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Fracture initiation pressures in permeable poorly consolidated sands
Ira. d. Rock Mech. Min. Sci. & Geomech. Ahstr. Vol. 19. pp. 255 to 266. 1982
0148-9062/82'060255-12503.00/0 Copyright © 1982 Pergamon Press Lid
Printed in Great Britain. All rights reserved
Fracture Initiation Pressures in Permeable Poorly Consolidated Sands P. H O R S R U D * R. R I S N E S I " R. K. B R A T L I *
This paper studies the stress distribution around an il~jection well drilled through a layer of poorly consolidated sand. As a plastically strained zone may f o r m around a hole drilled through this type of rock, both elastic and plastic theories must be applied. Solutions have been worked out in order to Jollow the development of stresses as the injection pressure increases fi'om zero up to the point where a fracture is initiated. H o w these stresses develop is dependent on the rock and ,fluid properties, mainly Poisson's ratio and the ratio between the horizontal and vertical stress. Several different stress distributions will appear as the injection pressure is increased. Each stress distribution is composed of one or more different stress solutions linked together. This is solved by numerical iteration. T h e injection pressure is increased until a .fracture is induced. Fracture initiation pressures for several different parameter sets have been examined. T h e results have then been contpared with fi'acture initiation pressures calculated by classical elastic theory. This comparison shows that the fi'acture initiation pressures can be lower than results calculated by elastic theory. Typically, with a Poisson's ratio o f 0.20 and a horizontal to vertical effective stress ratio of 0.40, the d!fference is in the order o f ntagnitude of I0%.
t, V = u = z = =
Constants defined in Appendix B. Radial displacement. Depth. Failure angle. = 1 - Cmj'C b = Blot constant. 7' = Constant defined in Appendix B. A" = Volumetric elastic strain. ¢;', ¢~, ~'-' = Elastic strain components. ¢,,P %,P E,P = Plastic strain components. E,, %, E: = Total strain components. h- = A scalar in the plastic flow rule. 2 = Lam6 parameter. # = Fluid viscosity. v = Poisson's ratio. 6 I, 0"2, (73 = Principal stress components. ar, a o, a: = Stress components in cylindrical co-ordinates. a, i = Radial stress at Rj. a, c = Radial stress at Re. a,o, a=o = Radial and vertical stress at R o. a h ( = a, o) = Horizontal stress at R o. ar = Uniaxial tensile strength.
NOMENCLATURE
Ai, Bi, Dl, O2 = Integration constants. CO = 2S¢otan~ = uniaxial compressive strength. Cma
Rock matrix compressibility.
Cb = Rock bulk compressibility. D,, Db = Constants defined in Appendix B. E = Young's modulus. Plastic flow function. f G = Shear modulus. h = Height of permeable layer. K = Horizontal to vertical effective stress ratio. k = Permeability, p = Fluid pressure. Pi = Fluid pressure at R~. Pc = Fluid pressure at plastic-elastic boundary. Po : Fluid pressure at Ro. Pf. el = Fracture initiation pressure calculated by elastic theory. ef, pl : Fracture initiation pressure calculated by plastic theory. O = Fluid flowrate (negative when injecting). r = Radical distance from center of wellbore. R i = Wellbore radius. Ro: Outer boundary radius. RI, R2, R3 = Radii of different plastic zones. R~ = Radius of entire plastic zone. Sco = Cohesive strength.
INTRODUCTION When
*Continental Shelf Institute, Hhkon Magnussons gt. IB, 7001 Trondhe]m, Norway. 5" Norsk Agip, Sandnes. Norway. 255
a liquid
is p u m p e d
into a borehole
at a high
enough
pressure, the stress at the borehole
s u r f a c e will
become
tensile. If this stress exceeds the tensile strength
o f t h e m a t e r i a l , a f r a c t u r e will o c c u r . T h i s t e c h n i q u e , c o m m o n l y k n o w n a s h y d r a u l i c f r a c t u r i n g , is a m e t h o d used by many
in t h e o i l i n d u s t r y
to improve
the pro-
256
P. Horsrud et al.
ductivity of a formation. However, for flooding projects it is essential to avoid initiating fractures. Determination of fracture initiation pressures is therefore important. One way of estimating the fracture gradient is to purposely initiate a fracture by increasing the wellbore pressure. This test, called the formation integrity test or leak-off test, is not always straightforward to interpret, especially in permeable formations. Unintended fracturing around injection wells can have severe consequences to a flooding project. Such fractures can penetrate deep into the formation, Although this will lead to improved injectivity it can jeopardize the whole injection project because of the reduction in sweep efficiency. Another result from such unintended fracturing could be influx of particles into the injectors at a system shut-down. This could eventually plug the injection wells. When killing a blow-out by creating a water bank around the blowing well, it is essential that no fracture is created between the injection well and the blowing well. Such a fracture would effictively short-circuit the flow between the wells, and make it impossible to block the blowing fluid by a water bank. The stress situation around boreholes has been investigated by a number of authors, in various degrees of refinement. The early work of Biot [1] laid the basis for much of the later work. He included the effect of the pore pressure in his investigation and thereby opened the way for studies of injection and fracturing in permeable rocks. Biot [2] later developed analytical solutions for non-linear stress-strain relations and applied them to fracturing around cylindrical and spherical cavities. Hubbert & Willis [3] studied theoretically hydraulic fracturing of rocks and concluded that the fractures should be perpendicular to the axis of the smallest principal stress. In tectonically relaxed areas with normal faulting this would imply that the fractures are vertical.
A criterion tor the initiation of vertical fractures was later presented by Haimson & Fairhurst [4]. They assumed the rock to be elastic and included also the effect of the pore pressure and the fluid flow into the formation. Eaton [5] suggested to use the horizontal stress level as the fracture pressure. He claims that in a permeable formation there is little difference between the fracture initiation pressure and extension pressure and does therefore not distinguish between the two. The same problem has been studied extensively by Geertsma [6,7]. He also discussed the consequences of formation failure around injection wells. In his analysis Geertsma assumed poroelastic rock, but he states that breakdown pressures will be lower than predicted by poroelastic theory if the rock surrounding the borehole has been exposed to non-elastic deformations. Medlin & Mass6 [8] studied fracture initiation in the laboratory and compared this with the results from a theory which was based on poroelastic behaviour. Their conclusions were restricted to results for hydrostatic stress conditions. For the non-penetrating fluid they found that when the stresses exceed the elastic range, failure pressures will be lower than predicted. For the penetrating fluid case their results were not completely consistent with the theory. We have earlier shown that plastically strained zones will be formed around the wellbore when a well is drilled through poorly consolidated sand [9]. In this analysis we also included the effect of the third principal stress, the vertical component resulting from the overburden. Typical examples of the resulting stress distributions are given in Figs 1 and 2. In this paper we will try to show how such a plastic zone can influence the stress distribution around the well during injection. The stress distribution can be followed as the injection pressure is increased and eventually reaches the level where a fracture will be initiated.
K
- 0.60
v
- 0.30
Co = 0.36 MPa (52.0 psi)
(MPa)
Ro
10.0 m
70
60"
50 ¸
40
-
30
0.2 Ri
0.4
0.6
0.8 ~
2
Po
1.4 rlm l
Rc
Fig. l. Example on STRESS 1
-
distribution.
Fracture Initiation Pressures in Permeable Sands
K
= 0.80
v
= 0.40
257
Co = 0.36 MPa {52.0 psi)
(MPal
Ro = 10.0 m
O~O
60
50
40
Po
30
0.2
0,4
0.6
0.8
Ri
1.0
1.2
Rc
1.4
rim)
Fig. 2. Example on STRESS 2--distribution.
The fracture initiation pressure given by this new model will be compared with results from classical elastic theory. MODEL DESCRIPTION The model considered is a vertical cylindrical open hole through a horizontal layer of porous and permeable rock. Axial symmetry around the well axis is assumed. The geometry of the problem is shown in Fig. 3. This illustrates a disc of sand with inner radius R~, outer radius Ro and height h. The radius of the plastic zone is R~. The rock is assumed isotropic and homogeneous, obeying a Coulombic failure criterion, and the pores
1 Po
Plastic
zone
zone
Fig. 3. Model configuration.
are completely filled with fluid. The material is subject to stresses in three dimensions, the principal stresses being radial, tangential and vertical, with the vertical principal stress parallel to the borehole axis. Stresses are given as total stresses, with contributions both from the fluid pressure in the pores and from forces in the solid material. Stresses and strains are considered positive when compressive. Deformations are assumed to be small, and the vertical strain component will contain only deformation caused by initial overburden loading of the material, i.e. plane strain. As the pressure in the wellbore is increased, fluid is assumed to flow radially into the formation in accordance with Darcy's law. Pressure increments are considered small enough to ensure that steady state conditions are established at all pressure levels, until the fracture initiation pressure has been reached. The pore fluid pressure at Ro is Po. When there is no fluid flow, P.o will be identical with the initial pore fluid pressure. If there is fluid flow in either direction, Po is equal to the initial pore fluid pressure only if R o is great compared to the radius of the wellbore. If R o is somewhat reduced, Po will el!her be sma!ler or greater than the initial pore fluid pressure, depending on the direction of the flow. Most of the changes in both stresses and pore pressure will usually take place close to the wellbore. At a reasonable distance from the wellbore, say a few meters, the stresses and pore pressure will be relatively constant, and the difference between Po and the initial pore fluid pressure will be small. The results presented in this paper are based on the assumption that the permeability is constant throughout the formation. It should, however, represent no fundamental problem to adopt other types of permeability distribution models.
