ARMA-77-0236

STRENGTH OFWELL COMPLETIONS

By

R. P. Nordgren

Shell Development Company Houston, Texas

ABSTRACT

The strength of completions in producing wells is studied by means of the theories of poroelasticity and poroplasticity. Solutions are obtained for an open hole completion and for a perforated completion modeled as a hemispherical cavity and as a long cylindrical cavity. Drawdown pressures for initial yield are given by simple formulas for the linear and parabolic forms of the extended yon Hises yield function. The theory presented here may lead to a practical criterion for drawdown pressures to avoid production problems.

INTRODUCTION

In many petroleum wells, the rate of production is limited by the risk of formation failure in the completion interval. A failure may halt production either permanently or until an expensive workover can be carried out. In order to minimize the risk

of failure, weak formations can be strengthened by gravel packing or through one of several available chemical treatments. However, such strengthening measures are expensive and may restrict production and complicate future recompletions to alternate in- tervals. At the other extreme, very strong formations may permit open hole completions with liners or screens. Such completions generally are more productive than the usual perforated completions.

Knowledge of completion strength is essential for selecting the optimum type of completion and evaluating the need for a strengthening treatment. The completion decision will be based largely on the maximum allowable production rate for each candidate completion type and strengthening treatment. The completion strength depends on the inherent strength of the treated or untreated formation rock and the stresses imposed on the rock during production. The stresses near the completion depend on many factors including flow rate, well pressure, reservoir pressure, and the original in- situ state of stress.

The present investigation is intended to provide a basic theoretical understanding of the mechanical strength of completions in production wells. To this end, an open hole completion and idealized perforated completions are analyzed according to the theories of poroelasticity and poroplasticity. The perforated completion is modeled as a hemispherical cavity and as a long cylindrical cavity. The simple geometries of the open hole and the hemispherical and cylindrical cavities permit approximate analytic solutions to be obtained. Also, the effect of reservoir depletion is investigated using an approximate solution for a circular disk- shaped reservoir.

An indication of the strength of each type of completion is obtained by determination of the onset of plastic behavior (initial yield) at the completion. Initial yield is determined from a solution of the poroelasticity equations for the

completion and a yield function for the formation rock. The yield functions chosen for the present study are the linear and parabolic forms of the axtended Hises yield function proposed by Drucker and Prager (1952) and Murrell (1963). These yield functions are partially supported by the experimental data of Antheunis et al. (1976) for a friable sandstone. For each type of completion, a simple formula is obtained for the critical well pressure at initial yield. Yield pressures are displayed in dimensionless form for the parabolic yield function. The criterion of initial yield is expected to provide a conservative criterion against completion failure.

In order to use one of the proposed yield criteria, it is necessary to know the material constants in the yield function. These constants should be determined by strength tests on core material whenever possible. As sugges- ted by Stein and Hilchie (1972) and Tixier et al. (1973), it may be possible to estimate the material constants from well-log parameters such as acoustic velocity. The details of such estimates and further consideration of

strength experiments lie outside the scope of the present investigation.

Our results should be applied to field problems with due caution. Further experimentation and field testing are still required to validate the theory or indicate when modifications may be necessary. In particular, the erosive effects of flow rate have not been considered in

the present study. Before presenting solutions for the comple-

tion problems, we will review briefly the theories of poroelasticity and poroplasticity. A nonmathematical discussion of the application of our results is given in the last section of the paper.

THEORY OF POROELASTICITY

The linear theory of elasticity is well established and the basic equations are developed in several texts (e.g., Timoshenko and Go0dier , 1951). Extension of the theory to account for the effect of pore pressure in a porous medium originates in the work of Blot (1941, 1955). SubsequentIy, the equations of poroelasticity have been derived and discussed by many authors. The form of the poroelasticity equations given by Geertsma (1966) is particu- larly convenient and will be used here.

With reference to rectangular Cartesian

coordinates x i (i - 1, 2, 3), the equations of equilibrium for any continuous medium, in the absence of body force, read

oil,j = 0 (1)* *Repeated indices imply smmaation and co•aa de- notes partial differentiation with respect to coordinates.

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vhere 044 is the sya•etric (total) stress tensor. For in•fatitestmal defor•ation of a continuous med-

ium, the strain-dispLace•ent relations are

1

eij= • (uid + uj ,i) (2) where e i. is the strain tensor and u. is the dis- placemen• vector. He follov the standard sign con- ventions of the theory of elasticity where Oll > 0 corresponds to tension and ell > 0 corresponas tb extension.

