Dogru2010.pdf

Equivalent Wellblock Radius for Partially Perforated Vertical Wells—Part I: Anisotropic Reservoirs With Uniform Grids Ali H. Dogru, Saudi Arabian Oil Company

Summary The well index in a numerical reservoir simulator relates the flow rate to the difference between well flowing pressure and simulator gridblock pressure. The standard method for computing well index, Peaceman’s formula (Peaceman 1978), requires an equivalent wellblock radius at which the gridblock pressure is equal to the pressure from an analytical solution for steady-state single-phase flow. Although Peaceman’s formula is accurate for fully penetrating vertical wells, it fails to account for the effect of vertical flow in partially penetrating wells. In this paper, we present a new analytical expression for the equivalent wellblock radius in a homogeneous, anisotropic reservoir with a uniform square grid around the well path. The new equation has the same structure as Peaceman’s equation but adds one new parameter to account for partial penetration and for vertical flow. The new formula reduces to Peaceman’s formula when the well is fully penetrating. Model simulation study showed that the new method reduced the error in the calculated flow rates from 30% to less than 1% at minimal cost. Introduction In numerical reservoir simulators, the flow rate for a perforated gridblock for a uniform grid is calculated from the product of a well index, and a pressure difference between well pressure and wellblock pressure for the constant unit fluid viscosity is given by

(

)

qk = WI k Pk − Pwf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) In Eq. 1, WIk is the well or perforation index for the gridblock k and is defined by WI k =

2 K z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) ⎛r ⎞ ln ⎜ o ⎟ ⎝ rw ⎠

In Eq. 2, ro is called the equivalent wellblock radius (shown in Fig. 1) and is calculated using Peaceman’s formula (Peaceman 1978) for a square uniform grid: ro = xe

−

 2

≈ 0.2x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

In Eqs. 1 and 2, Pk is the pressure for the gridblock k, Pwf represents specified flowing bottomhole pressure, and z is the thickness of the gridblock. The other variables are defined in the nomenclature. We need to note that Eq. 3 was developed for a homogeneous, isotropic reservoir for a single well producing at the center of the gridblock. It is also assumed that there are no neighboring wells and that the well is located away from the boundaries. Eq. 1 is derived for radial flow with no vertical-flow effects. Therefore, using ro defined by Eq. 3 in Eq. 2 for the rate calculations

Copyright © 2010 Society of Petroleum Engineers This paper (SPE 137051) was revised for publication from paper SPE 118845, first presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2–4 February 2009. Original manuscript received for review 14 August 2009. Revised manuscript received for review 13 January 2010. Paper peer approved 28 January 2010.

1034

is expected to yield erroneous rates where vertical flow is important. The magnitude of the errors depends on the magnitude of the vertical flow based on the vertical permeability and location of the perforation. For high-permeability, thick reservoirs with only partial perforation of the formation, such as those in the Middle East, the effect of vertical flow on the perforation indices will be pronounced. Peaceman (1978) introduced the concept of equivalent wellblock radius as the radius at which the computed steady-state flowing pressure from a continuum model is equal to the calculated pressure for the wellblock. Peaceman’s derivation is based on the assumption of radial single-phase flow to a vertical well. The original formula was later extended to account for nonsquare (but uniform) gridblocks and anisotropic permeability (Peaceman 1983). Lin (1995) introduced a new method to calculate equivalent well radius for the partially penetrating wells that are heterogeneous reservoirs for isotropic, anisotropic, or 3D. His method agrees with the fine-grid results within 1% error. However, his method requires a fine-grid simulation run to generate the parameters used in the calculations. Ding (1995) and Muggeridge et al. (2002) addressed the problem of scaleup procedure in the vicinity of the wellbore, which would yield better values for the productivity indices. Ding (1995) implemented his technique for 2D, 3D, horizontal, and deviated wells. Wolfsteiner et al. (2000) developed a semianalytical method in which Green’s functions are used to solve singlephase-flow equations along the well for a homogeneous reservoir. Reference well rates are determined for each block, after which block pressures are computed using a single-phase reservoir flow simulator. The well index is then computed from the relationship between flow rate and block pressure. Aavatsmark and Klausen (2003) developed a technique for the calculation of well index for slanted and slightly deviated wells for 3D grids. In this paper, our objective is to obtain an approximate but simple formula and procedure similar to Eq. 3 for the effective wellblock radius and perforation index for vertical wells that are partially perforated. We aim to show that the new method is easy to implement in a reservoir simulator and is cost effective. Specifically, we target full-field simulation models using large gridblocks. It is expected that the new method of using large gridblocks would yield results closer to that of a fine-grid model in rate and pressure prediction around the wells. The new method is not exact, yet offers accuracy improvement at minimal cost. First, we will present a formula similar to Peaceman’s (Peaceman 1978, 1983) but with one additional parameter to be computed for the equivalent wellblock radius. Initially, we will present the formula for a single perforation and later for multiple perforations. Derivations will be made for the homogeneous, anisotropic formation with square uniform grids. Nonsquare grids will follow. Then, we will develop a simple numerical method for calculating the equivalent well radius and the well index for a heterogeneous and anisotropic reservoir with square and nonsquare blocks. A sector from a full filed simulation model will be used to demonstrate the new method. All the derivations are placed in Appendix A for convenience. Rate Equation for a Partially Penetrating Vertical Well We begin by deriving a simple formula for the well index that is suitable for partially perforated vertical wells, as shown in Fig 2. December 2010 SPE Journal

y P

q

Pk

Δy

ro

kz

x

hp kr H

∆x

Δx Fig. 1—Definition of wellblock radius for horizontal flow (Peaceman 1978). rw

The derivation is similar to Peaceman’s (Peaceman 1978) but requires a new steady-state flow-rate equation that accounts for the effects of flow restriction around the wellbore and vertical-flow effects instead of the radial flow (Eq. 1) used by Peaceman (1978). Two basic forms of the rate equation can be used for partially penetrating wells. We will start with a steady-state-flow formula suggested by Kozeny (1953) and Craft and Hawkins (1959):

(

)

⎡ ⎛  h p ⎞ ⎤  e −  wf r q = 2 Kh p ⎢1 + 7 w cos ⎜ . . . . . . . . . . (4) ⎥ ⎝ 2 H ⎟⎠ ⎦⎥ 2h p ⎛ re ⎞ ⎢⎣ ln ⎜ ⎟ ⎝ rw ⎠ In this expression, q is the flow rate, the potential (datum corrected) at re is e, the potential at the wellbore is wf , hp is the length of open perforated interval, and H is the total formation thickness. The fluid potential  is defined by  = p −  gz , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) where  represents the average fluid density, g is the gravitational constant, and z is the vertical distance from a reference depth, which is usually the top of the formation. Muskat (1937, 1949) also studied the steady-state flow-rate model for partially penetrating wells. In this paper, we will use a more general equation for a steadystate-flow model that includes skin factor: q = 2 KH

(

e

−  wf

).

