Productivity enhancement in hydrofractured oil ...

Euro. Jnl of Applied Mathematics (1990), vol. 1, pp. 25-46

25

Productivity enhancement in hydrofractured oil reservoirsf PATRICK S. HAGAN1 and ROBERT W. COX2 s

1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Computer Research Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

(Received 26 June 1989)

Low permeability formations are often hydrofractured to increase the production rate of oil and gas. This process creates a thin, but highly permeable, fracture which provides an easy path for oil and gas to flow through the reservoir to the borehole. Here we examine the payoff of hydrofracturing by determining the increased production rate of a hydrofractured well. We find explicit formulas for the steady production rate in the three regimes of small, intermediate, and large (dimensionless) fracture conductivities. Previously, only the formula for the large fracture conductivity case was known. We assume that Darcy flow pertains throughout the reservoir. Then, the steadyfluidflow through the reservoir is governed by Laplace's equation with a second-order boundary condition along the fracture. Wefirstanalyze this boundary value problem for the case of small fracture conductivities. An explicit formula for the production rate is obtained for this case, essentially by combining singular perturbation methods with spectral methods in a function space which places the second-order boundary condition on the same footing as Laplace's equation. Next, we re-cast Laplace's equation as a variational principle which has the secondorder boundary condition as its natural boundary condition. This allows us to use simple trial functions to derive accurate estimates of the production rate in the intermediate conductivity case. Then, an asymptotic analysis is used to find the production rate for the large fracture conductivity case. Finally, the asymptotic and variationally-derived production rate formulas are compared to exact values of the production rate, which have been obtained numerically. It may be feasible to create more than a single fracture about a borehole. So we also develop similar asymptotic and variational formulas for the production rate of a well with multiple fractures. 1 Introduction A common problem in the oil industry is the discovery of oil or gas in tight (low permeability) rock formations, which can make production of the discovery uneconomically slow. This problem is often surmounted by hydrofracturing the well (Veatch 1983 a, b). In this process the borehole is perforated along a vertical segment at the same depths as the oil bearing strata, and a fracturing fluid is pumped down the borehole at extremely high pressures. This high-pressure fluid fractures the formation, usually creating a straight fracture centred about the borehole. The fracture is usually held open by 'packers' — sand or gravel is common (originally, nutshells were used!). As shown in figure 1, this fracture can be thought of as emanating radially from the borehole at, say, 0° and 180°. Typically, t Research supported by the US DOE under the Applied Mathematical Sciences Research Program KC-07-01-01-0 and by grant W-7405-ENG-36.