The linear constitutive equations for an isotropic poroelastic continuum may be•rritten in the followin• form (Geertsma, 1966):

Oij = 2C(eij + 1-• e tij) - (1-5) p •iJ (3)* where

3 (1-2•) e = ekk , • = •/CB' CB = 2(l+•)G

Here, • is the ratio of rock matrix compressibility C R to bulk compressibility C•, p is the pore pressure, G is the shear modulus •nd V is ?oisson's ratio for the bulk continuum. The last term in (3) represents the effect of pore pressure which is ana- logous to the effect of temperature in thermoelasticity as po/nted out by Lubinskt (1954). By (3),

see that a completely restrained body (ei• = 0) undergoes isotropic compressire stress as pore pressure increases. On the other hand, an unrestrained

Jacketed body (Oi4 - 0 ) undergoes expansion on increase of pore •ressure.

A slightly different interpretation of (3) is useful in applications such as ours when initial in-

situ stress is present. In this case, •it and • are regarded as changes in stress and pore pressure from their values in the initial state. Displace- ments are measured from the initial state and conse-

quently e.. vanishes in the initial state. Then

initLal stress t•sor to fo•m total stress tensor Oi.. Since the initial state of stress is in equil•rium, (1) holds for change in stress

On substitution of (2) and (3) into (1), *e have the displacement equations of equilibrium

1 (1-8) •,i = 0 (4) ui,JJ + (1-2v-•-• uj,ji

For a given pressure field •, the field equation (4) is to be solved for the displacement vector u o

Boundary conditions are imposed on u i or on stress vector

o i = • oij (5) Where •. is the unit vector normal to the boundary surface• The pore pressure • is obtained from solution of the equations for fluid flo• in a porous medium. In the simplest case of linear, single-phase flor, • satisfies the field equation (Geertsma, 1966)

ø 0 'kk = CT • + (1-8) e (6) where 0

c T •ere, k is the formation permeability, • the fluid viscosity, C• the fluid compressibility, and • the porosity. I• addition, boundary conditions are imposed on • or on the flow rate

*p• I•onecker delta •i' is defined as = J and

q=-•i p' i (7)

Also, • satisfies initial conditions at time zero. The right-hand side of (6) is usually negl•gible for the flo• problems of interest here. •aen e is negligible in (6), the flor problem is uncoupled and can be solved independently of the elasticity problem governed by equation

THEORY OF POROPLASTICITY

The constitutive equations of poroelasticity (3) furnish a linear isotropic relation between stress, strain, and pore pressure and there is no pe•aanent strain after a cycle of loading and unloading. Linear poroelasticity may be a good idealization of the actual behavior of porous rocks over a limited range of stress. However, at sufficiently large values of stress, departures from linearity occur and there are permanent or plastic strains after a cycle of loading and unloading. Generally, the rock fails at sufficiently large plastic strains or after a number of plastic loading cycles. In this section, we revtev the formulation of the theory of poroplasticity in which a yield function delineates the initiation of plastic yielding. In some problems, hitial yield provides a useful criterion against failure. Fuller discus- sions of poroplasticity have been given by Jaeger and Cook (1969), •adai (1950), and many others.

Upon introduction of a yield function f (Ott), the behavior of an elastic/plastic material can-•e separated into two regimes as follows:

0

Elastic if f < 0 or f = 0, f < 0 0 (8) Plastic if f = 0 and f - 0

and f > 0 is not admitted. Tn the incremental

theory of plasticity, the strain is divided into an elastic component e • e • • such that: iJ and a plastic component

m e ! e ! ! eij ij + ij (9) The elastic strain component satisfies equation (3) and the plastic strain component satisfies a flor rule as discussed by Heilder and Paslay (1970). In the present study, no use is made of the flow rule for e • • and it •r111 not be discussed further.

Ytetid functions used in this investigation are extensions of the classical yon Hises yield function to include the effect of mean stress. The extended

Hises yield function has the general form

f (,71 , J2 ) (10) •here the stress invariants J. and J_ are express-

able in terms of principal stresses •1' •2' •3 as J--1 = - 1_ (•1+02+05) _ P 3

1

+ The quantity J1 may be interpret•2as the mean eff- ective compresslye stress and J2 ' as the mean square shear stress.

