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

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.

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]

𝑄_𝑤=𝐹_𝐷2𝜋𝐾𝐻𝑃_𝑎−𝑃_𝑤/(𝜇𝐵)ln𝑅_𝑒/𝑅_𝑤−3/4,(2.1) where 𝑃_𝑎 is average reservoir pressure in the circular drainage area, 𝑃_𝑤 is flowing wellbore pressure, 𝐾 is permeability, 𝐻 is payzone thickness, 𝜇 is oil viscosity, 𝐵 is oil formation volume factor, 𝑅_𝑒 is radius of circular drainage area, 𝑅_𝑤 is wellbore radius, and 𝐹_𝐷 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, 𝑆_ps. Thus, (2.1) may be rewritten to include the pseudoskin factor due to partial penetration as [4, page 92]:

𝑄_𝑤=𝐹_𝐷2𝜋𝐾𝐻𝑃_𝑎−𝑃_𝑤/(𝜇𝐵)ln𝑅_𝑒/𝑅_𝑤−3/4+𝑆_ps.(2.2)

𝑆_ps 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,) 𝐿_𝑝=𝐿, and define partial penetration factor 𝜂:

𝜂=𝐿^𝑝𝐻=𝐻𝐿.(2.3)

Pseudoskin factor formula given by Brons and Marting is [5]

𝑆_ps=1𝜂−1lnℎ^𝐷−𝐺(𝜂),(2.4) where

ℎ_𝐷=𝐻𝑅_𝑤𝐾𝐾^ℎ_𝑣^1/2𝐺(𝜂)=2.948−7.363𝜂+11.45𝜂₂,(2.5)−4.675𝜂³.(2.6)

Pseudoskin factor formula given by Papatzacos is [6]

𝑆_ps=1𝜂−1ln𝜋ℎ2^𝐷+1𝜂ln2+𝜂𝜂Ψ¹−1Ψ₂−1^1/2,(2.7) where ℎ_𝐷 has the same meaning as in (2.5), and

Ψ₁=𝐻ℎ₁+0.25𝐿_𝑝Ψ₂,=𝐻ℎ₁+0.75𝐿_𝑝,(2.8) 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]

𝑆_ps=1𝜂−1ln𝐿^𝑝/𝑅_𝑤1−𝑅_𝑤/𝐿_𝑝−1.(2.9)

It must be pointed out that the well location in the reservoir has no effect on 𝑆_ps 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 calculate 𝑆_ps 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:

𝑄_𝑤=𝐹_𝐷2𝜋𝐾𝐻𝑃_𝑎−𝑃_𝑤/(𝜇𝐵)(1/2)ln2.2458𝐴/𝐶_𝐴𝑅^2𝑤,(2.10) where 𝐶_𝐴 is shape factor, and 𝐴 is drainage area.

Dietz evaluated shape factor 𝐶_𝐴 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

Figure 1 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 constant 𝐾_𝑥,𝐾_𝑦,𝐾_𝑧 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 time 𝑡=0, pressure is uniformly distributed in the reservoir, equal to the initial pressure 𝑃_𝑖.(3)The production occurs through a partially penetrating vertical well of radius 𝑅_𝑤, represented in the model by a uniform line sink.(4)A single phase fluid, of small and constant compressibility 𝐶_𝑓, constant viscosity 𝜇, and formation volume factor 𝐵, flows from the reservoir to the well at a constant rate 𝑄_𝑤. 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:

𝐾_𝑥𝜕²𝑃𝜕𝑥²+𝐾_𝑦𝜕²𝑃𝜕𝑦²+𝐾_𝑧𝜕²𝑃𝜕𝑧²=𝜙𝜇𝐶_𝑡𝜕𝑃𝜕𝑡+𝜇𝑞𝐵𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^,(3.1) in the box-shaped drainage volume:

Ω=(0,𝑎)×(0,𝑏)×(0,𝐻),(3.2) and we always assume

𝑏≥𝑎≫𝐻,(3.3) 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,

𝜕𝑃𝜕𝑁|||_Γ=0,(3.4) 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

𝑃||_𝑡=0=𝑃_𝑖.(3.5)

Define average permeability:

𝐾_𝑎=𝐾_𝑥𝐾_𝑦𝐾_𝑧^1/3.(3.6)

In order to simplify (3.1), we take the following dimensionless transforms:

𝑥_𝐷=𝑥𝐿𝐾𝐾^𝑎_𝑥^1/2,𝑦_𝐷=𝑦𝐿𝐾𝐾^𝑎_𝑦^1/2,𝑧_𝐷=𝑧𝐿𝐾𝐾^𝑎_𝑧^1/2𝑎_𝐷,=𝑎𝐿𝐾𝐾^𝑎_𝑥^1/2,𝑏_𝐷=𝑏𝐿𝐾𝐾^𝑎_𝑦^1/2𝐿_𝐷,=𝐾^𝑎𝐾_𝑧^1/2,𝐻_𝐷=𝐻𝐿𝐾𝐾^𝑎_𝑧^1/2𝑡_𝐷,=𝐾^𝑎𝑡𝜙𝜇𝐶_𝑡𝐿².(3.7)

The dimensionless wellbore radius is [8]

𝑅_𝑤𝐷=𝐾^𝑧/√𝐾_𝑥𝐾_𝑦^1/6𝐾^𝑥/𝐾_𝑦^1/4+𝐾_𝑦/𝐾_𝑥^1/4𝑅^𝑤2𝐿.(3.8)

Assume that 𝑞 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 𝑄_𝑤, and there holds [8]

𝑞=𝑄^𝑤𝐿_𝑝𝐷=𝑄^𝑤𝐿_𝐷.(3.9)

Dimensionless pressures are defined by

𝑃_𝐷=𝐾^𝑎𝐿𝑃_𝑖−𝑃𝑃𝜇𝑞𝐵,(3.10)_𝑤𝐷=𝐾^𝑎𝐿𝑃_𝑖−𝑃_𝑤𝜇𝑞𝐵.(3.11)

Then (3.1) becomes

𝜕𝑃_𝐷𝜕𝑡_𝐷−𝜕²𝑃_𝐷𝜕𝑥^2𝐷+𝜕²𝑃_𝐷𝜕𝑦^2𝐷+𝜕²𝑃_𝐷𝜕𝑧^2𝐷=𝛿𝑥^𝐷−𝑥^𝐷𝛿𝑦_𝐷−𝑦^𝐷𝛿𝑧_𝐷−𝑧^𝐷,(3.12) in the dimensionless box-shaped drainage volume

Ω_𝐷=0,𝑎_𝐷×0,𝑏_𝐷×0,𝐻_𝐷,(3.13) with boundary condition

𝜕𝑃_𝐷𝜕𝑁_𝐷||||_Γ𝐷=0,(3.14) and initial condition

𝑃_𝐷||_𝑡𝐷=0=0.(3.15)

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

𝜕𝑃𝜕𝑡−𝜕²𝑃𝜕𝑥²+𝜕²𝑃𝜕𝑦²+𝜕²𝑃𝜕𝑧²=𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^.(4.1)

Rewrite (3.14) below

𝜕𝑃𝜕𝑁|||_Γ=0,(4.2) and (3.15) becomes

𝑃||_𝑡=0=0.(4.3)

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 {𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)} [11, 12]:

𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)=1𝑎𝑏𝐻𝑑_𝑙𝑑_𝑚𝑑_𝑛cos𝑙𝜋𝑥𝑎cos𝑚𝑏𝜋𝑦cos𝑛𝐻𝜋𝑧,(4.4) where 𝑙,𝑚,𝑛 are nonnegative numbers, and

𝑑_𝑙=⎧⎪⎨⎪⎩11if𝑙=0,2if𝑙>0,(4.5) and 𝑑_𝑚,𝑑_𝑛 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]):

𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^=^∞_𝑙,𝑚_,𝑛=0𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^.(4.6)

In order to simplify the following derivations, we define the following notation:

^∞_𝑙,_𝑚,𝑛=0𝐹_𝑙𝑚𝑛(𝑥,𝑦,𝑧)=^∞_∞𝑙=0_∞𝑚=0_𝑛=0𝐹_𝑙𝑚𝑛(𝑥,𝑦,𝑧),(4.7) 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

_𝑙+𝑚_+𝑛>0𝐹_𝑙𝑚𝑛(𝑥,𝑦,𝑧)=^∞_∞𝑙≥0_∞𝑚≥0_𝑛≥0𝐹_𝑙𝑚𝑛(𝑥,𝑦,𝑧)(𝑙+𝑚+𝑛>0),(4.8) 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

𝑃𝑡,𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^=^∞_𝑙,𝑚_,𝑛=0𝑒_𝑙𝑚𝑛(𝑡)𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧),(4.9) where 𝑒_𝑙𝑚𝑛(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

^∞_𝑙,_𝑚,𝑛=0𝜕𝑒^𝑙𝑚𝑛(𝑡)𝜕𝑡𝑔^𝑙𝑚𝑛=(𝑥,𝑦,𝑧)−𝑒_𝑙𝑚𝑛(𝑡)Δ𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)_∞_𝑙,𝑚_,𝑛=0𝜕𝑒^𝑙𝑚𝑛(𝑡)𝜕𝑡+𝑒^𝑙𝑚𝑛(𝑡)𝜆_𝑙𝑚𝑛𝑔^𝑙𝑚𝑛=(𝑥,𝑦,𝑧)_∞𝑙,𝑚_,𝑛=0𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧),(4.10) where Δ is the three-dimensional Laplace operator

Δ=𝜕²𝜕𝑥²+𝜕²𝜕𝑦²+𝜕²𝜕𝑧²𝜆,(4.11)_𝑙𝑚𝑛=𝑙𝜋𝑎²+𝑚𝜋𝑏²+𝑛𝜋𝐻².(4.12)

From (4.3) and (4.9),

𝑒_𝑙𝑚𝑛(0)=0,(4.13) compare the coefficients of 𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧) at both sides of (4.10), we obtain

𝜕𝑒_𝑙𝑚𝑛(𝑡)𝜕𝑡+𝜆^𝑙𝑚𝑛𝑒_𝑙𝑚𝑛(𝑡)=𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^,(4.14) because 𝜆₀₀₀=0, from (4.14),

𝑒₀₀₀(𝑡)==𝑔₀₀₀𝑡𝑥^,𝑦^,𝑧^𝑡√𝑎𝑏𝐻.(4.15)

When 𝜆_𝑙𝑚𝑛≠0(𝑙+𝑚+𝑛>0), solve (4.14),

𝑒_𝑙𝑚𝑛(𝑡)=1−exp−𝜆^𝑙𝑚𝑛𝑡𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝜆_𝑙𝑚𝑛.(4.16)

Substitute (4.15) and (4.16) into (4.9) and obtain

𝑃𝑡,𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^=^∞_𝑙,𝑚_,𝑛=0=𝑡𝑒_𝑙𝑚𝑛(𝑡)𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)√𝑎𝑏𝐻𝑔⁰⁰⁰+(𝑥,𝑦,𝑧)_{𝑙+𝑚+𝑛>0}1−exp−𝜆_𝑙𝑚𝑛𝑡𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆_𝑙𝑚𝑛=𝑡𝑎𝑏𝐻+_{𝑙+𝑚+𝑛>0}𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆_𝑙𝑚𝑛−_{𝑙+𝑚+𝑛>0}exp−𝜆_𝑙𝑚𝑛𝑡𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆_𝑙𝑚𝑛.(4.17)

Define

𝐼₁=𝑡𝐼𝑎𝑏𝐻,(4.18)₂==Ψ𝑥,𝑦_{𝑙+𝑚+𝑛>0},𝑧;𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆_𝑙𝑚𝑛𝐼,(4.19)₃=_{𝑙+𝑚+𝑛>0}exp−𝜆_𝑙𝑚𝑛𝑡𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆_𝑙𝑚𝑛,(4.20) then

𝑃𝑡,𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^=𝐼₁+𝐼₂−𝐼₃.(4.21)

Recall (4.19), the average value of Ψ throughout of the total volume of the box-shaped reservoir is

Ψ_𝑎,𝑣=1𝑉_Ω=1Ψ(𝑥,𝑦,𝑧)𝑑𝑉𝑉^𝑎0^𝑏0^𝐻0=1Ψ𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^𝑑𝑥𝑑𝑦𝑑𝑧𝑉𝑔^𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝜆_𝑙𝑚𝑛^𝑎0^𝑏0^𝐻0_{𝑙+𝑚+𝑛>0}𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑑𝑥𝑑𝑦𝑑𝑧.(4.22)

Note that 𝑙+𝑚+𝑛>0 implies that at least one of 𝑙,𝑚,𝑛 must be greater than 0, without losing generality, we may assume

𝑙>0,(4.23) then

^𝑎0cos𝑙𝜋𝑥𝑎𝑑𝑥=0.(4.24) So,

^𝑎0^𝑏0^𝐻0_{𝑙+𝑚+𝑛>0}𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑑𝑥𝑑𝑦𝑑𝑧=0,(4.25) consequently,

Ψ_𝑎,𝑣=0.(4.26)

If time 𝑡 is sufficiently long, pseudo-steady-state is reached, 𝐼₃ decreases by exponential law, 𝐼₃ will vanish, that is,

𝐼₃≈0,(4.27) then

𝑃𝑡,𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^=𝑡𝑎𝑏𝐻+Ψ𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^.(4.28)

Substituting (4.28) into (4.1), we have

1𝑎𝑏𝐻−ΔΨ=𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^.(4.29)

Define

𝑓(𝑥,𝑦,𝑧)==−−ΔΨ1𝑎𝑏𝐻+𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^,(4.30) note that Ψ is equal to 𝐼₂ in (4.19), and

𝜕Ψ𝜕𝑁=0,onΓ.(4.31)

From Green's Formula [15],

0=_Γ𝜕Ψ𝜕𝑁𝑑𝑆=_ΩΔΨ𝑑𝑉=−_Ω𝑓(𝑥,𝑦,𝑧)𝑑𝑉,(4.32) that is,

_Ω𝑓(𝑥,𝑦,𝑧)𝑑𝑉=0,(4.33) where 𝑉 is volume of drainage domain Ω.

Define the following notation of internal product of functions 𝑓(𝑥,𝑦,𝑧) and 𝑔(𝑥,𝑦,𝑧):

⌊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,(4.34) 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

⌊𝑓(𝑥,𝑦,𝑧),1⌋=0,(4.35) and it is easy to prove

⌊𝑓(𝑥,𝑦,𝑧),𝑔₀₀₀⌋=0,(4.36) where 𝑔₀₀₀ means 𝑔_𝑙𝑚𝑛 when 𝑙=𝑚=𝑛=0.