26

P. S. Hagan and R. W. Cox

the vertical extent of the fracture roughly matches the top and bottom of the oil bearing strata. Although these fractures are very thin, usually no more than a few millimeters across, the permeability Kt in the fractures is radically larger than the permeability Aô of the surrounding formation. Thus the hydrofractures serve as easy avenues along which the oil can travel to reach the borehole. Advances in hydrofracturing technique promise to yield longer, wider, and more highly permeable fractures. It also appears possible to create more than a single fracture (Baker 1981, Veatch 1981, 1983 a, b, Swift 1979, Swift & Kusubov 1982). Here we determine the payoff of these techniques by determining the increase in the production rate due to the fracture. Specifically, we use asymptotic and variational techniques to analyze steady fluid flow in hydrofractured reservoirs. This analysis yields explicit formulas for the steady state production rate, the rate at which fluid flows from the reservoir into the borehole. These formulas then show how the production rate depends on the permeability, width, and length of the fracture. A key reservoir parameter is the dimensionless fracture conductivity!, a = aKt/2RE Ko, where ' a ' is the fracture's width and 2i?E is the fracture's length (see figure 1). In essence, this parameter measures how easy it is to transport fluid through the fracture compared to transporting it directly through the surrounding formation. Now a can range from below 0.03 to above 300. So we will obtain separate production rate formulas for the three regimes a 1) was known; for intermediate and small values of a, the production rate was found by numerical computation, by analogy with electrical conduction experiments, and by approximating the pressure as radially-symmetric (Friedman 1956, Gringarten, Ramey & Raghavan 1974, Mao 1977, McGuire & Sikora 1960, Prats 1961, Raymond & Binder 1967, Tinsley et al. 1969). Let us now be more specific. This paper considers the standard reservoir problem shown in figure 1, where there is a single well with borehole radius RB in the centre of a circular reservoir with an effective drainage radius /? D . The vertical thickness of the reservoir is h, and the reservoir is assumed to be bounded above and below by impermeable layers at depths z = za and z = zo + h. For simplicity, it is assumed that the fracture has a constant width a, that it is located symmetrically on the x-axis, — RE < x < R& and that the fracture extends vertically through the reservoir thickness h. Finally, in order to focus on the main effects of the fractures, we assume that the reservoir permeability Ko and the fracture permeability Kt are constant, we assume that there is a single fluid phase with a constant viscosity /i, and that Darcy flow pertains within the fracture as well as within the surrounding rock. This reservoir problem is constant in z between the impermeable layers at z = z0 and z = zo + h. Consequently, the pressure must also be constant in z. So there can be no fluid flow in the z direction, and the flow in the horizontal directions must be independent of the depth z. Thus we can ignore z, and need only solve the two-dimensional problem shown in figure 1 (b). The steady equations for the two-dimensional problem are derived in § 1.1. There we find t This parameter is also known as the 'dimensionless fracture flow capacity', and is often defined without the factor of two in the denominator. Also widely used is the relative fracture conductivity, aKJK,.

Productivity enhancement in hydrofractured oil reservoirs

27

.Drainage radius

Drainage radius r=RD

Drainage radius r=RD

(b) FIGURE 1. Side view (figure 1 (a)) and top view (figure 1 (b)) of a circular reservoir bounded by impermeable layers at depths z0 and zo + h. As shown, the hydrofracture has total length 2RE, width a, and height h.

that the fluid flow is governed by Laplace's equation everywhere in the reservoir except at

the fracture. Since the fracture is only a few millimetres across, we treat it as a zero-width boundary, and discover that the appropriate boundary condition along the fracture is second-order. Having a second-order boundary condition for a second-order equation (Laplace's equation) is certainly very awkward. However, in §1.2 we re-cast Laplace's equation as a variational principle which has this second-order boundary condition as its natural boundary condition. This immediately shows that the problem is well-posed, even with the second-order boundary condition. The variational principle also enables us to make a quick estimate of the production rate, which then guides later analysis.


28

Drainage radius R

FIGURE

2. Reservoir with two hydrofractures (m = 2).

In §2 singular perturbation methods are used to solve the steady equations and obtain an asymptotic formula for the production rate in the small fracture conductivity case, a. 4 1. In particular, the awkward second-order boundary condition is handled by working in a function space which puts it on the same footing as Laplace's equation. In §3 we briefly examine the large fracture conductivity case, a > 1, and derive the previously-known formula for the production rate for this case. We have been unable to solve the reservoir problem in the intermediate conductivity case. However, by exploiting the variational principle, in § 3 we obtain a formula which accurately estimates the production rate for the intermediate case. Finally, in §4 we compare the asymptotic and variational formulas found in §§2 and 3 with exact values of the production rate, which are obtained by solving the reservoir problem numerically. Soon it may be possible to create more than a single fracture (see figure 2) without unduly damaging the reservoir rock near the borehole (Swift & Kusubov 1982, Swift 1979). So in Appendix A we repeat the analysis for the case of multiple fractures. There we find asymptotic formulas for the production rate when a < 1 and when a > 1, and variational estimates when a = 0(1). 1.1 Governing equations