P. Horsrud et al.
258
Plastic theory
FRACTURING THEORY The orientation of a fracture ~s controlled by the in situ stress situation which again depends on geologic conditions and well depth. A fracture should be normal to the least principal stress [3]. In tectonically relaxed areas the vertical stress will normally be the greatest and the horizontal stress the smallest principal stress. Hence, fractures should be vertical. This analysis will therefore treat only the case of vertical fractures. A vertical fracture will occur when the tangential stress becomes tensile and exceeds the uniaxial tensile strength of the rock. The criterion for fracture initiation is thus p - ~r0 = ~rr
(1)
It is important to distinguish between fracture initiation pressure and fracture propagation pressure. The fracture initiation pressure is normally higher than the fracture propagation pressure.
Elastic theory Based on the elastic stress solutions presented in Appendix A, the fracture initiation pressure for a finite reservoir can be written as 2(1 - via,, - (1 - 2v)fl Poll - l / ( 2 1 n R d R 3 ] + (1 - V)ar P f , el =
2(1 - v) - (1 - 2v)fl [1 - 1/(2 In Ro/RO] (2)
where we have assumed Ro2 >> R2: ah is the horizontal stress at the external b o u n d a r y equal to the initial horizontal stress before the well is drilled. In this study it is assumed that the relationship between the effective horizontal and vertical stress is ah = Po + K(a--o - Po)
(3)
The parameter K is then the horizontal to vertical effective stress ratio. This is a general relation, and a recent paper by Breckels & van Eekelen [10] describes the relationships between horizontal stress and depth for different geologic areas based on fracture treatment data and formation integrity tests, Assuming an infinite reservoir, equation (2) can be rewritten as
Pf. el = Po +
2(ah - Po) + ~r 2-(1 -2v/1 -v)
{4)
This is consistent with the equations derived by Haimson & Fairhurst [4] and Medlin & Mass6 [8] when remembering that they presented their equations in terms of effective stresses. Comparing the results from elastic and plastic theories, equation (2) from the elastic theory is used, as this is applicable also to a finite model as long as the condition R 2 >> R 2 is fulfilled.
Plastically strained zones may form around the wellbore when a hole is drilled through a section of poorly consolidated sandstone [9]. The extent of this plastic zone will mainly depend on the rock's cohesive strength; low cohesive strength gives a deeper plastic zone. A more detailed discussion of the influence of the various parameters on the extent of tile critically stressed zone is given in [9]. Here are also given the complete stress solutions when there is no fluid flow between the formation and the wellbore (Q = 0). This will be the starting point for an analysis of the stress distribution at injection (Q < 0). There are two possible stress distributions when Q = 0, depending on the rock properties. These two distributions, with either cro or or: as the greatest principal stress in the outer part of the plastic zone, are illustrated in Figs 1 and 2. The two distributions are named STRESS 1 and STRESS 2, respectively. As the fluid pressure in the wellbore is increased, fluid will start to flow from the wellbore into the formation. Pressure is increased stepwise, and steady state conditions are assumed to be established at all pressure levels. During the increase in wellbore pressure there will be a continuous change in the stress distribution. There will generally be an increase in stress level and all stress components will be raised, but relative to the two others, the tangential stress, a 0, will decrease until eventually the breakdown pressure is reached and a fracture is opened. If we assume that the uniaxial tensile strength, ar, of the rock is zero, a fracture will be initiated when the fluid pressure in the wellbore is equal to the tangential stress at the wellbore wall. Expressed mathematically the fracture initiation condition is p = c~0 when r = R i
(5)
As the stresses are redistributed and interchanged as the pressure increases, a series of different stress distributions will have appeared before breakdown takes place. Figure 4 illustrates the possible stress distributions, starting with the two initial pictures when there is no fluid flow (Q = 0). The different possibilities in order of appearance as the wellbore pressure is increased are indicated, until finally the fracturing situation is reached. There are three possible stress distributions at breakdown; these are named FRAC 1, FRAC 2 and FRAC 3. The parameter set used will decide which line to follow in Fig. 4. The results presented in Fig. 4 were developed simply by following the stresses continuously with increasing wellbore pressure. When two of the stress components became equal or changed place, new solutions had to be developed. From STRESS 4 in Fig. 4 we see that the plastic zone contains four parts, with one stress solution for each part. The complete stress distribution is found by linking all solutions, both for the plastic and the elastic zone, together. Figure 4 indicates the increase in injection pressure as fracturing is approached. For simplicity we have
Fracture Initiation Pressures in Permeable Sands
o/
259
OZ O0
Ri
Rc
I Or I I I I R1
R~
Ri
Re
Rc
FRAC 3
FRAC 2
F'RAC I
'l
!,
I I
I
L_ . . . .
J Oz i
oo
Oe
~
. I[ Or
OrlI
II i
R1 R2
Ri R1R 2
Re
N~
R3 Rc
4_t
t_ t .........
Rc
STRESS 3
STRESS 4
STRESS 5
I oz
i
I Ri
R1
Rc
STRESS 2 I o
Ri
oz
!
I
oll
=
i I
Or
R1
Ri
Rc
R 1 Rc
STRESS
STRESS 2
I
INITIAL (Q - 0 )
INITIAL ( Q - O ) Fig. 4. S t r e s s d i s t r i b u t i o n s
at injection.
kept the radius of the critically stressed zone, R c, constant. The plastic zone may however increase or ,decrease with increasing injection pressure. The input parameters will control whether the plastic zone will increase of decrease. This will be treated later in this paper.
where ~: is some scalar which measures the degree of plastic deformation. The pressure distribution is given by Darcy's law for radial flow as
Solution method
when the permeability k is constant throughout the formation. The general solutions containing integration constants can now be worked out when applying these equations together with Hooke's law, the relation
Only the general procedure for finding the stress solutions in the plastic zone will be presented here. More details regarding this method are given in I-9]. The three basic equations used when deriving the stress solutions are: The Coulomb criterion:
P = P i + 2PQ - ~ In r/Ri
< =
e
Er
--
The equation of equilibrium:
dR
dr u
E~°
00" r
r~-r + at-a0=
0
The flow rule: ~3f
r
(11)
(7) Ee ~
El' =
(10)
and the strain-displacement relations
f = % - o 3 tan2~ + (tan2c~ - 1)p - 2Scotan~ = 0
(6)
+
(9)
(8)
Ezo - -
~=o -
flPo
2+2G Here e=o is the vertical deformation caused by the initial overburden loading, assuming that there is no horizontal deformation during this process.
260
P. Horsrud et al.
As the stress sohttions depend on which of the stress components will be the smallest, intermediate or greatest, each stress picture or stress distribution will contain one or more sets of stress solutions linked together. The stress solutions used in the elastic zone will be the same for all stress distributions. Stress solutions for four different cases are needed: (i) a , < ~ _
(12)
(iv) a, - a,, < o-_ Solutions for each of these cases are given in Appendix B.
To develop the complete stress distribution, the stress solutions in the plastic zone and the elastic stress solution must be linked together. The unknowns that must be determined are the integration constants in each stress solution in the plastic zone, the radius of the critically stressed zone, Re, and the radius of each stress solution in the critically stressed zone, R~, R 2 , R 3. The stress distributions at fracture initiation (FRAC) contain one additional unknown, the fracture initiation pressure, Pr, p~. To find the unknowns and link the solutions together we apply a set of boundary conditions, o~ = Pi when r = R i a~ (radial stress) is continuous u (displacement) is continuous
(13)
When the tangential stress is the intermediate principal stress and the displacement is continuous, it also results that or, (tangential stress) is continuous at the plastic-elastic boundary
(14)
In addition we have the fracture initiation condition (7o = Pi{ = O'r)
when r
= R i
(15)
which can be regarded as a boundary condition for the FRAC distributions. Which of the boundary conditions should be used depends on the number of unknowns and the types of stress solutions to be linked together.
The parameter group: /~-- Fluid viscosity: h - H e i g h t of formation ; k--Permeability: (Q---Fluid injection rate); will be discussed more extensively later as the influence of these parameters also depends on the permeability model assumed. Parentheses have been put around the fluid injection rate as this is an input parameter in the STRESS-distributions, but not in the FRAC-distributions. The influence of the parameters on the results varies greatly from parameter to parameter. In this presentation the parameters which are important to the conclusions made are varied. The rest of the parameters are kept constant. The values chosen should however represent a realistic field situation. Pressures are given as gradients since depth only enters indirectly. The values for the constant parameters are: fl= :~ = Ri = Pjz = o-:,]z = It = h = k =
1.0; 60.0~; 0.10 m; 10.18 kPa/m (0.45 psi/ft); 22.62 kPa/m (1.00 psi/ftt; 0.001 P a . s ( 1 . 0 c P ) : 1.00 m; 1 . 0 0 i t m 2 (1.0 D).
Here fl = 1.0 represents an incompressible rock matrix. The pressures are given as pressure gradients, but should in any case not be used for shallow depths (above 1000 m (3300 ft)). They can in principle be used for all depths below, but only as long as the input -parameters are constant for all depths. This will normally not be the case for parameters like pore pressure, overburden and Poisson's ratio. The material parameters which will be varied in this presentation are the stress ratio K, Poisson's ratio v and the uniaxial compressive strength C o. The model parameter R o, outer boundary radius, will also be varied. Flow parameters/L, h, k, Q
APPLICATION
OF
THEORY
Parameter t;alues
The numerical results depend on a long list of parameters: K Stress ratio; v--Poisson's ratio; fl Biot constant = 1 - Cmj'Cb; ---Failure angle; Co---Uniaxial compressive strength; Po Initial pore pressure; cr:o ---Initial overburden; R~ -Wellbore radius; R,,--Outer boundary radius.
In the STRESS-distributions the injection rate Q is an input parameter. This is equivalent to using the wellbore pressure P~ as an input parameter since the relation between Q and Pi is given through Darcy's law (equation 9). This relation is simple and easy to handle when the permeability is assumed constant in the whole formation. The only place where the flow parameters enter is then in the flow term (laQ/2nhk). In the FRAC-distributions Q is n o t - a n input parameter, but is adjusted according to Darcy~ law when the fracture initiation pressure is determined and the parameters #, h and k are given. In this case the flow parameters will therefore not affect the fracture initiation pressure.
