity of the theory, this can be achieved by adding an appropriate solution of the homogeneous equations of elasticity to the particular solution.

2.2.5 Complete formulation of the problem

For given AT the stresses are completely determined if stress boundary conditions are imposed at the wellbore and at infinity.

In the specification of the boundary conditions at the wellbore the following well model is used (Fig. 2.2). Two packers in a vertical wellbore are separated by a distance d. Between these packers a pressure Ap , uniform with height/ is exerted on the surrounding elastic formation. Ap = p -p. is the pressure differential needed to sustain a certain fluid injection rate q. The distance d can be larger than or equal to the reservoir height h. The change in stress Aa. . is calculated with respect to the initial stress state at Ap = 0 and AT = 0, and the reservoir at uniform pressure and

w

temperature. Effects of casing or perforations on the change in stress state are not considered. This model leads to the following set of boundary

IMPERMEABLE

PERMEABLE

~ri

_{r - 1}

GEOMETRY OF WELL WITH PACKERS PENETRATING A DISC-SHAPED RESERVOIR OF INFINITE RADIAL EXTENT AND SURROUNDED BY A N INFINITE IMPERMEABLE M E D I U M .

22 -Ao = Ap r w Aa = 0 rz Aa = 0 r Aa = 0 rz

} « - v |.|<f

(2.21)

} r .

V

|.|»f

Since we are considering an infinite elastic medium we have the additional boundary conditions: 1im A o . . = 0 (2.22) r-» ^ lint Aa. . = 0 Izl-» 1J ■ 2.2.6 Method of solution

To solve the problem the following decomposition of A o . . is made:

Aa.. = Aa..- + A a ? . (2.23)

where Aa. - is the solution to the "traction-free" wellbore problem: 13T Aa - = 0 rT Aa - = 0 rzT } r = r , -0» < z < +» (2.24) w

lim Aa..- = lim Aa. - = 0 r->» I z I ■*»

a n d A a . . is a s o l u t i o n t o t h e h o m o g e n e o u s e q u a t i o n s of e l a s t i c i t y d e s c r i b i n g t h e s t r e s s e s induced b y the r a d i a l loading o f the w e l l b o r e . F r o m ( 2 . 2 1 )

-» o

Ao = Ap Ao° - O rz (2.25) Ao° = O r Ao° = O rz

} «-«„. | . | » f

lim Ao. . = lim Ao.. = 0

Problem (2.25) is relatively well known and was first solved by Tranter . 7

His result was extended by Kehle to account for the shear stresses exerted on the formation by the frictional force of the packers. Kehle found that for realistic cases this effect can be neglected.

The solution to problem (2.24) has, to our knowledge, not been

published in the literature. We have determined Ao. - by decomposing it into the particular solution and a solution to the homogeneous equations of

elasticity,

Ao. .- = Ao. .- + Ao° - (2.26) ljT ljT 13T

such that (2.24) are satisfied.

As discussed earlier and shown in Appendix 2-B the particular solution Ao. - is obtained from an axisymmetric Goodier potential. In Appendix 2-C it is shown how the solutions for Ao. - and Ao.. are obtained from an

i]T 13

axisymmetric biharmonic function called the Love potential. The solutions appear as complicated Fourier integrals over modified Bessel functions. Table 2-1 summarises the various decompositions and methods of solution.

2.2.7 Asymptotic expansions of the stress solution

Tranter showed how his solution for Ao.. can be evaluated numerically. Unfortunately this method is rather cumbersome. Moreover, it cannot be

readily extended to evaluate Ao. - and Ao..- since these expressions contain

1 _ I J T ijT r

additional integrals over AT.

In Appendix 2-D it is shown that the integral transforms of the stress solution can be evaluated analytically if asymptotic expansions of the

24

-TABLE 2-1 - DECOMPOSITION OF STRESS TENSOR

Aa. . = Aa..- + Aa.. lj IJT i] Aa.. = +

Aa..-ljT 1}T Aa..-ljT

; Aa. . satisfies complete wellbore boundary conditions, Eq. (2.21)

Aa. - satisfies "tract

IJT

boundary conditions, Eq. (2.24)

; Aa. - satisfies "traction-free" wellbore

IJT

Component Differential equations (d.e.) Generating potential

Aa°.