The fluid flux Jo is related to the pressure p by Darcy's law

in the reservoir. For steady flow, fluid conservation requires that V • Jo = 0. Since the reservoir permeability Ko is constant, we obtain Laplace's equation V2/> = 0 for

(1.2)

which must pertain everywhere except along the fracture. Now the fracture is only a few millimetres wide. So we treat the fracture as a zero-width


29

boundary and account for its influence via an effective boundary condition. To derive this boundary condition, note that Darcy's law shows that JtTac(x) = -K,px(x,0)

for RB < x < RE,

(1.3)

is the rate at which fluid flows down the fracture towards the borehole. The difference a •^rac (•*)—^tracC* + dx) = — Ktpxx(x, 0) dx (1.4 a) /* must equal the rate that fluid seeps into the fracture between x and x + dx. Accounting for the fluid seeping into the fracture from both the wall y = 0+ and the wall y = 0" yields 4e P = ~f {/>»(*. 0+) - py(x, 0-)} dx.

(1.4 b)

Equating now yields the effective boundary condition along the fracture: />**+-£[/>„]-= °

at

y = 0> RB RE,

(1.15a) (1.15b)

32


Later we shall see that (1.15 d) is a good qualitative estimate of the production rate. Indeed, this estimate agrees with, and partially justifies, the approximate results in (Raymond & Binder 1967). The estimate (1.15) reveals three parameters: RJfi, RE/RD, and

Of these, R^/fi is always small, typically much smaller than 0.02. Therefore we will only consider the limit i?B//?->0. That is, from now on we set the borehole radius RB to zero. The fracture penetration RE/RO is usually between 0.1 and 0.8, so we will treat it as 0(1). Finally, as noted earlier, we will derive production rate formulas for a. -4 1, a = 0(1), and a > 1 since a, can range from below 0.03 to above 300. 2 Low fracture conductivity After setting the borehole radius RB to zero, the pressure is governed by the equation V(x2+y2) = 0 at s=\,

sl.

(2.9a) (2.9b)

We wish to solve the inner problem by expanding p{x,y) in the eigenfunctions of (2.8). However, the second-order boundary condition (2.8 b) has no adjoint boundary condition. So at first sight it seems that we cannot define an adjoint operator, and thus cannot define the adjoint eigenfunctions. This appears to make expanding p(x,y) extremely difficult, since it is only the orthogonality of the adjoint eigenfunctions with the ordinary eigenfunctions which allows the coefficients of an eigenfunction expansion to be calculated readily. These difficulties can be circumvented by giving the second-order boundary condition the 2

EJM 1

34


same status as the second-order equation (Friedman 1956). Specifically, given a function u(y), define the 'vector'

and let the inner product of any two such vectors be

r°° =

u(y)v(y)dy + u(0)v(0). Jo Inspection of (2.8 a), (2.8 b) shows that we need to consider the operator

(2.11)

(2AI)

io)\ Integrating by parts shows that L is self-adjoint, = (Lu, v> = - T uy{y) vy{y) dy,

(2.13)

Jo

in the inner product • Consequently its eigenfunctions are complete and orthogonal in this inner product. We shall solve the inner problem (2.8), (2.9) by expanding p(x, y) in the eigenfunctions of L at each x. Equation (2.13) shows that (u,Lu) is always non-positive, so the eigenvalues of L must also be non-positive. By solving Lu = —k2u, we find that the eigenfunction u(k,y) with eigenvalue —k2 is

where k is defined by tan 9^ = k, Q(0,0) = - 1. So from (2.16 b), Ax(k,0)

= -Qcosk.

(2.18)

The general solution of (2.17), (2.18) is 20 A(k,x) = — j cos kerkx + B(k) cosh kx.