Thus, 𝑓(𝑥,𝑦,𝑧) can be decomposed as [13, 14]:

𝑓(𝑥,𝑦,𝑧)=^∞_𝑙,𝑚_,𝑛=0𝑓,𝑔_𝑙𝑚𝑛𝑥^=,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)_{𝑙+𝑚+𝑛>0}=𝑓,𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)_{𝑙+𝑚+𝑛>0}=𝛿𝑥−𝑥^𝛿𝑦−𝑦^𝛿𝑧−𝑧^,𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧)_{𝑙+𝑚+𝑛>0}𝑔_𝑙𝑚𝑛𝑥^,𝑦^,𝑧^𝑔_𝑙𝑚𝑛(𝑥,𝑦,𝑧).(4.37)

The drainage volume is

𝑉=𝑎𝑏𝐻.(4.38)

Recall (4.28), the average pressure throughout the reservoir is

𝑃_𝑎,𝑣=1𝑉_Ω𝑃(𝑥,𝑦,𝑧)𝑑𝑥𝑑𝑦𝑑𝑧=𝑡𝑎𝑏𝐻+Ψ^𝑎,𝑣.(4.39)

The wellbore pressure at point (𝑥_𝑤,𝑦_𝑤,𝑧_𝑤) is

𝑃_𝑤=𝑡𝑎𝑏𝐻+Ψ^𝑤,(4.40) where Ψ_𝑤 is the value of Ψ at wellbore point (𝑥_𝑤,𝑦_𝑤,𝑧_𝑤).

Combining (4.39) and (4.40) gives

𝑃_𝑎,𝑣−𝑃_𝑤=Ψ_𝑎,𝑣−Ψ_𝑤,(4.41) which implies 𝑃_𝑎,𝑣−𝑃_𝑤 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 subscript 𝐷.

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

𝐽𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^;𝑙,𝑚,𝑛=^𝐿0Ψ𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^𝑑𝑧^=_{𝑙+𝑚+𝑛>0}ℑ_𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^;𝑙,𝑚,𝑛,(5.1) where

_𝑙+𝑚_+𝑛>0=ℑ_𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^;𝑙,𝑚,𝑛_{𝑙+𝑚+𝑛>0}×cos𝑚𝜋𝑎𝑏𝑦𝐻𝑑_𝑙𝑑1_𝑚𝑑_𝑛𝜆_𝑙𝑚𝑛cos𝑙𝜋𝑎𝑥×cos𝑚𝑏𝜋𝑦_cos𝑛𝐻𝜋𝑧𝑏cos𝑙𝜋𝑎𝑥^^𝐿0cos𝑛𝜋𝑧^𝐻𝑑𝑧^=_{𝑙+𝑚+𝑛>0}×⎧⎪⎨⎪⎩𝐻𝑎𝑏𝐻𝑑_𝑙𝑑1_𝑚𝑑_𝑛𝜆_𝑙𝑚𝑛cos𝑙𝜋𝑎𝑥cos𝑚𝑏𝜋𝑦cos𝑛𝐻𝜋𝑧𝐿𝜋c𝑛osco𝑚s𝜋𝑦𝑙_𝜋𝑎𝑥^cos𝑚𝜋𝑏𝑦^sin𝑛𝐻𝜋𝐿if𝑙≠0,𝑏cos𝑙𝜋𝑎𝑥^if𝑙=0.(5.2)

Define

𝐶𝐶={(𝑙,𝑚,𝑛)∶𝑙+𝑚+𝑛>0},(5.3)₁𝐶={(𝑙,𝑚,𝑛)∶𝑙=𝑚=0,𝑛>0},(5.4)₂𝐶={(𝑙,𝑚,𝑛)∶𝑙=0,𝑚>0,𝑛≥0},(5.5)₃={(𝑙,𝑚,𝑛)∶𝑙>0,𝑚≥0,𝑛≥0},(5.6) then it is easy to prove

𝐶₁𝐶=𝐶₁∪𝐶₂∪𝐶₃,∩𝐶₂=∅,𝐶₂∩𝐶₃=∅,𝐶₃∩𝐶₁=∅.(5.7)

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

𝐽==_{𝑙+𝑚+𝑛>0}ℑ_𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥^,𝑦^,𝑧^;𝑙,𝑚,𝑛_∞_𝑛=1ℑ_00𝑛+^∞_∞_𝑚=1_𝑛=0ℑ_0𝑚𝑛+^∞_∞𝑙=1_∞𝑚=0_𝑛=0ℑ_𝑙𝑚𝑛.(5.8)

Define the following notations:

𝐽_𝑧=^∞_𝑛=1𝐽ℑ_00𝑛,(5.9)_𝑦𝑧=^∞_∞_𝑚=1_𝑛=0𝐽ℑ_0𝑚𝑛,(5.10)_𝑥𝑦𝑧=^∞_∞𝑙=1_∞𝑚=0_𝑛=0ℑ_𝑙𝑚𝑛,(5.11) so

𝐽=𝐽_𝑧+𝐽_𝑦𝑧+𝐽_𝑥𝑦𝑧,(5.12) and the average value of 𝐽 at wellbore can be written as

𝐽_𝑎,𝑤=𝐽_𝑧,𝑎_𝑤+𝐽_{𝑦𝑧,𝑎}_𝑤+𝐽_{𝑥𝑦𝑧,𝑎}_𝑤.(5.13)

Rearrange (4.12) and obtain

𝜆_𝑙𝑚𝑛=𝑙𝜋𝑎²+𝑚𝜋𝑏²+𝑛𝜋𝐻²=𝜋𝐻²𝑛²+𝜇^2𝑙𝑚,(5.14) where

𝜇^2𝑙𝑚=𝑙𝐻𝑎²+𝑚𝐻𝑏²=𝐻𝑏²𝑚²+𝑙𝑏𝜇_𝑙0𝑎²,=𝑙𝐻𝜆_𝑙𝑚0𝑎,=𝜋𝐻²𝜆_0𝑚𝑛𝜇^2𝑙𝑚,=𝑚𝜋𝑏²+𝑛𝜋𝐻²=𝜋𝐻²𝑛²+𝑚𝐻𝜆_00𝑛𝑏²,=𝑛²𝜋²𝐻².(5.15)

There hold [16, page 47]

^∞_𝑛=1sin(𝑛𝑥)𝑛³=𝜋²𝑥6−𝜋4𝑥²+1𝑥2³(0≤𝑥≤2𝜋),(5.16)_∞_𝑛=11−cos(𝑛𝑥)𝑛⁴=𝜋²𝑥²12−𝜋1𝑥2³+4𝑥8⁴(0≤𝑥≤2𝜋).(5.17)

Recall (5.4) and (5.9), 𝐽_𝑧 is for the case 𝑙=𝑚=0,𝑛>0, and at wellbore of the off-center well,

𝐽𝑦=𝑦^≠0,𝑥^≠0,𝑥=𝑥^+𝑅_𝑤,0≤𝑧=𝑧^≤𝐿,_𝑧_𝑤=^∞_𝑛=1𝑎𝑏𝐻𝑑1_𝑛𝜆_00𝑛cos𝑛𝐻𝜋𝑧^𝐿0cos𝑛𝐻𝜋𝑧^𝑑𝑧^=2=𝑎2𝑏𝐻𝐻₂^∞_𝑛=1𝜋𝐻²𝑛²²cos𝑛𝐻𝜋𝑧𝑛𝐻𝜋sin𝑛𝐻𝜋𝐿𝑎𝑏𝜋³^∞_𝑛=1𝑛1₃sin𝑛𝐻𝜋𝐿cos𝑛𝐻𝜋𝑧.(5.18)

The average value of (𝐽_𝑧)_𝑤 along the well length is

𝐽_𝑧,𝑎_𝑤==𝐿11^𝐿0𝐽_𝑧𝑑𝑧𝐿^∞_𝑛=1𝜋2³𝑎𝐻𝑏²𝑛³sin𝑛𝐻𝜋𝐿^𝐿0cos𝑛𝜋𝑧=𝐻𝑑𝑧_∞=_𝑛=1𝜋³2𝑎𝐻𝑏𝐿²𝑛³sin𝑛𝐻𝜋𝐿𝑛𝐻𝜋sin𝑛𝐻𝜋𝐿_∞=_𝑛=1𝐻𝜋⁴₃2𝑎𝐻𝑏𝐿³𝑛⁴sin²𝑛𝐻𝜋𝐿=𝜋⁴𝐻𝑎𝑏𝐿^∞_𝑛=1𝑛1₄1−cos2𝑛𝐻𝜋𝐿₃=4𝜋𝐻⁴𝑎𝐿𝑏𝐿2𝐻𝜋𝐿²1𝜋2²−1𝜋22𝐻𝜋𝐿+4182𝐻𝜋𝐿²=2𝐻𝑎𝑏𝐿112−6𝐿𝐻+12𝐿𝐻²²3𝑎𝑏12−𝐻𝐿+2𝐿𝐻²²,(5.19) where we have used (5.17).