ID o

Aa

i jT

homogeneous d.e. of elasticity Love potential

homogeneous d.e. of elasticity Love potential

ljT d.e. of poro- and thermo-elasticity Goodier's displacement

Bessel functions are made. In the calculation of the integrals it is assumed that the axisymmetric function AT is constant over the reservoir height and zero outside of the reservoir, i.e.

AT = AT(r) . H ( | - |z|) (2.27)

where h is the reservoir height and H the step function defined as:

H(a-x) = 1 x < a

(2.28) = 0 x > a

If the Bessel functions are expanded in powers of r/h or r'/h (whichever is the smallest) the particular solution for the tangential stress becomes to lowest order:

A(J0T = " 2 ^2 ' dr'r'AT(r').N(z, j,r) + AAT(r).H(| - |z|)

r r w

+ f ƒ dr.r.AT(r.,. { , '

+ + 2

'"

>2 3/2

)

r (z + r' ) (z + r' ) (2.29) where z z_ N(z, -,r) = — „ , .^ + ~ . ■-_ (2.30) 1 ' 2 , 2 2,1/2 , 2 2,1/2 (z+ + r ) (z_ + r ) and z+ = | + z (2.31)

No precise criteria can be given for the region of validity of (2.29) as it stands. These criteria depend on the shape of AT and on the ratio of

its penetration depth to the reservoir height. For a specific AT the accuracy of (2.29) has to be investigated by comparison with a numerical evaluation of the exact solution for Aa.-. This will be discussed further in

ÖT Sections 2'. 4 and 2.5.

26

-If in the solution for Ao„- the Bessel functions containing r and r ÖT * w are expanded, the result becomes to lowest order in r /h and r/h:

A°*T = ^ A SH T <2'32>

where

A SHT = 4 ' d r'r' ^ (r' ) t 2 '+ ,3/2 + , 2' ,2,3/2* <2'33>

r (z + r' ) (z + r' )

w + —

As for (2.29) no general criteria can be given for the accuracy of (2.32). Further discussion appears in Sections 2.4 and 2.5.

If Tranter's solution for Laa is expanded to lowest order in r /h and

a w the term with the modified Bessel function K (kr) (see Appendix 2-D) is

neglected, we find: Ap r 2

Aa° = - ~Y (-f-) N(z, - , r) (2.34)

where N is the same function as in (2.30) but with h replaced by d. It can also be written as:

d u+ u -N(Z

' 2~'

r) =

2 TJl

+

2 Ï/2

(2>35) 1 [ u ^ + l]l//L [u_ + 1 ] V where d/2 + z u = (2.36) + r v '

At the wellbore (r = r ) the expression for Aan can be expanded while w a

r e t a i n i n g the K term, r e s u l t i n g in:

Ap

Aff

e

(r

w>

=

" ~f

N(z

' 2 ' V

( 2 . 3 7 ) AP w v a +

——u^

Z

' 2 > V

where:

d U+ U

-M(z, - . r) = ~ jji + — (2.38) [ u / + 1] ' [u_ + 1 ]J / 2

In Fig. 2.3 a plot of Aa. /Ap versus 2z/d is presented for various values of 2r /d. Poisson's ratio was taken as 0.2. The plot shows Aoa up to

w a

first order (Eq. (2.34) at r = r ) and up to second order (Eq. (2.37)). It is shown that for 2r /d < 0.1 sufficient accuracy is maintained if the second term is neglected. Furthermore, for 2r /d < 0.01 the function N/2

w behaves as a step function so that (2.37) becomes:

i o

«

( t

„' - - *\, W •= f

|.|»f

2r

} —r< 0.01 (2.39)

o 8 The result ACT (r ) = - Ap is the same as the well-known result obtained

0 w w

under plane strain conditions. In the latter case it is assumed that the loaded interval d is infinitely long.

We have shown that the solution for Aa. can be safely approximated by the first term in its asymptotic series expansion, as long as 2r /d < 0.1. However, since we are dealing with an asymptotic series the complete range of validity of this first term approximation may be substantially larger. This complete range of validity can only be determined by comparing (2.34) with a numerical evaluation of the exact expression for ACT.. Since however,

Ü

the condition 2r /d < 0.1 is satisfied for most practical cases we have not investigated this matter any further.