(2.19)

7TK

Clearly B{k) must be zero for all k; for otherwise the inner solution would blow up exponentially as x2+y2 became large, and it would not match the outer solution. So the inner solution must simply be | ]-coske-ktcos(ky + t)dk Jo * to leading order in a. Using (2.14 b), we can write this more explicitly as

(2.20)

Q Qr ^ | ^|- * * d J k + e eR ff P L ^ ^ d A

(2.21)

p(x,y) = --Q 71

77

Jo

*

^

\Jo * —1

where Re denotes the real part. 2.3 Matching The first integral in (2.21) diverges logarithmically as kÔ. We should anticipate this long wavelength divergence; for the inner problem is an elliptic problem containing no information about the outer boundary, and hence no information about the finiteness of the domain. This information is carried by the outer solution, so the divergence should be resolved in matching the inner solution (2.21) to the outer solution (2.9 b). To match, note that (2.21) becomes

p(x,y)

n

Q\

——ê'^dk

Jo

when x2+y2pl.

(2.22 a)

*

Moreover, note that the outer solution (2.9 b) can be written exactly as Co log ccsE v V + / ) = lim - Co (2.22 b) where y = 0.5772... is Euler's constant. We observe that the inner and outer solutions match at every non-zero value of k if and only if Co = 2Q/n. This clearly identifies the divergent integral as the outer solution. Thus the inner solution can now be written as

p(x,y) = C0logasE V(* 2 +/) + C0Re( P ^ e - ' ^ d A

(2.23 a)

Evaluating (2.23 a) at x = y = 0 shows that we must select Co = [y + logO/a^)]- 1 = [y + logiRJfl]-1

(2.23 b)

so that p(0,0) = - 1 . 2-2

36


It can be readily verified that (2.23) indeed solves the inner problem (2.8) and matches the outer solution (2.9). Therefore, regardless of how it was obtained, (2.23) must indeed be the inner solution. Since (2.23) reduces to the outer solution as x2+y2 becomes large, (2.23) must also be the uniformly valid solution to (2.1). So in terms of the original variables (see (2.7)) the pressure is given uniformly by = C0 log (r/RD) + Co Re ( P VJoo pk — \ to leading order in a. From (1.11), this yields the leading order production rate F = — hK0 Co = — hK0[y + log {RJfiF1.

(2.25)

For later convenience we rewrite this as R u ] + E1 + . . ) j with

for a « l

(2.26 a)

£-x = y-log|7r = 0.12563....

(2.26b)

The only significant difference between the asymptotic formula (2.26) and the variational estimate (1.15 d) is the relatively small term Ev Clearly this term must represent the work spent moving fluid in the 6 direction. 2.4 Higher order effects Continuing the asymptotic expansion to higher order is very tedious. The variational principle provides a simpler way to obtain a more accurate formula for the production rate. Define

p-=P

+

af,

(2.27)

where pex is the exact pressure, p is the leading order asymptotic solution given by (2.24), and axfr is the error. The first variation &Fmust be zero at pex. So substituting the leading order solution into the variational principle should yield the production rate Fto within an O(a2) error. To see if this is so, we combine (2.27) with the variational principle (1.12 a). This yields (with RB = 0) F=-hl /•

\Jo Jo

Jo

/

(2.28) The leading order solution (2.24) satisfies V2/? = 0, and also satisfies fiprr + -pe = 0 at 0 = 0, pe = 0 at d = br. r

(2.29a)


37

Since pex = p = — 1 at r = 0, and since pex = 0 at r = Ru, we also have ^ = 0 at r = 0, p + 2 + 2af) !»-»• '-*> + O(a 2 )). (2.30) Moreover, (2.24) reduces to J when r/RE$>a.

(2.31a)

So, /?pr(i{E,0) = C o a + . . . ,

and pe(r,0) = CoxRK/r+...

for

r > *B.

(2.31b)

Inspecting (2.31b) shows that, as expected, we can neglect the unknown xjr in (2.30) until O(a2). Equations (2.30) and (2.31) now yield the production rate through O(a) as F = — h K 0 C 0 { l - 2 < x C 0 / n + O( = - l + C°Re{log(z + V ( z 2 - l ) ) + l * W ( z 2 - l ) + .-.}, with

1

C° = {log(2tf D / J R E )-i4 + ...}- .