For a fully penetrating well, 𝐿=𝐻, then

𝐽_𝑧,𝑎_𝑤=0.(5.20)

Recall (5.5) and (5.10), 𝐽_𝑦𝑧 is for the case 𝑙=0,𝑚>0,𝑛≥0, and at wellbore of the off-center well,

𝐽𝑦=𝑦^≠0,𝑥^≠0,𝑥=𝑥^+𝑅_𝑤,0≤𝑧=𝑧^≤𝐿,_𝑦𝑧_𝑤=𝑎𝑏1𝐻^∞_∞𝑚=1_𝑛=0𝑑_𝑚𝑑_𝑛1𝜆_0𝑚𝑛cos²𝑚𝜋𝑏𝑦^cos𝑛𝐻𝜋𝑧^𝐿0cos𝑛𝐻𝜋𝑧^𝑑𝑧^=2𝑎𝑏𝐻^∞_∞𝑚=1_𝑛=0cos²𝑚𝜋𝑦^/𝑏cos(𝑛𝜋𝑧/𝐻)𝜋²𝑑_𝑛(𝑛/𝐻)²+(𝑚/𝑏)²^𝐿0cos𝑛𝜋𝑧^=2𝐻𝑑𝑧^𝑎𝑏𝐻^∞_𝑚=1^∞_𝑛=12(𝐻/𝑛𝜋)cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏+cos₂𝜋²(𝑛/𝐻)²+(𝑚/𝑏)²𝑚𝜋𝑏𝑦^𝑏²𝐿=2𝐻₃𝜋²𝑚²𝜋³𝑎𝑏𝐻^∞_𝑚=1^∞_𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏+cos₂𝑛𝑛²+(𝑚𝐻/𝑏)²𝑚𝜋𝑏𝑦^𝑏²𝐿𝜋=2𝐻₂𝐻³𝑚²+𝜋³2𝑎𝐻𝑏₂𝜋𝐻𝐿𝑏²³^∞_𝑚=1𝑚1₂cos²𝑚𝜋𝑏𝑦^𝜋³𝑎𝑏^∞_∞𝑚=1_𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏=2𝑏𝐿𝑛𝑛²+(𝑚𝐻/𝑏)²+𝜋²2𝑎𝐻𝐻₂^∞_𝑚=1𝑚1₂cos²𝑚𝜋𝑏𝑦^𝜋³𝑎𝑏^∞_∞𝑚=1_𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏𝑛𝑛²+(𝑚𝐻/𝑏)²,(5.21) where we use the following formulas [16, page 47]:

^∞_𝑚=1𝑚1₂cos(𝑚𝑥)=𝜋6²−𝜋2𝑥+𝑥4²(0≤𝑥≤2𝜋),(5.22)_∞𝑚₌₁𝑚1₂cos²(𝑚𝑥)=𝜋²6−𝜋2𝑥+𝑥2²(0≤𝑥≤𝜋).(5.23)

The average value of (𝐽_𝑦𝑧)_𝑤 along the well length is

𝐽_{𝑦𝑧,𝑎}_𝑤==2𝐿1𝑏𝐿^𝐿0𝐽_𝑦𝑧𝑑𝑧+𝑎2𝐻𝐻₂61−2𝑦𝑏^+2𝑦𝑏^22𝑎𝑏𝐿𝜋³^∞_∞𝑚=1_𝑛=12sin(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏𝑛𝑛²+(𝑚𝐻/𝑏)²^𝐿0cos𝑛𝜋𝑧=2𝑏𝐿𝐻𝑑𝑧+𝑎𝐻2𝐻₂61−2𝑦𝑏^+2𝑦𝑏^22𝑎𝑏𝐿𝜋³^∞_∞𝑚=1_𝑛=12𝐻sin²(𝑛𝜋𝐿/𝐻)cos²𝑚𝜋𝑦^/𝑏=2𝑏𝐿𝜋𝑛²𝑛²+(𝑚𝐻/𝑏)²+𝑎𝐻2𝐻₃61−2𝑦𝑏^+2𝑦𝑏^22𝑎𝑏𝐿𝜋⁴^∞_∞𝑚=1_𝑛=1[1−cos(2𝑛𝜋𝐿/𝐻)]cos²𝑚𝜋𝑦^/𝑏=2𝑏𝐿𝑛²𝑛²+(𝑚𝐻/𝑏)²𝑎𝐻61−2𝑦𝑏^+2𝑦𝑏^22+𝑎2𝑏𝐻𝐿𝜋³⁴^∞_∞𝑚=1_𝑛=1𝑚𝑏𝐻²cos²×1−cos2𝑛𝜋𝐿𝑚𝜋𝑏𝑦^𝐻,𝑛1²−1=2𝑏𝐿𝑛²+(𝑚𝐻/𝑏)²𝑎𝐻61−2𝑦𝑏^+2𝑦𝑏^22+2𝑎𝐻𝑏𝐿³𝜋⁴^∞_𝑚=1𝑚𝑏𝐻²cos²×_∞𝑚𝜋𝑏𝑦^_𝑛=1𝑛1²−cos(2𝑛𝜋𝐿/𝐻)𝑛²−1𝑛²+(𝑚𝐻/𝑏)²+cos(2𝑛𝜋𝐿/𝐻)=2𝑏𝐿𝑛²+(𝑚𝐻/𝑏)²𝑎𝐻61−2𝑦𝑏^+2𝑦𝑏^22+𝑎2𝑏𝐻𝐿𝜋³⁴^∞_𝑚=1𝑚𝑏𝐻²cos²×𝜋₂𝑚𝜋𝑏𝑦^−6−𝑏𝜋𝜋6²−𝜋22𝐻𝜋𝐿+412𝐻𝜋𝐿²+2𝑚𝑏𝜋𝐻coth𝑚𝐻𝑏𝜋−12𝑚𝑏𝐻²2𝑚𝐻cosh[(𝑚sin𝐻h𝜋(/𝑚𝑏)𝐻(1𝜋−/𝑏)2𝐿/𝐻)]−12𝑚𝑏𝐻²,(5.24) where we use the following formulas [16, page 47]:

^∞_𝑛=1cos(𝑛𝑥)𝑛²+𝛽²=𝜋2𝛽coshsin[𝛽h((𝜋𝛽𝜋−)𝑥)]−21𝛽²(0≤𝑥≤2𝜋),(5.25)_∞_𝑛=11𝑛²+𝛽²=𝜋2𝛽coth(𝛽𝜋)−21𝛽²(0≤𝑥≤2𝜋),(5.26) and we may simplify (5.24) further

