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Abstract The existing methods for evaluating the well productivity index are based on solution of transient problems. One approach is to consider the single well problem in infinite domain and subsequently apply the method of images. This puts restrictions on the geometry of the well and of the drainage volume. Another approach is to solve the transient problem in the bounded domain for late times. While in this case restrictions on the well geometry are less severe, the shape of the drainage volume is still limited to the simplest ones. In addition, for the constant rate case highly accurate wellbore pressures, for the constant pressure case highly accurate wellbore rates are required and that puts an extreme computational burden on the semi-analytical or numerical methods involved. Even with the most powerful methods and hardware available, the calculation of the productivity index of directionally drilled and partially penetrating wells, especially in more complex drainage volumes is a formidable task.

An additional problem is that in general, the productivity index for a well produced under constant pressure condition is different, although very close, from the productivity index of a well produced with constant rate.

In this work we present a new technique to evaluate single well productivity indices both for constant pressure and for constant rate conditions. The approach is based on the solution of two steady state boundary value problems with constant pressure prescribed on the wellbore. The two productivity indices, (for constant rate and constant wellbore pressure, respectively) are then computed as integral characteristics of the solutions of the corresponding time dependent boundary value problem. The two productivity indices are computed independently.

The method can be applied to any geometry of the reservoir (both 2-D and 3-D, regular or irregular), and any

direction and length of penetration of the well. Designer wells (with a freely prescribed path) can be also considered.

Introduction We consider a bounded reservoir with no flow outer conditions. The fluid is single phase, slightly compressible. A well producing with either constant pressure or constant rate is characterized by the productivity index defined as [19]:

)()()()(

tptptqtPI

aw = ,.(1)

where q(t) is the production rate, pw(t) is the flowing bottomhole pressure and pa(t) is the average reservoir pressure. We are particularly interested in the stabilized (late time) value of the PI. For the constant production rate stabilization means that the difference of average and wellbore pressure (the denominator) becomes time invariant. This flow regime is called pseudo-steady state (PSS). In the case of constant wellbore pressure, both the numerator and denominator keep changing with time, but their ratio stabilizes, leading to the flow regime called boundary-dominated (BD).

For constant thickness reservoirs the PI is a product of the

factor B

kh2 (in addition to the thickness, only fluid and

porous media properties are involved) and a dimensionless factor J, called the dimensionless productivity index. The dimensionless productivity index depends on the well-reservoir geometry and on the type of the flow regime. The PSS and the BD productivity indices are traditionally estimated by the following equation:

srCe

AJ

wA

Dietz

+

=

24ln

21

1

...(2)

where is Eulers constant, A is the drainage area, rw is the wellbore radius and s is the skin factor. The shape factor, CA for computing the PSS PI can be obtained from [6]. To compute the BD PI, one can use shape factors presented in [11]. In practice, for simple polygonal domains with the dimensionless radius RD > 1000 the same shape factor values are used in evaluation of both PSS PI and BD PI. However, it

SPE 89935

Analytical Method of Evaluating Productivity Index for Constant Production Rate or Constant Wellbore Pressure A. Ibragimov1, D. Khalmanova1, P.P. Valko2 and J. Walton1, Texas A&M University, 1Department of Mathematics, 2Department of Petroleum Engineering

2 SPE 89935

is known that the BD PI of a well is, in general, different from the PSS PI. In particular, the empirical evidence from numerical calculations is that the PSS PI is always greater or equal to the BD PI [11].

The applicability of (2) is contingent on the method of images - a two-dimensional drainage area to which the method of images can be applied, must be of a shape, which, when translated infinitely many times in all directions, can cover the entire 2-D plane. Evaluation of the shape factors CA both for PSS PI and BD PI for the drainage area shapes not presented in the literature requires one to solve a transient problem of fluid flow in porous media.

The majority of solutions for evaluating the productivity index in three-dimensional problems, i.e. for directionally drilled wells, follow the same principle as the two-dimensional methods in that they are based on a semi-analytical solution for a particular case, from which one finds a convenient approximate formula which is then applied to similar reservoir/well configurations. The semi-analytical solution is often based on the superposition of analytical solutions for a transient problem in an unbounded reservoir. For the solution of the problem to be unique, additional assumptions must be made. Usually the restrictions are imposed on the distribution of the pressure on the wellbore. Under one such restriction, the wellbore pressure is assumed to be constant on the wellbore at any particular time instant. Under another restriction, the pressure flux is constant at a particular time instant.

The solution in a bounded reservoir is then expressed in terms of an infinite time dependent series, similarly to the technique used [16]. Then a comprehensive computing procedure is applied to determine the stabilized values of the time dependent series [14,17,19, 3]. Alternatively, numerical Laplce transform inversion can be also applied.