⎛r ⎞ ln ⎜ e ⎟ + Sc ⎝ rw ⎠

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

In Eq. 6, Sc is called completion (perforation) skin factor. Completion skin factor can be calculated from well tests or formulas. Several authors have suggested different formulas for the completion skin. We will use skin factors suggested by Odeh (1980) and Brons and Marting (1961). The expression for flow from restriction can have a more complicated form than Eqs. 4 and 6; however, our objective is to provide a reasonable approximation that can correct the radial-flow model presented by Peaceman (Eq. 1). Comparing the flow-rate equation proposed by Kozeny (1953) and Craft and Hawkins (1959) (Eq. 4) with Eq. 6, we see that Eq. 6 includes the effect of vertical permeability Kz through the skin factor expressed by Odeh (1980) and Brons and Marting (1961) . In this paper, for convenience, we will use the form suggested by Eq. 4 by introducing a new parameter called fp, the partialpenetration factor. With the new parameter, Eq. 4 becomes q = 2 Kh p f p

(

e

−  wf

⎛r ⎞ ln ⎜ e ⎟ ⎝ rw ⎠

December 2010 SPE Journal

).

. . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

re

z Fig. 2—Partially-penetrating-well model.

The term hp fp in Eq. 7 represents the effective formation thickness, the portion of the total formation thickness H that contributes to flow. Using the Kozeny/Craft-Hawkins (Kozeny 1953; Craft and Hawkins 1959) model, the partial-penetration factor fp becomes fP = 1 + 7

⎛  hp ⎞ rw cos ⎜ ⎟ . . . . . . . . . . . . . . . . . . . . . . . . . . (8) 2h p ⎝ 2 H⎠

Because the skin factor Sc can be obtained from a well test or can be estimated by formulas (Odeh 1980; Brons and Marting 1961), Eq. 6 can also be written in the form of Eq. 7 by simply equating Eq. 6 to Eq. 7 and solving for fp: fp =

H ln ( re / rw ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) h p ln ( re / rw ) + Sc

In Eqs. 8 and 9, it is seen that, for a fully penetrating well, hp = H, Sc = 0, and, thus, fp = 1. Therefore, fp varies between 1 and a larger positive number b, 1 ≤ fp ≤ b, where b assumes values based on parameters in Eq. 8 and the value of skin factor obtained from the well test or by formulas (Odeh 1980; Brons and Marting 1961). Implementation in a Reservoir Simulator. For a gridblock k, which contains a perforation of a partially penetrating vertical well, we can write the rate equation using the form suggested in Eq. 7 rather than Eq. 1: qk = 2 K z k f

( pk

k

−  wf

).

⎛r ⎞ ln ⎜ o ,3 D ,k ⎟ ⎝ rw ⎠

. . . . . . . . . . . . . . . . . . . . . . . (10)

In Eq. 10, zk is the thickness of the gridblock containing the open perforation and fpk is the partial-penetration factor for the same gridblock, which can be calculated by Eq. 8 or Eq. 9. The term k in Eq. 10, fluid potential for gridblock k, is calculated by the simulator, and ro, 3D, k is the unknown equivalent wellblock radius on which k resides. The flow rate qk, calculated by Eq. 10, assumes that completion k, in addition to horizontal flow, receives vertical flow from above and below. Therefore, the term zk fpk is the effective gridblock thickness that accounts for both horizontal and vertical flow; hence, zk fpk > zk for partially penetrating well with fpk > 1. Rearranging Eq. 10, the new well index or perforation index for a vertical well becomes 1035

y ΦB

Δy

Z=0

k=1 k=2

x ΦB

ΦB k-1

x

Φk

qk

k

Δx

Δz

k+1

Kz

ΦB

KX

Δx

ΦB

k=N-1

Fig. 3—Constant potential boundary conditions in areal direction.

k=N z

Δx

WI =

2 K z k f pk , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11) ⎛ ro ,3 D ,k ⎞ ln ⎜ ⎝ rw ⎟⎠

where ro,3D,k is still unknown and will be developed in the following subsection. Calculation of Equivalent Wellblock Radius. In this subsection, we will define the equivalent wellblock radius using a square uniform gridblock in the areal directions and uniform grid in the vertical direction for an anisotropic reservoir. Consider a portion of a reservoir where a well is located at the center of the 3D box, as shown in Figs. 3 and 4. Let this portion of the reservoir be divided into a finite difference grid with x = y in the areal directions. The reservoir thickness H can be divided into Nz vertical layers with constant thickness z. We assume, also, that Kx and Ky are constants and are equal and that Kz is constant but may not be equal to Kx (so Kx = Ky ≠ Kz). Porosity is assumed to be constant. Because horizontal grid scales in reservoir models are typically much larger than vertical scales, it is common for flow to be primarily 2D, even at the column of gridblocks neighboring the wellblocks. When this is the case, it is reasonable to assume that the reservoir potential is uniform in the vertical direction at neighboring gridblocks while still allowing for vertical flow in the column of wellblocks. In addition, we assume that the time dependence of potential can be neglected. This assumption is generally realistic for field-scale simulation models because the potentials in the neighboring cells do not vary significantly over the timestep, which is usually a couple of weeks to a month. Also, generally, wells reach pseudosteady state after the initial transient portion (fluid expansion) of the production. The transient portion is usually very short compared with the long production lives of the reservoirs. This assumption is easily realized for high-permeability, thick, and large oil reservoirs. We assume then, that the cells neighboring the well cells have constant potential B. The equivalent wellblock radius (Fig. 5) for 2D (horizontal and vertical) flow for the cell k is derived by writing a steady-state volume-balance equation for the cell k. Assuming constant fluid viscosity with unit value (1 cp), volume balance for the cell k becomes

Fig. 4—Constant potential boundary, vertical planes.

where qh ,k = 4Tx (  B −  k ) and qv ,k = Tz (  k −1 −  k ) + Tz (  k +1 −  k ). . . . . . . . . . . . . . . . . (13b) To determine the equivalent wellblock radius ro,3D,k for the gridblock k (Fig. 5), using the flow-rate definition suggested by Eq. 7, we can write the total flow rate into the gridblock k in the following form: qk = 2 K z f p ,k