A sim?• form of (10) is the linear relation betveen J?•'- and• proposed by Drucker and Prager (1952), namely

I •21/2 a• 1) (11) = - (•o +

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The material constant T is a measure of shear strength at zero mean stressøand • measures the increase in shear strength with mean compressire stress. Eq- uation (11) can be readily generalized to a multi- linear convex yield function which is useful in applications as will be seen later.

Another simple am• useful form of (10) is the parabolic yield function proposed by Murrell (1963), namely

F= 3 2- C J1 (12) where C is a material constant which measures the increase of shear strength with mean compressire strength. Combinations of (11) aD• (12) also are possible.

Full consideration of the experimental problem of selecting a yield function and determining the associated material constants is beyond the scope of the present investigation. Discussion of this difficult problem is given by Jaeger and Cook (1969) and throughout the literature of rock mechanics. As an illustration of the difficulties involved, we use the Mises invariants (10) to interpret the experimental data for a reservoir rock presented by Antheunis et al (1976). The experiments consisted of conventional triaxial compression and extension tests as well as collapse tests on thick walled cylinders under radial and axial pressures. Figure 1 shows an interpretation of this data* for Material

A which had a Brinell ha¾•ess number of 9 to 11. In calculating• 3 and J2-'- the stress state in the triaxial specimefis is assumed to be uniform. The stress state in the thick walled cylinders at collapse is assumed to be given by the Lam• solution of elasticity theory (Timoshenko and Goodlet, 1951). This solution neglects any inelastic behavior prior to collapse. The magnitude of inelastic effects in the collapse test is not known. As seen from Figure 1, the experimental data can be fit reasonably well by a bilinear form of (11) or by combination of (11) and (12), except fo• the triaxial extension data. The extension fractures usually observed in the triaxial extension test (Jaeger and Cook, 1969) may constitute a different type of failure mechanism than the shearing failures usually observed in the triaxial compression test and the collapse test and expected in completion failure. The triaxiat extension test was the main basis for adoption by Anthen- his et al. (1976) of the Mohr-Coulomb yield function and a critical-strain failure criterion instead of the extended Mises yield function. We believe that both approaches merit serious consideration. Con- siderable further research will be required to reach a completely acceptable general failure criterion.

IN-SITU STRESS AND RESERVOIR PRESSURE

The state of stress in the earth has been the subject of considerable research as discussed by Jaeger and Cook (1969), Howard and Fast (1970), and Macpherson amd Berry (1972). Here we give a brief review of'the subject for depths of interest in petroleum production.

On the basis of density measurements, it is generally agreed that the vertical component of total (compressire) in-situ stress S. increases with depth at a gradient of approxtmatelyVl.0 psi/ft with considerable variation at shallow depths. The mini- mum horizontal component of total in-situ stress, SH1 , has been estimated from measurements of fracture extension and shut-in pressure. In tectonically

*Specifically, the data was taken from Table 1 and Figures 1 and 3 of Antheunis et al (1976).

relaxed regions, S•m is generally between 0.6 to 1.0 times S V. The oth• horizontal stress S. 2 is difficult to determine and it is often assume• that and S.. are equal, in which case they are denote•Zby S•. • tectonically active regions, the horizontal sEresses may exceed S v. Around salt domes, geolo- gical evidence indicates that the principal in-situ stresses do not act in horizontal and vertical di-

rections and they may have widely different values. Initial reservoir pressures p• also may vary

considerably. Shallower reservoirõ generally follow the hydrostatic pressure gradient of 0.45-0.48 psi/ ft. In the Gulf Coast region and some other regions, the pressure gradient often departs from the hydrostatic gradient at a particular depth and increases abruptly with depth. In extremely deep

wells, p• may approach S ß Aftar production, t•e pressure in a depleted

reservoir may be reduced to as little as 1000 psi. Reservoir depletion also changes the components of total stress through the poroelastic effect discussed in connection with equation (3). An example of this change is considered in Appendix A for a deeply buried circular disk-shaped reservoir. When the radius of the reservoir is much greater than its height and much less than its depth and the reservoir is at the depleted constant pressure p, we fiD•

(Appendix A) that the new in-situ stresses S V' and S H' near the well are appro•m-tely

ß

s v' = Sv, s•' = s•- n(p i - p) (13) where

n - (l-S) (1-2v) /

and SV, S H are the original in-situ compresslye stresses.