For a directionally drilled well, skin s in equation (2) is substituted by the geometric skin sg, which captures the effects associated with the deviation of the well from a fully penetrating vertical one. A vertical fully penetrating well corresponds to sg = 0. In most cases the methods for computing the productivity index of a deviated or horizontal well in a three dimensional reservoir are aimed at obtaining an appropriate value of a shape factor CA and geometric skin factor sg in equation (2). The effects of the geometry of the external boundaries and the relative position of the well in the reservoir are included in the shape factor CA [15].

As seen from this brief review, the existing methods and techniques of evaluation of the productivity index impose serious restrictions on the geometry of the reservoir. In particular, the vertical dimension of the reservoir has to be small in comparison to its lateral dimensions to allow one to neglect the flow in the vertical direction or include its effect in the geometrical skin, sg. Another restriction is due to the use of the method of images, which requires the drainage area shape to be convex and suitable for covering the whole plane when translated infinitely many times.

Of course, equation (2) is merely a formalism to present and retrieve precomputed values. In general, to evaluate a productivity index of a well, one needs to solve a transient problem on a time interval long enough for the PI to stabilize, which makes the approach computationally very demanding .

In this paper we present an alternative method of evaluating both PSS and BD productivity indices of a well. It is based on the solution of corresponding time independent (steady-state) boundary value problems as opposed to transient methods. The methodology presented here becomes particularly useful in evaluating productivity indices for a well of arbitrary configuration in a general three-dimensional flow situation, since the mentioned steady-state problems are solved in a finite domain corresponding to the actual reservoir shape. The productivity indices of a well (PSS and BD) are expressed analytically in terms of the solutions to the mentioned steady-state problem, which can be obtained numerically using well established finite difference or finite element software packages. The software package used in this work is FEMLAB [7].

Mathematical Statement of the Initial Boundary Value Problems. Model of the Productivity Index. Let a point in Rn be denoted by x = (x1,...,xn), n = 2,3. Let denote the reservoir with the exterior boundary e. Let w denote the boundary of corresponding to the wellbore. Let u(x,t) be the pressure at point x at time t in the reservoir.

Then u(x,t) satisfies the usual diffusivity equation

tuLu = (3)

Definition L in terms of the material properties of the medium and the flowing fluid is given in the nomenclature. The reservoir is bounded and isolated, therefore, the no-flow boundary condition is specified on e:

0=

e

uK .(4)

Depending on the regime of production, three types of boundary conditions will be considered on w:

constant rate of production:

=

w

qu ,G ...(5) constant wellbore pressure:

wuuw= ...(6)

constant wellbore pressure inside a thin-skin zone:

0)( 3 =

+

wwuu

u K .....(7)

The initial pressure distribution is a positive function on :

0)0,( uxu = ...(8) Here, -q is the constant volumetric rate of production, uw

(uw3 in Eq. 7) is the constant wellbore pressure and models the thin-skin effect. The relationship between and skin s will be analyzed in one of the subsequent sections.

SPE 89935 3

This leads to three initial boundary value problems (A.1)-(A.4), (A.17)-(A.20) and (A.32)-(A.35), describing the three basic production methods: constant rate, constant wellbore pressure and constant wellbore pressure with skin, respectively.

Questions regarding the uniqueness and existence of solutions of the posed initial boundary value problems are discussed in Appendix A.

Definition of the Diffusive Capacity Let u(x,t) be a solution [8] of the parabolic equation (2) with boundary condition (3) specified on e and either a), b) or c) on w. Let T>0 be such that u(x,t) > 0 for all x in and 0 < t < T. The Diffusive Capacity corresponding to the solution u(x,t) is the ratio:

= uu

u

tuJw

w K),( ...(9) Note that in the boundary condition (7) uw3 is a positive

constant denoting the average of the pressure on the wellbore (measured inside the wellbore). The corresponding diffusive capacity is defined as

= uu

u

tuJw

w

3

),( K (10) The mathematical concept diffusive capacity, J(u,t) has the

meaning of the dimensionless productivity index of the well. In the following two sections we present the values of the pseudo-steady-state and boundary-dominated productivity indices. To that end, we provide the conditions providing for J(u,t) to be constant with time and present the constant values of J(u,) for each of the problems (A.1)-(A.4) and (A.17)-(A.20), respectively.