( B −  k ) . ln

. . . . . . . . . . . . . . . . . . . . . . . . (14)

x

ro ,3 D ,k

Next, we solve for (  B −  k ) from Eq. 14 and substitute into Eq. 12. The following equation is obtained: ⎛ x ⎞  ⎛ qv , k ⎞ ln ⎜ ⎟ = f p ,k ⎜ 1 − q ⎟ . . . . . . . . . . . . . . . . . . . . . . . (15) ⎝ ⎝ ro ,3 D ,k ⎠ 2 k ⎠ The equivalent wellblock radius for the partially penetrating wells can be solved from Eq. 15: ⎡  f ⎛ q ⎞⎤ ro ,3 D ,k = x exp ⎢ − p ,k ⎜ 1 − v ,k ⎟ ⎥. . . . . . . . . . . . . . . . . . (16a) qk ⎠ ⎦ 2 ⎝ ⎣ y ΦB

Δy

ΦK

r o,3D

x

4Tx (  B −  k ) + Tz (  k −1 −  k ) + Tz (  k +1 −  k ) = qk. . . . . . (12) The first term in Eq. 12 represents the total horizontal flow rate qh,k. The remaining terms in Eq. 12 describe the total vertical-flow rate qv,k into the gridblock, and qk is the total withdrawal rate form the perforation. Transmissibilities Tx and Tz are defined in the Nomenclature. Eq. 12 can be abbreviated as qh ,k + qv ,k = qk, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13a) 1036

∆x

Δx Fig. 5—Definition of wellblock radius for the vertical-flow effects, areal view. December 2010 SPE Journal

The vertical-flow rate qv,k, defined by Eq. 13b, is obtained by solving a vertical 1D potential equation analytically for (zk). This derivation is placed in Appendix A. Eq. 16 with Eq. 11 completely defines the new perforation (well) index for any perforation for a partially penetrating vertical well. It is important to note that the new equivalent wellblock radius described in Eq. 16a reduces to Peaceman’s formula (Peaceman 1978) (Eq. 3) if the vertical-flow rate is zero (qv = 0). For this case, the partial-penetration factor fp,k = 1, and, hence, Eq. 16 becomes ⎛ ⎞ ro ,3 D ,k = x exp ⎜ − ⎟ ≈ 0.2x . Further examination of Eq. 16 sug⎝ 2⎠ ⎛ q ⎞ gests that the term ⎜ 1− v ,k ⎟ is a correction to fp,k; hence, we can qk ⎠ ⎝ introduce ⎛ q ⎞ f p ,k = f p ,k ⎜ 1 − v ,k ⎟ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16b) qk ⎠ ⎝ and ⎛ f ⎞ ro ,3 D ,k = x exp ⎜ − p ,k ⎟ . . . . . . . . . . . . . . . . . . . . . . . . . (16c). 2 ⎠ ⎝ ⎛ q ⎞ The correction term ⎜ 1− v ,k ⎟ in Eq. 16b varies between unity (no qk ⎠ ⎝ ⎛ q ⎞ vertical flow) and zero (purely vertical flow) [i.e., 0 ≤ ⎜ 1 − v ,k ⎟ ≤ 1]. qk ⎠ ⎝ Similarly, the new modified partial penetration factor f p ,k varies between zero and unity (i.e., 0 ≤ f p ,k ≤ 1 with 1 ≤ f p ,k ≤ b, where b is a real constant). Effect of Partial Penetration on the Equivalent Well Radius. As discussed earlier, the equivalent wellblock radius for horizontal flow with square grids is approximately equal to 0.2x. For partially penetrating wells, this is no longer true. Fig. 6 shows a numerical experiment carried out using a square grid. The model was assumed to have three equal-sized grids in the x direction and

200

three equal-size grids in the y direction, with x = y = 820 ft. In the vertical direction, the reservoir is 100 ft thick (H = 100 ft) and is divided into five equal-thickness layers (z = 20 ft). Reservoir permeability was set to 100 md in all directions. A vertical well was placed in the center of the central gridblock. Reservoir gridblocks surrounding the central cell were assumed to have constant potential b = 3,000 psi and constant flowing well potential wf = 1,000 psi. Two cases were considered—a fully penetrating well (1) with all five layers open to flow and (2) with only the top layer open to flow. For both cases, dimensionless potential drop D = 2 K z f p ,k  k −  wf r where r ∈( rw ,x ) was plotted against q x with at r = x and at r = rw. Here, k is the potential at the top layer of the central cell. The values of k for all the gridblocks were obtained from a numerical solution of a 1D potential equation (see Appendix A). Results of the numerical experimentation are shown in Fig. 6. As seen, for a fully penetrating well (blue line), the dimensionless-pressure-drop line intersects the zero line (D = 0) at r/x = 0.2, indicating that flow is parallel and radial (ro = 0.2x, ro = 164 ft). On the other hand, the partial-penetration case (red) shows that the pressure-drop line intersects the horizontal line where D = 0 at r/x = 0.6, ro = 504 ft. Similarly, variation of the corrected partial-penetration factor f p ,k is shown to change from 1.0 for a fully penetrating well to 0.3 for a partially penetrating vertical well with hp/H = 0.2. For this case, perforation (well) index calculated by the new method was 1.5 times the Peaceman well (perforation) index.

(

)

Vertical Profile for Equivalent Wellblock Radius. Equivalent wellblock radius for a partially penetrating well can show variation in the vertical direction, depending on the formation properties and discretization. For good vertical permeability and typical vertical gridblock sizes, this may be insignificant. Fig. 7 shows the variation of equivalent wellblock radius with vertical distance and comparison to Peaceman’s 2D equivalent well radius. In Fig. 7, the same reservoir model used in the previous example was used. This time, the reservoir was divided into 11

H=100 ft , kx =kz =100 md Dx=820 ft, Dz=20 ft, b=3,000 psi Pwf =1,000 psi

100

Partial Penetration: Top Completion open

2 π K Δz fp(Pk –Pw)/q

f p ,k= 1 0

f p , k = 0.3

Fully Penetrating Well

-100

Partially Penetrating Well

-200

ro,2D

ro,3D

-300 0.0001

rw /Δ x

0.001

0.01

r /Δ x

0.1

0.2

0.6

1.0

re / Δx

Fig. 6—Effect of partial penetration on equivalent wellblock radius. December 2010 SPE Journal

