# Pseudo-Steady-State Productivity Formula for a Partially Penetrating Vertical Well in a Box-Shaped Reservoir

Authors: Jing Lu; Djebbar Tiab
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long-producing times. Taking a partially penetrating well as a uniform line sink in three dimensional space, by the orthogonal decomposition of Dirac function and using Green's function to three-dimensional Laplace equation with homogeneous Neumann boundary condition, this paper presents step-by-step derivations of a pseudo-steady-state productivity formula for a partially penetrating vertical well arbitrarily located in a closed anisotropic box-shaped drainage volume. A formula for calculating pseudo skin factor due to partial penetration is derived in detailed steps. A convenient expression is presented for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle, and for arbitrary position of the well within the rectangle.


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## Body

## 1. Introduction

Well productivity is one of primary concerns in oil field development and provides the basis for oil field development strategy. To determine the economical feasibility of drilling a well, the engineers need reliable methods to estimate its expected productivity. Well productivity is often evaluated using the productivity index, which is defined as the production rate per unit pressure drawdown. Petroleum engineers often relate the productivity evaluation to the long-time performance behavior of a well, that is, the behavior during pseudo-steady-state or steady-state flow.For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long producing times. In these reservoirs, also called volumetric reservoirs, there can be no flow across the impermeable outer boundary, such as a sealing fault, and fluid production must come from the expansion and pressure decline of the reservoir. This condition of no flow boundary is also encountered in a well that is offset on four sides.Flow enters the pseudo-steady-state regime when the pressure transient reaches all boundaries after drawdown for a sufficiently long-time. During this period, the rate of pressure decline is almost identical at all points in the reservoir and wellbore. Therefore, the difference between the average reservoir pressure and pressure in the wellbore approaches a constant with respect to time. Pseudo-steady-state productivity index is defined as the production rate divided by the difference of average reservoir pressure and wellbore pressure, hence the productivity index is basically constant [1, 2].In many oil reservoirs the producing wells are completed as partially penetrating wells. If a vertical well partially penetrates the formation, the streamlines converge and the area for flow decreases in the vicinity of the wellbore, which results in added resistance, that is, a pseudoskin factor. Only semianalytical and semi-empirical expressions are available in the literature to calculate pseudoskin factor due to partial penetration.Rarely do wells drain ideally shaped drainage areas. Even if they are assigned regular geographic drainage areas, they become distorted after production commences, either because of the presence of natural boundaries or because of lopsided production rates in adjoining wells. The drainage area is then shaped by the assigned production share of a particular well. An oil reservoir often has irregular shape, but a rectangular shape is often used to approximate an irregular shape by petroleum engineers, so it is important to study well performance in a rectangular or box-shaped reservoir [1, 2].


## 2. Literature Review

The pseudo-steady-state productivity formula of a fully penetrating vertical well which is located at the center of a closed isotropic circular reservoir is [3, page 63](2.1)Qw=FD2πKH(Pa-Pw)/(μB)ln(Re/Rw)-3/4,
where Pa is average reservoir pressure in the circular drainage area, Pw is flowing wellbore pressure, K is permeability, H is payzone thickness, μ is oil viscosity, B is oil formation volume factor, Re is radius of circular drainage area, Rw is wellbore radius, and FD is the factor which allows the use of field units and practical SI units, and it can be found in [3, page 52, Table 5.1].Formula (2.1) is only applicable for a fully penetrating vertical well at the center of a circular drainage area with impermeable outer boundary.If a vertical well is partially penetrate the formation, the streamlines converge and the area for flow decreases in the region around the wellbore, and this added resistance is included by introducing the pseudoskin factor,Sps. Thus, (2.1) may be rewritten to include the pseudoskin factor due to partial penetration as [4, page 92]:(2.2)Qw=FD2πKH(Pa-Pw)/(μB)ln(Re/Rw)-3/4+Sps.Sps can be calculated by semianalytical and semiempirical expressions presented by Brons, Marting, Papatzacos, and Bervaldier [5–7].Assume that the well-drilled length is equal to the well producing length, (i.e., perforated interval,)Lp=L, and define partial penetration factor η:(2.3)η=LpH=LH.Pseudoskin factor formula given by Brons and Marting is [5](2.4)Sps=(1η-1)[ln(hD)-G(η)],
where(2.5)hD=(HRw)(KhKv)1/2,(2.6)G(η)=2.948-7.363η+11.45η2-4.675η3.Pseudoskin factor formula given by Papatzacos is [6](2.7)Sps=(1η-1)ln(πhD2)+(1η)ln[(η2+η)(Ψ1-1Ψ2-1)1/2],
where hD has the same meaning as in (2.5), and(2.8)Ψ1=Hh1+0.25Lp,Ψ2=Hh1+0.75Lp,
and h1 is the distance from the top of the reservoir to the top of the open interval.Pseudoskin factor formula given by Bervaldier is [7](2.9)Sps=(1η-1)[ln(Lp/Rw)(1-Rw/Lp)-1].It must be pointed out that the well location in the reservoir has no effect onSps calculated by (2.4), (2.7), and (2.9).By solving-three-dimensional Laplace equation with homogeneous Dirichlet boundary condition, Lu et al. presented formulas to calculateSps in steady state [8].To account for irregular drainage shapes or asymmetrical positioning of a well within its drainage area, a series of shape factors was developed by Dietz [9]. Formula (2.1) can be generalized for any shape into the following formula:(2.10)Qw=FD2πKH(Pa-Pw)/(μB)(1/2)ln[2.2458A/(CARw2)],
where CA is shape factor, and A is drainage area.Dietz evaluated shape factorCA for various geometries, in particular, for rectangles of various aspect ratios with single well in various locations. He obtained his results graphically, from the straight line portion of various pressure build-up curves. Earlougher et al. [10] carried out summations of exponential integrals to obtain dimensionless pressure drops at various points within a square drainage area and then used superposition of various square shapes to obtain pressure drops for rectangular shapes. The linear portions of the pressure drop curves so obtained, corresponding to pseudo-steady-state, were then used to obtain shape factors for various rectangles.The methods used by Dietz and Earlougher et al. are limited to rectangles whose sides are integral ratios, and the well must be located at some special positions within the rectangle.Lu and Tiab presented formulas to calculate productivity index and pseudoskin factor in pseudo-steady-state for a partially penetrating vertical well in a box-shaped reservoir, they also presented a convenient expression for calculating the shape factor of an isotropic rectangle reservoir [1, 2]. But in [1, 2], they did not provide detail derivation steps of their formulas.The primary goal of this paper is to present step-by-step derivations of the pseudo-steady-state productivity formula and pseudoskin factor formula for a partially penetrating vertical well in an anisotropic box-shaped reservoir, which were given in [1, 2]. A similar procedure in [8] is given in this paper, point sink solution is first derived by the orthogonal decomposition of Dirac function and Green's function to Laplace equation with homogeneous Neumann boundary condition, then using the principle of superposition, point sink solution is integrated along the well length, uniform line sink solution is obtained, and rearrange the resulting solution, pseudo-steady-state productivity formula and shape factor formula are obtained. A convenient expression is derived for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle and for arbitrary position of the well within the rectangle.