Pseudo-Steady-State Productivity Index For short vertical and slanted wells the wellbore pressure can be assumed to be uniformly distributed on the wellbore at each moment of time. If a horizontal well is assumed to have an infinite conductivity, then the wellbore pressure is uniformly distributed on the wellbore. One should note that such argument fails for long slanted wells. In the Appendix A it is shown that if the PI of a well produced with constant rate is constant, that is, if the pseudo-steady-state is reached and pressure is uniformly distributed along the well, then the value of PSS PI is determined by the formula:

=dVu

VJ PSS1

,.(11)

where u1 is a solution of the following steady-state boundary value problem on :

,11 VLu = ....(12)

01 =

e

uG ,.....(13)

01 =wu ......(14) In addition, it is shown that u1 is, in some sense, uniquely defined by the value of the initial reserves (provided that the pseudo-steady state is reached).

Boundary-Dominated Productivity Index In Appendix A it is shown that to determine the value of the boundary-dominated productivity index, it is sufficient to solve the following steady-state eigenvalue problem:

,kkkL = .(15) 0=

e

k

G ,.......(16)

0=wk ...(17) Let 0 be the minimal eigenvalue of problem (15)-(17) and

0 the corresponding eigenfunction. Then the boundary-dominated productivity index is equal to:

VJ II 0= ,...(18) Both problems (12)-(14) and (15)-(17) are time

independent (steady-state), therefore the computation time is greatly reduced as compared to solving a transient problem. Productivity Index for 3D Problems A number of numerical computations were conducted for various well configurations in three-dimensional domains. Two domains were considered for the numerical study. Drainage volume D1 (a cylindrical domain) is shown on Fig. 1. A side view of the drainage volume D2 and its horizontal cross-section are shown in Fig. 2 and Fig. 3 respectively. The area of the horizontal cross-section of the domain D2 is equal to the domain of the horizontal cross-section of the cylinder D1 . The ratio of the radius of the outer cylinder to the wellbore radius is denoted by RD. For both D1 and D2, we set RD= 1000. The length c is chosen so that the area of the horizontal cross-section of D2 is 80% of the area of the full circle. Reservoir model D2 was chosen to illustrate the utility of the method for domains that are not appropriate for the method of images.

A number of well configurations were considered for both reservoir geometries D1 and D2. In each case, a well was modeled as a right circular cylinder. For domain D2, the direction of any well is such that the projection of the well on the top of the reservoir corresponded to the schematic configuration shown in Fig. 3. Then the vertical cross-section of the reservoir containing the well is a rectangle for both D1 and D2. Figs. 4,5,6,7 and 8 show such cross-section for every considered well configuration. In configurations (A), (B), (D)

4 SPE 89935

and (E), the center of the symmetry of the well coincides with the center of symmetry of the cross-section. In configuration (C) the well is drilled from the middle of the top side of the cross-section.

The reservoir is assumed isotropic and homogeneous, so that with proper non-dimensionalization, the volume parameter becomes V = h (R2D - 1) / 2 and the linear operator is the Laplacian, L = .

Directionally Drilled Wells. Fig. 9 illustrates the behavior of the pseudo steady state and boundary dominated indices for a directionally drilled fully penetrating well. The well passes through the center of symmetry of the reservoir cross-section for both domains. As expected, both JI and JII productivity indices increase with the length of penetration.

Fig. 10 shows how JI and JII change with direction of the well of the fixed length passing through the center of symmetry of the domain. In all cases, the penetration length of the well is equal to h so that for = 0, the vertical well fully penetrates the reservoir. The graphs of JI and JII as functions of the angle of the well direction, shown in Figs. 11 and 12 reveal that the optimal direction of a well of the fixed penetration length is not the vertical one. It is an indication of the effect of the vertical flow of fluid from the bottom of the reservoir toward the slanted well. Clearly, this effect is impossible to quantify by a reduced two dimensional problem for a fully penetrating vertical well. Horizontal Well. Due to large size of the reservoir, the PSS and BD PIs do not differ significantly from each other for the considered three dimensional domains and well configurations. Therefore, the numerical study for horizontal wells was restricted to the cylindrical domain D1.

Methods presented in [14,8] rely heavily on the assumption that the vertical dimension of the reservoir is small compared to the penetration length of the well. Moreover, as noted in [15], the precision of the evaluation of the productivity index for horizontal wells decreases drastically as the distance from the well to vertical boundaries of the reservoir becomes comparable to the distance to the top and/or the bottom of the reservoir, if the reduction to the two-dimensional problem is used. This subsection presents computational results for such settings when the assumption of the small reservoir thickness and the well being clearly inside the drainage area are relaxed.

The last setting considered is a horizontal well with configuration (E), located at distance d below the plane of symmetry of domain D1. The graphs of the computed pseudo steady state productivity index JI as a function of distance d from the center of the reservoir for various penetration lengths L are shown in Fig.14.