1037

equal-thickness layers (9.09 ft) and the well was assumed to be completed in Layers 4, 5, 6, 7, and 8, which were open to flow. The flow-rate profile is also shown in the figure (edge perforations get more flow than the middle ones). As seen, the Peaceman well radius is approximately 0.2x, whereas the new well radius averages approximately 0.40x, showing slight variation in the vertical direction (higher at the edges; this is exaggerated in the figure to display the concept). For this case, the new method calculated the well index to be 1.15 times the Peaceman well index. Analytical Expressions for the Equivalent Wellblock Radius. In the next two subsections, we will present analytical expressions for the equivalent wellblock radius for a single perforation located at the top of the formation or located at any point except the upper and lower boundary for the uniform square grids. Analytical expression for a single perforation completed at the bottom of the formation can be determined easily from the solution presented for the top of the formation. Single Perforation Located at the Top of the Formation. Using the analytical solution for the fluid potentials (Appendix A), we ⎛ q ⎞ can express the ⎜ 1− v ,k ⎟ term in simple analytical functions that q ⎠ ⎝ k

are easy to compute. By substituting the analytical solution in Eq. 16, we obtain the formula for the equivalent wellblock radius for perforation k = 1. ⎤ ⎡  f p ,k ⎥ ⎢− 2 ⎥ ⎢ ⎫⎞ ⎥ ⎢ ⎛ ⎧ cosh a ⎪⎟ ⎥ . ro ,3 D ,1 = x exp ⎢ ⎜ ⎪ a z ⎢ ⎜ ⎪ ⎪⎟ ⎥ ⎢× ⎜ 1 − ⎨ ⎬⎟ ⎥ ⎡ ⎤ − − h a H − z cos H z ( ) ( ) 2 1 ⎦ ⎪⎟ ⎥ ⎢ ⎜ ⎪ ⎣ ⎢ ⎜ ⎪ ⎪⎟ ⎥ sinh aH ⎢⎣ ⎝ ⎩ ⎭⎠ ⎥⎦ . . . . . . . . . . . . . . . . . . . . . . (17a)

( )

(

WI k =

2 K z ⎛r ⎞ ln ⎜ o ,3 D ,k ⎟ r ⎝ w ⎠

⎡ r  z ⎞ ⎤ ⎢1 + 7 w cos ⎛⎜ ⎥. . . . . . . . . . . (17b) ⎝ 2 H ⎟⎠ ⎥ 2z ⎢ ⎣ ⎦

In Eq. 17a, z1 is depth of the perforation and z2 is the depth of the adjacent gridblock in the vertical direction. Kx The parameter a for a square grid is defined as a = 2 . K z x 2 For high vertical permeability and top completions, the parameter fp,k is calculated from the skin factor obtained from a well test or by the formula of Kozeny/Craft-Hawkins (Kozeny 1953; Craft and Hawkins 1959). If the Brons and Marting method (Brons and Marting 1961) is chosen for the calculation of the skin factor, a simple iteration may be needed because the formula contains weak dependency on the external radius (in this case, the equivalent wellblock radius). For more-general cases, historical production rate of a well can be used to determine the type of the correlation suitable for the particular well. In this case, the well rate calculated by the new method can be compared against the measured rate by changing the correlations. The correlation that yields the closest match is chosen. Single Perforation Located at Any Point Inside the Formation, 0 < zs < H. Using the analytical solution presented in Appendix A, we obtain an expression for the equivalent wellblock radius, zk = zs: ⎡ ⎛ ⎧ cosh ⎡ a ( z k −1 ) ⎤ ⎫ ⎞ ⎤ ⎣ ⎦⎪ ⎟⎥ ⎢ ⎪ ⎜ ⎢ ⎪ ⎟⎥ ⎪ ⎜ f − cosh az k ⎪ ⎟⎥ ⎢ ⎜ 1 − L ⎪⎨ ⎬ ⎢ z ⎪ asinh az s ⎪ ⎟ ⎥ ⎜ ⎥ ⎢ ⎜ ⎪ ⎟⎥ ⎪ ⎢ ⎟ ⎜ ⎪⎭ ⎥ ⎪⎩ f ⎟ , = x exp ⎢⎢ − p ,k ⎜ 2 ⎜ ⎧ cosh ⎡ a ( H − z k +1 ) ⎤ ⎫⎟ ⎥ ⎢ ⎣ ⎦ ⎪⎟ ⎥ ⎜ ⎪ ⎥ ⎢ ⎜ ⎪ ⎡ ⎤ ⎪⎟ ⎢ ⎜ − f R ⎪ − cosh ⎣ a ( H − z k ) ⎦ ⎪⎟ ⎥ ⎢ ⎜ z ⎨ asinh ⎡ a ( H − z ) ⎤ ⎬⎟ ⎥ ⎪ s ⎦⎪ ⎥ ⎢ ⎣ ⎟ ⎜ ⎪ ⎪⎟ ⎥ ⎢ ⎜ ⎪ ⎪ ⎢⎣ ⎝ ⎩ ⎭⎠ ⎥⎦ . . . . . . . . . . . . . . . . . . . . . . (18a)

(

ro ,3 D ,k

)

The perforation (well) index becomes

(

)

)

ro/Δx 1 2 3 4

New Method

5 6

Peaceman's Method

7 8 9 10 11

Fig. 7—Vertical variation of equivalent well radius. 1038

December 2010 SPE Journal

k=1

q1

x

Pressure 4760 4770 4780 4790

k=2 q k-1 qk Δz z

q k+1

q Nc

k=N-1 k=N Δx

z

Fig. 8—Multiple well perforations.

where zs is the depth of the perforation. The parameters fL and fR are defined by fL =

( az ) cosh ⎡⎣ a ( H − z )⎤⎦ . . . . . . . . . . . . . . . (18b) ⎧sinh ( az ) cosh ⎡ a ( H − z ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎪⎩+ cosh ( az ) sinh ⎡⎣ a ( H − z ) ⎤⎦ ⎪⎭ sinh

s

Fig. 9—Coarse-grid model with well located at the center.

s

s

s

s

s

Nc

(

 t ( z ) = ∑  z ≤ z j j =1

and

(

)

az s cosh ⎡⎣ a ( H − z s ) ⎤⎦ fR = . . . . . . . . . . . . . . (18c) ⎧sinh az s cosh ⎡ a ( H − z s ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎪⎩+ cosh az s sinh ⎡⎣ a ( H − z s ) ⎤⎦ ⎪⎭ cosh

(

(

)

)

Partial-penetration factor fp,k can be determined by the methods described earlier in this paper. Our suggestion is to use Odeh’s method (Odeh 1980) for perforations not located at the top or bottom because it yields results closer to the fine-grid numerical solution. To use the Odeh (1980) or the Brons and Marting (1961) method, we set re = x in Eq. 9. ⎛ x ⎞ ln ⎜ ⎟ ⎝ rw ⎠ H fp = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19) z ⎛ x ⎞ ln ⎜ ⎟ + Sc ⎝ rw ⎠ Sc in Eq. 19 is defined by formulas (Craft and Hawkins 1959; Odeh 1980; Brons and Marting 1961). The perforation index WIk is calculated by Eq. 11. For a general case, the measured rate (well test) can be used to select the best correlation for this well by matching the calculated rate by this method with the measured rate by changing the correlation. Multiple Perforations. The analytical solution for the potential equation can be obtained by superposing the analytical solution for each perforation (Fig. 8). Let t(z) represent the total potential drop at location z from multiple sources located at the well qj, j = 1, Nc, where Nc is the total number of completions. By superposing the potential drops, Nc