2.2.8 Simplified solution method

If 2r/h < 0.01 then from the previous subsection — N(z,-,r) becomes a step function. Under this condition the particular solution (2.29) becomes in the neighbourhood of the wellbore (r ~ r ):

A a0T = " \ ! dr'r'AT(r') + AAT(r) + ASH-(z) |z| < | r r w = A SH i( z , | z | > | (2.40) with AS - defined in (2.33).

28 -o.o - 0 2 -0.4 -0.6 -1.0 _ - 5 U = L = « 2rw /h 0.01 2rw/h = 0.1 2rw/h= 1.0 UP TO FIRST ORDER UP TO SECOND ORDER 0.8 1.2 1.6 1.8 2 . 0 2 Z / h

Ao-0°/Apw FOR VARIOUS VALUES OF 2rw /h

We now observe that for |z| < h/2 Eq. (2.40) consists of the well-known plane strain expression (2.18) plus the term AS -.

Taking, for convenience, the distance between the packers as being equal to the reservoir height (h=d), we have for the complete tangential stress field near the wellbore:

Aan = Aan- + Ao„- + Laa 9 6T 0T 6 r 2 r = (7i ) { A SHT " A pw} " ~2 S d r , r , A T(r' ) (2.41) r r w + AAT(r) + A SH T |z| < | r 2 " = (—) AS - + AS - |z| > r r HT HT ' ' 2 for { 2r/h < 0.01 r - r w

In Appendix 2-E it is shown that (2.41) could have been obtained from the plane strain expression for the particular solution (2.18) and from the simple plane strain solution to the homogeneous equations of elasticity provided that the combined plane strain solution is subjected to the boundary condition

Aa r

lim { = AS - - » < z < + » (2.42) _£-►00

From this observation AS - can be interpreted as an apparent change in the all-round horizontal far field stress.

A similar situation exists for the vertical stress. From Appendix 2-E:

Aaz- = AAT(r) + ASV- (z) |z| < |

(2.43) =ASv-(z) | z | > |

30

-where

ASv-(z) = - 2ASHÏ(z) (2.44)

From comparison with (2.18) AS - can be interpreted as an apparent change in the vertical far-field stress. When plane strain solutions are used this change must be incorporated as the boundary condition:

l i m r-*oo A(7 z = ASt -VT - co < z < + » (2.45)

In Chapter 3 the above interpretation is used in calculating the poro-elastic stresses around a vertical fracture. The poro-poro-elastic stresses at the fracture wall are calculated in plane strain. Eq. (2.42) with a slightly adapted version of (2.33) is then used to account for deviations from plane strain.

2.2.9 Numerical method to evaluate Aa^z 0T

In Appendix 2-F a numerical method is presented for the evaluation of the particular solution Ao.-. This method is useful for investigating the range of validity of the asymptotic expression (2.29). It can also be used when AT does not satisfy (2.27) but has a general dependence on the vertical coordinate. The method consists of the numerical evaluation of certain

integrals. To eliminate the oscillatory behaviour of the integrand, the solution for Aafl- is represented in terms of complete elliptic functions

rather than in terms of Bessel functions. The integrals are then evaluated 9

using a standard integration routine from the IMSL library .

2.3 SOLUTION FOR THE PRESSURE AND TEMPERATURE FIELDS

2.3.1 Temperature field

The temperature field is determined from Lauwerier's solution in cylindrical coordinates . This solution applies to the injection of an incompressible fluid at constant rate. One-dimensional vertical heat

conduction in cap and base rock is taken into account, horizontal conduction is neglected and inside the reservoir the temperature is assumed to be

constant in the vertical direction.