## 3. Partially Penetrating Vertical Well Model

Figure1 is a schematic of a partially penetrating well. A partially penetrating vertical well of length L drains a box-shaped reservoir with height H, length (x direction) a, and width (y direction) b. The well is parallel to the z direction with a length L≤H, and we assume b≥a.Figure 1
Partially Penetrating Vertical Well Model.The following assumptions are made.(1)
The reservoir is homogeneous, anisotropic, and has constantKx,Ky,Kz permeabilities, thickness H, and porosity ϕ. All the boundaries of the box-shaped drainage volume are sealed.(2)
The reservoir pressure is initially constant. At timet=0, pressure is uniformly distributed in the reservoir, equal to the initial pressure Pi.(3)
The production occurs through a partially penetrating vertical well of radiusRw, represented in the model by a uniform line sink.(4)
A single phase fluid, of small and constant compressibilityCf, constant viscosity μ, and formation volume factor B, flows from the reservoir to the well at a constant rate Qw. Fluids properties are independent of pressure.(5)
No gravity effect is considered. Any additional pressure drops caused by formation damage, stimulation, or perforation are ignored, we only consider pseudoskin factor due to partial penetration.The partially penetrating vertical well is taken as a uniform line sink in three dimensional space. The coordinates of the two end points of the uniform link sink are (x′,y′,0) and (x′,y′,L). We suppose the point (x′,y′,z′) is on the well line, and its point convergence intensity is q.By the orthogonal decomposition of Dirac function and using Green's function to Laplace equation with homogeneous Dirichlet boundary condition, Lu et al. obtained point sink solution and uniform line sink solution to steady-state productivity equation of a partially penetrating vertical well in a circular cylinder reservoir [8]. For a box-shaped reservoir and a circular cylinder reservoir, the Laplace equation of a point sink is the same, in order to obtain the pressure at point (x,y,z) caused by the point (x′,y′,z′), we have to obtain the basic solution of the following Laplace equation:(3.1)Kx∂2P∂x2+Ky∂2P∂y2+Kz∂2P∂z2=ϕμCt∂P∂t+μqBδ(x-x′)δ(y-y′)δ(z-z′),
in the box-shaped drainage volume:(3.2)Ω=(0,a)×(0,b)×(0,H),
and we always assume(3.3)b≥a≫H,
and δ(x-x′), δ(y-y′), δ(z-z′) are Dirac functions.All the boundaries of the box-shaped drainage volume are sealed, that is,(3.4)∂P∂N|Γ=0,
where ∂P/∂N|Γ is the exterior normal derivative of pressure on the surface of box-shaped drainage volume Γ=∂Ω.The reservoir pressure is initially constant(3.5)P|t=0=Pi.Define average permeability:(3.6)Ka=(KxKyKz)1/3.In order to simplify (3.1), we take the following dimensionless transforms:(3.7)xD=(xL)(KaKx)1/2,yD=(yL)(KaKy)1/2,zD=(zL)(KaKz)1/2,aD=(aL)(KaKx)1/2,bD=(bL)(KaKy)1/2,LD=(KaKz)1/2,HD=(HL)(KaKz)1/2,tD=KatϕμCtL2.The dimensionless wellbore radius is [8](3.8)RwD=(Kz/KxKy)1/6[(Kx/Ky)1/4+(Ky/Kx)1/4]Rw2L.Assume thatq is the point convergence intensity at the point sink (x′,y′,z′), the partially penetrating well is a uniform line sink, the total productivity of the well is Qw, and there holds [8](3.9)q=QwLpD=QwLD.Dimensionless pressures are defined by(3.10)PD=KaL(Pi-P)μqB,(3.11)PwD=KaL(Pi-Pw)μqB.Then (3.1) becomes(3.12)∂PD∂tD-(∂2PD∂xD2+∂2PD∂yD2+∂2PD∂zD2)=δ(xD-xD′)δ(yD-yD′)δ(zD-zD′),
in the dimensionless box-shaped drainage volume(3.13)ΩD=(0,aD)×(0,bD)×(0,HD),
with boundary condition(3.14)∂PD∂ND|ΓD=0,
and initial condition(3.15)PD|tD=0=0.