For all practical purposes, one can conclude that the optimal location of a horizontal well in a cylindrical reservoir D1 is in the horizontal plane of symmetry of the reservoir. Note that for long wells, however, the PSS PI slightly increases for small values of d. This may be an indication of interesting feature of the diffusive capacity as a geometrical characteristic defined through the first eigenvalue 0. The latter is sensitive to the location of the well relative to the planes and lines of symmetry of the domain. In three

dimensional domains, there are more than one such planes and lines of symmetry and, therefore, there may be several well configurations yielding maximal productivity index. Skin Factor for Boundary-Dominated Productivity Index te reservoir produced with a constant wellbore pressure from a well with a thin skin zone is modeled by the initial boundary-value problem (A.32) (A.35). To determine the time-invariant value of the productivity index, given by equation (A.39), one needs to solve the steady-state problem (A.36) (A.38).

Skin Factor in Circular Reservoir. Assume that the reservoir is homogenenous and isotropic and the well is fully penetrating and its cross-section is perfectly circular. Then the problem (A.32)-(A.35) can be reduced to a two-dimensional problem. After non-dimensionalization, the Sturm-Liouville problem (A.36)-(A.38) reduces to:

,

kkkrrr =

1

SPE 89935 5

Difference between PSS and BD PIs Property B.2 in Appendix B provides means to investigate the difference between JI and JII as a function of the shape of the exterior boundary of the domain. This difference is determined by the constant C, which, in its turn, is determined by the character of the distribution on of the function 0 and its corresponding eigenvalue 0. The first eigenpair of the problem (A.17)-(A.20) is directly linked to the symmetry and curvature of the exterior boundary. To illustrate the effect of the curvature and the symmetry of the exterior boundary, consider domains, shown in Fig. 15 and 16. Each of these domains has the interesting property that if we cut it into two, one part may have an unproportionally large portion of the perimeter while having a small portion of the area. Comprehensive mathematical description and numerous examples of domains of this type can be found in [17]. For either shape in Figs. 15-16, the domain parameters b and change so that the ratio of the area of the domain to the radius of the well is held constant and corresponds to RD= 1000. The circular well is located in the center of gravity. The symmetrical domain is presented in order to illustrate the importance of symmetry: the difference between JI and JII for a symmetrical domain is significantly less than for a nonsymmetrical domain with the same curvature of the exterior boundary. The values of the PSS and BD PIs as well as the difference between them are presented in Table 1. Several Important Examples of the Transient Productivity Index in Constant Wellbore Pressure Regime Property C.2 (see Appendix C) allows one to analyze the transient behavior of the productivity index of a well produced with a constant wellbore pressure. In this section we give several examples of the transient productivity index and their physical meaning.

Example 1. Suppose that a well is produced with constant

rate, the productivity index is constant and the well has infinite conductivity. Suppose further, that at some time t0 > 0, the production regime was changed to a constant wellbore pressure production. Now the initial condition (A.20) is given by the pressure distribution in at t0. Using integration by parts, one can show that for any k = 0,1..., dkck > 0 and property C.2 implies that J(u,t)JII. In other words, when the production regime changes from pseudo-steady-state to boundary-dominated, the productivity index monotonically decreases to the boundary-dominated PI.

Example 2. For the purpose of analysis it is frequently

assumed that at t = 0 the pressure in the reservoir is distributed uniformly, i.e., u0(x) = ui, where ui is a positive constant. Then ck=uidk and the productivity index is monotonically decreasing to the boundary-dominated PI.

It may seem that JII is the minimal value of the productivity index in constant wellbore pressure regime. As a counter example of the latter statement, consider the initial pressure distribution yielding the productivity index which is less than the boundary dominated PI.

Example 3. Let u0(x) = 1000(x)-30(x). Then the diffusive capacity J(u,t) < 0V.