H 2 TD = erfc Wr D } R < 1, |z| < | 2(l-RD ) „ 2 A D TD + D = erfc { ~ — R < 1, |z| > f (2.46) 2/(rD(l-RD2,) = 0 R > 1, -o» < z < » where and TD TD RD R c = = = = T - T r e s T. . - T m j r e s 4a t M s s 2 2 h M r r R c M . 1/2 lM h?rJ r (2.47) r

T = initial reservoir temperature res

T. . = injection temperature

in]

t = injection time h = reservoir height

a = thermal diffusivity of cap and base rock M = heat capacity of cap and base rock

M = heat capacity of fluid filled reservoir rock M = heat capacity of injection fluid

= injection rate

R is the radius of the temperature front and obeys the simple convective heat balance:

32

-heat absorbed by injection fluid = -heat given off by the reservoir or

2

qt M AT = it R h M AT (2.48)

n w c r '

At R the temperature difference tends to zero (Eq. (2.46)).

The assumption of one dimensional vertical heat conduction is justified if:

M R

Pe =

ÏTTT

>:> x ( 2

-

4 9

>

s s

where Pe is the dimensionless Peclet number. Condition (2.49) means that the radial velocity of the temperature front is much greater than the vertical velocity of the temperature transients in cap and base rock. Therefore, with isotropic thermal conductivities and approximately equal thermal

diffusivities in the reservoir and cap and base rock, radial temperature transients may be neglected. Using the expression for R in (2.47), (2.49)

c becomes: M q W » 1 (2.50) M ffha . s s

Condition (2.50) is satisfied for most field conditions.

In Fig. 2.4 T inside the reservoir has been plotted vs R for

different values of r . This plot shows that for r < 0.05 the step function is a good approximation to the temperature profile inside the reservoir. This is the convective limit in which the amount of heat given off by cap and base rock is small with respect to the amount of heat given off by the reservoir.

2.3.2 Pressure field

For simplicity, we first consider fluid flow without discontinuities in fluid mobility. The line source solution for injection of slightly compressible fluid at constant rate is given by :

LAUWERIER TEMPERATURE PROFILES INSIDE RESERVOIR

34 -where Ap = p ( r , t ) - p k.

X

h q

n

= fluid mobility (X = — ) M = reservoir height = injection rate = hydraulic diffusivity (r? = k 0MCfc t = i n j e c t i o n time 00 - g Q

Ei = exponential integral (- Ei(-x) = ƒ ds) x

2

For r /47jt < 0.02 ( 2 . 5 1 ) can be approximated by:

2 * ^ - - l n f - ( 2 . 5 2 ) q R e w i t h R = 1.5 /(Tjt) ( 2 . 5 3 ) e

Eq. (2.52) has the form of the steady-state solution with a time-dependent exterior radius. We therefore call (2.52) the quasi steady-state

approximation.

Introducing the dimensionless quantities:

rD = R - (2'54> e = 2jrhXAp. ^D q Eq. (2.52) becomes: A pD = - In rQ (2.55)

Fig. 2.5 compares (.2.55) with the line source solution (2.51). It is shown that for r > 0.6 the accuracy of (2.55) becomes less than 13%. However, for calculating poro-elastic stresses (2.55) is sufficiently accurate, since most of the pressure rise and therefore most of the stress build-up occurs close to the well.

An expression for the pressure field in the presence of

7.0

6.0

-UNE-SOURCE QUASI STEADY-STATE

1.2 1.4

UNE-SOURCE AND QUASI STEADY-STATE SOLUTION

36

-that incompressible fluid displaces slightly compressible oil in a piston­ like manner (Fig. 2.6). The flooded zone consists of a cold region with constant fluid mobility and a warm region with constant fluid mobility. The following exact solution has been obtained:

2 2 27Th . l n c 1 _ F 1 1 , FV„ . , Fx

Ap, = r— In — + r— In — - ~ r— exp (T~7) .Ei(- 7 - )

q *! X r X R 2 X ^ 4rjt' v 4rjt' 2 2 R R R

T

4 p

2

=

x ;

l n

r - 2 ^

e x p

w

) > E l (

" ^

J (2

*

56) R 2 2 2?rh . 1 1 . _ F . r „ — Ap3 = - - T- exp (T^)-Ei(-T^)

with Ap., X., i=l,2,3 the pressure differences and mobilities in the cold flooded zone, the warm flooded zone and the oil zone, respectively, TJ is the hydraulic diffusivity in the oil zone. R and R„ are the radii of the