𝐽_{𝑦𝑧,𝑎}_𝑤=2𝑎𝑏𝐻𝐿16−2𝑦𝑏^+𝑦^22𝑏²+𝑎2𝑏𝐻𝐿𝜋³⁴^∞_𝑚=1cos²𝑚𝜋𝑏𝑦^𝑚𝑏𝐻²×𝜋²𝐿𝐻−𝜋²𝐿²𝐻²+𝑏𝜋=2𝑏𝐿2𝑚𝐻cosh[(𝑚sin𝐻h𝜋(/𝑚𝑏)𝐻(1𝜋−/𝑏)2𝐿/𝐻)]−coth𝑚𝐻𝑏𝜋𝑎𝐻16−2𝑦𝑏^+2𝑦𝑏^22+𝑎2𝑏𝐻𝐿𝜋³⁴^∞_𝑚=1cos²×⎧⎪⎨⎪⎩𝜋₂𝑚𝜋𝑏𝑦^𝑚1²𝐿𝑏²𝐻³−𝜋²𝐿²𝑏²𝐻⁴+𝑏³𝜋×cosh[(𝑚𝐻𝜋/𝑏)(12−𝑚2𝐻𝐿³/𝐻)]=2𝑏𝐿sinh(𝑚𝐻𝜋/𝑏)−coth𝑚𝐻𝑏𝜋𝑎𝐻16−2𝑦𝑏^+2𝑦𝑏^22+𝑎2𝜋𝑏²1−𝐻𝐿𝜋6²−𝜋²𝑦^2𝑏+𝜋²𝑦^2′+𝑏₂2𝑏²𝑎𝐿𝜋³^∞_𝑚=1cos²×cosh[(𝑚𝐻𝜋/𝑏)(1𝑚−𝜋𝑏2𝑦𝐿^/𝐻)𝑚]1³=2𝑏sinh(𝑚𝐻𝜋/𝑏)−coth𝑚𝐻𝑏𝜋+𝑎𝑏₂16−2𝑦𝑏^+2𝑦𝑏^22𝑎𝐿𝜋³^∞_𝑚=1cos²𝑚𝜋𝑦^/𝑏𝑚³cosh[(𝑚sin𝐻h𝜋(/𝑚𝑏)𝐻(1𝜋−/𝑏)2𝐿/𝐻)]−coth𝑚𝐻𝑏𝜋.(5.27)

For a fully penetrating well, 𝐿=𝐻, then

𝐽_{𝑦𝑧,𝑎}_𝑤=2𝑎𝑏16−2𝑦𝑏^+𝑦^22𝑏².(5.28)

Define

𝑓(𝑥)=sinh[𝛼(1−𝑥)]sinh(𝛼𝑥),(5.29) since the derivative of 𝑓(𝑥) is

𝑓^(𝑥)==𝛼𝛼csionshh[(𝛼𝛼𝑥(1)s−in2h𝑥)[]𝛼,(1−𝑥)]−𝛼cosh[𝛼(1−𝑥)]sinh(𝛼𝑥)(5.30) consequently,

𝑓^12=0.(5.31)

When 𝑥=0 and 𝑥=1,

𝑓(0)=𝑓(1)=0.(5.32)

When 𝑥=1/2,𝑓(𝑥) reaches maximum value, let

𝑥=𝐿𝐻,(5.33) and the producing length 𝐿 is a variable, define

𝐹(𝐿)=cosh[𝛽𝜋(1−2𝐿/𝐻)]−cosh(𝛽𝜋)=−2sinh[𝛽𝜋(s1in−h𝐿(𝛽/𝐻𝜋))]sinh[𝛽𝜋𝐿/(𝐻)]sinh(𝛽𝜋),(5.34) thus when 𝐿=𝐻/2,|𝐹(𝐿)| reaches maximum value,

||𝐹(𝐿)||_max=|||𝐹𝐻=2sinh2₂|||(𝛽𝜋/2)=sinh2(s𝛽in𝜋h)₂(𝛽𝜋/2)=2sisnihnh(𝛽(𝜋𝛽𝜋/2/)2)cosh(𝛽𝜋/2)cosh(𝛽𝜋/2)<1,(5.35) so 𝐹(𝐿) is a bounded function, let

𝛽=𝑚𝐻𝑏,(5.36) then

𝐽_{𝑦𝑧,𝑎}_𝑤=2𝑎𝑏16−2𝑦𝑏^+𝑦^2×_∞2𝑏²+𝑎𝐿𝑏²𝜋³_𝑚₌₁cos²𝑚𝜋𝑦^/𝑏=2𝑏𝑚³cosh[(𝑚sin𝐻h𝜋(/𝑚𝑏)𝐻(1𝜋−/𝑏)2𝐿/𝐻)]−coth𝑚𝐻𝑏𝜋×𝑎_∞16−2𝑦𝑏^+2𝑦𝑏^22+𝑎𝐿𝑏²𝜋³_𝑚₌₁cos²𝑚𝜋𝑦^/𝑏≈2𝑏𝑚³−2sinh[(𝑚𝐻𝜋s/i𝑏n)h(1(𝑚−𝐻𝐿𝜋/𝐻/𝑏))]sinh(𝑚𝐿𝜋/𝑏)×_𝑀𝑎16−2𝑦𝑏^+2𝑦𝑏^22+𝑎𝐿𝑏²𝜋³_𝑚₌₁cos²𝑚𝜋𝑦^/𝑏𝑚³−2sinh[(𝑚𝐻𝜋s/i𝑏n)h(1(𝑚−𝐻𝐿𝜋/𝐻/𝑏))]sinh(𝑚𝐿𝜋/𝑏).(5.37)

Since 0<𝐿/𝐻<1, from (5.34) and (5.35), there holds

^∞_𝑚=101|||||cos₂𝑚𝜋𝑦^/𝑏≤𝑚³−2sinh[(𝑚𝐻𝜋s/i𝑏n)h(1(𝑚−𝐻𝐿𝜋/𝐻/𝑏)_∞)]sinh(𝑚𝐿𝜋/𝑏)|||||_𝑚₌₁₀₁1𝑚³=𝜁(3)−¹⁰⁰_𝑚₌₁1𝑚³=4.9502×10⁻⁵,(5.38) where 𝜁(3) is 𝑅𝑖𝑒𝑚𝑎𝑛𝑛-𝜁 function:

𝜁(3)=^∞_𝑚₌₁1𝑚³=1.202057,(5.39) thus

^∞_𝑚=1≈𝑚1₃2sinh[(𝑚𝐻𝜋/s𝑏i)n(h1(−𝑚𝐻𝐿/𝜋𝐻/𝑏))₁₀₀]sinh(𝑚𝐿𝜋/𝑏)_𝑚₌₁𝑚1₃2sinh[(𝑚𝐻𝜋/s𝑏i)n(h1(−𝑚𝐻𝐿/𝜋𝐻/𝑏))]sinh(𝑚𝐿𝜋/𝑏).(5.40)

So, in (5.37), 𝑀=100 is sufficient to reach engineering accuracy.

Recall (5.6) and (5.11), 𝐽_𝑥𝑦𝑧 is for the case 𝑙>0,𝑚≥0,𝑛≥0, and at wellbore of the off-center well,

𝑦=𝑦^≠0,𝑥^≠0,𝑥=𝑥^+𝑅_𝑤,0≤𝑧=𝑧^≤𝐿,(5.41) then

𝐽_𝑥𝑦𝑧_𝑤=×𝑎𝑏1𝐻_∞_∞𝑙=1_∞𝑚=0_𝑛=0cos(𝑛𝜋𝑧/𝐻)cos𝑙𝜋𝑥^/𝑎cos𝑙𝜋𝑥^+𝑅_𝑤/𝑎cos²𝑚𝜋𝑦^/𝑏×_𝐿0𝑑_𝑙𝑑_𝑚𝑑_𝑛𝜆_𝑙𝑚𝑛cos𝑛𝜋𝑧^=1𝐻𝑑𝑧^𝑎𝑏𝐻^∞_∞𝑙=1_𝑚=0cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎cos²𝑚𝜋𝑏𝑦^_𝑛=14(𝐻/𝑛𝜋)sin(𝑛𝑑𝜋_𝑚𝐿𝜆/𝐻_𝑙𝑚𝑛)cos(𝑛𝜋𝑧/𝐻)+𝑑_𝑚2𝜆𝐿_𝑙𝑚0.(5.42)