Drilling and completion operations also may cause changes in the stress state near the well. The stresses near cemented and perforated completions are difficult to quantify even before production begins. The expansion properties of cements are subject to uncertainty amd even the basic mecha- nisms involved in shaped-charge perforations are not well understood. Further, time-dependent relaxation phenomena may take place after the'well is complet- ed. Thus, we assume that the stress state near the well is the in-situ state or the state (13) for depleted reservoirs.

HEMISPHERICAL PERFOBATION

We consider a perforated completion in the form of a hemispherical cavity. This model seems most applicable to relatively weak formations where a long perforation tunnel could not remain open. Fur- ther, the hemisphere has intrinsic appeal as the "strongest" shape for a cavity or arch.

A poroelastic solution is readily obtained from known solutions in elasticity and thermoelasticity for a spherical cavity. For application to a hemispherical cavity, this solution requires zero shear stress and zero normal displacement on the diametri- cal plane forming the hemisphere from the sphere. We assume that such is the case for the completion, thereby neglecting any cement-perforation bond which may. have been disturbed by the perforation operation anyway. The poroelastic solution is examined in terms of the yield functions to determine well pressure for initial yield.

First, we recall the solution for a spherical cavity in an unlimited elastic medium under uniaxial compresslye stress S at a large distance from the cavity. With reference to cylindrical coordimates

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^

(r, 8, z) with z in the direction of S, the solution for stresses at the intersection of the plane z = 0 with the cavity reads (Timoshenko and Goodier, 1951)

^

O r = 0, O z • -81S, 08 = -82S (14) where

27-15v 15v-3 81 = 14-10• ' 82 = 14-1•

By the poroelasticity-thermoelasticity analogy (Lubinski, 1954) and a thermoelasticity solution (Timoshenko and Goodlet, 1951), the solution for this same spherical cavity under internal pressure p with isotropic stress S at infinity and steady redial flow at rate 2q (q into a hemisphere) gives radial and tangential stresses at the cavity as

= i 3 npq (15) Or -Pw' Ot = • Pw - • S + 2•ka

where a is the cavity radius. For the case of a spherical cavity under in-situ principal stresses (S H , SH2, Sv) , on superposing (14) and (15), we fin• the following stresses at the intersection of the principal stress axis for SH2 with the cavity:

1 1 Or = -Pw' 08 = • Pw - S8' Oz = • Pw-Sz (16)

where

3 nqU S8 = • SH2 + 82(Sv-SH2) + 81(SH1-SH2) - 2•ka

3 nq• Sz = • SH2 + 81(Sv-SH2) + 82(SHl-SH2) - 2•ka

In forming (16), we think of SH2 as being the isotropic stress, S v - Sw? as being uniaxial stress in the vertical direc•on, and S.1 - S'2 as being uniaxial stress in the other hor•zonta• direction. When SH1 and S • are different, two cases should be considered in w•at follows with SKi and SH2 values interchanged in (16) to find the worst case. Note that the principal stresses S , S.-, S._ need not be vertical and horizontal for tee •1 z hemispherical perforation.

By (10), the Mises stress invariants associated with (16) are

-- 1

J1 = • (Sz + S8) - Pw (17)

-- 1

J2 "• J12 + • (Sz - S8)2 In order to investigate initial yield, the linear yield criterion (11) at the cavity can be written using (17) as

f = A1 (Pw - Pl)(Pw where

3 = A1 = • _ e2, B1 2eTo Cl = ¬ (Sz_So)2 _ • 2 o

- p2 )

(P•] (B12-4Aic1)l/2]/2A1 1 --

p = • (Sz+S 8) + [-B 1 +

The roots Pl and P2 are real if

(18)

(19)

3To2 >_ A 1 (S z - se) 2 (20) If (20) holds, then the material is elastic (f < O) at the cavity if

Pl < Pw < P2 for A 1 > 0, (e < /3/2) (21)

Pw < Pl or P2 < Pw for A 1 < 0

If A] < 0, then by (20) there will always be two real ?oot8. In this case, the root P2 corresponds to •. < 0 in general and only the p. root is of practical interest. The condition p• < p. 1 does not place a lower limit on the yield pressure

Pw' If A_ > 0 and there are no real roots, as

indicatedlby violation of (20), then f cannot. change sign and by (18), f > 0. However, this is not admissable and the perforation must already have yielded.