J J C F

temperature front and of the flood front, respectively. If: -r— < 0.02 4rjt -(2.57) 2 f - < 0.02

the'following expression is a good approximation to (2.56) R X, R„ Xn R Ap,„ = In — + r~ In -— + r— In —-ID r X„ R X_ R„ 2 c 3 F X R X R 2 3 F X, R A 1 1 e A p

3 D

=

r

3 l n

r

2jrhX

with Ap.„ = Ap., j = 1,2,3 JD q *}

INJECTION FLUID INCOMPRESSIBLE OIL SLIGHTLY COMPRESSIBLE

SCHEMATIC R E P R E S E N T A T I O N OF P R E S S U R E P R O F I L E WITH V E R T I C A L D I S P L A C E M E N T F R O N T S .

38

-2.4 ANALYTICAL SOLUTION FOR THERMO-ELASTIC STRESS VARIATIONS Calculation of stresses for temperature field with step profile

It was shown in Section 2.3.1 that a step profile is a good

approximation to the temperature distribution inside the reservoir provided that 2 4a t M rD = — | 1- < 0.05 (2.59) h M r

Furthermore, if we neglect the temperature change in cap and base rock due to conduction, we simply have:

AT(r,z) = AT.H(^ - |z|).H(R -r) (2.60)

with H the step function defined in (2.28).

If this simple profile is substituted into the thermo-elastic part of :he remaining integrals can be easily

this thermo-elastic component is given by:

Aa_- the remaining integrals can be easily evaluated. From (2.29) and (2.32)

ST

=

ST

+ A

Vr

r 2 A r = (7*) A SH T ƒ dr'ATfr'Jr'.Nfz,-^) + y i ( r ) H ( j - |z|) 2r r w A » z z_ + — ƒ dr'r' AT(r'). { - + } (2.61) (z 2 + r-V/ 2 (z 2 + r '2)3 / 2 where: AS„_ = — ƒ dr'AT(r')r' { r — - r + ^ ^ ,n] (2.62) HT 4 v ' , 2 .2 3/2 , 2 ,2 3/2 v ' r (z + r' ) (z + r' ) w + -and N as defined in (2.30).

0TD A „ A T T r

D

=

t

c 2D = R c hD ~ 2R c (2.63) z

After integration there results:

r „ 2 , r ~ 2 — w D 1 w D 1 i i Aa

*TD "

(

^ - >

AS

HTD

+

I <^> ^ D ' W " Ï

N

< W

1 ) +

^V^D'*

r < 1 (2.64a) 2 w D 2 1 w D (

— >

A S

HTD

-1

(

—r> ^ v w

r D

*

x (2

-

64b) D r_ where: AS„m„ = T (N(z .h.r „) - N(z„,h .1)) (a) HTD 4 v D D wD D D ' z z N(V W \ 2 + D\l/2 + 2 „ % 1/2 ( 2'6 5 ) ( b )

(z + r ) (z + r )

v D+ D ' v D- D ' z „ = h„ + z ( c ) D+ D - D v '

A t the wellbore (2.64a) becomes:

^ T D ^ w D5 " 2 A SH T D + H ( hD - lsDl> ( 2'6 6 )

The solution for Ao. incorporates the traction-free boundary condition at the wellbore. The extra term Aa. due to a load Ap applied at the wellbore

a w is not included.

40

-A few wellbore radii out into the reservoir -Aaflrn„ = (r ^/r„) AS„„„

0TD wD' D HTD becomes very small and Ao.m„ becomes identical to the particular solution

0TD

Range of validity of asymptotic expressions

To investigate the range of validity of (2.64) we have first compared the analytical solution for Ao. (Eq. (2.64) minus Ao. ) with a numerical evaluation of Ao. . The latter is obtained using the method described in Appendix 2-F.

The results are shown in Figs. 2.7a-2.7f. Here Aortm„ was made

3 0TD

dimensionless with respect to |AT| rather than AT and therefore, since the dimensionless change in stress is negative for r < 1, a cooled reservoir is represented. For comparison, the analytical plane strain solution is also shown (see Eq. (2.73) below).