Fracture Initiation Pressures in Permeable Sands K
= 0.40
v
= 0.20
261
Co = 14,33MPa (2078.5 psi) Q
= - 26043.8 c m 3 / s
(MPa)
70
60
0t
=
00
50 oa
40'
Po
30
0.2
0.4
0.6
~
0.8
4
r(m)
Rc
Fig. 5. Example on FRAC l~distribution.
initiated at the wellbore wall. The figure also illustrates that the type of stress distribution is in each case dependent on the properties of the formation. Three different values for the stress ratio have been used, 0.40, 0.60 and 0.80. Numerical results for these cases are presented in Tables l, 2 and 3, respectively. In each case three values for Poisson's ratio has been used, 0.20, 0.30 and 0.40. The uniaxial compressive strength C o, is varied between values representing a poorly consolidated material and up to values representing a somewhat consolidated material. Two values have been used for the radius of the outer boundary, 10 and 100m, but keeping Po constant as previously men-
This situation may change if a different and more complicated permeability distribution model is assumed. However, this should be done in a separate study. Results The main results will be presented in this section. One important theoretical result has, however, already been presented in Fig. 4, which illustrates the stress distributions. This is important in the sense that it shows in detail how the stresses around a well develop as a function of increasing injection pressure. The stress development can thus be followed until a fracture is
K
= 0.40 = 0.40
o
Co = 14.33 MPa (2078.5 psi)
(MPa}
Q
= - 11395.9 cm3/s
70.
-
-
GZO
-
-
oh
-
-
Po
60,
50.
40,
P 30'
0~2 Ri
0~4
0~6
0~5
1~0
1~2
Re
Fig. 6. Example on FRAC 2--distribution.
1~4 r(m}
262
P. Horsrud et al. TABLE I. K
-
-
0.40 (csh = 15.16 kPa, m) R c ;11
C<>
R<,
Q = t)
v
(MPa)
(m)
R~
P~, pt:
Pf,
(m)
(m)
(kPa!m)
{kPa/ml
0.20 0.20 0.20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 l l 1 I 1
1.02 0.22 0.15 1.02 0.22 0.15
6.70 2.32 0.95 36.37 8.26 2.27
15,42 16.00 16.53 15.58 16,11 16.64
17.65 17,65 17.65 17.81 17.81 17.81
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14.33 0,36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
I I 1 1 l 1
1.11 0.23 0.16 1.11 0.23 0.16
4.39 1.64 0.65 22.23 5.20 1.36
15.45 15.82 16.19 15.50 15.86 16.25
16.86 16.86 16.86 16.95 16,95 16.95
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0.36 7.17 14.33
10.0 t0.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
2 2 2 2 2 2
7,77 0.40 0.17 7.77 0.40 0.17
7.91 4.08 0.76 78.78 33.68 1.77
13.84 13.10 12,96 13.55 12.71 12.70
16.02 16.02 16.02 16.07 16.07 16.07
FRAC-distribution
TABLE 2. K = 0.60 (a h = 17.64 kPa/m)
CO
Ro
R c at Q = 0
Rc
Pr.ptlz
Pf,d,/z
v
(MPa)
(m)
FRAC-distribution
(m)
(m)
(kPa/m)
(kPa/m)
0,20 0,20 0,20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 t00.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 1 1 1
0.85 0.20 0.14 0.85 0.20 0,14
1.25 0.74 0.44 3,36 1.59 0.77
19.57 20.06 20.57 19.73 20.23 20.75
21.39 21.39 21.39 21.63 21.63 21.63
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 1 1 1
0.85 0.20 0.14 0.85 0.20 0,14
1.07 0.61 0.34 2.78 1.26 0.56
19.23 19.59 19.94 19.31 19.67 20.05
20.20 20.20 20.20 20.34 20.34 20.34
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0.36 7. I7 14.33
10.0 10.0 10.0 t00.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 2 1 1 2
0.85 0.19 0.14 0.85 0.19 0.14
(I.61 0.31 0.16 1.29 0.51 0.20
18.49 18.65 18.59 18,50 18.68 18.64
18.95 18.95 18.95 19.01 19.01 19.01
TABLE 3. K = 0.80 (o"h - 20.13 kPa/m) R e at Q = 0 (m)
Re {m)
Pr, pLz (kPa/m)
Pr,ev/: (kPa/m)
I 1 I l 1 1
0.93 0.22 0.16 0.93 0.22 0.16
0.65 0.45 0.31 1.31 0.79 0.47
23.62 24.10 24.59 23.85 24.35 24.86
25.13 25.13 25.13 25.44 25.44 25.44
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 I 1 1
0.93 0.22 0.16 0.93 0.22 0.16
0.58 0.39 0.25 1.17 0.67 0.37
23.00 23.34 23.67 23,12 23.48 23.83
23.53 23.53 23.53 23.72 23.72 23.72
FRAC FRAC FRAC FRAC FRAC FRAC
I 1 I 1 I 1
0.92 0.22 0.16 (/.92 0.22 0. I6
0,39 0.24 0.13 0.69 0.35 0.I6
21.93 22.06 22.12 21.96 22.12 22.27
21.87 21.87 21.87 21.95 21.95 21.95
v
C<, (MPa)
R, (m)
0.20 0.20 0.20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14,33 0.36 7.17 14.33
10.0 I0.0 10.0 100,0 100.0 100.0
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0,36 7.17 14.33
10.0 10,0 10.0 100.0 100.0 I00.0
FRAC-distribution
Fracture Initiation Pressures in Permeable Sands
K
=0.40
v
= 0.40
o
263
Co = 2 3 . 8 8 MPa (3464.1 psi)
(MPa)
Q
= - 1 6 7 0 4 . 4 cm 3 Is
70
- - .
Ozo
60
50'
of
f
oh
09
Or
40,
-
30'
o12
0:4
ole
o;0
1:o
1;2
-
Po
~;4 rim)
Ri Rc
Fig. 7. Example on FRAC 3--distribution.
tioned. The stress distribution at fracture initiation for each case is also included in the tables. The calculated results are Re, the radius of the critically stressed zone at fracture initiation; Pf, p~, the fracture initiation pressure given by plastic theory; and Pr.e~, the fracture initiation pressure given by elastic theory. Pr, e~is calculated from equation (2). Included in the table is also the radius of the critically stressed zone at zero injection rate, which will be the situation after the well is drilled, but before injection has started. One example on each of the three FRAC-distributions is presented graphically in Figs 5, 6 and 7. The depth chosen is 3048 m (10,000 ft).
DISCUSSION In Tables 1, 2 and 3 it is shown that the differences between fracture initiation pressures calculated by plastic theory and elastic theory varies considerably. The difference depends on the initial state of stress (stress ratio K) and the rock properties (Poisson's ratio and uniaxial compressive strength Co). For a poorly consolidated sand with a Poisson's ratio of 0.20 the fracture initiation pressure calculated by the plastic theory is in the order of magnitude of 10~o lower than the fracture initiation pressure calculated by the elastic theory. This difference decreases as the stress ratio increases. For high values of the stress ratio and Poisson's ratio, it may be found that the fracture initiation pressure calculated by the plastic theory is higher than the value calculated by the elastic theory. The most common type of stress distribution at fracture initiation is FRAC 1. Examples on each of the three FRAC-distributions are given in Figs 5, 6 and 7.
When the stress ratio is low the fracture initiation pressure is lower than predicted by elastic theory even when the rock is somewhat consolidated. In addition we would expect the degree of consolidation near the wellbore to be rather small as the rock is weakened through plastic deformation. When the rock enters into the plastic state, intergranular cementation bonds will break, resulting in a decrease in the cohesive strength. As it is the strength of the material close to the wellbore (in the plastic zone) which governs the fracture initiation pressure, a reduced strength in this zone will result in a decrease of the fracture initiation pressure. The normal trend of reduced fracture initiation pressure with reduced strength is however not always followed. With a stress ratio of 0.40 and a Poisson's ratio of 0.40 the situation is different. From Table 1, the fracture initiation pressure seems to decrease with increasing strength. The total picture is however somewhat different. This is illustrated in Fig. 8, where the fracture initiation pressure is plotted as a function of compressive strength for this particular set of properties. The fracture initiation pressure decreases slightly with increasing strength before increasing rapidly again. The same phenomenon occurs with K = 0.60 and v = 0.40, see Table 2, although to a much smaller degree. There is no obvious physical explanation to this phenomenon. It is also difficult to say if this is a real phenomenon or if it is caused by simplifying assumptions in the theoretical model, for instance neglecting the tensile strength of the rock. It seems to be dependent on the relation between Poisson's ratio and the in-situ stresses, because the phenomenon disappears when the stress ratio K increases. From Fig. 6 it is also evident that the fracture will not propagate throughout the formation at this pressure. The fracture would only extend a small distance
264
P. Horsrud et al.
1
Pf,pl (kPa/m)
16.0
Pf,el
15.0
Oh
K
0.40
~'
0.40
Ro
10.0 m
14.0
1 3 . 0 t ~ 5.0
10.0
15.0
20.0
25.0
30,0
Co(NIPa)~
Fig. 8. Fracture initiation pressure its a function of compressive strength.
into the formation, because the fracture initiation pressure is lower than the far-field horizontal stress. This is also reflected in the increase in the tangential stress towards the plastic-elastic boundary which is typical for this type of stress distribution. Listed in the tables is the radius of the critically stressed zone at fracture initiation. Also included in the tables is the radius of the plastic zone after the well is drilled, but before injection has started, thus it is possible to attain an idea of how the plastic zone develops during injection. An examination of this point reveals that both increase and decrease of the plastic zone will occur during injection. For low stress ratios there is an increase as the fracture initiation pressure is approached, while the opposite also takes place for high stress ratios. We also find that Rc strongly depends on the strength of the material, an increase in cohesive strength reduces the extent of the critically stressed zone. In this analysis we have assumed that the uniaxial tensile strength of the rock, a t , is zero. If the rock is not totally unconsolidated this will be a questionable assumption. This implies that the fracture initiation pressure calculated with high compressive strengths are too low, both for results from plastic theory and elastic theory. But even if too low absolute values are predicted, one would not expect the relation between results from plastic and elastic theory to have changed significantly. Elastic theory would still predict too high fracture initiation pressures in most cases. In the description of the model it was stated that when R o is varied, Po should be changed accordingly if fluid flow is involved. However, the results in Tables 1, 2 and 3 are calculated with a constant Po- The effect this has on the fracture initiation pressure when varying Ro between 10 and 100 m is small, indicating that at a reasonable distance from the wellbore the pore pressure is relatively constant and has little influence on the stresses.