The average value of (𝐽_𝑥𝑦𝑧)_𝑤 along the well length is 𝐽_{𝑥𝑦𝑧,𝑎}_𝑤=𝑎𝑏1𝐻^∞_∞𝑙=1_𝑚=0cos𝑙𝜋𝑎𝑥^cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎cos²𝑚𝜋𝑏𝑦^_𝑛=14(𝐻/𝑛𝜋)sin(𝑛𝜋𝐿/𝐻)∫^𝐿0cos(𝑛𝜋𝑧/𝐻)𝑑𝑧=1𝑑_𝑚𝜆_𝑙𝑚𝑛𝐿+𝑑_𝑚2𝜆𝐿_𝑙𝑚0𝑎𝑏𝐻^∞_∞𝑙=1_𝑚=0cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎cos²𝑚𝜋𝑏𝑦^_𝑛=14(𝐻/𝑛𝜋)²sin²(𝑛𝜋𝐿/𝐻)=𝐻₄𝑑_𝑚𝜆_𝑙𝑚𝑛𝐿+𝑑_𝑚2𝜆𝐿_𝑙𝑚0𝑎𝑏𝐻𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤𝑎^∞_𝑚=0cos²×_∞𝑚𝜋𝑏𝑦^_𝑛=12[1−cos(2𝑛𝜋𝐿/𝐻)]𝑑_𝑚𝑛²𝑛²+𝜇^2𝑙𝑚𝐿+2𝜋²𝐿=𝐻₃𝑑_𝑚𝐻²𝜇^2𝑙𝑚𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎^∞_𝑚=0𝑑2_𝑚cos²𝑚𝜋𝑏𝑦^_𝑛=1𝜇1^2𝑙𝑚𝐿1−cos2𝑛𝐻𝜋𝐿×𝑛1²−1𝑛²+𝜇^2𝑙𝑚+𝜋²𝐿=𝐻₃𝐻²𝜇^2𝑙𝑚𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎^∞_𝑚=0𝑑_𝑚𝜇2^2𝑙𝑚𝐿cos²𝑚𝜋𝑏𝑦^_𝑛=1𝑛1²−cos(2𝑛𝜋𝐿/𝐻)𝑛²−1𝑛²+𝜇^2𝑙𝑚+cos(2𝑛𝜋𝐿/𝐻)𝑛²+𝜇^2𝑙𝑚+𝜋²𝐿²=𝐻₃𝐻²𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×𝜋₂𝑎^∞_𝑚=0𝑑_𝑚𝜇2^2𝑙𝑚𝐿cos²𝑚𝜋𝑏𝑦^−6−𝜋𝜋6²−𝜋22𝐻𝜋𝐿+142𝐻𝜋𝐿²2𝜇_𝑙𝑚coth𝜇^𝑙𝑚𝜋−1+𝜋2𝜇^2𝑙𝑚2𝜇_𝑙𝑚cosh𝜇^𝑙𝑚𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−2𝜇1^2𝑙𝑚+𝜋²𝐿²𝐻²,(5.43) where we use (5.22) and (5.25).

Let 𝑥=0, recast (5.26), we obtain

1𝛽²+2^∞_𝑛=11𝑛²+𝛽²=𝜋𝛽coth(𝛽𝜋),_∞_𝑛=01𝑛²+𝛽²𝑑_𝑛=𝜋𝛽coth(𝛽𝜋).(5.44) So,

𝐽_{𝑥𝑦𝑧,𝑎}_𝑤=𝐻³𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×𝜋₂𝑎^∞_𝑚=0𝑑_𝑚𝜇2^2𝑙𝑚𝐿cos²𝑚𝜋𝑏𝑦^𝐿𝐻−𝜋²𝐿²𝐻²+𝜋²𝐿²𝐻²+2𝜇𝜋_𝑙𝑚cosh𝜇^𝑙𝑚𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚=𝐻𝜋₃𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎_𝑚₌₀𝑑_𝑚𝜇𝜋^3𝑙𝑚𝐿cos²𝑚𝜋𝑏𝑦^×cosh𝜇^𝑙𝑚𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚+2𝜋𝐿𝜋²𝐿𝐻^∞_𝑚=0cos²𝑚𝜋𝑦^/𝑏=𝐻₃𝑑_𝑚𝜇^2𝑙𝑚𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎_𝑚₌₀𝑑_𝑚𝜇𝜋^3𝑙𝑚𝐿cos²𝑚𝜋𝑏𝑦^×cosh𝜇^𝑙𝑚𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚+𝑎𝑏𝜋₃𝜋𝐻³𝑙coth𝑙𝑏𝑎𝜋+𝜋𝐻²^∞_𝑚=0cos2𝑚𝜋𝑦^/𝑏=𝐻₃𝑑_𝑚𝜇^2𝑙𝑚𝑎𝑏𝐿𝜋³^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×_∞𝑎_𝑚₌₀𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑏𝑦^×cosh𝜇^𝑙𝑚𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚+𝐻𝜋₃𝑎𝑏𝜋⁴^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤+𝜋₂𝑎,𝑎𝐻𝑏𝜋³𝑙³coth𝑙𝑏𝑎𝜋𝐻^∞_𝑚=0cos2𝑚𝜋𝑦^/𝑏𝑑_𝑚𝜇^2𝑙𝑚.(5.45)

Since

𝜋𝐻²^∞_𝑚=0cos2𝑚𝜋𝑦^/𝑏=𝑏₂𝑑_𝑚𝜇^2𝑙𝑚𝐻³^∞_𝑚=0cos2𝑚𝜋𝑦^/𝑏=𝑏₂𝑑_𝑚𝑚²+(𝑏𝑙/𝑎)²𝐻³𝜋𝑏𝑎𝑙cosh𝜋𝑏𝑙1−2𝑦^/𝑏/𝑎≈𝑏sinh(𝜋𝑏𝑙/𝑎)𝐻³𝜋𝑙𝑎exp−2𝜋𝑦^𝑙𝐻₃𝑎,𝑎𝑏𝜋⁴|||||^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤𝑎𝐻𝑏³𝜋𝑙𝑎exp−2𝜋𝑦^𝑙≤1𝑎|||||𝜋³ln1−exp−2𝜋𝑦^≈0,/𝑎(5.46) thus

𝐽_{𝑥𝑦𝑧,𝑎}_𝑤≈𝐻³𝑎𝑏𝐿𝜋³^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×cosh𝜇_𝑙𝑚𝑎^∞_𝑚=0𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑏𝑦^𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚+1𝜋𝜋^∞_𝑙=11𝑙coth𝑙𝑏𝑎𝜋cos𝑙𝜋𝑎𝑥^cos𝑙𝜋𝑥^+𝑅_𝑤≈𝐻₃𝑎𝑎𝑏𝐿𝜋³^∞_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×cosh𝜇_𝑙𝑚𝑎^∞_𝑚=0𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑏𝑦^𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚−1𝜋2𝜋ln4sin𝜋2𝑅𝑎^𝑤sin𝜋2𝑥^+𝑅_𝑤2𝑎,(5.47) where we use the following formula [16, page 46]:

^∞_𝑛=1cos(𝑛𝑥)𝑛=−ln2sin𝑥2,(5.48) and the following simplifications:

coth𝑙𝑏𝜋𝑎≈1,_∞_𝑙=11𝑙coth𝑙𝑏𝑎𝜋cos𝑙𝜋𝑎𝑥^cos𝑙𝜋𝑥^+𝑅_𝑤≈_∞𝑎_𝑙=11𝑙cos𝑙𝜋𝑎𝑥^cos𝑙𝜋𝑥^+𝑅_𝑤=−1𝑎2ln4sin𝜋2𝑅𝑎^𝑤sin𝜋2𝑥^+𝑅_𝑤2𝑎.(5.49)