## 4. Point Sink Solution

For convenience in the following reference, we use dimensionless transforms given by (3.7)–(3.10), every variable, drainage domain, initial and boundary conditions should be taken as dimensionless, but we drop the subscript D.Consequently, (3.12) is expressed as(4.1)∂P∂t-(∂2P∂x2+∂2P∂y2+∂2P∂z2)=δ(x-x′)δ(y-y′)δ(z-z′).Rewrite (3.14) below(4.2)∂P∂N|Γ=0,
and (3.15) becomes(4.3)P|t=0=0.We want to solve (4.1) under the boundary condition (4.2) and initial condition (4.3), and to obtain point sink solution when the time t is so long that the pseudo-steady-state is reached.If the boundary condition is (4.2), there exists the following complete normalized orthogonal system {glmn(x,y,z)} [11, 12]:(4.4)glmn(x,y,z)=1abHdldmdncos(lπxa)cos(mπyb)cos(nπzH),
where l,m,n are nonnegative numbers, and(4.5)dl={1ifl=0,12ifl>0,
and dm,dn have similar definitions.According to the complete normalized orthogonal systems of the Laplace equation's basic solution, Dirac function has the following expression for homogeneous Neumann boundary condition ([13, 14]):(4.6)δ(x-x′)δ(y-y′)δ(z-z′)=∑l,m,n=0∞glmn(x,y,z)glmn(x′,y′,z′).In order to simplify the following derivations, we define the following notation:(4.7)∑l,m,n=0∞Flmn(x,y,z)=∑l=0∞∑m=0∞∑n=0∞Flmn(x,y,z),
which means in any function F(x,y,z), the subscripts l,m,n of any variable must count from 0 to infinite.And define(4.8)∑l+m+n>0Flmn(x,y,z)=∑l≥0∞∑m≥0∞∑n≥0∞Flmn(x,y,z)(l+m+n>0),
which means in any function F(x,y,z), the subscripts l,m,n of any variable must be no less than zero, and at least one of the three subscripts l,m,n must be positive to guarantee l+m+n>0. And the upper limit of the subscripts l,m,n is infinite.Let(4.9)P(t,x,y,z;x′,y′,z′)=∑l,m,n=0∞elmn(t)glmn(x,y,z),
where elmn(t) are undetermined coefficients.Substituting (4.9) into left-hand side of (4.1), and substituting (4.6) into right-hand side of (4.1), we obtain(4.10)∑l,m,n=0∞{∂elmn(t)∂tglmn(x,y,z)-elmn(t)Δ[glmn(x,y,z)]}=∑l,m,n=0∞{∂elmn(t)∂t+elmn(t)λlmn}glmn(x,y,z)=∑l,m,n=0∞glmn(x′,y′,z′)glmn(x,y,z),
where Δ is the three-dimensional Laplace operator(4.11)Δ=∂2∂x2+∂2∂y2+∂2∂z2,(4.12)λlmn=(lπa)2+(mπb)2+(nπH)2.From (4.3) and (4.9),(4.13)elmn(0)=0,
compare the coefficients of glmn(x,y,z) at both sides of (4.10), we obtain(4.14)∂elmn(t)∂t+λlmnelmn(t)=glmn(x′,y′,z′),
because λ000=0, from (4.14),(4.15)e000(t)=g000(x′,y′,z′)t=tabH.Whenλlmn≠0(l+m+n>0), solve (4.14),(4.16)elmn(t)=[1-exp(-λlmnt)]glmn(x′,y′,z′)λlmn.Substitute (4.15) and (4.16) into (4.9) and obtain(4.17)P(t,x,y,z;x′,y′,z′)=∑l,m,n=0∞elmn(t)glmn(x,y,z)=(tabH)g000(x,y,z)+∑l+m+n>0[1-exp(-λlmnt)]glmn(x′,y′,z′)glmn(x,y,z)λlmn=tabH+∑l+m+n>0glmn(x′,y′,z′)glmn(x,y,z)λlmn-∑l+m+n>0exp(-λlmnt)glmn(x′,y′,z′)glmn(x,y,z)λlmn.Define(4.18)I1=tabH,(4.19)I2=Ψ(x,y,z;x′,y′,z′)=∑l+m+n>0glmn(x′,y′,z′)glmn(x,y,z)λlmn,(4.20)I3=∑l+m+n>0exp(-λlmnt)glmn(x′,y′,z′)glmn(x,y,z)λlmn,
then(4.21)P(t,x,y,z;x′,y′,z′)=I1+I2-I3.Recall (4.19), the average value of Ψ throughout of the total volume of the box-shaped reservoir is(4.22)Ψa,v=(1V)∫ΩΨ(x,y,z)dV=(1V)∫0a∫0b∫0HΨ(x,y,z;x′,y′,z′)dxdydz=(1V)(glmn(x′,y′,z′)λlmn)∫0a∫0b∫0H∑l+m+n>0glmn(x,y,z)dxdydz.Note thatl+m+n>0 implies that at least one of l,m,n must be greater than 0, without losing generality, we may assume(4.23)l>0,
then(4.24)∫0acos(lπxa)dx=0.
So,(4.25)∫0a∫0b∫0H∑l+m+n>0glmn(x,y,z)dxdydz=0,
consequently,(4.26)Ψa,v=0.If timet is sufficiently long, pseudo-steady-state is reached, I3 decreases by exponential law, I3 will vanish, that is,(4.27)I3≈0,
then(4.28)P(t,x,y,z;x′,y′,z′)=tabH+Ψ(x,y,z;x′,y′,z′).Substituting (4.28) into (4.1), we have(4.29)1abH-ΔΨ=δ(x-x′)δ(y-y′)δ(z-z′).Define(4.30)f(x,y,z)=-ΔΨ=-(1abH)+δ(x-x′)δ(y-y′)δ(z-z′),
note that Ψ is equal to I2 in (4.19), and(4.31)∂Ψ∂N=0,onΓ.From Green's Formula [15],(4.32)0=∫Γ∂Ψ∂NdS=∫ΩΔΨdV=-∫Ωf(x,y,z)dV,
that is,(4.33)∫Ωf(x,y,z)dV=0,
where V is volume of drainage domain Ω.Define the following notation of internal product of functionsf(x,y,z) and g(x,y,z):(4.34)⌊f(x,y,z),g(x,y,z)⌋=∫Ωf(x,y,z)g(x,y,z)dxdydz=∫Ωf(x,y,z)g(x,y,z)dV,
where ⌊f,g⌋ means the internal product of functions f and g.From (4.33), we know that the internal product of f(x,y,z) and constant number 1 is zero(4.35)⌊f(x,y,z),1⌋=0,
and it is easy to prove(4.36)⌊f(x,y,z),g000⌋=0,
where g000 means glmn when l=m=n=0.Thus,f(x,y,z) can be decomposed as [13, 14]:(4.37)f(x,y,z)=∑l,m,n=0∞⌊f,glmn(x′,y′,z′)⌋glmn(x,y,z)=∑l+m+n>0⌊f,glmn(x′,y′,z′)⌋glmn(x,y,z)=∑l+m+n>0⌊δ(x-x′)δ(y-y′)δ(z-z′),glmn(x′,y′,z′)⌋glmn(x,y,z)=∑l+m+n>0glmn(x′,y′,z′)glmn(x,y,z).The drainage volume is(4.38)V=abH.Recall (4.28), the average pressure throughout the reservoir is(4.39)Pa,v=(1V)∫ΩP(x,y,z)dxdydz=tabH+Ψa,v.The wellbore pressure at point(xw,yw,zw) is(4.40)Pw=tabH+Ψw,
where Ψw is the value of Ψ at wellbore point (xw,yw,zw).Combining (4.39) and (4.40) gives(4.41)Pa,v-Pw=Ψa,v-Ψw,
which implies Pa,v-Pw is independent of time.