Physically, this example may be interpreted as follows: assume that the reservoir has been depleted by a set of wells. Suppose that the old wells are shut down and a new well is drilled and produced. Then the productivity index of the new well will monotonically increase to the boundary-dominated productivity index value. References [1] Adams, R.A.,Sobolev Spaces, Academic Press, NY, 1975. [2] Blasingame, T.A., Doublet, L.E., Valko, P.P., Development and application of the multiwell productivity index (MPI), SPE Journal, v.5, No.1 (2000). [3] Cinco-Ley, H., Ramey, H.J., Jr., Miller, F.G., Pseudo-skin factors for partially penetrating directionally drilled wells, paper SPE 5589, presented at 50th Annual Fall Meeting, Dallas, Texas, September 28-October 1, 1975. [4] Cinco, H., Miller, F.G., Ramey, H.J., Jr., Unsteady-state pressure distribution created by a directionally drilled well, JPT, November 1975, pp. 1392--1400. [5] Courant, R., Hilbert, D., Methods of Mathematical Physics, Interscience Publisher, New York, 1953. [6] Dietz, D.N., Determination of average reservoir pressure from build-up Surveys, JPT, August 1965, pp. 955--959. [7] FEMLAB online support knowledge base URL: http://www.comsol.com/support/knowledgebase. [8] Friedman, A., Partial Differential Equations of Parabolic Type, Prentice Hall, New Jersey, 1964. [9] Goode, P.A., Kuchuk, F.J., Inflow performance of horizontal wells, SPE Reservoir Engineering, August 1991, pp. 319--323. [10] Hawkins, M., F., Jr., A note on the skin effect, Petr. Trans. AIME, 207, (1956), pp. 356--357. [11] Helmy, W. and Wattenbarger, R.A., New shape factors for wells produced at constant pressure, paper SPE 39970, presented at SPE Gas Technology Symposium, Calgary, Canada, March 15-18, 1998. [12] Hurst, W., Establishment of skin effect and its impediment to fluid flow into a well bore, Petroleum Engineer, 25, (1953), pp. B-6 B-16. [13] Hurst, W., Clark, J.D., Brauer, E.B., The skin effect in producing wells, JPT., November 1969, pp. 1483--1489. [14] Ibragimov, A.I., Baganova, M.N., Study of transient flow filtration towards a single horizontal well, in the book:

6 SPE 89935

Fundamental Bases of New Technologies in Oil and Gas Industry, Moscow, Nauka, (2000), pp.192-198. [15] Larsen, L., General productivity models for wells in homogeneous and layered Reservoirs, paper SPE 71613, presented at 2001 SPE Annual Conference and Exhibition, New Orleans, Lousiana, September 30-October 3, 2001. [16] Matthews, C.S. Brons, F. and Hazebroek, P., A method for determination of average pressure in a bounded reservoir, Trans. AIME, 201, (1954), pp. 182--191. [17] Maz'ya, V.G. Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997. [18] Pucknell, J.K, Clifford, P.J., Calculations of total skin factors, paper SPE 23100, presented at Offshore Europe Conference, Aberdeen, UK, September 3-6, 1991. [19] Raghavan, R., Well Test Analysis, Prentice Hall, New York, 1991 [20] Yildiz, T., Assessment of total skin factors in perforated wells, paper SPE 82249, presented at SPE European Formation Damage Conference, Hague, Netherlands, May 13-14, 2003. Nomenclature A - Dimensionless drainage area A(x) - Symmetric positive definite matrix, CA - Shape factor h m Reservoir thickness (constant) J - Dimensionless productivity index k m2 Permeability L - Linear elliptic operator,

Lu=div(A(x) grad u) p Pa Pressure PI m3/(s Pa) Productivity index q m3/s Production rate re m Outer radius rw m Wellbbore radius RD Dimensionless outer radius r - Radial coordinate s - Skin factor t - Time u - Dimensionless dependent variable V - Dimensionless volume of W - Surface area of the wellbore a - Coefficient, modeling skin - Boundary - Eigenvalue Pa s Fluid viscosity - Eigenfunction - Reservoir

Ku

- Co-normal derivative on , defined as:

if nK is outer normal on , then

nuAu KK = )(

v - Average of function v on

wv - Average of function v on w Subscripts III - Nonzero skin BD I - PSS II - BD e - Exterior k, i - Integer indices w - Well Superscripts - Corresponding to a value of the parameter Appendix A. PSS and BD Productivity Indices Pseudo-steady-state productivity index Let represent a bounded reservoir with a single well of an arbitrary configuration and u(x,t) be pressure of fluid at point x at time t > 0. Then the pressure in the reservoir produced with a constant rate satisfies the following problem:

,tuLu = .(A.1)

01 =

e

uG ,...(A.2)

=

w

qu ,G .....(A.3)

0)0,( uxu = .(A.4) Note that the integral boundary condition (A.3) makes the problem (A.1)-(A.4) ill-posed: there is an infinite number of solutions. However, it is possible to define such a class of solutions in which a solution is unique. Under the assumption of the infinite conductivity of the well, the wellbore pressure is modeled as uniformly distributed on the wellbore at each moment of time t, i.e., the wellbore pressure satisfies the following condition:

BtCuw

+= .(A.5) There is a unique solution of problem (A.1)-(A.4) that satisfies the condition (A.5).