CONCLUSIONS The stress distribution around an injection well when the injection pressure is increased up to the fracture initiation pressure has been studied. Our analysis shows that a zone of plastically strained material will exist around a well drilled through poorly consolidated sand. As the injection pressure is increased, this plastic zone may grow or it may decrease, depending on the rock properties. But even if the zone decreases, the material cannot return to its original state. In a poorly consolidated sand, fractures will therefore be initiated in a plastically strained material. The fracture initiation pressures calculated when taking this into consideration can be lower than those calculated by elastic theory. The difference depends on the initial state of stress (stress ratio) and the rock properties (Poisson's ratio and uniaxial compressive strength) in the plastic zone. The difference is greatest when the sand is poorly consolidated and the stress ratio and Poisson's ratio are in the lower region. The difference is then typically in the order of magnitude of
10%. Besides contributing to the basic understanding of the rock behaviour around a well, there are possibly practical consequences. Such an analysis may be helpful when interpreting fracturing tests (formation integrity test, leak-off test) from permeable formations. It should also be evaluated if a leak-off test can be used to estimate the compressive strength of the rock. Once the fracture initiation pressure is determined, the strength can be estimated by back-calculation. If unconsolidated layers could thus be identified, this would be very helpful in, for instance, sand control.
Acknowledgements- The authors wish to express their gratitude to Norsk Agip, Norsk Hydro and Statoil for the financial support making this work possible. Received 14 December 1981;revised 8 July 1982.
Fracture Initiation Pressures in P e r m e a b l e Sands REFERENCES
265
the stress solutions in an elastic material are found as
1. Biot M. A. General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164 (1941). 2. Biot M. A. Exact simplified non-linear stress and fracture analysis around cavities in rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. ! ! , 261-266 (1974). 3. Hubbert M. K. & Willis D. G. Mechanics of hydraulic fracturing. Trans. Am. Inst. Min. Engrs 210, 153-163 (1957). 4. Haimson B. & Fairhurst C. Initiation and extension of hydraulic fractures in rocks. Soc. Petrol. Engrs J. 310-318 (1967). 5. Eaton B. A. Fracture gradient prediction and its application in oil field operations. J. Peo'ol. Technol. 1353-1360 (1969). 6. Geertsma J. Problems of rock mechanics in petroleum production engineering. Proc. 1st Com3r. ISRM, Vol. 1, pp. 585-594 (1966). 7. Geertsma J. Some rock mechanical aspects of oil and gas welt completions. Proc. European Offshore Petroleum Conf. & Exhibition, Vol. I, pp. 301-310 (1979). 8. Medlin W. L. & Mass6 L. Laboratory investigation of fracture initiation pressure and orientation. Soc. Petrol. Engrs J. 19, 129-144 (1979). 9. Risnes R., Bratli R. K. & Horsrud P. Sand stresses around a wellbore. Proc. SPE Middle East Tech. Conf, pp. 709-727 (1981). 10. Breckels I. M. & van Eekelen H. A. M. Relationship between horizontal stress and depth in sedimentary basins, paper SPE 10336, 56th Annual Fall Tech. Cot~ & Exhibition (1981). 11. Jaeger J. C. & Cook N. G. W. Fundamentals of Rock Mechanics, pp. 212-214. Chapman & Hall, London (19761.
ar = at,, + (fir,, -- arl) Ro - R i I - 2v
(
R~
I-
2(~-2_ ,,)fl ~Ry-o_~-R~[ l - ( R 2 ) 2]
-(Po-Pi'
InIg,,/rl l + ln(R,,/Ri) )
a,,=a,,,+
-
~-R~!--z
1
-(P°- P3 2(~--g l [ I n ( R o / r ) - 1]} + In (Ro/R i~
(A8I
R~ a_ = a.o. + 2v(a,o-o',i) Ro2 _ R 2
}-2,, f
fi
- (P" - P') 1 - v
R,~
( R:, - g~
+ in(Ro/----R i) With a plastic zone between the wellbore and the elastic zone, (A7) must be replaced by
APPENDIX A Elastic Stress Solutions
d +
;:)
+/3
when
r = R~
a,=
when
r : Ro
(A9)
This analysis follows Biot's theory and the basic equations can be found in for instance Jaeger & Cook E11]. The displacement equation is in the case given as
I;. + 2~t~
a, = a,c
=
o
a,o
The stress solutions are now given by (A8) when replacing a,~, R~, P~ with a,o R,, P, respectively. In developing (AS) we also used the relations
(All
Ev
(1 + v)(l - 2v) E
when there are no volume forces. The pressure gradient is given by Darcy's law in radial form
a
~
--
(AI0)
--
2(1 + v)
dp
pQ dr - 2xhkr
(A2)
APPENDIX
The stress-strain relations are
B
Plastic Stress Solutions
a, = ).A" + 2 G ~ + tip % = 2A" + 2GE~'~+ tip
The general solutions for the four different cases are: (A31
(i) o~ < o , < a,,
% = ).A e + 2Ge'.." + tip
l[
a, = p - t
where A" = E~ + e'~j + ~
(A4)
a°= P-t
If we assume only radial displacement after initial loading, the strain displacement relations are
(1 +
u
+
(A5)
F
r)(l
1- v
V
Alp
27thkJ + 2(t + 1)z V -
2v)
I
I i A~_r,
1) P Q ]
C°-(t+
a= = [2v(l - f l ) + f l ] p - t
du dr er3 -
l[
.e ]
C° - 2xhk j + 2(t +
2Co-(t
(BI)
+2) 2xhkj
(a--o - flPo) + 2v(t + 2)(t + 1) A~_ r'
where
e'.' = e." where e:o is the vertical strain caused by the initial overburden loading when assuming no horizontal deformation during this loading. From (A3) we then find
A~ = integration constant; t = t a n 2 ~ - 1; V = (t + 1)2 + 1 - v(t + 2) 2 . (ii) o r < a o = a~
o:. - / / P o e:°-
2 + 2G
(A6)
a, = p - t
By solving (AlL combining the result with the other equations and applying the boundary conditions O-r =
O'ri
when
r
=
R i
(A7) O, =
O-ro
when
r =
R o
o,,=o:=p-t
o
2~nhk + 2(t + 1)(1 + v)Blr'
(B2)
C o - ( t + 1) 2~/~k~k + 2(t + l)e(l + v)Btr'
where B l = integration constant.
266
P. Horsrud et al.
(iii) o', < a o < a .
(t
1)D~ , + ~t ~- l ) [ v ( t + 2 ) + U
1
a, =
(t +[,,l)I
~ [ t ( t + 1) - vt(t + 2) + (1 - 2v)(t + 2)fl]p
+V
13 O-~'in r V
+ 1 ,,[(1 V
v)(t +
-
1)
l l-2v -
v]Co
+
V I
(t
+
1)
+
I-v
-
,-~
[v(t + 2) - ),]D2r
V
1 !C.
I ) 2 ( a : , , -- tiP,,)
(t +
t+l ---[v(t+2)+y]Dlr V t+l
t +1
(1 -- v)(t ~ 1 ) - - v
I l-2l'
Da D~ + [vtt + 2 ) + 13 V + V + [v(t + 2 ) +
I ] / )~ -!In r
:-I
where
1
x (a.o - flPo) + ~ [ v ( t + 2) + ? ] D ~ r = - '
D! = Integration c o n s t a n t ; D2
=
Integration c o n s t a n t ;
),2
:
(t + 1)2 _ 2v(t + 1) + 1:
1
+ ~ [ v ( t q- 2) - y ] D 2 r - : - '
I
2(1 - 2 v ) t ( 1
-fi)
O a
1
a o = ~ [ v t 2 + (1 - 2v)((t + 1)2 + l)fl]p
(B3)
t(t + 1) - vt(t + 2) + (1 + [v(t + 2 ) + 7
2
D.
]~-
Db
Db
1
l -
V
2v
l-v
-
Db
~]D2r-;-] (iv) 0-~ :
a= = - ~ - ] t ( t v
[_
t +
2l,
i '2(;
C.