## 5. Uniform Line Sink Solution

For convenience, in the following reference, every variable, drainage domain, initial and boundary conditions should be taken as dimensionless, but we drop the subscriptD.The producing portion of the partially penetrating well is between point(x′,y′,0) and point (x′,y′,L), recall (4.4) and (4.19), in order to obtain uniform line sink solution, we integrate Ψ with respect to z′ from 0 to L, then(5.1)J(x,y,z;x′,y′,z′;l,m,n)=∫0LΨ(x,y,z;x′,y′,z′)dz′=∑l+m+n>0ℑlmn(x,y,z;x′,y′,z′;l,m,n),
where(5.2)∑l+m+n>0ℑlmn(x,y,z;x′,y′,z′;l,m,n)=∑l+m+n>0(1abHdldmdnλlmn)cos(lπxa)×cos(mπyb)cos(nπzH)×cos(mπy′b)cos(lπx′a)∫0Lcos(nπz′H)dz′=∑l+m+n>0(1abHdldmdnλlmn)cos(lπxa)cos(mπyb)cos(nπzH)×{(Hπn)cos(lπx′a)cos(mπy′b)sin(nπLH)ifl≠0,Lcos(mπy′b)cos(lπx′a)ifl=0.Define(5.3)C={(l,m,n):l+m+n>0},(5.4)C1={(l,m,n):l=m=0,n>0},(5.5)C2={(l,m,n):l=0,m>0,n≥0},(5.6)C3={(l,m,n):l>0,m≥0,n≥0},
then it is easy to prove(5.7)C=C1∪C2∪C3,C1∩C2=∅,C2∩C3=∅,C3∩C1=∅.Recall (5.1) and (5.2), and use (5.3)–(75), J(x,y,z;x′,y′,z′;l,m,n) can be decomposed as(5.8)J=∑l+m+n>0ℑlmn(x,y,z;x′,y′,z′;l,m,n)=∑n=1∞ℑ00n+∑m=1∞∑n=0∞ℑ0mn+∑l=1∞∑m=0∞∑n=0∞ℑlmn.Define the following notations:(5.9)Jz=∑n=1∞ℑ00n,(5.10)Jyz=∑m=1∞∑n=0∞ℑ0mn,(5.11)Jxyz=∑l=1∞∑m=0∞∑n=0∞ℑlmn,
so(5.12)J=Jz+Jyz+Jxyz,
and the average value of J at wellbore can be written as(5.13)Ja,w=(Jz,a)w+(Jyz,a)w+(Jxyz,a)w.Rearrange (4.12) and obtain(5.14)λlmn=(lπa)2+(mπb)2+(nπH)2=(πH)2(n2+μlm2),
where(5.15)μlm2=(lHa)2+(mHb)2=(Hb)2[m2+(lba)2],μl0=lHa,λlm0=(πH)2μlm2,λ0mn=(mπb)2+(nπH)2=(πH)2[n2+(mHb)2],λ00n=n2π2H2.There hold [16, page 47](5.16)∑n=1∞sin(nx)n3=π2x6-πx24+x312(0≤x≤2π),(5.17)∑n=1∞1-cos(nx)n4=π2x212-πx312+x448(0≤x≤2π).Recall (5.4) and (5.9), Jz is for the case l=m=0,n>0, and at wellbore of the off-center well,(5.18)y=y′≠0,x′≠0,x=x′+Rw,0≤z=z′≤L,(Jz)w=∑n=1∞(1abHdnλ00n)cos(nπzH)∫0Lcos(nπz′H)dz′=(2abH)∑n=1∞(H2π2n2)cos(nπzH)(Hnπ)sin(nπLH)=(2H2abπ3)∑n=1∞(1n3)sin(nπLH)cos(nπzH).The average value of(Jz)w along the well length is(5.19)(Jz,a)w=(1L)∫0LJzdz=(1L)∑n=1∞(2H2π3abn3)sin(nπLH)∫0Lcos(nπzH)dz=∑n=1∞(2H2π3abLn3)sin(nπLH)[(Hnπ)sin(nπLH)]=∑n=1∞(2H3π4abLn4)sin2(nπLH)=(H3π4abL)∑n=1∞(1n4)[1-cos(2nπLH)]=(H3π4abL)(2πLH)2[π212-π12(2πLH)+148(2πLH)2]=(4HLab)(112-L6H+L212H2)=(2HL3ab)(12-LH+L22H2),
where we have used (5.17).For a fully penetrating well,L=H, then(5.20)(Jz,a)w=0.Recall (5.5) and (5.10), Jyz is for the case l=0,m>0,n≥0, and at wellbore of the off-center well,(5.21)y=y′≠0,x′≠0,x=x′+Rw,0≤z=z′≤L,(Jyz)w=(1abH)∑m=1∞∑n=0∞(1dmdnλ0mn)cos2(mπy′b)cos(nπzH)∫0Lcos(nπz′H)dz′=(2abH)∑m=1∞∑n=0∞{cos2(mπy′/b)cos(nπz/H)π2dn[(n/H)2+(m/b)2]∫0Lcos(nπz′H)dz′}=(2abH)∑m=1∞{∑n=1∞2(H/nπ)cos(nπz/H)sin(nπL/H)cos2(mπy′/b)π2[(n/H)2+(m/b)2]+cos2(mπy′b)(b2Lπ2m2){∑n=1∞1m2‍}}=(2H3π3abH)∑m=1∞{∑n=1∞2cos(nπz/H)sin(nπL/H)cos2(mπy′/b)n[n2+(mH/b)2]+cos2(mπy′b)(b2LπH3m2){∑n=1∞1m2‍}}=(2H2π3ab)(πLb2H3)∑m=1∞(1m2)cos2(mπy′b)+(2H2π3ab)∑m=1∞∑n=1∞{2cos(nπz/H)sin(nπL/H)cos2(mπy′/b)n[n2+(mH/b)2]}=(2bLπ2aH)∑m=1∞(1m2)cos2(mπy′b)+(2H2π3ab)∑m=1∞∑n=1∞{2cos(nπz/H)sin(nπL/H)cos2(mπy′/b)n[n2+(mH/b)2]},