For a fully penetrating well, 𝐿=𝐻, (5.47) is simplified as

𝐽_{𝑥𝑦𝑧,𝑎}_𝑤=−21𝜋ln4sin𝜋2𝑅𝑎^𝑤sin𝜋2𝑥^+𝑅_𝑤2𝑎.(5.50)

Recall (5.13), then

𝐽_𝑎,𝑤=𝐽_𝑧,𝑎_𝑤+𝐽_{𝑦𝑧,𝑎}_𝑤+𝐽_{𝑥𝑦𝑧,𝑎}_𝑤=2𝐻𝐿+3𝑎𝑏𝑏₂12−𝐻𝐿+2𝐿𝐻²²+2𝑎𝑏16−2𝑦𝑏^+2𝑦𝑏^22𝑎𝐿𝜋³^𝑀_𝑚=1cos²𝑚𝜋𝑦^/𝑏+𝐻₃𝑚³cosh[(𝑚sin𝐻h𝜋(/𝑚𝑏)𝐻(1𝜋−/𝑏)2𝐿/𝐻)]−coth𝑚𝐻𝑏𝜋𝑎𝑏𝐿𝜋³^𝑁_𝑙=1cos𝑙𝜋𝑥^𝑎cos𝑙𝜋𝑥^+𝑅_𝑤×cosh𝜇_𝑙𝑚𝑎×^𝑀_𝑚=0𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑏𝑦^𝜋(1−2𝐿/𝐻)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚−1𝜋2𝜋ln4sin𝜋2𝑅𝑎^wsin𝜋2𝑥^+𝑅_𝑤2𝑎.(5.51)

Recall (4.28) and (4.40), the average wellbore pressure along the uniform line sink is

𝑃_𝑎,𝑤=𝑡𝑎𝑏𝐻+𝐽^𝑎,𝑤,(5.52) then (4.41) becomes

𝑃_𝑎,𝑣−𝑃_𝑎,𝑤=𝐽_𝑎,𝑣−𝐽_𝑎,𝑤,(5.53) which implies 𝑃_𝑎,𝑣−𝑃_𝑎,𝑤 is independent of time.

6. Productivity Formula and Shape Factor Formula

Note that (5.53) is in dimensionless form, that is,

𝑃_{𝑎,𝑣𝐷}−𝑃_{𝑎,𝑤𝐷}=𝐽_{𝑎,𝑣𝐷}−𝐽_{𝑎,𝑤𝐷}.(6.1)

Formulas (4.26), (4.41), (5.1), (5.2), and (5.53) are in dimensionless forms, recall (4.26) and obtain

Ψ_{𝑎,𝑣𝐷}=0,𝐽_{𝑎,𝑣𝐷}=0,(6.2) which implies

𝑃_{𝑎,𝑣𝐷}−𝑃_{𝑎,𝑤𝐷}=−𝐽_{𝑎,𝑤𝐷}=−𝐾^𝑎𝐿𝑃_𝑎−𝑃_𝑤𝜇𝑞𝐵.(6.3)

In order to simplify the above formulas, let

𝑌_𝑒=𝑎,𝑋_𝑒=𝑏,𝑌_𝑤=𝑥^,𝑋_𝑤=𝑦^,(6.4) then

𝑌_𝑒𝐷=𝑎_𝐷,𝑋_𝑒𝐷=𝑏_𝐷,𝑌_𝑤𝐷=𝑥^𝐷,𝑋_𝑤𝐷=𝑦^𝐷.(6.5)

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

𝑄_𝑤=𝐹_𝐷2𝜋𝐾_𝑥𝐾_𝑦^1/2𝐻𝑃_𝑎−𝑃_𝑤/(𝜇𝐵)Λ+𝑆_ps,(6.6) where 𝑃_𝑎 is average reservoir pressure throughout the box-shaped drainage volume, 𝑃_𝑤 is average wellbore pressure, and

Λ=4𝜋𝑋^𝑒𝐷𝜂𝑌_𝑒𝐷16−2𝑋𝑋^𝑤𝐷_𝑒𝐷+𝑋^2𝑤𝐷−ln4||sin𝜋2𝑌_𝑤𝐷2𝑋^2𝑒𝐷+𝑅_𝑤𝐷/2𝑌_𝑒𝐷||sin𝜋𝑅_𝑤𝐷/2𝑌_𝑒𝐷𝑆𝜂(6.7)_ps=4𝜋𝐻^𝐷𝐿_𝐷3𝜂𝑋_𝑒𝐷𝑌_𝑒𝐷21−𝜂+𝜂2²+𝜋²𝜂2𝑌𝑋^2𝑒𝐷_𝑒𝐷𝐿_𝐷^𝑀_𝑚=1cos²𝑚𝜋𝑋_𝑤𝐷/𝑋_𝑒𝐷×cosh𝑚𝜋𝐻_𝐷𝑚³/𝑋_𝑒𝐷(1−2𝜂)+si2n𝐻h_3𝐷𝑚𝜋𝐻_𝐷/𝑋_𝑒𝐷−coth𝑚𝑋𝜋𝐻^𝐷𝑒𝐷𝜋²𝜂𝑋_𝑒𝐷𝑌_𝑒𝐷𝐿_𝐷^𝑁_𝑙=1cos𝑙𝜋𝑌^𝑤𝐷𝑌_𝑒𝐷cos𝑙𝜋𝑌^𝑤𝐷+𝑅_𝑤𝐷×_𝑀𝑌_𝑒𝐷_𝑚₌₀𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑋𝑋^𝑤𝐷_𝑒𝐷cosh𝜇^𝑙𝑚𝜋(1−2𝜂)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚𝜋,(6.8) where 𝜂 is partial penetration factor defined in (2.3), 𝑆_ps is pseudoskin factor due to partial penetration, and

𝜇_𝑙𝑚=𝑙𝐻𝑎²+𝑚𝐻𝑏²^1/2.(6.9)

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

𝑆_ps=0.(6.10)

If a fully penetrating vertical well located in a closed isotropic rectangular reservoir,

𝑆_ps=0,𝐿_𝑝=𝐻,𝐾_𝑥=𝐾_𝑦=𝐾_𝑧=𝐾.(6.11)

Then (6.6) reduces to

𝑄_𝑤=2𝜋𝐾𝐻𝑃^𝑎−𝑃_𝑤/(𝜇𝐵)Θ−ln4sin𝜋𝑅_𝑤/2𝑌_𝑒sin𝜋𝑌_𝑤/𝑌_𝑒,(6.12) where

Θ=4𝜋𝑋^𝑒𝑌_𝑒61−12𝑋𝑋^𝑤_𝑒+21𝑋𝑋^𝑤_𝑒².(6.13)

Note that for a rectangle, its area is 𝐴=𝑋_𝑒𝑌_𝑒, recall (2.10), equate (2.10) to (6.12),

2𝜋𝐾𝐻𝑃_𝑎−𝑃_𝑤/(𝜇𝐵)Θ−ln4sin𝜋𝑅_𝑤/2𝑌_𝑒sin𝜋𝑌_𝑤/𝑌_𝑒=2𝜋𝐾𝐻𝑃^𝑎−𝑃_𝑤/(𝜇𝐵)(1/2)ln2.2458𝑋_𝑒𝑌_𝑒/𝐶_𝐴𝑅^2𝑤.(6.14)

A new expression to calculate the Dietz shape factor is obtained by solving 𝐶_𝐴 in (6.14),

𝐶_𝐴=88.6657𝑓¹sin²𝜋𝑓₃exp𝑓₄,(6.15) where

𝑓₁=𝑋^𝑒𝑌_𝑒,𝑓₂=𝑋^𝑤𝑋_𝑒,𝑓₃=𝑌^𝑤𝑌_𝑒𝑓,(6.16)₄=8𝜋𝑓₁16−𝑓2²+𝑓2²².(6.17)

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.