Along with the problem (A.1)-(A.4) consider the steady-state boundary value problem:

,11 VLu = ...(A.6)

01 =

e

uG ,.(A.7)

SPE 89935 7

01 =wu .....(A.8) Sufficient condition. If (i) the initial pressure distribution u0 in problem (A.1)-(A.4) is equal to qu1, where u1 is the solution of the steady-state boundary value problem (A.6)-(A.8) and (ii) the solution u of (A.1)-(A.4) satisfies the condition (A.5), then the diffusive capacity J(u,t)=:JI is constant and determined by the following formula:

=dxu

VJ I1

,.....(A.9)

Proof. If the initial pressure distribution is equal to qu1 then u=qu1-qt/V is a solution. It is not hard to see that u satisfies the condition (A.5). Then, from the definition of the diffusive capacity (equation (8)), using the divergence theorem, one concludes that J(u,t) is constant and defined by JI in equation (A.9).

If the pseudo-steady state is reached, then the average wellbore pressure satisfies the following equation:

tVqCuw = ,..(A.10)

where C is a constant. Using the equation (6), the problem for pressure in the

reservoir produced with a constant rate from a well with infinite conductivity, can be recast as:

,tuLu = (A.11)

0=

e

uG ,..(A.12)

,tVqCu

w= ...(A.13)

0)0,( uxu = .(A.14) The solution of the problem (7)-(10) must satisfy the

condition:

=

w

qu ,G ....(A.15) Suppose that u(x,t) is a solution of (A.11)-(A.14) and, in

addition, u satisfies the condition (A.15). If the initial pressure distribution u0 is different from qu1, but the diffusive capacity is constant, can it be different from JI? The answer to this question is given by the following. Necessary condition. If the conditions hold:

(i) u is a solution of (A.11)-(A.15), (ii) u is a subject to the condition (A.15), (iii) J(u,t)=J(u) is independent of time,

then

=+ 0)( *10 Cdxquu ,.(A.16) where u1 is the solution of the problem (A.6)-(A.8) and C* is a constant independent of q and domain .

The latter assertion can be proved by using the maximum principle for parabolic equation. Boundary Dominated Productivity Index Let represent a bounded reservoir with a single well of an arbitrary configuration and u(x,t) be pressure of oil at point x at time t>0. Then the pressure in the reservoir produced with a constant wellbore pressure satisfies the following problem:

,tuLu = .(A.17)

0=

e

uG ,...(A.18)

wuuw= ..(A.19)

0)0,( uxu = ,.(A.20) where uw is a positive constant. There is a unique solution of problem (A.17)-(A.20).

Along with the problem (A.17)-(A.20) consider the related Sturm-Liouville problem (k = 1,2,.):

,kkkL = (A.21)

0=

e

k

G ,......(A.22)

0=wk ......(A.23) Let 0 be the minimal eigenvalue of problem (A.21)-(A.23) and 0 corresponding eigenfunction.

Sufficient condition. If the initial condition (A.8) u0 is equal to 0, then the diffusive capacity J(u,t)=:JII is constant and determined by the following formula:

.0VJ II = ,..(A.24) Proof. Let u2 be a solution of the problem:

,22 tuLu = (A.25)

02 =

e

uG ,..(A.26)

02 =wu ..(A.27) wuuxu = 02 )0,( ,.(A.28)

8 SPE 89935

Then u=u2+uw solves (A.17)-(A.20). The diffusive capacity for problem (A.17)-(A.20) J(u,t) reduces to:

.),(1

),(),(2

2

2

==dxtxu

V

dSu

tuJtuJ w K .(A.29)

If in (A.20) u0= 1 then the solution of the problem (A.25)-(A.28) is given by:

,)(),( 002textxu = ...(A.30)

and the diffusive capacity J(u2,t) is constant and determined by:

.),(: 02 VtuJJ II == .(A.31) In fact, the diffusive capacity J(u2,t) is constant provided that the initial distribution u2(x,0) is equal to any eigenfunction i, i = 1,2,.However, only the eigenfunction corresponding to the minimal eigenvalue 0 does not change sign on , therefore, in terms of the initial pressure distribution in the reservoir, 0 is the only physically realistic initial condition. Boundary-Dominated Productivity Index with Skin Factor Let represent a bounded reservoir with a single well of an arbitrary configuration and u(x,t) be pressure of fluid at point x at time t > 0. Then the pressure in the reservoir produced with a constant wellbore pressure from a well with a nonzero skin satisfies the following problem:

,tuLu = .(A.32)

0=

e

uG ,...(A.33)

0)( 3 =

+

wwuu

u K ...(A.34)

0)0,( uxu = ,.(A.35) where uw3 is a positive constant, denoting the constant wellbore pressure and is a given constant. There is a unique solution of problem (A.17)-(A.20).

Along with the problem (A.17)-(A.20) consider the related Sturm-Liouville problem ( k = 1,2,...):

, kkkL = ..(A.36)

0=

e

k

G ,.....(A.37)

0=

+

w

kk

K ....(A.38)

Let 0 be the minimal positive eigenvalue of problem (A.36)-(A.38) and 0 corresponding eigenfunction.