(

)

 t ( z ) = ∑  z , z j , . . . . . . . . . . . . . . . . . . . . . . . . . . . (20) j =1

where December 2010 SPE Journal

×

(

⎪sinh ⎨ ⎪⎩+ cosh

( az ) aK x ysinh (

(

)

(

az j cosh ⎡⎣ a H − z j az j cosh ⎡⎣ a H − z j ⎤⎦

(

)

(

)

)⎤⎦

⎫ ⎪ ⎬ ⎡ ⎤ az j sinh ⎣ a H − z j ⎦ ⎪ ⎭

)

(

)

cosh

⎡ ⎣

z

(

{

)

az j ⎤⎦

(

)

(

)

q j sinh ⎡⎣ a H − z j ⎤⎦ cosh az j ⎡ ⎤⎫ ⎪ sinh az j cossh ⎣ a H − z j ⎦ ⎪ ⎨ ⎬ + cosh az j sinh ⎡⎣ a H − z j ⎤⎦ ⎪ ⎩⎪ ⎭ cosh ⎡⎣ a ( H − z ) ⎤⎦ . aK z x ysinh ⎡⎣ a H − z j ⎤⎦ . . . . . . . . . . . . . . . . . . . . . . . (21)

+ z > z j

×

q j sinh

)⎧

)⎧

( (

(

) )

( (

) )

)}

The equivalent wellblock radius ro,3D,k is obtained from Eq. A-24 q by evaluating the term v ,k as defined by Eqs. A-27 and A-28 in qk Appendix A. The  in Eq. 21 is the Dirac delta function, which assumes the value of unity if the argument is true, otherwise zero. Numerical Experiments and Verification of the New Equivalent Wellblock Radius. A 3×3×11 test model (Fig. 9) was used to test the new method and compare with Peaceman’s method. A vertical well was placed in the center of the central gridblock. Gridblocks surrounding the central well were assumed to have constant potential, 4,780 psi. In the areal plane, square grids were used with 820-ft sides. Vertical layer thickness was assumed to be constant for all of the 11 layers (9.09 ft). Two completion scenarios were considered—(1) completion interval at the top of the formation, Layers 1 and 2, and (2) completion interval at the middle of the formation, Layers 5 and 6. The flowing bottomhole pressure was set to 4,424 psi. Reservoir oil viscosity was 3 cp. Reservoir permeability was set to 500 md in x and y directions. The vertical permeability was 250 md. A simulator (Dogru et al. 2002) was run to predict the oil rate for three cases: • Base case, coarse grid using Peaceman’s well index (Fig. 9) • Locally refined grids for the central cell where the well is located (81×81×11 grid with x = 10.12 ft) using Peaceman’s well index (Figs. 10 and 11) 1039

Fig. 10—Locally refined areal grid.

• New method, coarse grid with new well index (Fig. 9) For the top-completion-interval scenario (1), model runs showed that oil-production rate predicted by the coarse grid, Case 1, was 22% less than the oil rate calculated by locally refined grids, Case 2. Coarse grid with new well index, Case 3, yielded rates very close to the locally refined case, within 0.6% difference. For this scenario, the new well index was 1.31 times the Peaceman’s well index. We have used Muskat’s method to calculate the skin factor. For the middle-completion-interval scenario (2), Layers 5 and 6, the completed interval receives flow from the top and the bottom of the completion interval, and, hence, the effect of vertical flow is more pronounced. Model runs showed that the difference between Case 1 and Case 2 was approximately 30% (29.5%). Case 3 with the new well index resulted in oil rate very close to that of Case 2, with a difference of 0.7%. The new method well index was 1.43 times the Peaceman’s well index. We have used the Odeh (1980) method to calculate the completion skin factor. Conclusions • A new practical formula and procedure have been developed for the equivalent wellblock radius and perforation index for partially penetrating vertical wells for square areal grids. The new formula is easy to implement in a numerical reservoir simulator. Analytical solutions developed in this paper can be used for homogeneous and anisotropic reservoirs with uniform grids. • The theory presented here is valid only if the potential around the wellblocks remains constant and approximately steady-state flow conditions exist. Well indices calculated by this method can be repeated at every timestep during the simulation or once at the beginning of the simulation, depending on the conditions. Constant-potential approach for the neighboring gridblocks is generally justifiable for large, high-permeability, thick reservoirs for full-field-simulation models. • New formulas for the equivalent wellblock radius and well indices depend only on the grid properties and not on the well boundary condition (specified flow rate or bottomhole pressure). • Numerical experiments have shown that Peaceman’s method (Peaceman 1978, 1983) underpredicts the well index, while the new well index can correct the error in flow rates. For a model problem, the new method reduced the error from 30% to less than 1%. An alternative to the new method is to use locally refined grids around the wells. Depending on the degree of refinement, this approach can be very costly for full-field-simulation models with many wells. • Accuracy of equivalent wellblock radius and perforation indices calculated by the new method depends on the accuracy of the rate formula. If the perforation skin is used in the rate formula, it needs to be calculated accurately. It can be estimated from a well test or from correlations. In either case, it is advisable to calibrate the new rate equation used to existing historical production data. Alternatively, published formulas for the skin factor can be used. • The new method can be expanded to horizontal wells or multilateral wells by following the same methodology of this paper. 1040

Fig. 11—Locally refined vertical grid.

Nomenclature fp = partial-penetration factor H = total formation thickness, cm k = gridblock index Kx = permeability in x direction, darcies Ky = permeability in y direction, darcies Kz = permeability in z direction, darcies N = number of gridblocks P = pressure, atm q = production rate, cm3/s re = drainage radius, cm ro = wellblock radius, cm rw = well radius, cm T = transmissibility Tx = Kxyz/x Ty = Kyxz/y Tz = Kzyx/z WI = well index  = potential, atm Subscripts B = boundary c = completion e = external (drainage) f = flowing or fraction j = completion number k = cell number (index) L = left p = perforated or partial t = total w = well wf = flowing well x = areal x direction y = areal y direction z = vertical direction Acknowledgments The author would like to thank Hussein Kazemi and Dean Oliver for valuable discussions and suggestions. The author would also like to thank Jorge Pita and Larry Fung for reviewing the manuscript and Tom Dreiman for model building. References Aavatsmark, I. and Klausen, R.A. 2003. Well Index in Reservoir Simulation for Slanted and Slightly Curved Wells in 3D Grids. SPE J. 8 (1): 41–48. SPE-75275-PA. doi: 10.2118/75275-PA. Aziz, K. and Settari, A. 1979. Petroleum Reservoir Simulation. Essex, UK: Elsevier Applied Science Publishers. Babu, D.K. and Odeh, A.S. 1989. Productivity of a Horizontal Well. SPE Res Eng 4 (4): 417–421. SPE-18298-PA. doi: 10.2118/18298-PA. December 2010 SPE Journal