where we use the following formulas [16, page 47]:(5.22)∑m=1∞(1m2)cos(mx)=π26-πx2+x24(0≤x≤2π),(5.23)∑m=1∞(1m2)cos2(mx)=π26-πx2+x22(0≤x≤π).The average value of(Jyz)w along the well length is(5.24)(Jyz,a)w=(1L)∫0LJyzdz=(2bLaH)(16-y′2b+y'22b2)+(2H2abLπ3)∑m=1∞∑n=1∞{2sin(nπL/H)cos2(mπy′/b)n[n2+(mH/b)2]∫0Lcos(nπzH)dz}=(2bLaH)(16-y′2b+y'22b2)+(2H2abLπ3)∑m=1∞∑n=1∞{2Hsin2(nπL/H)cos2(mπy′/b)πn2[n2+(mH/b)2]}=(2bLaH)(16-y′2b+y'22b2)+(2H3abLπ4)∑m=1∞∑n=1∞{[1-cos(2nπL/H)]cos2(mπy′/b)n2[n2+(mH/b)2]}=(2bLaH)(16-y′2b+y'22b2)+(2H3abLπ4)∑m=1∞∑n=1∞(bmH)2cos2(mπy′b)×[1-cos(2nπLH),1n2-1n2+(mH/b)2]=(2bLaH)(16-y′2b+y2'2b2)+(H32abLπ4)∑m=1∞(bmH)2cos2(mπy′b)×∑n=1∞[1n2-cos(2nπL/H)n2-1n2+(mH/b)2+cos(2nπL/H)n2+(mH/b)2]=(2bLaH)(16-y′2b+y'22b2)+(2H3abLπ4)∑m=1∞(bmH)2cos2(mπy′b)×{π26-[π26-π2(2πLH)+14(2πLH)2]-[(bπ2mH)coth(mHπb)-12(bmH)2]+[(bπ2mH)cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-12(bmH)2]},
where we use the following formulas [16, page 47]:(5.25)∑n=1∞cos(nx)n2+β2=(π2β){cosh[β(π-x)]sinh(βπ)}-12β2(0≤x≤2π),(5.26)∑n=1∞1n2+β2=(π2β)coth(βπ)-12β2(0≤x≤2π),
and we may simplify (5.24) further(5.27)(Jyz,a)w=(2bLaH)(16-y′2b+y'22b2)+(2H3abLπ4)∑m=1∞cos2(mπy′b)(bmH)2×{π2LH-π2L2H2+(bπ2mH){cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}}=(2bLaH)(16-y′2b+y'22b2)+(2H3abLπ4)∑m=1∞cos2(mπy′b)(1m2)×{{cosh[(mHπ/b)(1-2L/H)]sinh(mHπb)-coth(mHπb)}π2Lb2H3-π2L2b2H4+(b3π2mH3)×{cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}{∑n=1∞1m2‍}}=(2bLaH)(16-y′2b+y'22b2)+(2baπ2)(1-LH)(π26-π2y′2b+π2y2′2b2)+(b2aLπ3)∑m=1∞cos2(mπy′b)(1m3)×{cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}=(2ba)(16-y′2b+y2'2b2)+(b2aLπ3)∑m=1∞[cos2(mπy′/b)m3]{cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}.For a fully penetrating well,L=H, then(5.28)(Jyz,a)w=(2ba)(16-y′2b+y'22b2).Define(5.29)f(x)=sinh[α(1-x)]sinh(αx),
since the derivative of f(x) is(5.30)f′(x)=αcosh(αx)sinh[α(1-x)]-αcosh[α(1-x)]sinh(αx)=αsinh[α(1-2x)],
consequently,(5.31)f′(12)=0.Whenx=0 and x=1,(5.32)f(0)=f(1)=0.Whenx=1/2,f(x) reaches maximum value, let(5.33)x=LH,
and the producing length L is a variable, define(5.34)F(L)=cosh[βπ(1-2L/H)]-cosh(βπ)sinh(βπ)=-2sinh[βπ(1-L/H)]sinh[βπL/(H)]sinh(βπ),
thus when L=H/2,|F(L)| reaches maximum value,(5.35)|F(L)|max=|F(H2)|=2sinh2(βπ/2)sinh(βπ)=2sinh2(βπ/2)2sinh(βπ/2)cosh(βπ/2)=sinh(βπ/2)cosh(βπ/2)<1,
so F(L) is a bounded function, let(5.36)β=mHb,
then(5.37)(Jyz,a)w=(2ba)(16-y′2b+y2'2b2)+(b2aLπ3)×∑m=1∞[cos2(mπy′/b)m3]{cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}=(2ba)(16-y′2b+y2'2b2)+(b2aLπ3)×∑m=1∞[cos2(mπy′/b)m3]{-2sinh[(mHπ/b)(1-L/H)]sinh(mLπ/b)sinh(mHπ/b)}≈(2ba)(16-y′2b+y2'2b2)+(b2aLπ3)×∑m=1M[cos2(mπy′/b)m3]{-2sinh[(mHπ/b)(1-L/H)]sinh(mLπ/b)sinh(mHπ/b)}.Since0<L/H<1, from (5.34) and (5.35), there holds(5.38)∑m=101∞|[cos2(mπy′/b)m3]{-2sinh[(mHπ/b)(1-L/H)]sinh(mLπ/b)sinh(mHπ/b)}|≤∑m=101∞1m3=ζ(3)-∑m=11001m3=4.9502×10-5,
where ζ(3) is Riemann-ζ function:(5.39)ζ(3)=∑m=1∞1m3=1.202057,
thus(5.40)∑m=1∞(1m3){2sinh[(mHπ/b)(1-L/H)]sinh(mLπ/b)sinh(mHπ/b)}≈∑m=1100(1m3){2sinh[(mHπ/b)(1-L/H)]sinh(mLπ/b)sinh(mHπ/b)}.So, in (5.37), M=100 is sufficient to reach engineering accuracy.Recall (5.6) and (5.11), Jxyz is for the case l>0,m≥0,n≥0, and at wellbore of the off-center well,(5.41)y=y′≠0,x′≠0,x=x′+Rw,0≤z=z′≤L,
then(5.42)(Jxyz)w=(1abH)×∑l=1∞∑m=0∞∑n=0∞{[cos(nπz/H)cos(lπx′/a)cos[(lπ(x′+Rw))/a]cos2(mπy′/b)dldmdnλlmn]×∫0Lcos(nπz′H)dz′}=(1abH)∑l=1∞∑m=0∞cos(lπx′a)cos[lπ(x′+Rw)a]cos2(mπy′b)×{∑n=1∞[4(H/nπ)sin(nπL/H)cos(nπz/H)dmλlmn]+2Ldmλlm0}.The average value of(Jxyz)w along the well length is