(3.4a) (3.4b)

where sE is RE/RD as before. Thus the production rate is F = — hKa {log (RJRJ

+ log 2 - | 4 + ...}" 1

for

a£>l

(3.5)

to leading order in a. 1. Here the log 2 term represents the work done moving fluid in the ^-direction, and is not particularly small. Note that in writing (3.5) we have omitted all O(s%) and smaller terms. Even when the fracture extends across 80% of the reservoir (sE = 0.80), this omission results in less than a 1 % correction. As in §2.4, the variational principle can be exploited to obtain F through 0{oTl). Omitting the details, this yields for a > l , where

Ix = \TT-2 + 4/TT = 0.8440....

(3.6a) (3.6b)

Note the strong similarity with the small a formula (2.34). 3.2 Intermediate fracture conductivities

Since we have been unable to solve the boundary value problem when a, is 0(1), here we exploit the variational principle to estimate the production rate. Consider the class of trial functions ro p(r, 6) = A{r) + I B\r)cos 2*0, (3.7a)


39

where we restrict the coefficients Bk(r) to be linear for r < i?E: Bk(r) = ylcr/RE

for 0 < r < i?E.

(3.7 b)

This class of trial functions represents a compromise: a narrower class would yield simpler estimates, while a broader class would yield more accurate estimates. Substituting this class of trial functions into the variational principle (1.12) and minimizing is a routine, if tedious, calculation. After eliminating O(s™) terms for simplicity, it yields the estimated production rate

^W+JX^y-1. where

a = —--—-

(3.8a, (3.8 b)

and

^ = -^—=1.68035.... (3.8c) v —8 This estimate's accuracy can be gauged in the next section by comparing it to exact values of the production rate. 4 Numerical results. Conclusions

In this paper we set out to obtain simple production rate formulas which can be used to gauge the potential effectiveness of hydrofracturing a given reservoir. With RB = 0, this work led to the asymptotic formula

with

£ 1 = y-log|7r = 0.12563...,

(4.1b)

F ~ ^ ^ 0 | l o g ^ l + ^ | e ) + log2-y E -I 1 /a + ...}" 1 ,

(4.2a)

/ 1 = i7r-2 + 4/7r = 0.8440...,

(4.2 b)

for the regime a 1. We also obtained the variational estimate

with

a=

,

N

7 r ( a + AC)

with *, = - 5 ^ = 1.68035...,

(4.3b)

7 T 8

which should be valid for all a. Essentially exact values of the production rate have been obtained by using a finite element method to solve the variational principle numerically. These exact production rates are compared to formulas (4.1)—(4.3) in figures 4(a)-4(c). This comparison clearly establishes the accuracy of all three formulas. The precision of the asymptotic results when


40

I

I

I

I I I I III

001

|

I

.

|

I I I I I II

111

.11,

1 '

| | I II

10

1 a (a)

1

1 1 I t 1 1 11

100

1

1 1 1 1 1 11

•

-

1 1

6 — ~

I I I | | ||

01

I

D ID "E'" D

I

_ A Cfl U* JU

/-b v-a

-

\ v

—

-/_--^^"^-Exact

-

/

i

i

i

i

2 —

1 1 1 11 III

1

1 II 1

01

-

1

001

10 a (b)

FIGURE 4 (a, b). For legend see facing page.

100


41

.1 4

Mill

001

I

10

1

01

I I I II I

100

a (c) FIGURE 4. (a) The dimensionless production rate F/(hK0//i) for RJRD = 0.25. The solid curve shows the numerically-determined exact production rate, while curve a is the asymptotic formula for a P 1 (equation (4.2)), curve (b) is the asymptotic formula for a