Brons, F. and Marting, V.E. 1961. The Effect of Restricted Fluid Entry on Well Productivity. J Pet Technol 13 (2): 172–174; Trans., AIME, 222. SPE-1322-G. doi: 10.2118/1322-G. Craft, B.C. and Hawkins, M.F. 1959. Petroleum Reservoir Engineering. Upper Saddle River, New Jersey: Prentice Hall Press. Ding, Y. 1995. Scaling-up in the Vicinity of Wells in Heterogeneous Field. Paper SPE 29137 presented at the SPE Reservoir Simulation Symposium, San Antonio, Texas, USA, 12–15 February. doi: 10.2118/29137-MS. Dogru, A.H., Sunaidi, H.A., Fung, L.S., Habiballah, W.A., Al-Zamel, N., and Li, K.G. 2002. A Parallel Reservoir Simulator for Large-Scale Reservoir Simulation. SPE Res Eval & Eng 5 (1): 11–23. SPE-75805PA. doi: 10.2118/75805-PA. Donnez, P. 2007. Essentials of Reservoir Engineering, 201–205. Paris: Editions TECHNIP. Kozeny, J. 1953. Hydraulik: Ihre Grundlagen und Praktische Anvendung. Vienna, Austria: Springler-Verlag. Lin, C.Y. 1995. New Well Models for Partially Penetrating Wells in Heterogeneous Reservoirs Using Non-Uniform grids. Paper SPE 29122 presented at the SPE Reservoir Simulation Symposium, San Antonio, Texas, USA, 12–15 February. doi: 10.2118/29122-MS. Muggeridge, A.H., Cuypers, M., Bacquet, C., and Barker, J.W. 2002. Scale-up of well performance for reservoir flow simulation. Petroleum Geoscience 8 (2): 133–139. Muskat M. 1949. Physical Principles of Oil Production. New York: International Series in Pure Applied Physics, McGraw-Hill Book Co. Muskat, M. 1937. The Flow of Homogeneous Fluids Through Porous Media. New York: McGraw-Hill Book Co. Odeh, A. 1980. An Equation for Calculating Skin Factor Due to Restricted Entry. J Pet Technol 32 (6): 964–965. SPE-8879-PA. doi: 10.2118/8879PA. Peaceman, D.W. 1978. Interpretation of Well-Block Pressures in Numerical Reservoir Simulation. SPE J. 18 (3): 183–194; Trans., AIME, 265. SPE-6893-PA. doi: 10.2118/6893-PA. Peaceman, D.W. 1983. Interpretation of Well-Block Pressures in Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability. SPE J. 23 (3): 531–543; Trans., AIME, 275. SPE-10528PA. doi: 10.2118/10528-PA. Wolfsteiner, C., Durlofsky, L.J., and Aziz, K. 2000. Approximate Model for Productivity of Nonconventional Wells in Heterogeneous Reservoirs. SPE J. 5 (2): 218–226. SPE-56754-PA. doi: 10.2118/62812-PA.

Appendix A—Derivation of Equivalent Wellblock Radius and Analytical Solutions for the Potential Equation Analytical Solution for the Potential Equation. Differential Equation for Potential in Vertical (z) Direction. Our objective is to derive a 1D potential (pressure) equation in the vertical direction for the discretized system. Considering the finite-difference discretization shown in Fig. A-1 and assuming constant permeability

in areal directions (x, y) and constant vertical permeability in the vertical (z) direction with constant gridblock potential B for the neighboring cells where the perforation is located, steady-state volume balance for the cell k, excluding the source term, is given by 4Tx (  B −  k ) + Tz (  k −1 −  k ) + Tz (  k +1 −  k ) = 0 , . . . . . (A-1) where Tx and Tz (Tx = Kxyz/x, Tz = Kzxz/x) are the transmisibilities between the cell k and its areal and vertical neighbors. By examining Eq. A-1, we see that potential  varies only in the z direction. We can rearrange Eq. A-1 by dividing each side by Kzxy and further dividing each by z; the following equation is obtained:

( k −1 − 2 k +  k +1 ) − 4 ⎛

4K x Kx ⎞ ⎜⎝ K x 2 ⎟⎠  k = − K x 2 (  B ) . z z . . . . . . . . . . . . . . . . . . . . . . (A-2)

z 2

Recognizing that lim

z → 0

( k −1 − 2 k +  k +1 ) = d 2 , z 2

Eq. A-2 yields a second-order ordinary differential equation for the potential (z): d 2 − a = b , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3) dz 2 where ⎛ Kx ⎞ a = 4⎜ 2 ⎝ K z x ⎟⎠ and b=−

4K x ( B ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4) K z x 2

Analytical Solution. A general solution for Eq. A-3 can be written in the following form: b  ( z ) = − + C1 exp a

ΦW

Φk

Δy

k=1 Φk−1

ΦE

Φk Φs

z

Φk+1

Δx

y

k =Nz

x

Fig. A-1—Grid system. December 2010 SPE Journal

Δz

( az ) + C exp ( − az ). 2

. . . . . . . . . . (A-5a)

The derivative can be obtained by differentiating Eq. A-5 with respect to z: ∂ ( z ) = C1 a exp ∂z

ΦN

dz 2

( az ) − C

2

(

)

a exp − az . . . . . . . . . (A-5b)

The unknown coefficients C1 and C2 are determined from the boundary conditions. Two boundary conditions are specified rate at the perforation and no-flow boundary condition at the boundary. We can consider three cases—(1) single perforation located at the top of the formation; (2) single perforation located at any point along the wellbore, away from the boundaries; and (3) single perforation located at the bottom of the well. The last case can be derived from (1) easily. We will show the first two cases only. Perforation Located at the Top of the Formation, zs = 0. Let zs be the location of the source (perforation). Specified Rate. ⎛ ∂ ⎞ q = K z x y ⎜ at z s = 0. . . . . . . . . . . . . . . . . . . . . (A-6) ⎝ ∂z ⎟⎠ z = 0 s

1041

⎛ ∂ ⎞ = 0 at z = H. . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7) ⎜⎝ ⎟ ∂z ⎠ z = H Applying Eqs. A-6 and A-7 to Eq. A-5b, we can determine the two unknowns C1 and C2, and, using the definitions cosh(x) and sinh(x), we obtain the complete analytical solution: b qcosh ⎡⎣ a ( H − z ) ⎤⎦ (z) = − − . . . . . . . . . . . . . . . . . . (A-8) a aK z x 2sinh aH

(

)