(1 - 2v)(v(t + 2) -
(t + 1)(t + 2)(a.,, - flPo}
-~- ~ [V(/" 4- 2 ) J r - "1t] D l r 7-1 -- ~ ]-/'(t + 2 } -
t+lF
H-~
v)(t+ (=E;i
(I - 2v)t(l - fl)
#Q
t + 1 - 2v
2~zhk
(°:°
-
flPo):
m
a o < cz~ ~7r =
+ 1) - vt(t + 2) + (1 - 2v)(t + 2)fi
1)
O"K
(B4)
(7 0 - - (5 h
a. - aK(t + I ) t+
1
pQ
+ v(t + 2 ) V
+ I v ( t + 2 ) + 7 2] ~ - l n r _ - v v t C o
1 1 4- - - v - - - -
2v)(t + 2 ) f l ]
t p + (t + 1)[v(t + 2) + 1] D° V
where crK - c o n s t a n t .
tp + Co
0148-9062/82'060255-12503.00/0 Copyright © 1982 Pergamon Press Lid
Printed in Great Britain. All rights reserved
Fracture Initiation Pressures in Permeable Poorly Consolidated Sands P. H O R S R U D * R. R I S N E S I " R. K. B R A T L I *
This paper studies the stress distribution around an il~jection well drilled through a layer of poorly consolidated sand. As a plastically strained zone may f o r m around a hole drilled through this type of rock, both elastic and plastic theories must be applied. Solutions have been worked out in order to Jollow the development of stresses as the injection pressure increases fi'om zero up to the point where a fracture is initiated. H o w these stresses develop is dependent on the rock and ,fluid properties, mainly Poisson's ratio and the ratio between the horizontal and vertical stress. Several different stress distributions will appear as the injection pressure is increased. Each stress distribution is composed of one or more different stress solutions linked together. This is solved by numerical iteration. T h e injection pressure is increased until a .fracture is induced. Fracture initiation pressures for several different parameter sets have been examined. T h e results have then been contpared with fi'acture initiation pressures calculated by classical elastic theory. This comparison shows that the fi'acture initiation pressures can be lower than results calculated by elastic theory. Typically, with a Poisson's ratio o f 0.20 and a horizontal to vertical effective stress ratio of 0.40, the d!fference is in the order o f ntagnitude of I0%.
t, V = u = z = =
Constants defined in Appendix B. Radial displacement. Depth. Failure angle. = 1 - Cmj'C b = Blot constant. 7' = Constant defined in Appendix B. A" = Volumetric elastic strain. ¢;', ¢~, ~'-' = Elastic strain components. ¢,,P %,P E,P = Plastic strain components. E,, %, E: = Total strain components. h- = A scalar in the plastic flow rule. 2 = Lam6 parameter. # = Fluid viscosity. v = Poisson's ratio. 6 I, 0"2, (73 = Principal stress components. ar, a o, a: = Stress components in cylindrical co-ordinates. a, i = Radial stress at Rj. a, c = Radial stress at Re. a,o, a=o = Radial and vertical stress at R o. a h ( = a, o) = Horizontal stress at R o. ar = Uniaxial tensile strength.
NOMENCLATURE
Ai, Bi, Dl, O2 = Integration constants. CO = 2S¢otan~ = uniaxial compressive strength. Cma
Rock matrix compressibility.
Cb = Rock bulk compressibility. D,, Db = Constants defined in Appendix B. E = Young's modulus. Plastic flow function. f G = Shear modulus. h = Height of permeable layer. K = Horizontal to vertical effective stress ratio. k = Permeability, p = Fluid pressure. Pi = Fluid pressure at R~. Pc = Fluid pressure at plastic-elastic boundary. Po : Fluid pressure at Ro. Pf. el = Fracture initiation pressure calculated by elastic theory. ef, pl : Fracture initiation pressure calculated by plastic theory. O = Fluid flowrate (negative when injecting). r = Radical distance from center of wellbore. R i = Wellbore radius. Ro: Outer boundary radius. RI, R2, R3 = Radii of different plastic zones. R~ = Radius of entire plastic zone. Sco = Cohesive strength.
INTRODUCTION When
*Continental Shelf Institute, Hhkon Magnussons gt. IB, 7001 Trondhe]m, Norway. 5" Norsk Agip, Sandnes. Norway. 255
a liquid
is p u m p e d
into a borehole
at a high
enough
pressure, the stress at the borehole
s u r f a c e will
become
tensile. If this stress exceeds the tensile strength
o f t h e m a t e r i a l , a f r a c t u r e will o c c u r . T h i s t e c h n i q u e , c o m m o n l y k n o w n a s h y d r a u l i c f r a c t u r i n g , is a m e t h o d used by many
in t h e o i l i n d u s t r y
to improve
the pro-
256
P. Horsrud et al.
ductivity of a formation. However, for flooding projects it is essential to avoid initiating fractures. Determination of fracture initiation pressures is therefore important. One way of estimating the fracture gradient is to purposely initiate a fracture by increasing the wellbore pressure. This test, called the formation integrity test or leak-off test, is not always straightforward to interpret, especially in permeable formations. Unintended fracturing around injection wells can have severe consequences to a flooding project. Such fractures can penetrate deep into the formation, Although this will lead to improved injectivity it can jeopardize the whole injection project because of the reduction in sweep efficiency. Another result from such unintended fracturing could be influx of particles into the injectors at a system shut-down. This could eventually plug the injection wells. When killing a blow-out by creating a water bank around the blowing well, it is essential that no fracture is created between the injection well and the blowing well. Such a fracture would effictively short-circuit the flow between the wells, and make it impossible to block the blowing fluid by a water bank. The stress situation around boreholes has been investigated by a number of authors, in various degrees of refinement. The early work of Biot [1] laid the basis for much of the later work. He included the effect of the pore pressure in his investigation and thereby opened the way for studies of injection and fracturing in permeable rocks. Biot [2] later developed analytical solutions for non-linear stress-strain relations and applied them to fracturing around cylindrical and spherical cavities. Hubbert & Willis [3] studied theoretically hydraulic fracturing of rocks and concluded that the fractures should be perpendicular to the axis of the smallest principal stress. In tectonically relaxed areas with normal faulting this would imply that the fractures are vertical.
A criterion tor the initiation of vertical fractures was later presented by Haimson & Fairhurst [4]. They assumed the rock to be elastic and included also the effect of the pore pressure and the fluid flow into the formation. Eaton [5] suggested to use the horizontal stress level as the fracture pressure. He claims that in a permeable formation there is little difference between the fracture initiation pressure and extension pressure and does therefore not distinguish between the two. The same problem has been studied extensively by Geertsma [6,7]. He also discussed the consequences of formation failure around injection wells. In his analysis Geertsma assumed poroelastic rock, but he states that breakdown pressures will be lower than predicted by poroelastic theory if the rock surrounding the borehole has been exposed to non-elastic deformations. Medlin & Mass6 [8] studied fracture initiation in the laboratory and compared this with the results from a theory which was based on poroelastic behaviour. Their conclusions were restricted to results for hydrostatic stress conditions. For the non-penetrating fluid they found that when the stresses exceed the elastic range, failure pressures will be lower than predicted. For the penetrating fluid case their results were not completely consistent with the theory. We have earlier shown that plastically strained zones will be formed around the wellbore when a well is drilled through poorly consolidated sand [9]. In this analysis we also included the effect of the third principal stress, the vertical component resulting from the overburden. Typical examples of the resulting stress distributions are given in Figs 1 and 2. In this paper we will try to show how such a plastic zone can influence the stress distribution around the well during injection. The stress distribution can be followed as the injection pressure is increased and eventually reaches the level where a fracture will be initiated.
K
- 0.60
v
- 0.30
Co = 0.36 MPa (52.0 psi)
(MPa)
Ro
10.0 m
70
60"
50 ¸
40
-
30
0.2 Ri
0.4
0.6
0.8 ~
2
Po
1.4 rlm l
Rc
Fig. l. Example on STRESS 1
-
distribution.
Fracture Initiation Pressures in Permeable Sands
K
= 0.80
v
= 0.40
257
Co = 0.36 MPa {52.0 psi)
(MPal
Ro = 10.0 m
O~O
60
50
40
Po
30
0.2
0,4
0.6
0.8
Ri
1.0
1.2
Rc
1.4
rim)
Fig. 2. Example on STRESS 2--distribution.
The fracture initiation pressure given by this new model will be compared with results from classical elastic theory. MODEL DESCRIPTION The model considered is a vertical cylindrical open hole through a horizontal layer of porous and permeable rock. Axial symmetry around the well axis is assumed. The geometry of the problem is shown in Fig. 3. This illustrates a disc of sand with inner radius R~, outer radius Ro and height h. The radius of the plastic zone is R~. The rock is assumed isotropic and homogeneous, obeying a Coulombic failure criterion, and the pores
1 Po
Plastic
zone
zone
Fig. 3. Model configuration.
are completely filled with fluid. The material is subject to stresses in three dimensions, the principal stresses being radial, tangential and vertical, with the vertical principal stress parallel to the borehole axis. Stresses are given as total stresses, with contributions both from the fluid pressure in the pores and from forces in the solid material. Stresses and strains are considered positive when compressive. Deformations are assumed to be small, and the vertical strain component will contain only deformation caused by initial overburden loading of the material, i.e. plane strain. As the pressure in the wellbore is increased, fluid is assumed to flow radially into the formation in accordance with Darcy's law. Pressure increments are considered small enough to ensure that steady state conditions are established at all pressure levels, until the fracture initiation pressure has been reached. The pore fluid pressure at Ro is Po. When there is no fluid flow, P.o will be identical with the initial pore fluid pressure. If there is fluid flow in either direction, Po is equal to the initial pore fluid pressure only if R o is great compared to the radius of the wellbore. If R o is somewhat reduced, Po will el!her be sma!ler or greater than the initial pore fluid pressure, depending on the direction of the flow. Most of the changes in both stresses and pore pressure will usually take place close to the wellbore. At a reasonable distance from the wellbore, say a few meters, the stresses and pore pressure will be relatively constant, and the difference between Po and the initial pore fluid pressure will be small. The results presented in this paper are based on the assumption that the permeability is constant throughout the formation. It should, however, represent no fundamental problem to adopt other types of permeability distribution models.
P. Horsrud et al.
258
Plastic theory
FRACTURING THEORY The orientation of a fracture ~s controlled by the in situ stress situation which again depends on geologic conditions and well depth. A fracture should be normal to the least principal stress [3]. In tectonically relaxed areas the vertical stress will normally be the greatest and the horizontal stress the smallest principal stress. Hence, fractures should be vertical. This analysis will therefore treat only the case of vertical fractures. A vertical fracture will occur when the tangential stress becomes tensile and exceeds the uniaxial tensile strength of the rock. The criterion for fracture initiation is thus p - ~r0 = ~rr
(1)
It is important to distinguish between fracture initiation pressure and fracture propagation pressure. The fracture initiation pressure is normally higher than the fracture propagation pressure.