Solution 7. The average permeability is 𝐾_𝑎=(0.1×0.4×0.025)^1/3=0.1𝜇m².(7.1)
Using dimensionless transforms given by (3.7) through (3.10), we obtain 𝑋_𝑒𝐷=80010×00..11^1/2=80.0,𝑌_𝑒𝐷=20010×00..41^1/2𝑋=10.0,_𝑤𝐷=10010×00..11^1/2=10.0,𝑌_𝑤𝐷=5010×00..14^1/2𝐿=2.5,_𝐷=0.10.025^1/2=2.0,𝐻_𝐷=2010×00.0.125^1/2𝑅=4.0,_𝑤𝐷=0.025(0.1×0.4)^1/2^1/6×0.10.4^1/4+0.4𝜂=10.00.1^1/4×(20×.110)=0.0075𝜇20.0=0.5,1−2𝜂=0,_𝑙𝑚=4𝑙10²+4𝑚80²^1/2=4𝑙²25+4𝑚0²0^1/2.(7.2)
Recalling (6.8), pseudoskin factor due to partial penetration can be expressed as 𝑆_ps=Ψ₁+Ψ₂+Ψ₃,(7.3) where Ψ₁=4𝜋𝐻^𝐷𝐿_𝐷Ψ₂3𝜂𝑋_𝑒𝐷𝑌_𝑒𝐷21−𝜂+𝜂2²,=2𝑋^2𝑒𝐷𝜋²𝜂𝑌_𝑒𝐷𝐿_𝐷¹⁰⁰_𝑚=1𝑚1³cos²×cosh𝑚𝐻_𝐷𝑚𝜋𝑋𝑋^𝑤𝐷_𝑒𝐷𝜋/𝑋_𝑒𝐷(1−2𝜂)sinh𝑚𝐻_𝐷𝜋/𝑋_𝑒𝐷−coth𝑚𝐻^𝐷𝜋Ψ₃𝑋_𝑒𝐷,=2𝐻^3𝐷𝜋²𝜂𝑋_𝑒𝐷𝑌_𝑒𝐷𝐿_𝐷¹⁰⁰_𝑙=1cos𝑙𝜋𝑌^𝑤𝐷𝑌_𝑒𝐷cos𝑙𝜋𝑌^𝑤𝐷+𝑅_𝑤𝐷×₁₀₀𝑌_𝑒𝐷_𝑚₌₀𝑑_𝑚1𝜇^3𝑙𝑚cos²𝑚𝜋𝑋𝑋^𝑤𝐷_𝑒𝐷cosh𝜇^𝑙𝑚𝜋(1−2𝜂)sinh𝜇_𝑙𝑚𝜋−coth𝜇^𝑙𝑚𝜋.(7.4)
Consequently, Ψ₁=4×𝜋×4.0×2.0Ψ=0.301×0407.5,₂×80×10×21−12+2×12²=2×80²×𝜋²×0.5×10×2.0₁₀₀_𝑚₌₁𝑚1³×cos²10×cosh(0)80×𝜋×𝑚Ψ=−10.92,₃sinh(4.0×𝜋×𝑚/80.0)−coth840.0.0×𝜋×𝑚=2×4.0³𝜋²×0.5×80×10×2×¹⁰⁰_𝑙=1cos²𝑙×𝜋×2.5×10₁₀₀_𝑚₌₁24×𝑙²/25+𝑚²/400^3/2×cos²×⎛⎜⎜⎝cosh(0)𝑚×8𝜋0×10sinh𝜋×√4×𝑙²/25+𝑚²/400−coth𝜋×4×𝑙²+12525+4𝑚0²0⎞⎟⎟⎠𝑆=−1.086,_ps8×𝑙³×sinhc(o𝜋s×h(20×)𝑙/5)−coth𝜋×2×5𝑙=0.01047+(−10.92)+(−1.086)≈−12.00.(7.5)
Recalling (6.7), Λ is calculated by Λ=4×𝜋×8−ln(04.5×sin×(𝜋×610.−00275×1/28×+120)×1×8²sin(𝜋×(0.0075+2×2.5)/2×10))=27.52.0.5(7.6)
We use (6.6), 𝐹_𝐷=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 𝑃𝐼=86.4×2×𝜋×(0.1×0.4)^1/2×20/(5×1.25)27.52+(−12.00)=22.39Sm³/D/MPa.(7.7)

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 𝐾ℎ_ℎ𝐷=𝐾_𝑥𝐾_𝑦^1/2=(0.1×0.4)^1/2=0.2𝜇m²,=𝐻𝑅_𝑤𝐾𝐾^ℎ_𝑣^1/2=200.1×00.0.225^1/2𝐺(𝜂)=2.948−7.363𝜂+11.45𝜂₂=565.685,−4.675𝜂³=2.948−7.363×0.5+11.45×0.5²−4.675×0.5³=1.545,(7.8) thus from (2.4), we have 𝑆_ps=1=𝜂1−1lnℎ^𝐷−𝐺(𝜂)=4.07.953.−1[ln(565.685)−1.545](7.9)
If we use Papatzacos's pseudoskin factor formula, then Ψℎ₁=0,₁=𝐻ℎ₁+0.25𝐿_𝑝=20Ψ0+0.25×10=8.0,₂=𝐻ℎ₁+0.75𝐿_𝑝=200+0.75×10=2.667,(7.10) thus from (2.7), we have 𝑆_ps=1𝜂−1ln𝜋ℎ2^𝐷+1𝜂ln2+𝜂𝜂Ψ¹−1=1Ψ₂−1^1/2=5.00.056.−1ln𝜋×5625.685+01.5ln2+0.50.528.6.067−−11^1/2(7.11)
If we use Bervaldier's pseudoskin factor formula, then 𝐿_𝑝=10(m),𝑅_𝑤=0.1(m),(7.12) thus from (2.9), we have 𝑆_ps=1𝜂−1ln𝐿^𝑝/𝑅_𝑤1−𝑅_𝑤/𝐿_𝑝=1−1=3.06.552.−1l1n−(100./10/.110)−1(7.13)
But the pseudoskin factor in Example One calculated by (6.8) is 𝑆_ps=−12.0.(7.14)
Formulas (2.4), (2.7), and (2.9) cannot account for the effect of well location inside a finite drainage volume on 𝑆_ps. But (6.8) is applicable to a well arbitrarily located in a box-shaped reservoir, 𝑆_ps is a function of well location parameters 𝑋_𝑤 and 𝑌_𝑤, and 𝑆_ps is also a function of reservoir size parameters 𝑋_𝑒 and 𝑌_𝑒. This is the reason why significant differences exist between 𝑆_ps calculated by (6.8) and 𝑆_ps 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 with 𝑋_𝑒/𝑌_𝑒=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 with 𝑋_𝑒/𝑌_𝑒=4, use (6.16), 𝑓₁=𝑋^𝑒𝑌_𝑒=4,𝑓₂=𝑋^𝑤𝑋_𝑒=0.5,𝑓₃=𝑌^𝑤𝑌_𝑒=0.5,(7.15) then 𝑓₄=(8𝜋×4)×16−02.5+0.25²=4.1888.(7.16)
Use (6.15), the shape factor is 𝐶_𝐴=88.6657×4×sin²(𝜋×0.5)=5.3783.exp(4.1888)(7.17)
The corresponding shape factor given by Dietz [9] is 𝐶_𝐴=5.38, and 𝐶_𝐴=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.