From the figures there is excellent agreement between the analytical and numerical solution within the cooled region. For smaller dimensionless reservoir heights the results become less accurate close to the temperature front (Fig. 2.7g). Close to the front outside the cooled region the relative error can become as large as 20%. Within the cooled region the absolute stress level is higher so that with approximately the same absolute error the relative error inside is much smaller than the relative error outside the cooled region.

In Ref. 5 Perkins and Gonzalez investigated the change in tangential stress induced by a cylindrical inclusion of constant temperature. They did not consider the presence of a wellbore. They provided an expression for the change in tangential stress averaged over the cooled region. This expression was obtained by curve fitting to numerical results.

We calculate the average change in tangential stress by putting r = 0 in (2.64a)/ integrating over z and dividing by h. The result is:

A

^ D

5

* 1 " Ï {I* + < 1 / V

2

1

1 / 2

"

1

/

h

D

} r

wD

=

°

( 2

-

6 7 )

•

In Fig. 2.8 the result obtained by Perkins and Gonzalez is compared with (2.67). The maximum difference is found to be 6%. For completeness

(2.64a) with r „ = 0 and z„ = 0 is also shown. wD D

At the wellbore we have for the particular solution:

AaamrJr „) = AS„m„ + H(h„ - I z I ) (2.68)

- 0 . 6 -0.8 -1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 -_ h0=10. zD=0.0 NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

DIMENSIONLESS RADIAL DISTANCE FROM THE WELLBORE

1.8 2.0

FIG. 2.7a CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE

1.0 0.8 0.6 -0.4 0.2 0.0 -- 0 . 2 - 0 . 4 - 0 . 6 - 0 . 8 T" T T -1.0 hD = 10 rWD = 0.06 rD =0.1 NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL)

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

DIMENSIONLESS VERTICAL DISTANCE FROM THE MIDDLE OF THE RESERVOIR

FIG. 2.7b CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE

- 42 1.0 0.8 0.6 0.4 0.2 0.0 - 0 . 2 - 0 . 4 - 0 . 6 0.8 T t . 0 0 . 0 - 1 . 0 "-NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL)

_1_ _1_

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 DIMENSIONLESS RADIAL DISTANCE FROM THE WELLBORE

1.8 2.0

FIG. 2.7C CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE

1.0 0.8 h 0.6 Q.4 0.2 0.0 T -1.0 h0 ■ 1.0 0.0' 0.1 rw 0 = 0 . 0 0 6 j _ NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 DIMENSIONLESS VERTICAL DISTANCE FROM THE MIDDLE

OF THE RESERVOIR

FIG. 2 . 7 d CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE

NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL)

J_ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

DIMENSIONLESS RADIAL DISTANCE FROM THE WELLBORE

1.8 2.0

FIG. 2 . 7 e CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROFILE

1.0 0.8 0.6 0.4 1 1 1 NUMERICAL SOLUTION

PLANE STRAIN SOLUTION (ANALYTICAL) ASYMPTOTIC SOLUTION (ANALYTICAL) h0 «0.1

rWD ■ 0.0006

r„ =0.1

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 DIMENSIONLESS VERTICAL DISTANCE FROM THE MIDDLE

OF THE RESERVOIR

FIG. 2.7f CHANGE IN TANGENTIAL STRESS INDUCED BY A STEP PROHLE

44 -< Z LU O 20.0 16.0 12.0 -OC O cc o: Ld 8.0 < 4.0 0.0 h„ =1.0 rw0= 0.006 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FIG. 2.7 g RELATIVE ERROR OF ASYMPTOTIC SOLUTION

co CO 1 • 1 CO _ I < h-z UJ O z < (— en to UJ _ i z o CO z UJ 2 O z UJ o z < X

o

I . U 0.9 0.8 0.7 0.6 0.5 n A 1 1—i i i i 111 1 1—i i i i 111 1 1—r—r f / t: / /.•/ ' . ■ / '// / y i / t A »/: ~ '/.'" '/ * / / ■ / — */ * s •'' ,''~Ss^ ••■' M i l ] -_ — FITTED CURVE OF P4G