Elastic theory Based on the elastic stress solutions presented in Appendix A, the fracture initiation pressure for a finite reservoir can be written as 2(1 - via,, - (1 - 2v)fl Poll - l / ( 2 1 n R d R 3 ] + (1 - V)ar P f , el =
2(1 - v) - (1 - 2v)fl [1 - 1/(2 In Ro/RO] (2)
where we have assumed Ro2 >> R2: ah is the horizontal stress at the external b o u n d a r y equal to the initial horizontal stress before the well is drilled. In this study it is assumed that the relationship between the effective horizontal and vertical stress is ah = Po + K(a--o - Po)
(3)
The parameter K is then the horizontal to vertical effective stress ratio. This is a general relation, and a recent paper by Breckels & van Eekelen [10] describes the relationships between horizontal stress and depth for different geologic areas based on fracture treatment data and formation integrity tests, Assuming an infinite reservoir, equation (2) can be rewritten as
Pf. el = Po +
2(ah - Po) + ~r 2-(1 -2v/1 -v)
{4)
This is consistent with the equations derived by Haimson & Fairhurst [4] and Medlin & Mass6 [8] when remembering that they presented their equations in terms of effective stresses. Comparing the results from elastic and plastic theories, equation (2) from the elastic theory is used, as this is applicable also to a finite model as long as the condition R 2 >> R 2 is fulfilled.
Plastically strained zones may form around the wellbore when a hole is drilled through a section of poorly consolidated sandstone [9]. The extent of this plastic zone will mainly depend on the rock's cohesive strength; low cohesive strength gives a deeper plastic zone. A more detailed discussion of the influence of the various parameters on the extent of tile critically stressed zone is given in [9]. Here are also given the complete stress solutions when there is no fluid flow between the formation and the wellbore (Q = 0). This will be the starting point for an analysis of the stress distribution at injection (Q < 0). There are two possible stress distributions when Q = 0, depending on the rock properties. These two distributions, with either cro or or: as the greatest principal stress in the outer part of the plastic zone, are illustrated in Figs 1 and 2. The two distributions are named STRESS 1 and STRESS 2, respectively. As the fluid pressure in the wellbore is increased, fluid will start to flow from the wellbore into the formation. Pressure is increased stepwise, and steady state conditions are assumed to be established at all pressure levels. During the increase in wellbore pressure there will be a continuous change in the stress distribution. There will generally be an increase in stress level and all stress components will be raised, but relative to the two others, the tangential stress, a 0, will decrease until eventually the breakdown pressure is reached and a fracture is opened. If we assume that the uniaxial tensile strength, ar, of the rock is zero, a fracture will be initiated when the fluid pressure in the wellbore is equal to the tangential stress at the wellbore wall. Expressed mathematically the fracture initiation condition is p = c~0 when r = R i
(5)
As the stresses are redistributed and interchanged as the pressure increases, a series of different stress distributions will have appeared before breakdown takes place. Figure 4 illustrates the possible stress distributions, starting with the two initial pictures when there is no fluid flow (Q = 0). The different possibilities in order of appearance as the wellbore pressure is increased are indicated, until finally the fracturing situation is reached. There are three possible stress distributions at breakdown; these are named FRAC 1, FRAC 2 and FRAC 3. The parameter set used will decide which line to follow in Fig. 4. The results presented in Fig. 4 were developed simply by following the stresses continuously with increasing wellbore pressure. When two of the stress components became equal or changed place, new solutions had to be developed. From STRESS 4 in Fig. 4 we see that the plastic zone contains four parts, with one stress solution for each part. The complete stress distribution is found by linking all solutions, both for the plastic and the elastic zone, together. Figure 4 indicates the increase in injection pressure as fracturing is approached. For simplicity we have
Fracture Initiation Pressures in Permeable Sands
o/
259
OZ O0
Ri
Rc
I Or I I I I R1
R~
Ri
Re
Rc
FRAC 3
FRAC 2
F'RAC I
'l
!,
I I
I
L_ . . . .
J Oz i
oo
Oe
~
. I[ Or
OrlI
II i
R1 R2
Ri R1R 2
Re
N~
R3 Rc
4_t
t_ t .........
Rc
STRESS 3
STRESS 4
STRESS 5
I oz
i
I Ri
R1
Rc
STRESS 2 I o
Ri
oz
!
I
oll
=
i I
Or
R1
Ri
Rc
R 1 Rc
STRESS
STRESS 2
I
INITIAL (Q - 0 )
INITIAL ( Q - O ) Fig. 4. S t r e s s d i s t r i b u t i o n s
at injection.
kept the radius of the critically stressed zone, R c, constant. The plastic zone may however increase or ,decrease with increasing injection pressure. The input parameters will control whether the plastic zone will increase of decrease. This will be treated later in this paper.
where ~: is some scalar which measures the degree of plastic deformation. The pressure distribution is given by Darcy's law for radial flow as
Solution method
when the permeability k is constant throughout the formation. The general solutions containing integration constants can now be worked out when applying these equations together with Hooke's law, the relation
Only the general procedure for finding the stress solutions in the plastic zone will be presented here. More details regarding this method are given in I-9]. The three basic equations used when deriving the stress solutions are: The Coulomb criterion:
P = P i + 2PQ - ~ In r/Ri
< =
e
Er
--
The equation of equilibrium:
dR
dr u
E~°
00" r
r~-r + at-a0=
0
The flow rule: ~3f
r
(11)
(7) Ee ~
El' =
(10)
and the strain-displacement relations
f = % - o 3 tan2~ + (tan2c~ - 1)p - 2Scotan~ = 0
(6)
+
(9)
(8)
Ezo - -
~=o -
flPo
2+2G Here e=o is the vertical deformation caused by the initial overburden loading, assuming that there is no horizontal deformation during this process.
260
P. Horsrud et al.
As the stress sohttions depend on which of the stress components will be the smallest, intermediate or greatest, each stress picture or stress distribution will contain one or more sets of stress solutions linked together. The stress solutions used in the elastic zone will be the same for all stress distributions. Stress solutions for four different cases are needed: (i) a , < ~ _
(12)
(iv) a, - a,, < o-_ Solutions for each of these cases are given in Appendix B.
To develop the complete stress distribution, the stress solutions in the plastic zone and the elastic stress solution must be linked together. The unknowns that must be determined are the integration constants in each stress solution in the plastic zone, the radius of the critically stressed zone, Re, and the radius of each stress solution in the critically stressed zone, R~, R 2 , R 3. The stress distributions at fracture initiation (FRAC) contain one additional unknown, the fracture initiation pressure, Pr, p~. To find the unknowns and link the solutions together we apply a set of boundary conditions, o~ = Pi when r = R i a~ (radial stress) is continuous u (displacement) is continuous
(13)
When the tangential stress is the intermediate principal stress and the displacement is continuous, it also results that or, (tangential stress) is continuous at the plastic-elastic boundary
(14)
In addition we have the fracture initiation condition (7o = Pi{ = O'r)
when r
= R i
(15)
which can be regarded as a boundary condition for the FRAC distributions. Which of the boundary conditions should be used depends on the number of unknowns and the types of stress solutions to be linked together.
The parameter group: /~-- Fluid viscosity: h - H e i g h t of formation ; k--Permeability: (Q---Fluid injection rate); will be discussed more extensively later as the influence of these parameters also depends on the permeability model assumed. Parentheses have been put around the fluid injection rate as this is an input parameter in the STRESS-distributions, but not in the FRAC-distributions. The influence of the parameters on the results varies greatly from parameter to parameter. In this presentation the parameters which are important to the conclusions made are varied. The rest of the parameters are kept constant. The values chosen should however represent a realistic field situation. Pressures are given as gradients since depth only enters indirectly. The values for the constant parameters are: fl= :~ = Ri = Pjz = o-:,]z = It = h = k =
1.0; 60.0~; 0.10 m; 10.18 kPa/m (0.45 psi/ft); 22.62 kPa/m (1.00 psi/ftt; 0.001 P a . s ( 1 . 0 c P ) : 1.00 m; 1 . 0 0 i t m 2 (1.0 D).
Here fl = 1.0 represents an incompressible rock matrix. The pressures are given as pressure gradients, but should in any case not be used for shallow depths (above 1000 m (3300 ft)). They can in principle be used for all depths below, but only as long as the input -parameters are constant for all depths. This will normally not be the case for parameters like pore pressure, overburden and Poisson's ratio. The material parameters which will be varied in this presentation are the stress ratio K, Poisson's ratio v and the uniaxial compressive strength C o. The model parameter R o, outer boundary radius, will also be varied. Flow parameters/L, h, k, Q
APPLICATION
OF
THEORY
Parameter t;alues
The numerical results depend on a long list of parameters: K Stress ratio; v--Poisson's ratio; fl Biot constant = 1 - Cmj'Cb; ---Failure angle; Co---Uniaxial compressive strength; Po Initial pore pressure; cr:o ---Initial overburden; R~ -Wellbore radius; R,,--Outer boundary radius.
In the STRESS-distributions the injection rate Q is an input parameter. This is equivalent to using the wellbore pressure P~ as an input parameter since the relation between Q and Pi is given through Darcy's law (equation 9). This relation is simple and easy to handle when the permeability is assumed constant in the whole formation. The only place where the flow parameters enter is then in the flow term (laQ/2nhk). In the FRAC-distributions Q is n o t - a n input parameter, but is adjusted according to Darcy~ law when the fracture initiation pressure is determined and the parameters #, h and k are given. In this case the flow parameters will therefore not affect the fracture initiation pressure.