(5.43)(Jxyz,a)w=(1abH)∑l=1∞∑m=0∞cos(lπx′a)cos[lπ(x′+Rw)a]cos2(mπy′b)×{∑n=1∞[4(H/nπ)sin(nπL/H)∫0Lcos(nπz/H)dzdmλlmnL]+2Ldmλlm0}=(1abH)∑l=1∞∑m=0∞cos(lπx′a)cos[lπ(x′+Rw)a]cos2(mπy′b)×{∑n=1∞[4(H/nπ)2sin2(nπL/H)dmλlmnL]+2Ldmλlm0}=(H4abHπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]∑m=0∞cos2(mπy′b)×{∑n=1∞2[1-cos(2nπL/H)]dmn2(n2+μlm2)L+2π2LdmH2μlm2}=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]{∑m=0∞(2dm)cos2(mπy′b)×{∑n=1∞(1μlm2L)[1-cos(2nπLH)]×(1n2-1n2+μlm2)+π2LH2μlm2}=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]{∑m=0∞(2dmμlm2L)cos2(mπy′b)×{∑n=1∞[1n2-cos(2nπL/H)n2-1n2+μlm2+cos(2nπL/H)n2+μlm2]}+(π2L2H2)}=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]∑m=0∞(2dmμlm2L)cos2(mπy′b)×{π26-[π26-(π2)(2πLH)+(14)(2πLH)2]-[(π2μlm)coth(μlmπ)-(12μlm2)]+[(π2μlm)cosh[μlmπ(1-2L/H)]sinh(μlmπ)-12μlm2]}+(π2L2H2)},
where we use (5.22) and (5.25).Letx=0, recast (5.26), we obtain(5.44)1β2+2∑n=1∞1n2+β2=(πβ)coth(βπ),∑n=0∞1(n2+β2)dn=(πβ)coth(βπ).
So,(5.45)(Jxyz,a)w=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]∑m=0∞(2dmμlm2L)cos2(mπy′b)×{(π2LH-π2L2H2+π2L2H2)+(π2μlm)[cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)]}=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]×{∑m=0∞(πdmμlm3L)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}+(2L)(π2LH)∑m=0∞cos2(mπy′/b)dmμlm2}=(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]×{∑m=0∞(πdmμlm3L)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}+(abπ3H3l)coth(lbπa)+(π2H)∑m=0∞cos(2mπy′/b)dmμlm2}=(H3abLπ3)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]×{∑m=0∞(1dmμlm3)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}}+(H3abπ4)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a,(abπ3H3l)coth(lbπa)+(π2H)∑m=0∞cos(2mπy′/b)dmμlm2].Since(5.46)(π2H)∑m=0∞[cos(2mπy′/b)dmμlm2]=(b2H3)∑m=0∞{cos(2mπy′/b)dm[m2+(bl/a)2]}=(b2H3)(πabl){cosh[πbl(1-2y′/b)/a]sinh(πbl/a)}≈(bH3)(πal)exp(-2πy′la),(H3abπ4)|∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a](bH3)(πal)exp(-2πy′la)|≤(1π3)ln[1-exp(-2πy′/a)]≈0,
thus(5.47)(Jxyz,a)w≈(H3abLπ3)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]∑m=0∞(1dmμlm3)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}+(1π)∑l=1∞(1l)coth(lbπa)cos(lπx′a)cos[lπ(x′+Rw)a]≈(H3abLπ3)∑l=1∞cos(lπx′a)cos[lπ(x′+Rw)a]∑m=0∞(1dmμlm3)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}-(12π)ln{4sin(πRw2a)sin[π(2x′+Rw)2a]},
where we use the following formula [16, page 46]:(5.48)∑n=1∞cos(nx)n=-ln[2sin(x2)],
and the following simplifications:(5.49)coth(lbπa)≈1,∑l=1∞(1l)coth(lbπa)cos(lπx′a)cos[lπ(x′+Rw)a]≈∑l=1∞(1l)cos(lπx′a)cos[lπ(x′+Rw)a]=-12ln{4sin(πRw2a)sin[π(2x′+Rw)2a]}.For a fully penetrating well,L=H, (5.47) is simplified as(5.50)(Jxyz,a)w=-(12π)ln{4sin(πRw2a)sin[π(2x′+Rw)2a]}.Recall (5.13), then(5.51)Ja,w=(Jz,a)w+(Jyz,a)w+(Jxyz,a)w=(2HL3ab)(12-LH+L22H2)+(2ba)(16-y′2b+y2'2b2)+(b2aLπ3)∑m=1M[cos2(mπy′/b)m3]{cosh[(mHπ/b)(1-2L/H)]sinh(mHπ/b)-coth(mHπb)}+(H3abLπ3){∑l=1Ncos(lπx′a)cos[lπ(x′+Rw)a]×∑m=0M(1dmμlm3)cos2(mπy′b)×{cosh[μlmπ(1-2L/H)]sinh(μlmπ)-coth(μlmπ)}}-(12π)ln{4sin(πRw2a)sin[π(2x′+Rw)2a]}.Recall (4.28) and (4.40), the average wellbore pressure along the uniform line sink is(5.52)Pa,w=tabH+Ja,w,
then (4.41) becomes(5.53)Pa,v-Pa,w=Ja,v-Ja,w,
which implies Pa,v-Pa,w is independent of time.