Close examination of the ratio b/a reveals that this term represents the average potential around the perforation cells:

where qL and qR are unknowns that are solved by using the following two equations: qL + qR = qk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-17) and  L ( z s ) =  R ( z s ) at z = zs. . . . . . . . . . . . . . . . . . . . . . . . (A-18) Solving Eqs. A-15 through A-18, we obtain qL =

s

s

k

s

s

b B = . a

s

Specified Bottomhole Pressure. In the case of specified potential at the wellbore wf , the boundary conditions for a single perforation at the top of the formation will be

(

qR =

)

Substituting Eqs. A-5a and A-5b into Eq. A-9 and using Eq. A-7, we obtain ⎞ ⎛ ⎟ ⎜ ⎛b ⎞ WI k ⎜ +  wf ⎟ ⎟ ⎝a ⎠ b ⎜ ⎟ (z) = − − ⎜ 2 a ⎜ ⎧ K z x asinh asinh ( H − z s ) ⎫ ⎟ ⎪ ⎪ ⎜⎨ ⎬⎟ ⎜⎝ ⎪+WI cosh ⎡⎣ asinh ( H − z s ) ⎤⎦ ⎪ ⎟⎠ k ⎩ ⎭

(

)

× cosh ⎡⎢ a H − z ⎤⎥ . k ⎦ ⎣

. . . . . . . . . . . . . . . . . . .(A-10)

Perforation Located Away From the Top and Bottom Boundaries, 0 < zs < H. Let qL and qR be the left and right portions (fraction) of the total rate q. Left side of the source, ⎛ ∂ ⎞ = 0 at z = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-11) ⎝⎜ ∂z ⎠⎟ z = 0 ⎛ ∂ ⎞ qL = − K z x 2 ⎜ at zfrom left → z, . . . . . . . . . . . . . . (A-12) ⎝ ∂z ⎟⎠ z→ zs and right side of the source (perforation), ⎛ ∂ ⎞ qR = K z x 2 ⎜ at zfrom right → zs . . . . . . . . . . . . . . . (A-13) ⎝ ∂z ⎠⎟ z→ zs ⎛ ∂ ⎞ = 0 at z = H. . . . . . . . . . . . . . . . . . . . . . . . . . . (A-14) ⎜⎝ ⎟ ∂z ⎠ z = H Using Eqs. A-11 and A-14 in Eq. A-5b, we determine the unknown coefficients C1 and C2. The analytical solution becomes

( ) ( )

qL cosh az b L (z ) = − − , a aK z x 2sinh az s for 0 < z < zs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-15) qR cosh ⎡⎣ a ( H − z ) ⎤⎦ b , R (z ) = − − a aK z x 2sinh ⎡⎣ a ( H − z s ) ⎤⎦ for zs ≤ z < H, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-16)

s

and

⎛ ∂ ⎞ K z x 2 ⎜ = WI k  k −  wf . . . . . . . . . . . . . . . . . . . (A-9) ⎝ ∂z ⎟⎠ z = 0

1042

( az ) cosh ⎡⎣ a ( H − z )⎤⎦ q . . . . . . . . . . . . (A-19) ⎧sinh ( az ) cosh ⎡ a ( H − z ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎪⎩+ cosh ( az ) sinh ⎡⎣ a ( H − z ) ⎤⎦ ⎪⎭ sinh

sinh ⎡⎣ a ( H − z s ) ⎤⎦ cosh

(

)

(

az s

)

⎧sinh az s cosh ⎡ a ( H − z s ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎪⎩+ cosh az s sinh ⎡⎣ a ( H − z s ) ⎤⎦ ⎪⎭

(

qk . . . . . . . . . . . (A-20)

)

Eqs. A-15 and A-16 together with Eqs. A-19 and A-20 completely define the analytical solution for a single perforation located between the top and bottom of the formation. For convenience, we will use fL and fR as the fractions of the total rate to simplify Eqs. A-19 and A-20. By using this definition, Eq. A-19 and A-20 can be shortened because qL = fLqk and qR = fRqk. Using the definition b of  B = − , Eqs. A-15 and A-16 can be written as a L (z ) = B −

( az ) aK x sinh ( az ) qL cosh 2

z

s

for 0 < z ≤ zs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-21a) and R (z ) = B −

qR cosh ⎡⎣ a ( H − z ) ⎤⎦

aK z x 2sinh ⎡⎣ a ( H − z s ) ⎤⎦

for zs ≤ z < H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-21b) Equivalent Wellblock Radius. Writing the volume balance for the gridblock k and denoting the vertical-flow rate into k by qv,k, 4K z (  B −  k ) + qv ,k = qk . . . . . . . . . . . . . . . . . . . . . . (A-22a) Because the flow becomes approximately radial in the (x, y) space into the perforation k, we can write the radial-flow-rate equations as qk =

2 K z fP (  B −  k ) . . . . . . . . . . . . . . . . . . . . . . . (A-22a) ln z / ro ,3 D ,k

Solving for (B− k) from Eq. A-22b and substituting them into the volume-balance equation for the cell k in Eq. A-22a (Donnez 2007), we obtain ⎛ x ⎞ f ⎛ q ⎞ =  p ⎜ 1 − v ,k ⎟ . . . . . . . . . . . . . . . . . . . . . . (A-23) ln ⎜ ⎟ 2⎝ qk ⎠ ⎝ ro ,3 D ,k ⎠ From Eq. A-23, we obtain the equivalent gridblock radius: ⎡  f ⎛ q ⎞⎤ ro ,3 D ,k = x exp ⎢ − P ⎜ 1 − v ,k ⎟ ⎥ . . . . . . . . . . . . . . . . . . (A-24) qk ⎠ ⎦ ⎣ 2 ⎝ December 2010 SPE Journal

⎛ q ⎞ Perforation at the Top. We would need to express the ⎜ 1− v ,k ⎟ qk ⎠ ⎝ term analytically because fp,k is available by Eq. 8 or Eq. 9. By the definition of qv,k, vertical flow from the cell below the perforation (Cell 2) into the perforation (Cell 1) is given by ⎡ ( z 2 ) −  ( z1 ) ⎤⎦ qv ,k = K z x 2 ⎣ . . . . . . . . . . . . . . . . . . . . . (A-25) z Substituting Eq. A-9 into Eq. A-25, we obtain an expression ⎛ q ⎞ for ⎜ 1− v ,k ⎟ , and, using it in Eq. A-24, we obtain the formula for q ⎠ ⎝ k

the equivalent gridblock radius:

ro ,3 D ,k

perforated interval. To simplify the derivation, however, let us assume that all the perforations are contiguous, 1, 2, 3, …, NC. The analytical solution for the potential equation can be obtained by superposing the analytical solution for each perforation. Let t(z) represent the total potential drop at location z from multiple sources located at the well qj, j = 1, Nc, where Nc is the total number of completions. By superposition,