If A. > 0 and there are real roots, then (21)

places a •ower limit on Pw at which initial yield occurs. The upper limit on p in (21) may not have physical significance since f•ilure by fracturing could occur before this limit is reached. Fractur-

ing failure is not of interest in the present

study and no use will be made of the upper limit P2 in (21). Also, further investigation shows that initial yield may not occur at the cavity face as assumed above if the flow rate is sufficiently high. Such flow rates are much higher than usually encoun- tered in practice and will not be considered here.

The foregoing results for the linear yield function (11) can be extended to the convex piece- wise linear form. The linear segments are simply considered individually and the greatest lower limit

p] (for A 1 > 0 cases) is the lower limit on Pw fdr elastic behavior.

The parabolic yield function (12) also leads to solution of a quadratic equation for yield pressure. We find that the perforation is elastic if

Pl < Pw < P2 and C •q' [Sz-SO] (22) where

Plt 1 2 2 3 1/2 P = • (Sz+S e) - • c ß • [c 2 - • (Sz-Se)2] If the restriction on C is not satisfied, then yield

has already occurred. Again, the upper limit on Pw may not be physically significant.

The effect of the flow rate q on the yield pressure is easily examined. By (16) and (18), A•, B., C. are independent of q. Then, by (16), (19)• a•d (i2), the change in yield pressure with q for both yield functions is given by

nvq (23) Pl = - 3•k•

Thus, yield pressure decreases as q increases. This somewhat surprising result reflects the fact that compressire normal stresses decrease as pore pressure decreases through the poroelastic effect as discussed in connection with equation (3). This

decrease i•/•ompressive stress decreases the shear measure J2-' more than • 1 in (17), thereby decreasing •he tendency to yield.

The preceding results for yield pressure can be cast into a convenient dimensionless form for

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graphical display. As seen from (16) and (22), the

dimensionless yield pressures pl/C and p2/C for the parabolic yield function are functions of the dimensionless variables

S V SH1 SH2 •q • ' S V ' S V 2•kaC' • (24)

Numerical results for the parabolic (Murrell) yield function are shown in Figure 2 for q = 0, S = S - = Hi H2 S., • = 0.2, and various values of S./S V. These results can be extended to include flow rate using

(23). The upper limits P2 are shown dashed in Figure 2 for mathematical completeness and they may not be physically meaningful as already discussed. Results similar to Figure 2 can be plotted for the linear yield function by (16) and (19). Application of the results of Figure 2 to completion problems is discussed in the last section of the paper.

LONG CYLINDRICAL PERFORATION

We consider a long cylindrical perforation lying in a horizontal plane. A state of plane strain is assumed in planes normal to the perforation. This idealization neglects effects of the cemented well bore and the end of the perforation.* Also, we neglect the poroelastic effect of pressure change accompanying fluid flow since this pressure field is not readily idealized. As in the previous section, we seek to determine the well pressure for initial yield.

Using the Lam• and Kirsch solutions in the theory of elasticity (Timoshenko and GoodYet, 1951), the components of stress in cylindrical coordinates at the perforation face are

o r = -pw,O0 = Pw-S0 , o z -S z (25)

where

S O = 2S v - (Sv-SH1)(1-2cos 20)

S z = SH2 + 2• (Sv-SH1) cos 20

Here the in-situ stress S_ 2 is assumed to be aligned with the perforation. The invariants of stress (10) for (25) are

-- 1

J1 = • (Sz + 2S8) - Pw (26) = 1 J2 (Pw - S8 )2 + • (Sz - S8 )2

and the linear yield function (11) becomes

f = A2 (Pw - Pl )(pw - P2 ) (27) where

A 2 = 1-• 2, B 2 = i •2 c 2 = • (Sz-So) -

• - T + 1 (28) o '• • (Sz-So)

(p:] - p = S O + [-B 2 + (B22 - 4A2c2)l/2]/2A2 *Results of a finite-element numerical analysis

indicate that the plane strain idealization is valid away from the ends if the perforation length is at least four times its diameter.