## 6. Productivity Formula and Shape Factor Formula

Note that (5.53) is in dimensionless form, that is,(6.1)Pa,vD-Pa,wD=Ja,vD-Ja,wD.Formulas (4.26), (4.41), (5.1), (5.2), and (5.53) are in dimensionless forms, recall (4.26) and obtain(6.2)Ψa,vD=0,Ja,vD=0,
which implies(6.3)Pa,vD-Pa,wD=-Ja,wD=-[KaL(Pa-Pw)μqB].In order to simplify the above formulas, let(6.4)Ye=a,Xe=b,Yw=x′,Xw=y′,
then(6.5)YeD=aD,XeD=bD,YwD=xD′,XwD=yD′.Combining (3.6), (3.9), (5.13), (6.3), the pseudo-steady-state productivity formula for a partially penetrating vertical well in an anisotropic closed box-shaped reservoir is obtained(6.6)Qw=FD2π(KxKy)1/2H(Pa-Pw)/(μB)Λ+Sps,
where Pa is average reservoir pressure throughout the box-shaped drainage volume, Pw is average wellbore pressure, and(6.7)Λ=(4πXeDηYeD)(16-XwD2XeD+XwD22XeD2)-ln{4|sin[π(2YwD+RwD)/2YeD]|sin[πRwD/2YeD]}η(6.8)Sps=(4πHDLD3ηXeDYeD)(12-η+η22)+(2XeD2π2ηYeDLD)∑m=1M[cos2(mπXwD/XeD)m3]×{cosh[(mπHD/XeD)(1-2η)]sinh(mπHD/XeD)-coth(mπHDXeD)}+{(2HD3π2ηXeDYeDLD)∑l=1Ncos(lπYwDYeD)cos[lπ(YwD+RwD)YeD]×∑m=0M(1dmμlm3)cos2(mπXwDXeD){cosh[μlmπ(1-2η)]sinh(μlmπ)-coth(μlmπ)}},
where η is partial penetration factor defined in (2.3), Sps is pseudoskin factor due to partial penetration, and(6.9)μlm=[(lHa)2+(mHb)2]1/2.In the above equations,M=N=100 is sufficient to reach engineering accuracy.For a fully penetrating well,L=H, then (6.8) reduces to(6.10)Sps=0.If a fully penetrating vertical well located in a closed isotropic rectangular reservoir,(6.11)Sps=0,Lp=H,Kx=Ky=Kz=K.Then (6.6) reduces to(6.12)Qw=2πKH(Pa-Pw)/(μB)Θ-ln{4sin[πRw/(2Ye)]sin(πYw/Ye)},
where(6.13)Θ=(4πXeYe)[16-12(XwXe)+12(XwXe)2].Note that for a rectangle, its area isA=XeYe, recall (2.10), equate (2.10) to (6.12),(6.14)2πKH(Pa-Pw)/(μB)Θ-ln{4sin[πRw/(2Ye)]sin(πYw/Ye)}=2πKH(Pa-Pw)/(μB)(1/2)ln[2.2458XeYe/(CARw2)].A new expression to calculate the Dietz shape factor is obtained by solvingCA in (6.14),(6.15)CA=88.6657f1sin2(πf3)exp(f4),
where(6.16)f1=XeYe,f2=XwXe,f3=YwYe,(6.17)f4=(8πf1)(16-f22+f222).Formula (6.6) is recommended to calculate productivity index in pseudo-steady-state, because it does not require the shape factor, and it is applicable to an off-center partially penetrating vertical well in pseudo-steady-state arbitrarily located in an anisotropic box-shaped reservoir.So, the step-by-step derivations of pseudo-steady-state productivity formula and shape factor formula which were published in [1, 2] have been given in the above sections.