AV/rOAtTTH AK1AI VTtPAl PT^I II T "™ AVLKAuCU ANALT llwAL KLOUUI A hi A 1 V T t f * A 1 D F C I ft T CftO • — A

1 ' ■ » ■ ■ ■ ■ ! | ■ ' t ■ 1 ■ 1 1 ■ 1 1 1 1 ■ 1 l l 1 1 _J - 1 - L U . l l | | l _ J _ - l . 1 1 1

10 ,-J 10" 10°

1/hc

10' 102 105

with AS from (2.65a). The particular solution can be regarded as the

superposition of the solution for a cylindrical inclusion with radius R and temperature AT and the solution for an inclusion with radius r and

w

temperature -AT. Since the analytical solution for Aoa__ is accurate at

r > R for h > 10 or 2R /h < 0.1, we can similarly conclude that (2.68) is accurate at least for 2r /h < 0.1.

w

For Aff„ we have at the wellbore: 0TD

S T D (r

wD> "

A £

W

r

w D >

+ A f f

ÖTD

( r

wD

) = 2 A S

HTD

+ « ' V ^ D '

<

2

'

69

>

From the similarity between (2.69) and (2.68) Aafl is also accurate at

least for 2 r /h < 0.1. This is confirmed by the fact that the functional w

relationship between r and h in (2.69) is the same as that between r and d

w w in the first term of Ao. (Eq. (2.37)). A conservative criterion for the

o

accuracy of the analytical solution for Aafl is 2r /d < 0.1.

a w

We conclude that when the use of temperature distribution (2.60) is justified, an accurate description of the thermo-elastic stress change is given by the analytical expression (2.64) with the restriction that

2r /h < 0.1. Most likely this condition can be relaxed. However, since it is satisfied for most practical cases we have not attempted to specify it any further.

The plane strain and vertical strain limits

Following the discussion in Section 2.2.4, two limiting cases can be considered, h ■* » which is the plane strain case and h -* 0 which is the vertical strain case.

For simplicity, we take 2r /h < 0.01 so that according to Fig. 2.3

1 w "

- N(z .h_,r _) becomes a step function. From (2.64) and (2.65) we have the 1 D O WO

46 -h » 1 (plane strain) ASHTD = ° 2D = ° ( 3 ) ^ H T D = ° lZDl ' hD ( b ) } hQ > 100 ASHTD = ° '2DI * hD ( C ) ( 2'7 0 1 AS„mT, = T |zj t h„ (d) HTD 4 ' D' D ' } 10 < h < 100 AS„„„ = " 7 |zj 4. hn (e) HTD 4 ' D' D 0TD = 2 A" -0TD " 4

STD

=

"

1 4 Z

D = °

I

2

DI

f h

D

} 10 < h < 100

I

Z

D I *

h

D

(a) r 2 }(-T2) « 1 (b) (2.71) D (c) AaafT,„(r „) = 1 z„ = 0 (a) 0TD wD D AöflmT,(r „) = 1.5 |z ! t h„ (b) (2.72) 0TD wD ' D' D } 10 < h < 100 Aa. (r « ) = - ; |z„| * hn (c) 0TDv wD' 2 ' D' D v

From (2.70a) we have that at z = 0 (2.64) reduces to the plane strain result: A^ ( p s ) - A„(PS> - i ,fwDx2 * I rn < 1 A°0TD " A<7ÖTD ~ 2 ( ~ > + 2 ° D (2.73) * " 2 ( 2 — > rD " 1 rD

h « 1 (vertical strain) AS. HTD 2 AS = 0 HTD Ao 0TD Aa 0TD = 1 = 0 Aartm^(r „) = 2 ÖTD wD

*WW

=

°

< h. > h. < h. > h. < h. > h. (2.74) r

„

_wD 2 ( — ) « 1 D (2.75) (2.76)

We see from these limits that inside the reservoir the stress changes double as the state of deformation goes from plane strain to vertical

strain. In the plane strain case the stress change at the vertical reservoir boundary is 50% larger than a bit further inside the reservoir for

10 < h < 100.

It is interesting to observe that a few wellbore radii out in the reservoir the stress discontinuity across the vertical reservoir boundary equals A AT whereas at the wellbore it equals 2A AT.