The roots Pl and pA are real if •2zz • (1-• 2) (Sz-Ss) 2 (29)

If (29) holds, then the material is elastic (f < 0) if

Pl < Pw < P2 for A 2 > 0, (• < 1) (30)

Pw < Pl or P2 < Pw for A 2 < 0

The discussion of (29) and (30) is identical with the discussion following equation (21) and will not be repeated. In checking for yield, S• and S must be evaluated at 0 = 0 and 0 - •/2 in (25). z

For the parabolic yield function (12) by (26), the perforation is elastic if

Pl < Pw < P2 and -C I 2(Sz-S 8) ! 3C (31) where

P = S 8 - • C; [C 2 - (Sz-S 8 - 21-C)211/2//3 If the second of (31) is not satisfied, then f > 0 which is not admissable and the perforation has already yielded.

Dimensionless results for yield pressure (31) under the parabolic (Murrell) yield function are shown in Figure 3 for S -- S = S • V = 0 2, and H1 2 H ' various values of SH/S v. Again 2, the dashed lines corresponding to the larger root p• in (31) may not be physically meaningful. AppIication of the results is discussed in the last section of the

paper.

OPEN HOLE COMPLETION

We confine attention to an open circular borehole in line with the vertical in-situ principal

stress S V and carry out an analysis similar to the previous section. According to the Lam• and Kirsch solutions for plane strain in the theory of elasticity (Timoshenko and GoodYet, 1951), cylindrical components of stress at the borehole are given by

ør = -Pw' O0 = Pw-So ' Oz = -Sz (32)

S 0 = 2SH2 - (SH2 - SH1)(1 - 2cos 20)

S z = S v + 2• (SH2 - SH1) cos 28 The analysis procedure is exactly the same as

for the cylindrical perforation in the previous section. Specifically, (32) evaluated at 8 = 0 and •/2 is used in (28). Then (28) to (30) apply for the linear yield function and (31) applies for the parabolic yield function.

Dimensionless results for yield pressure under the parabolic (Murrell) yield function are given in Figure 4 which is discussed in the next section.

APPLICATIONS

Formulas for the well pressure at initiation of yield have been developed in the preceding sec- tions for a hemispherical perforation, a long cylindrical perforation and an open hole completion. Specifically, the lower yield pressure p. for the hemispherical perforation is given by equation (19)

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for the linear yield function (11) and by (22) for

the parabolic yield function (12) where S and S 8 are defined by (16). For the long cylindrZical perforation, the lower yield pressure Pl is given by (28) for the linear yield function aSd by (31) for the parabolic yield function where S_ and Sa are defined by (25). The same formulas •28) an• (31) apply to the open hole completion with S_ and S• defined by (32) instead of (25). As•noted b•fore, the upper yield pressure P2 may not be physically meaningful since failure by fracturing could occur before this limit is reached. The lower

yield pressure Pl is expected to provide a limit on well pressure to-avoid completion failure in each case. However, this limit must be applied with caution since field verification is lacking.

The dimensionless plots of yield pressure for the parabolic (Murrell) yield function in Figures 2 to 4 give considerable insight into the effects of various parameters on the completion problem. Fur- ther, relative strengths of the various completions can be compared. For example, a hemispherical perforation in a normally pressured reservoir (p•/S w • 0.45) with Sw/S•r -- 0.8 would yield immediatel• i• the strength"pa•ameter C is such that C/S v < 0.45 according to Figure 2. If C/S v = 0.6, th•n drawdown to a pressure of p. /S,, = 0.21 is possible without yielding, assumingWS•Sv remains at 0.8. However, reservoir depletion •ff•ctively decreases S• as seen from equation (13) for the case of a d•eply buried thin circular reservoir. If Sw/S v changes to 0.7, then yield is predicted at p..,7S•'= 0.30 for C/S v • 0.6. Similar conclusions can"be-reached for the'long cylindrical perforation from Figure 3. Comparison of Figures 2 and 3 shows that the yield pressure for the hemispherical perforation is lower than for the long cylindrical perforation. Thus, the hemispherical perforation is stronger as might be expected.

For the case considered above (p _/S.. = 0.45

and S•/S V • 0.8), the open hole compl•tign yields immediately if C/S• < 0.33 according to Figure 4. If C/S.. = 0.6, the'yield pressure is p /S.. = 0.16

w ¾ for .S..YS.. = 0.8 and p /S.. • 0.05 for S./Sj. = 0.7. v w v [.1 v In th•s case, the open hole completion is stronger

than the hemispherical perforation. However, this will not always be the case as seen from Figures 2 and 4.