⎡ ⎛ ⎧cosh ⎡ a ( H − z 2 ) ⎤ ⎫ ⎞ ⎤ ⎪ ⎣ ⎦ ⎪⎟ ⎥ ⎢ ⎜ ⎨ ⎬ ⎥ ⎢ ⎜ ⎡ ⎪⎩− cosh ⎣ a ( H − z1 ) ⎤⎦ ⎪⎭ ⎟ ⎥  f p ,k ⎢ ⎜ ⎟ . 1− = x exp ⎢ − ⎟⎥ 2 ⎜ a zsinh aH ⎢ ⎜ ⎟⎥ ⎢ ⎜ ⎟⎥ ⎠ ⎥⎦ ⎝ ⎢⎣ . . . . . . . . . . . . . . . . . . . . . (A-26)

×

)

)

(

)

⎡ ⎡ cosh az az k ⎤ ⎤ k −1 − cosh ⎢ fL ⎢ ⎥⎥ ⎢ ⎢ ⎥⎥ a z aK z  sinh k z ⎦⎥ ⎢ ⎣ ⎢ ⎛ ⎧cosh ⎡ a ( H − z ) ⎤ ⎫ ⎞ ⎥ k +1 ⎦ ⎪ q ⎣ ⎥ ⎜ ⎪⎨ 1 − v ,k = 1 − ⎢⎢ ⎬ ⎟ , . . . . . (A-28) qk ⎜ ⎪− cosh ⎡ a ( H − z k ) ⎤ ⎪ ⎟ ⎥ ⎢ ⎣ ⎦⎭ ⎟ ⎥ ⎜ ⎩ ⎢ + f R ⎜ aK zsinh ⎡ a H − z ⎤ ⎟ ⎥ z k )⎦ ⎥ ⎣ ( ⎢ ⎟ ⎜ ⎢ ⎟⎥ ⎜ ⎢⎣ ⎠ ⎥⎦ ⎝

(

)

where fL =

sinh

(

(

az

cosh ⎡⎣ k)

)

a ( H − zk ) ⎤⎦

⎧sinh az k cosh ⎡ a ( H − z k ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎡ ⎤⎪ cosh sinh + − az a H z ( ⎪⎩ k k )⎦ ⎭ ⎣

(

)

and sinh ⎡⎣ a ( H − z k ) ⎤⎦ cosh( az ) k . fR = ⎧sinh az k cosh ⎡ a ( H − z k ) ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎪⎩+ cosh az k sinh ⎡⎣ a ( H − z k ) ⎤⎦ ⎭⎪

(

(

)

)

The terms fL and fR are the fractions of the total flow rate qk. qL = f L qk , qR = f R qk , f L + f R = 1. . . . . . . . . . . . . . . . . . . . (A-29) Substituting Eq. A-28 into Eq. A-24, we obtain the equivalent well radius. Multiple Perforations. Let the well have NC number of completions for the perforations j = 1, NC. We need to note that all perforations do not have to be continuous. We may have a set of perforations followed by a nonperforated interval and again followed by a new December 2010 SPE Journal

q j sinh

(

⎧sinh ⎪ ⎨ + cosh ⎩⎪

(

)

(

a z j cosh ⎡⎣ a H − z j az j cosh ⎡⎣ a H − z j ⎤⎦

(

)

(

)

)⎤⎦

⎫ ⎪ ⎬ az j sinh ⎡⎣ a H − z j ⎤⎦ ⎪ ⎭

)

( az ) aK x ysinh (

(

cosh

⎡ ⎣

)

z

(

⎛  −  k  k +1 −  k ⎞ = K z x ⎜ k −1 + . . . . . . . . . . . . . . (A-27) ⎝ ⎠⎟ z z

Substituting the analytical solution obtained for this case, Eq. A-15 and Eq. A-16, into Eq. A-27, we obtain

) )

(

)

+ z > zj az j ⎤⎦ q j sinh ⎡⎣ a H − z j ⎤⎦ sinnh az j × ⎧sinh az j cosh ⎡ a H − z j ⎤ ⎫ ⎪ ⎣ ⎦ ⎪ ⎨ ⎬ ⎡ ⎤ i nh a H − z + si cosh az ⎪⎩ j j ⎦⎪ ⎣ ⎭ coosh ⎡⎣ a ( H − z ) ⎤⎦ . . . . . . . . (A-31) × aK z x ysinh ⎡⎣ a H − z j ⎤⎦ ×

2

(

(

Nc

 t ( z ) = ∑  z ≤ z j j =1

(

)

where

Perforation in the Middle. For a perforation not located at the formation top or bottom, total vertical flow into the perforation k (from above and below) is given by qv , k

(

Nc

 ( z ) = ∑  z , z j , . . . . . . . . . . . . . . . . . . . . . . . . . . (A-30) t j =1

(

(

)

)

)

{

(

(

(

(

)

)

)

)}

Solution for any location z is then computed from b  ( z ) =  B + t  ( z ) = − +  ( z ) . . . . . . . . . . . . . . . . (A-32) a Wellblock Radius for Gridblock k. The expression for equivalent wellblock radius for the gridblock k is Eq. 16. To obtain an analytical expression for the equivalent wellblock q radius ro,3D,k, we need to evaluate the v ,k term, because qk ⎡  −  k  k +1 −  k ⎤ qv ,k = K z x 2 ⎢ k −1 + ⎥⎦, . . . . . . . . . . . . . (A-33) z z ⎣ where  k −1 −  k =  t ( z k −1 ) −  t ( z k ) and  k +1 −  k =  t ( z k +1 ) −  t ( z k ). . . . . . . . . . . . . . . . . . (A-34) The analytical solution is obtained by substituting Eq. A-31 into Eq. A-34 to obtain analytical expressions for k−1 − k and k+1 − k. The resulting expression is substituted into Eq. A-33 and Eq. 16 to obtain the equivalent wellblock radius. Ali H. Dogru is the chief technologist of computational modeling at Saudi Aramco’s Advanced Research Center. His primary areas of interest are parallel reservoir simulation, reservoir engineering, large-scale parameter estimation, and high-performance computing. He holds a PhD degree in petroleum engineering with a minor in applied mathematics from The University of Texas at Austin. His industrial experience is with Core Labs and Mobil R&D in Dallas, Texas, and Saudi Aramco. He was chairman of the SPE JPT Special Series Committee 2004–08. Dogru worked with various other SPE committees, including Editorial Review and SPE Fluid Mechanics. Currently, he is a director of the SPE R&D Technical Section Committee and a member of JPT Special Series Committe. Dogru received the 2008 SPE Reservoir Description and Dynamics award. 1043