Fracture Initiation Pressures in Permeable Sands K
= 0.40
v
= 0.20
261
Co = 14,33MPa (2078.5 psi) Q
= - 26043.8 c m 3 / s
(MPa)
70
60
0t
=
00
50 oa
40'
Po
30
0.2
0.4
0.6
~
0.8
4
r(m)
Rc
Fig. 5. Example on FRAC l~distribution.
initiated at the wellbore wall. The figure also illustrates that the type of stress distribution is in each case dependent on the properties of the formation. Three different values for the stress ratio have been used, 0.40, 0.60 and 0.80. Numerical results for these cases are presented in Tables l, 2 and 3, respectively. In each case three values for Poisson's ratio has been used, 0.20, 0.30 and 0.40. The uniaxial compressive strength C o, is varied between values representing a poorly consolidated material and up to values representing a somewhat consolidated material. Two values have been used for the radius of the outer boundary, 10 and 100m, but keeping Po constant as previously men-
This situation may change if a different and more complicated permeability distribution model is assumed. However, this should be done in a separate study. Results The main results will be presented in this section. One important theoretical result has, however, already been presented in Fig. 4, which illustrates the stress distributions. This is important in the sense that it shows in detail how the stresses around a well develop as a function of increasing injection pressure. The stress development can thus be followed until a fracture is
K
= 0.40 = 0.40
o
Co = 14.33 MPa (2078.5 psi)
(MPa}
Q
= - 11395.9 cm3/s
70.
-
-
GZO
-
-
oh
-
-
Po
60,
50.
40,
P 30'
0~2 Ri
0~4
0~6
0~5
1~0
1~2
Re
Fig. 6. Example on FRAC 2--distribution.
1~4 r(m}
262
P. Horsrud et al. TABLE I. K
-
-
0.40 (csh = 15.16 kPa, m) R c ;11
C<>
R<,
Q = t)
v
(MPa)
(m)
R~
P~, pt:
Pf,
(m)
(m)
(kPa!m)
{kPa/ml
0.20 0.20 0.20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 l l 1 I 1
1.02 0.22 0.15 1.02 0.22 0.15
6.70 2.32 0.95 36.37 8.26 2.27
15,42 16.00 16.53 15.58 16,11 16.64
17.65 17,65 17.65 17.81 17.81 17.81
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14.33 0,36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
I I 1 1 l 1
1.11 0.23 0.16 1.11 0.23 0.16
4.39 1.64 0.65 22.23 5.20 1.36
15.45 15.82 16.19 15.50 15.86 16.25
16.86 16.86 16.86 16.95 16,95 16.95
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0.36 7.17 14.33
10.0 t0.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
2 2 2 2 2 2
7,77 0.40 0.17 7.77 0.40 0.17
7.91 4.08 0.76 78.78 33.68 1.77
13.84 13.10 12,96 13.55 12.71 12.70
16.02 16.02 16.02 16.07 16.07 16.07
FRAC-distribution
TABLE 2. K = 0.60 (a h = 17.64 kPa/m)
CO
Ro
R c at Q = 0
Rc
Pr.ptlz
Pf,d,/z
v
(MPa)
(m)
FRAC-distribution
(m)
(m)
(kPa/m)
(kPa/m)
0,20 0,20 0,20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 t00.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 1 1 1
0.85 0.20 0.14 0.85 0.20 0,14
1.25 0.74 0.44 3,36 1.59 0.77
19.57 20.06 20.57 19.73 20.23 20.75
21.39 21.39 21.39 21.63 21.63 21.63
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 1 1 1
0.85 0.20 0.14 0.85 0.20 0,14
1.07 0.61 0.34 2.78 1.26 0.56
19.23 19.59 19.94 19.31 19.67 20.05
20.20 20.20 20.20 20.34 20.34 20.34
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0.36 7. I7 14.33
10.0 10.0 10.0 t00.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 2 1 1 2
0.85 0.19 0.14 0.85 0.19 0.14
(I.61 0.31 0.16 1.29 0.51 0.20
18.49 18.65 18.59 18,50 18.68 18.64
18.95 18.95 18.95 19.01 19.01 19.01
TABLE 3. K = 0.80 (o"h - 20.13 kPa/m) R e at Q = 0 (m)
Re {m)
Pr, pLz (kPa/m)
Pr,ev/: (kPa/m)
I 1 I l 1 1
0.93 0.22 0.16 0.93 0.22 0.16
0.65 0.45 0.31 1.31 0.79 0.47
23.62 24.10 24.59 23.85 24.35 24.86
25.13 25.13 25.13 25.44 25.44 25.44
FRAC FRAC FRAC FRAC FRAC FRAC
1 1 1 I 1 1
0.93 0.22 0.16 0.93 0.22 0.16
0.58 0.39 0.25 1.17 0.67 0.37
23.00 23.34 23.67 23,12 23.48 23.83
23.53 23.53 23.53 23.72 23.72 23.72
FRAC FRAC FRAC FRAC FRAC FRAC
I 1 I 1 I 1
0.92 0.22 0.16 (/.92 0.22 0. I6
0,39 0.24 0.13 0.69 0.35 0.I6
21.93 22.06 22.12 21.96 22.12 22.27
21.87 21.87 21.87 21.95 21.95 21.95
v
C<, (MPa)
R, (m)
0.20 0.20 0.20 0.20 0.20 0.20
0.36 7.17 14.33 0.36 7.17 14.33
10.0 10.0 10.0 100.0 100.0 100.0
FRAC FRAC FRAC FRAC FRAC FRAC
0.30 0.30 0.30 0.30 0.30 0.30
0.36 7.17 14,33 0.36 7.17 14.33
10.0 I0.0 10.0 100,0 100.0 100.0
0.40 0.40 0.40 0.40 0.40 0.40
0.36 7.17 14.33 0,36 7.17 14.33
10.0 10,0 10.0 100.0 100.0 I00.0
FRAC-distribution
Fracture Initiation Pressures in Permeable Sands
K
=0.40
v
= 0.40
o
263
Co = 2 3 . 8 8 MPa (3464.1 psi)
(MPa)
Q
= - 1 6 7 0 4 . 4 cm 3 Is
70
- - .
Ozo
60
50'
of
f
oh
09
Or
40,
-
30'
o12
0:4
ole
o;0
1:o
1;2
-
Po
~;4 rim)
Ri Rc
Fig. 7. Example on FRAC 3--distribution.
tioned. The stress distribution at fracture initiation for each case is also included in the tables. The calculated results are Re, the radius of the critically stressed zone at fracture initiation; Pf, p~, the fracture initiation pressure given by plastic theory; and Pr.e~, the fracture initiation pressure given by elastic theory. Pr, e~is calculated from equation (2). Included in the table is also the radius of the critically stressed zone at zero injection rate, which will be the situation after the well is drilled, but before injection has started. One example on each of the three FRAC-distributions is presented graphically in Figs 5, 6 and 7. The depth chosen is 3048 m (10,000 ft).
DISCUSSION In Tables 1, 2 and 3 it is shown that the differences between fracture initiation pressures calculated by plastic theory and elastic theory varies considerably. The difference depends on the initial state of stress (stress ratio K) and the rock properties (Poisson's ratio and uniaxial compressive strength Co). For a poorly consolidated sand with a Poisson's ratio of 0.20 the fracture initiation pressure calculated by the plastic theory is in the order of magnitude of 10~o lower than the fracture initiation pressure calculated by the elastic theory. This difference decreases as the stress ratio increases. For high values of the stress ratio and Poisson's ratio, it may be found that the fracture initiation pressure calculated by the plastic theory is higher than the value calculated by the elastic theory. The most common type of stress distribution at fracture initiation is FRAC 1. Examples on each of the three FRAC-distributions are given in Figs 5, 6 and 7.
When the stress ratio is low the fracture initiation pressure is lower than predicted by elastic theory even when the rock is somewhat consolidated. In addition we would expect the degree of consolidation near the wellbore to be rather small as the rock is weakened through plastic deformation. When the rock enters into the plastic state, intergranular cementation bonds will break, resulting in a decrease in the cohesive strength. As it is the strength of the material close to the wellbore (in the plastic zone) which governs the fracture initiation pressure, a reduced strength in this zone will result in a decrease of the fracture initiation pressure. The normal trend of reduced fracture initiation pressure with reduced strength is however not always followed. With a stress ratio of 0.40 and a Poisson's ratio of 0.40 the situation is different. From Table 1, the fracture initiation pressure seems to decrease with increasing strength. The total picture is however somewhat different. This is illustrated in Fig. 8, where the fracture initiation pressure is plotted as a function of compressive strength for this particular set of properties. The fracture initiation pressure decreases slightly with increasing strength before increasing rapidly again. The same phenomenon occurs with K = 0.60 and v = 0.40, see Table 2, although to a much smaller degree. There is no obvious physical explanation to this phenomenon. It is also difficult to say if this is a real phenomenon or if it is caused by simplifying assumptions in the theoretical model, for instance neglecting the tensile strength of the rock. It seems to be dependent on the relation between Poisson's ratio and the in-situ stresses, because the phenomenon disappears when the stress ratio K increases. From Fig. 6 it is also evident that the fracture will not propagate throughout the formation at this pressure. The fracture would only extend a small distance
264
P. Horsrud et al.
1
Pf,pl (kPa/m)
16.0
Pf,el
15.0
Oh
K
0.40
~'
0.40
Ro
10.0 m
14.0
1 3 . 0 t ~ 5.0
10.0
15.0
20.0
25.0
30,0
Co(NIPa)~
Fig. 8. Fracture initiation pressure its a function of compressive strength.
into the formation, because the fracture initiation pressure is lower than the far-field horizontal stress. This is also reflected in the increase in the tangential stress towards the plastic-elastic boundary which is typical for this type of stress distribution. Listed in the tables is the radius of the critically stressed zone at fracture initiation. Also included in the tables is the radius of the plastic zone after the well is drilled, but before injection has started, thus it is possible to attain an idea of how the plastic zone develops during injection. An examination of this point reveals that both increase and decrease of the plastic zone will occur during injection. For low stress ratios there is an increase as the fracture initiation pressure is approached, while the opposite also takes place for high stress ratios. We also find that Rc strongly depends on the strength of the material, an increase in cohesive strength reduces the extent of the critically stressed zone. In this analysis we have assumed that the uniaxial tensile strength of the rock, a t , is zero. If the rock is not totally unconsolidated this will be a questionable assumption. This implies that the fracture initiation pressure calculated with high compressive strengths are too low, both for results from plastic theory and elastic theory. But even if too low absolute values are predicted, one would not expect the relation between results from plastic and elastic theory to have changed significantly. Elastic theory would still predict too high fracture initiation pressures in most cases. In the description of the model it was stated that when R o is varied, Po should be changed accordingly if fluid flow is involved. However, the results in Tables 1, 2 and 3 are calculated with a constant Po- The effect this has on the fracture initiation pressure when varying Ro between 10 and 100 m is small, indicating that at a reasonable distance from the wellbore the pore pressure is relatively constant and has little influence on the stresses.