## 7. Examples and Discussions

The following examples are given to calculate well productivity index, pseudoskin factor due to partial penetration, and shape factor.Example One
Use (6.6) to calculate productivity index of a partially penetrating vertical well in pseudo-steady-state in a closed box-shaped anisotropic reservoir. The wellbore, reservoir, and fluid properties data practical SI units are given in Table 1.Table 1
Wellbore, Reservoir, and Fluid Properties Data.
Reservoir length,Xe800 mReservoir width,Ye200 mPayzone thickness,H20 mWell location inx direction, Xw100 mWell location iny direction, Yw50 mProducing well length,Lp10 mWellbore radius,Rw0.1 mPermeability inx direction, Kx0.1μm2Permeability iny direction, Ky0.4μm2Permeability inz direction, Kz0.025μm2Oil viscosity,μ5.0mPa.sFormation volume factor,B1.25Rm3/Sm3Solution 7.
The average permeability is(7.1)Ka=(0.1×0.4×0.025)1/3=0.1(μm2).
Using dimensionless transforms given by (3.7) through (3.10), we obtain
(7.2)XeD=(80010)×(0.10.1)1/2=80.0,YeD=(20010)×(0.10.4)1/2=10.0,XwD=(10010)×(0.10.1)1/2=10.0,YwD=(5010)×(0.10.4)1/2=2.5,LD=(0.10.025)1/2=2.0,HD=(2010)×(0.10.025)1/2=4.0,RwD=(0.025(0.1×0.4)1/2)1/6×((0.10.4)1/4+(0.40.1)1/4)×0.1(2×10)=0.0075η=10.020.0=0.5,1-2η=0,μlm=[(4l10)2+(4m80)2]1/2=(4l225+m2400)1/2.
Recalling (6.8), pseudoskin factor due to partial penetration can be expressed as
(7.3)Sps=Ψ1+Ψ2+Ψ3,
where
(7.4)Ψ1=(4πHDLD3ηXeDYeD)(12-η+η22),Ψ2=(2XeD2π2ηYeDLD)∑m=1100[1m3cos2(mπXwDXeD)]×{cosh[(mHDπ/XeD)(1-2η)]sinh(mHDπ/XeD)-coth(mHDπXeD)},Ψ3=(2HD3π2ηXeDYeDLD)∑l=1100cos(lπYwDYeD)cos(lπ(YwD+RwD)YeD)×{∑m=0100(1dmμlm3)cos2(mπXwDXeD){cosh[μlmπ(1-2η)]sinh(μlmπ)-coth(μlmπ)}}.
Consequently,(7.5)Ψ1=(4×π×4.0×2.03×0.5×80×10)×(12-12+12×22)=0.01047,Ψ2=(2×802π2×0.5×10×2.0)×(∑m=1100((cosh(0)sinh(4.0×π×)-∐coth(4.080.0×π×m))1m3×cos2(1080×π×m)×(cosh(0)sinh(4.0×π×m/80.0)-coth(4.080.0×π×m))))=-10.92,Ψ3=(2×4.03π2×0.5×80×10×2)×∑l=1100cos2(l×π×2.510)×(∑m=1100(2(4×l2/25+m2/400)3/2)×cos2(m×π×1080)×(cosh(0)sinh(π×4×l2/25+m2/400)-coth(π×4×l225+m2400))+(1258×l3)×(cosh(0)sinh(π×2×l/5)-coth(π×2×l5)⋃))=-1.086,Sps=0.01047+(-10.92)+(-1.086)≈-12.00.
Recalling (6.7), Λ is calculated by
(7.6)Λ=(4×π×80.5)×(16-12×8+12×82)-ln(4×sin(π×0.0075/2×10)×sin(π×(0.0075+2×2.5)/2×10))0.5=27.52.
We use (6.6), FD=86.4 for practical SI units, the productivity index (the production rate per unit pressure drawdown) in pseudo-steady-state of the given well is
(7.7)PI=86.4×2×π×(0.1×0.4)1/2×20/(5×1.25)27.52+(-12.00)=22.39(Sm3/D/MPa).Example Two
Using the formulas given by Brons and Marting, Papatzacos, Bervaldier, calculate pseudoskin factor of the well in Example One.Solution 7.
If we use Brons and Marting's pseudoskin factor formula, then(7.8)Kh=(KxKy)1/2=(0.1×0.4)1/2=0.2(μm2),hD=(HRw)(KhKv)1/2=(200.1)×(0.20.025)1/2=565.685,G(η)=2.948-7.363η+11.45η2-4.675η3=2.948-7.363×0.5+11.45×0.52-4.675×0.53=1.545,
thus from (2.4), we have
(7.9)Sps=(1η-1)[ln(hD)-G(η)]=(10.5-1)[ln(565.685)-1.545]=4.793.
If we use Papatzacos's pseudoskin factor formula, then(7.10)h1=0,Ψ1=Hh1+0.25Lp=200+0.25×10=8.0,Ψ2=Hh1+0.75Lp=200+0.75×10=2.667,
thus from (2.7), we have
(7.11)Sps=(1η-1)ln(πhD2)+(1η)ln[(η2+η)(Ψ1-1Ψ2-1)1/2]=(10.5-1)ln(π×565.6852)+(10.5)ln[(0.52+0.5)(8.0-12.667-1)1/2]=5.006.
If we use Bervaldier's pseudoskin factor formula, then(7.12)Lp=10(m),Rw=0.1(m),
thus from (2.9), we have
(7.13)Sps=(1η-1)[ln(Lp/Rw)1-Rw/Lp-1]=(10.5-1)[ln(10/0.1)1-0.1/10-1]=3.652.
But the pseudoskin factor in Example One calculated by (6.8) is
(7.14)Sps=-12.0.
Formulas (2.4), (2.7), and (2.9) cannot account for the effect of well location inside a finite drainage volume on Sps. But (6.8) is applicable to a well arbitrarily located in a box-shaped reservoir, Sps is a function of well location parameters Xw and Yw, and Sps is also a function of reservoir size parameters Xe and Ye. This is the reason why significant differences exist between Sps calculated by (6.8) and Sps calculated by (2.4), (2.7), and (2.9).Example Three
A fully penetrating vertical well is located at the center of an isotropic rectangular reservoir withXe/Ye=4, calculate the shape factor and compare with the corresponding shape factors given by Dietz and Earlougher et al.Solution 7.
Since the well is located at the center of the rectangular reservoir withXe/Ye=4, use (6.16),
(7.15)f1=XeYe=4,f2=XwXe=0.5,f3=YwYe=0.5,
then
(7.16)f4=(8π×4)×(16-0.52+0.522)=4.1888.
Use (6.15), the shape factor is
(7.17)CA=88.6657×4×sin2(π×0.5)exp(4.1888)=5.3783.
The corresponding shape factor given by Dietz [9] is CA=5.38, and CA=5.3790 given by Earlougher et al. [10]. Thus, there does not exist significant difference between the shape factor values calculated by our proposed formula and given by Dietz and Earlougher et al., which indicates that our proposed formula is reliable and reasonable accurate.
More examples are given in [1, 2] to calculate productivity index and pseudoskin factor due to partial penetration by using the proposed formulas, the values of shape factors obtained by the methods of Dietz, Earlougher, and the proposed shape factor formula are compared. The proposed formulas are shown to be reliable and reasonable accurate by the examples in [1, 2], because the proposed equations are derived by solving analytically the involved three-dimensional Laplace equation, they are a fast analytical tool to evaluate well performance in pseudo-steady-state.


## 8. Summary and Conclusions

The summary and conclusions of this paper are given below.(1)
A pseudo-steady-state productivity formula for an off-center partially penetrating vertical well in a closed box-shaped reservoir is presented.(2)
A formula for calculating pseudoskin factor due to partial penetration is presented; the pseudoskin factor of a vertical well in a box-shaped reservoir is a function of well location and reservoir size.(3)
The proposed formulas are reliable and reasonable accurate, because the proposed formulas are derived by the orthogonal decomposition of Dirac function and Green's function to Laplace equation with homogeneous Neumann boundary condition, they are a fast analytical tool to evaluate well performance in pseudo-steady-state.



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*Source: 907206-2010-04-08.xml*