Sufficient condition. If the initial condition (A.8) u0 is equal to 0, then

the diffusive capacity J(u,t)=:JIII is constant and determined by the following formula:

.0VJ III= ,.(A.39)

Proof. Let u2 be a solution of the problem:

,33 tu

Lu = .....(A.40)

03 =

e

uG ,..........(A.41)

0)( 33 =+

w

uu K .........(A.42)

303 )0,( wuuxu = ,...(A.43) Then u=u3+uw3 solves (A.17)-(A.20). The diffusive capacity for problem (A.17)-(A.20) J(u,t) reduces to:

.),(1

),(),(3

3

3

==dxtxu

V

dSu

tuJtuJ w K .(A.44)

If in (A.20) u0= 0 then the solution of the problem (A.25)-(A.28) is given by:

,)(),( 002textxu

= .(A.45) and the diffusive capacity J(u3,t) is constant and determined by:

.),(:)( 03 VtuJJ III == ..(A.46)

When parameter in the boundary condition (A.34) is positive, the minimal eigenvalue 0 is positive and the corresponding eigenfunction 0 does not change sign on . When is negative, the minimal eigenvalue may be negative which would yield the negative value for the diffusive capacity. The latter is an indication of the injection into the well, therefore we restrict our attention to positive eigenvalues only.

SPE 89935 9

Appendix B. Comparison of the constant productivity indices

Boundary-Dominated PI vs. Boundary-Dominated PI with Skin Assume that the initial distributions in problems (A.1)-(A.4), (A.17)-(A.20) and (A.32)-(A.35) are such that the diffusive capacities for these problems (JI, JII and JIII(), respectively) are time invariant and given by equations (A.9), (A.31) and (A.46), respectively.

Let H1,2() be the usual Sobolev space [1]. Denote by H01,2(,w) the closure in the H1,2() norm of smooth functions that vanish on w, and by H01,2(,w,) the closure in the H1,2() norm of smooth functions such that

0)( =+

wuu K [1].

The following are well known variational principles yielding the first eigenvalues 0 and 0 of problems (A.21)-(A.23) and (A.36)-(A.38), respectively.

,inf2),(

0 2,10

=dxu

udxuA

eHu ..(B.1)

,

1

inf2

2

),,(0 2,1

0

+=

dxu

dSuudxuAw

eHu

.(B.2)

These two principles lead to the following Property B.1. When 0, JIII()JII. Pseudo-Steady-State PI vs. Boundary-Dominated PI Another important comparison can be made between the pseudo-steady-state and boundary-dominated productivity indices. Property B.2. If the solution of the problem (A.1)-(A.4) satisfies the condition (A.5) and the initial conditions in problems (A.1)-(A.4) and (A.17)-(A.20) are such that the corresponding diffusive capacities are time-independent, then

,IIIII JCJJ ...(B.3) where .00 /max =C Proof. Let u1 in H01,2(,w) be a solution of (A.6)-(A.8). We will show that

.1 01

dxu .(B.4) From (B.1) it follows that

.)()(

21

110

dxu

dxuuA (B.5)

Applying the divergence theorem to the denominator in (B.5) and making use of (A.6)-(A.8) we obtain:

.12

1

10

dxu

dxu

V .(B.6)

The latter can be rewritten as

.1)(1

12

1

21

0 dxudxu

dxu

V

.(B.7)

Inequality (B.4) now follows from Hlders inequality [1]. Let 0 be a solution of (A.21) - (A.23). After multiplication of both sides of (A.6) by 0, followed by integration over , using the symmetry of A, from the divergence theorem one concludes that

= dxVdxudxu 0010100 1max . The latter can be recast as the second part of the inequality (B.3), using the positivity of u1 and 0. Appendix C. Transient Productivity Indices

Case of Constant Flux on the Wellbore Consider solutions of problem (A.1)-(A.4) with the constant flux on the wellbore, i.e. let u be a solution of the following problem:

,tuLu = ...(C.1)

01 =

e

uG ,..(C.2)

Wqu =

K .....(C.3)

0)0,( uxu = (C.4) The solution to (C.1)-(C.4) is given (up to an additive constant) by u(x,t) = qv(x)-qt/V+h(x,t), where v(x) is a solution of the steady-state problem:

,1V

Lv = .(C.5)

0=

e

vG ,.....(C.6)

10 SPE 89935

Wv

w

1=

G ,..(C.7) and h(x,t) is a solution of the corresponding problem with homogeneous boundary conditions:

,thLh = ...(C.8)