CONCLUSIONS The stress distribution around an injection well when the injection pressure is increased up to the fracture initiation pressure has been studied. Our analysis shows that a zone of plastically strained material will exist around a well drilled through poorly consolidated sand. As the injection pressure is increased, this plastic zone may grow or it may decrease, depending on the rock properties. But even if the zone decreases, the material cannot return to its original state. In a poorly consolidated sand, fractures will therefore be initiated in a plastically strained material. The fracture initiation pressures calculated when taking this into consideration can be lower than those calculated by elastic theory. The difference depends on the initial state of stress (stress ratio) and the rock properties (Poisson's ratio and uniaxial compressive strength) in the plastic zone. The difference is greatest when the sand is poorly consolidated and the stress ratio and Poisson's ratio are in the lower region. The difference is then typically in the order of magnitude of
10%. Besides contributing to the basic understanding of the rock behaviour around a well, there are possibly practical consequences. Such an analysis may be helpful when interpreting fracturing tests (formation integrity test, leak-off test) from permeable formations. It should also be evaluated if a leak-off test can be used to estimate the compressive strength of the rock. Once the fracture initiation pressure is determined, the strength can be estimated by back-calculation. If unconsolidated layers could thus be identified, this would be very helpful in, for instance, sand control.
Acknowledgements- The authors wish to express their gratitude to Norsk Agip, Norsk Hydro and Statoil for the financial support making this work possible. Received 14 December 1981;revised 8 July 1982.
Fracture Initiation Pressures in P e r m e a b l e Sands REFERENCES
265
the stress solutions in an elastic material are found as
1. Biot M. A. General theory of three-dimensional consolidation. J. appl. Phys. 12, 155-164 (1941). 2. Biot M. A. Exact simplified non-linear stress and fracture analysis around cavities in rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. ! ! , 261-266 (1974). 3. Hubbert M. K. & Willis D. G. Mechanics of hydraulic fracturing. Trans. Am. Inst. Min. Engrs 210, 153-163 (1957). 4. Haimson B. & Fairhurst C. Initiation and extension of hydraulic fractures in rocks. Soc. Petrol. Engrs J. 310-318 (1967). 5. Eaton B. A. Fracture gradient prediction and its application in oil field operations. J. Peo'ol. Technol. 1353-1360 (1969). 6. Geertsma J. Problems of rock mechanics in petroleum production engineering. Proc. 1st Com3r. ISRM, Vol. 1, pp. 585-594 (1966). 7. Geertsma J. Some rock mechanical aspects of oil and gas welt completions. Proc. European Offshore Petroleum Conf. & Exhibition, Vol. I, pp. 301-310 (1979). 8. Medlin W. L. & Mass6 L. Laboratory investigation of fracture initiation pressure and orientation. Soc. Petrol. Engrs J. 19, 129-144 (1979). 9. Risnes R., Bratli R. K. & Horsrud P. Sand stresses around a wellbore. Proc. SPE Middle East Tech. Conf, pp. 709-727 (1981). 10. Breckels I. M. & van Eekelen H. A. M. Relationship between horizontal stress and depth in sedimentary basins, paper SPE 10336, 56th Annual Fall Tech. Cot~ & Exhibition (1981). 11. Jaeger J. C. & Cook N. G. W. Fundamentals of Rock Mechanics, pp. 212-214. Chapman & Hall, London (19761.
ar = at,, + (fir,, -- arl) Ro - R i I - 2v
(
R~
I-
2(~-2_ ,,)fl ~Ry-o_~-R~[ l - ( R 2 ) 2]
-(Po-Pi'
InIg,,/rl l + ln(R,,/Ri) )
a,,=a,,,+
-
~-R~!--z
1
-(P°- P3 2(~--g l [ I n ( R o / r ) - 1]} + In (Ro/R i~
(A8I
R~ a_ = a.o. + 2v(a,o-o',i) Ro2 _ R 2
}-2,, f
fi
- (P" - P') 1 - v
R,~
( R:, - g~
+ in(Ro/----R i) With a plastic zone between the wellbore and the elastic zone, (A7) must be replaced by
APPENDIX A Elastic Stress Solutions
d +
;:)
+/3
when
r = R~
a,=
when
r : Ro
(A9)
This analysis follows Biot's theory and the basic equations can be found in for instance Jaeger & Cook E11]. The displacement equation is in the case given as
I;. + 2~t~
a, = a,c
=
o
a,o
The stress solutions are now given by (A8) when replacing a,~, R~, P~ with a,o R,, P, respectively. In developing (AS) we also used the relations
(All
Ev
(1 + v)(l - 2v) E
when there are no volume forces. The pressure gradient is given by Darcy's law in radial form
a
~
--
(AI0)
--
2(1 + v)
dp
pQ dr - 2xhkr
(A2)
APPENDIX
The stress-strain relations are
B
Plastic Stress Solutions
a, = ).A" + 2 G ~ + tip % = 2A" + 2GE~'~+ tip
The general solutions for the four different cases are: (A31
(i) o~ < o , < a,,
% = ).A e + 2Ge'.." + tip
l[
a, = p - t
where A" = E~ + e'~j + ~
(A4)
a°= P-t
If we assume only radial displacement after initial loading, the strain displacement relations are
(1 +
u
+
(A5)
F
r)(l
1- v
V
Alp
27thkJ + 2(t + 1)z V -
2v)
I
I i A~_r,
1) P Q ]
C°-(t+
a= = [2v(l - f l ) + f l ] p - t
du dr er3 -
l[
.e ]
C° - 2xhk j + 2(t +
2Co-(t
(BI)
+2) 2xhkj
(a--o - flPo) + 2v(t + 2)(t + 1) A~_ r'
where
e'.' = e." where e:o is the vertical strain caused by the initial overburden loading when assuming no horizontal deformation during this loading. From (A3) we then find
A~ = integration constant; t = t a n 2 ~ - 1; V = (t + 1)2 + 1 - v(t + 2) 2 . (ii) o r < a o = a~
o:. - / / P o e:°-
2 + 2G
(A6)
a, = p - t
By solving (AlL combining the result with the other equations and applying the boundary conditions O-r =
O'ri
when
r
=
R i
(A7) O, =
O-ro
when
r =
R o
o,,=o:=p-t
o
2~nhk + 2(t + 1)(1 + v)Blr'
(B2)
C o - ( t + 1) 2~/~k~k + 2(t + l)e(l + v)Btr'
where B l = integration constant.
266
P. Horsrud et al.
(iii) o', < a o < a .
(t
1)D~ , + ~t ~- l ) [ v ( t + 2 ) + U
1
a, =
(t +[,,l)I
~ [ t ( t + 1) - vt(t + 2) + (1 - 2v)(t + 2)fl]p
+V
13 O-~'in r V
+ 1 ,,[(1 V
v)(t +
-
1)
l l-2v -
v]Co
+
V I
(t
+
1)
+
I-v
-
,-~
[v(t + 2) - ),]D2r
V
1 !C.
I ) 2 ( a : , , -- tiP,,)
(t +
t+l ---[v(t+2)+y]Dlr V t+l
t +1
(1 -- v)(t ~ 1 ) - - v
I l-2l'
Da D~ + [vtt + 2 ) + 13 V + V + [v(t + 2 ) +
I ] / )~ -!In r
:-I
where
1
x (a.o - flPo) + ~ [ v ( t + 2) + ? ] D ~ r = - '
D! = Integration c o n s t a n t ; D2
=
Integration c o n s t a n t ;
),2
:
(t + 1)2 _ 2v(t + 1) + 1:
1
+ ~ [ v ( t q- 2) - y ] D 2 r - : - '
I
2(1 - 2 v ) t ( 1
-fi)
O a
1
a o = ~ [ v t 2 + (1 - 2v)((t + 1)2 + l)fl]p
(B3)
t(t + 1) - vt(t + 2) + (1 + [v(t + 2 ) + 7
2
D.
]~-
Db
Db
1
l -
V
2v
l-v
-
Db
~]D2r-;-] (iv) 0-~ :
a= = - ~ - ] t ( t v
[_
t +
2l,
i '2(;
C.
(1 - 2v)(v(t + 2) -
(t + 1)(t + 2)(a.,, - flPo}
-~- ~ [V(/" 4- 2 ) J r - "1t] D l r 7-1 -- ~ ]-/'(t + 2 } -
t+lF
H-~
v)(t+ (=E;i
(I - 2v)t(l - fl)
#Q
t + 1 - 2v
2~zhk
(°:°
-
flPo):
m
a o < cz~ ~7r =
+ 1) - vt(t + 2) + (1 - 2v)(t + 2)fi
1)
O"K
(B4)
(7 0 - - (5 h
a. - aK(t + I ) t+
1
pQ
+ v(t + 2 ) V
+ I v ( t + 2 ) + 7 2] ~ - l n r _ - v v t C o
1 1 4- - - v - - - -
2v)(t + 2 ) f l ]
t p + (t + 1)[v(t + 2) + 1] D° V
where crK - c o n s t a n t .
tp + Co