From Figures 2 to 4, we see that a high ratio

of horizontal to vertical in-situ stress (S./S V > 1) has an adverse effect on yield pressures. Pot example, if S../S.. • 1.4 and p /S.. -- 0.45, then yield occurs immediately if C/S• • 0v. 75 for the hemispherical perforation and if CfS V < 1.17 for the open hole completion. Open hole completion may not be advisable in such cases.

The case of abnormally high reservoir pressure also can be examined with the dimensionless plots of Figures 2 to 4. For example, if p /S.. -- 0.8 and

S./S. -- 0.9, then yield occurs im•Wedi•tely if C/S V <n 0.•18 for the hemispherical perforation and C/S v < 0.10 for the open hole completion. In general• less formation strength is needed for initial production from abnormally pressured reservoirs than from normally pressured reservoirs. However, depletion may have a larger effect in abnormally pressured

reservoirs by decreasing S H as indicated by equation (13).

Finally, we note that dimensionless plots such as Figures 2 to 4 provide a convenient method of presenting field data on completion strength.

REFERENCES

ANTHEUNIS, D., GEERTSMA, J. AND VRIEZEN, P. B., 1976, Mechanical Stability of Perforation Tunnels in Friable Sand-

Stones, Paper Presented at the 31st Annual Petroleum Mechanical Engr. Conference of the ASME, Sept. BIOT, M. A., 1941, General Theory of Three-Dimensional Consolidation, J. Appl. Phys., v. 12, pp. 155-164. •IOT, M. A., 1955, Theory 6f Elasticity and Consolidation for a Porous Aniso-

tropic Solid, •. Appl. Phys., v. 26, pp. 182-185. DRUCKER, D.C. and PRAGER, W, 1952, Soil Mechanics and Plastic Analysis or Limit Design, •uart. Appl. Math., v. 10, pp. 157-165. EASEN, G., NOBLE, B., and SNEDDON, I. N., 1955, On Certain Integrals of Lipschitz- Hankel Type Involving Products of Bessel Functions, Phil Trans Royal Soc., London, A 247, pp. 529-551. GEERTSMA, J., 1966, Problems of Rock Mechanics in Petroleum En•lneering, First Intl. Congress Intl. Soc. Rock Mech., Lisbon, Portugal. GEERTSMA, J., 1973, Land Subsidence Above Compacting Oil and Gas Reservoirs, J. Pert. Tech., pp. 734-744. HOWARD, G. C. and FAST, C. R., 1970, Hydraulic Fracturing, Soc. Pert. Engrs. of AIME, New York. JAEGER, J. C. and COOK, N. G. W., 1969, Fundamentals of Rock Mechanics, Chapman and Mall, Ltd., London. LUBINSKI, A., 1954, The Theory of Elasticity for Porous Bodies Displaying a STrong Pore Structure, Proc. 2nd U.S. Natl. Congress Appl. Mech., p. 247. MACPHERSON, L. A. and BERRY, L. N.,1972, Prediction of Fracture Gradients from Log Derived Elastic Moduli, The Log Analyst, Oct., v. 13, no. 13, pp. 12-19. MURRELL, S. A. F., 1963, A Criterion for Brittle Fracture of Rocks and Concrete under Triaxial Stress and the Effect of Pore Pressure on the Criterion, Proc. Fifth Rock Mechanics Symp., Univ. of Minn., C. Fairhurst (ed.), Oxford, Pergamon, pp. 563-577. NADAI, A., 1950, Theory of Flow and Fracture of Solids, Second Edition, v. 1, McGraw-Hill Book Co., New York. SNEDDON, I. N., 1951, Fourier Transforms, McGraw-Hill Book Co., Inc., New York. STEIN, N. and HILCHIE, D. W., 1972, Estimating the Maximum Producing Rate Possible from Friable Sandstones without

using Sand Control, J. Petr. Tech., Sept., pp. 1157-1160. TIMOSHENKO, S. and GOODIER, J. N., 1951, Theory o_•fElasticity (Second Edition), McGraw-Hill Book Co., New York. TIXIER, M.P., LOVELESS, G. W., and ANDERSON, R. A., 1975, Estimation of Formation Strength from the Mechanical Properties Log, J. Petr. Tech., March, pp. 283-293. WEIDLER, J. B., Jr. and PASLAY, P. R., 1970, Constitutive Relations for Inelastic Granular Medium, •. Engr. Mech., Div. ASCE, v. 96, pp. 395-406.