0=

e

hG ,.....(C.9)

0=

w

hG ,...(C.10)

)()()0,( 0 xqvxuxh = (C.11) The solution to (C.8)-(C.11) is given by

=

=0

)(),(n

tnn

nexctxh , where n and n are solutions of the related Sturm-Liouville problem and cn are the coefficients of the Fourier expansion of h(x,0) in terms of n. The diffusive capacity J(u,t) is given by

+=

hhvvqtuJ

ww

),( .(C.12)

In equation (C.12) wv and v are constant and wh and h are functions of time. Clearly, the difference

=

=

0)(

n

tnnwnw

nechh converges to a constant as t, therefore, J(u,t) converges to a constant value J as t. However, J is not necessarily equal to JI.

Property C.1. JI is not a unique value of the PSS PI.

Constant Wellbore Pressure Case For simplicity consider the following problem

,tuLu = (C.13)

0=

e

uG ,..(C.14)

0=wu ...(C.15) 0)0,( uxu = ,(C.16)

Along with (C.13)-(C.16) consider related Sturm-Liouville problem

,kkkL = (C.17)

0=

e

k

G ,..(C.18)

0=wk ..(C.19) Denote dk=k(x)dx and ck=k(x)u0(x)dx. Then the diffusive capacity can be written as

=

=

=

0

0),(k

tkk

kt

kkk

k

k

edc

edcVtuJ

(C.20)

Property C.2. If for any k = 0,1..., dkck > 0 then J(u,t)JII.

SPE 89935 11

Shape JI JII II

III

JJJ

100

percent

0.0 0.1227 0.1065 4.69

0.4 0.0539 0.4370 23.34

0.6 0.0137 0.0100 37.00

See Fig. 15

0.8 0.0071 0.005 39.22

0.8 0.0990 0.1222 19.00 See. Fig. 16 0.95 0.056 0.0311 82.00

Table 1. Difference between PSS and BD PIs.

h

h

Fig.5. Schematic of the Well Configuration (B).

h

Fig. 4. Schematic of the Well Configuration (A).

h

RD

h

Fig. 1. Schematic of the Reservoir Model D1

c

Fig. 3 Schematic of the Horizontal Cross-section of Domain D2

Fig. 2. Schematic of the Reservoir Model D2

12 SPE 89935

0 10 20 30 40 50 60 70 800.16

0.165

0.17

0.175

0.18

0.185

0.19 h=100, RD=1000

PI

JIJII

0 10 20 30 40 50 60 700.152

0.154

0.156

0.158

0.16

0.162

0.164

0.166

0.168

0.17

h=100, RD=1000

PI

JIJII

h

h

Fig. 6. Schematic of the Well Configuration (C).

h

Ld

Fig. 8. Schematic of the Well Configuration (E).

hL

Fig. 7. Schematic of the Well Configuration (D).

0 10 20 30 40 50 60 70

0.2

0.25

0.3

0.35

0.4 h=100, RD=1000

PI

JIJIIJIJII

Domain D1

Domain D2

Fig. 10. Productivity Indices for Domain D1, Well Configuration (A).

Fig. 11. Productivity Indices for Domain D2, Well Configuration (C).

SPE 89935 13

0 20 40 60 80 100 120 140 160 180 2000.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

d

PI

h=500, RD=1000

L=1700

L=1500

L=1300

L=900

L=500

Fig.14. PSS PI For Domain D2, Well Configuration (E).

a0 x

y=b(1-(x/a)1-)

b

y

Fig. 16. Symmetrical Domain Violating Isoperimetric Inequality.

y

y=b(1-(x/a)1-)

b

a x0

Fig. 15. Domain Violiating Isoperimetric Inequality.

Fig. 12. Productivity Indices for Domain D1, Well Configuration (C).

0 10 20 30 40 50 60 70 800.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18 RD=1000

PI

JIJIIJIJII

Fig. 13. Productivity Indices for Domain D1, Well Configuration (D).

500 700 900 1100 1300 1500

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4 h=100, RD=1000

L

PI

JIJII

14 SPE 89935

Fig. 18. Skin s as a function of for RD=10000.

-100 -80 -60 -40 -20 0 20 40 60 80 100-20

0

20

40

60

80

100

s( )

RD=1000

Fig. 17. Skin s as a function of .

-100 -80 -60 -40 -20 0 20 40 60 80 100-20

0

20

40

60

80

100

s( )

RD=1000

Fig. 20. Eigenfunction for negative . RD=1000.

0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

1.5

2

2.5

r

0-10 (

r)

Fig.19. Eigenfunction for positive . RD=1000.

0 100 200 300 400 500 600 700 800 900 10000.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

r

010 (

r)