In Fig. 2.9 Aa. (r ) is shown for 2r /h = 0.01 and for various values of h„.

2.5 ANALYTICAL SOLUTION FOR PORQ-ELASTIC STRESS VARIATIONS

Calculation of stresses for quasi steady-state pressure profile

When no discontinuities in fluid mobility are present, the pressure distribution can be approximated by (from 2.55):

with

Ap„ = - In rn

rD D

r

D

=

t

48

-DIMENSIONLESS THERMO-ELASTIC STRESS CHANGE AT WELLBORE

R = 1.5 /(rjt) e

When (2.77) is substituted into the poro-elastic counterpart of (2.61) the integrals can be easily evaluated.

We introduce the dimensionless variables: z , h ZD = R ' ftD = 2R e e . - 2ffhX A- „ „„ A%>D = a T ^ A°öp <2-7 9> HpD qA Hp

After integration there results:

r 2 +

Ï Q< V V

1

* ~ Ï °< V

W •

(r

D ±

1} (2

'

80)

=

(

r f >

AS

H

P

D

+

\

N

< W V

{

" Ï ~2

+

^ A

A

VW

+

I>

} D rD D ( rD> 1) w h e r e APD!r w D> 1 1 AS

H

P

D

=

—1 '

N(

WW

+

I

Q

(

Z

D'

h

D'

1}

" I « « V V W

( 2 . 8 1 ) z z Q(z , h , r ) = a s i n h + a s i n h rD rD Z~ , = n^ + Zr , D+ D - D

50

-/ 2

with asinh x = ln(x + /(x + 1)), N defined in (2.65b) and H the step function in (2.28).

At the wellbore (2.80) becomes:

S

P

D

( r

wD> "

A

eD

(r

wD>-

H(

V K I '

+ 2 A S

H

P

D " <

2

'

82

>

As for the thermo-elastic case, a few wellbore radii out into the oir Aaa _ = (r _/r_) As„ _ become

0pD wD D HpD identical to the particular solution Ao,

reservoir Aaa „ = (r n/rn) As„ „ becomes very small and Aan _ becomes

0pD wD D HpD J 0pD

0pD* Range of validity of asymptotic expressions

As in the previous section, we have first compared the analytical solution for Aafl with the numerical one using the method given in Appendix

2-F. The results are shown in Figs. 2.10a-2.10f. The figures show that there is excellent agreement between the analytical and numerical solutions both within and outside the reservoir. In fact, the curves for the numerical and analytical solution coincide completely in Figs. 2.10a and 2.10b and almost completely in Figs. 2.10c-2.10f. For comparison the analytical plane strain solution is also shown (see Eq. (2.87) below).

. o

For Ao. _ we have at the wellbore: _0pD

A o ° (r \ = AS„ „ (2.83) 0pD wD HpD

with AS„ „ from (2.81). HpD

To investigate the properties of (2.83) it is convenient to form the dimensionless quotient: Ac.° (r ) Ao ° (r \ ASn „ dp vi flpD wD7 _ HpD A Ap(r ) Ap fr ) Ap„(r „) p w rD wD rD wD 7 N(z„,h„,r „) - T Ufz^fh,,^ „) 4 D D wD 4 D D wD (2.84)

where from (2.81) and (2.77):

m y y i ) - Q(

zD

>

hD

>

rwD

)

u (

W

r

«D

} =

Ï T 7 T

{ 2

'

8 5 )

wD

Fig. 2.11 shows the function - U for 2r /h = 0.01 and for various values of

2 1

rwD » 0.003 hD «1.0 z„ = 0.0 \

V .

_i_ PRESSURE STRESS (NUMERICAL) STRESS (ANALYTICAL) STRESS (PLANE STRAIN)

_l_ _1_

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

FIG. 2.10a DIMENSIONLESS PORO-ELASTIC STRESS CHANGES

O o O o -O O O -O rwD « 0.003 r„ ■ 1.0 0.1 0.0 0.2 0.4 0.6 0.8 1.0 PRESSURE STRESS (NUMERICAL) STRESS (ANALYTICAL) . . . . ■ J . ' B J H M M W W I 1.2 1.4 1.6 1.8 2.0