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APPENDIX A - CIRCULAR DISK AT CONSTANT PRESSURE IN

AN UNLIMITED ELASTIC MEDIUM

We consider a disk of radius R and thickness 2h

raised to a uniform pressure p. The disk is part of an unlimited homogenous isotropic poroelastic medium. We follow a method given by Timoshenko and Goodier (1951) to obtain a solution near r = 0.

A particular solution of (4) for displacements u. can be written as

u i = •,i (A-i) where

V2• = CmP, C = •/2G (A-2) m

We consider a uniform pressure field of the form

p = const., if 0 < r < R, I zl < h p = 0, otherwise (A-3)

In the present problem, • = •(r,z) and by the method

of Hankel transfor!n_•(S•eddon, 1951), we find that •(r,z) =• • (•z) Jo (•r) d • (A-4)

o

where

C•R [•-•h • = •3 cosh ½z- 1] Ji(½R), [z[ < h Cm9 sinh •h J1 (•R), > h

The displacements can be obtained in integral form from (A-4) using (A-i) and the stresses follow from equation (3). The displacements u and u and the stresses o and o are continuou• at [z• = h as required. Zõince t• stresses and displacements approach zero as r + • and ]z[ + • , the particular solution is the complete solution for this problem.

The integrals in the solution can be expressed in terms of elliptic integrals and numerical tabula- tions are available (Eason et al., 1955). We do not pursue this here since our main interest is the neighborhood of r = 0 where the completion is located. The stresses obtained can be considered as corrections to the in-situ stresses in this

region. On expanding the Bessel functions about r = 0 and evaluating known integrals, we obtain the following approximate expressions for change in stresses:

^

o = o e =-up [1- 1 r i f (z) ] ^ 1

= - - uP f (z) øz 2 where

h+z h-z

f (z) = 2 + 2[(h+z) + R2] 1/2 2[(h-z) 2 + R2] 1/2 with an error of the form

•(:• if h > R (A-6)*

*The symbol g (x) means that g (x)/x - constant as

For a thin reservoir (h << R), (A-5) reduces to

r 2R (A-7)

In this case, pressure depletion h•s a significant effect on the horizontal stresses •_ and om^,but a negligible effect on the %ertical'stressUo . This

z

is reasonable since expansion can readily occur vertically, but not horizontally.

The limiting case of a thick disk also can be considered and the plane strain solution is obtained. Also, it is possible to include the effect of " a free surface without essential difficulty. The effect of radially variable pressure can be considered by means of the Duhamel integral. Alter- natively, the effect of variable pressure and a free surface can be treated by the method of Green's function as done by Geertsma (1973) for an infinitesimally thin disk.

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750 I

500

250

PARABOLIC

C :500

1/3 ß

1/3 ß ß1/2

.1

ß3

LINEAR

To= 50, a= 1.0

To:100, a :0.8

TRIAXIAL TEST

X COMPRESSION

'1- EXTENSION

ß COLLAPSE TEST

AXIAL PRESS.

RADIAL PRESS.

o I I o 500 lOOO

J"-I KG/CM2

Fig. 1 - Experimental data (Material A) of Antheunis et al. (1976) interpreted by elasticity theory and the extended Mises yield function.

Pw

1.0

0.5 i

1.0 S./Sv

•0.9

I

0.5 .5

C/Sv

Fig. 2 - Dimensionless yield pressure versus dimensionless strength for a hemispherical perforation umder the parabolic (Murrell) yield functiom (q • 0, • - 0.2).

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1.0

Pw 0.5

Sv

1.0 Sm•/Sv

m

• mmmm mmmm m mmm m m m m m m mmm mmmm mmm m m m

m m

o

o 0.5 1.o 1.5

C/Sv

Fig. 3 - Dimensionless yield pressure versus dimensionless strength for a long cylindrical perforation under the parabolic (Hurtell) yield function (q = O, • = 0.2).

1.0

Pw 0.5

Sv

1.0: S./Sv •_,4 I 1.6 mm m mm ,

0'70• 0.$

o o 0.5 1.o 1.5

C/Sv

Fig. 4 - Dimensionless yield pressure versus dimensionless strength for an open hole completion under the parabolic (Murrell) yield function (q = O, v - 0.2).

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