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Mathematical Problems in Engineering
Volume 2010, Article ID 907206, 35 pages
http://dx.doi.org/10.1155/2010/907206
Research Article

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

1Department of Petroleum Engineering, The Petroleum Institute, P.O. Box 2533, Abu Dhabi, UAE
2Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, T-301 Sarkeys Energy Center, 100 East Boyd Street, Norman, OK 73019-1003, USA

Received 8 September 2009; Accepted 9 February 2010

Academic Editor: Alexander P. Seyranian

Copyright © 2010 Jing Lu and Djebbar Tiab. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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]

𝑄𝑤=𝐹𝐷𝑃2𝜋𝐾𝐻𝑎𝑃𝑤/(𝜇𝐵)𝑅ln𝑒/𝑅𝑤3/4,(2.1) 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]:

𝑄𝑤=𝐹𝐷𝑃2𝜋𝐾𝐻𝑎𝑃𝑤/(𝜇𝐵)𝑅ln𝑒/𝑅𝑤3/4+𝑆ps.(2.2)

𝑆ps can be calculated by semianalytical and semiempirical expressions presented by Brons, Marting, Papatzacos, and Bervaldier [57].

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:

𝐿𝜂=𝑝𝐻=𝐿𝐻.(2.3)

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

𝑆ps=1𝜂1ln𝐷𝐺(𝜂),(2.4) where

𝐷=𝐻𝑅𝑤𝐾𝐾𝑣1/2,𝐺(2.5)(𝜂)=2.9487.363𝜂+11.45𝜂24.675𝜂3.(2.6)

Pseudoskin factor formula given by Papatzacos is [6]

𝑆ps=1𝜂1ln𝜋𝐷2+1𝜂𝜂lnΨ2+𝜂11Ψ211/2,(2.7) where 𝐷 has the same meaning as in (2.5), and

Ψ1=𝐻1+0.25𝐿𝑝,Ψ2=𝐻1+0.75𝐿𝑝,(2.8) and 1 is the distance from the top of the reservoir to the top of the open interval.

Pseudoskin factor formula given by Bervaldier is [7]

𝑆ps=1𝜂𝐿1ln𝑝/𝑅𝑤1𝑅𝑤/𝐿𝑝1.(2.9)

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

𝑄𝑤=𝐹𝐷𝑃2𝜋𝐾𝐻𝑎𝑃𝑤/(𝜇𝐵)(𝐶1/2)ln2.2458𝐴/𝐴𝑅2𝑤,(2.10) where 𝐶𝐴 is shape factor, and 𝐴 is drainage area.

Dietz evaluated shape factor 𝐶𝐴 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 𝐿𝐻, and we assume 𝑏𝑎.

907206.fig.001
Figure 1: Partially Penetrating Vertical Well Model.

The following assumptions are made.

(1)The reservoir is homogeneous, anisotropic, and has constant 𝐾𝑥,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 time 𝑡=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 radius 𝑅𝑤, represented in the model by a uniform line sink.(4)A single phase fluid, of small and constant compressibility 𝐶𝑓, 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:

𝐾𝑥𝜕2𝑃𝜕𝑥2+𝐾𝑦𝜕2𝑃𝜕𝑦2+𝐾𝑧𝜕2𝑃𝜕𝑧2=𝜙𝜇𝐶𝑡𝜕𝑃𝜕𝑡+𝜇𝑞𝐵𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧,(3.1) in the box-shaped drainage volume:

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

𝑏𝑎𝐻,(3.3) and 𝛿(𝑥𝑥), 𝛿(𝑦𝑦), 𝛿(𝑧𝑧) are Dirac functions.

All the boundaries of the box-shaped drainage volume are sealed, that is,

𝜕𝑃|||𝜕𝑁Γ=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=𝑃𝑖.(3.5)

Define average permeability:

𝐾𝑎=𝐾𝑥𝐾𝑦𝐾𝑧1/3.(3.6)

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

𝑥𝐷=𝑥𝐿𝐾𝑎𝐾𝑥1/2,𝑦𝐷=𝑦𝐿𝐾𝑎𝐾𝑦1/2,𝑧𝐷=𝑧𝐿𝐾𝑎𝐾𝑧1/2,𝑎𝐷=𝑎𝐿𝐾𝑎𝐾𝑥1/2,𝑏𝐷=𝑏𝐿𝐾𝑎𝐾𝑦1/2,𝐿𝐷=𝐾𝑎𝐾𝑧1/2,𝐻𝐷=𝐻𝐿𝐾𝑎𝐾𝑧1/2,𝑡𝐷=𝐾𝑎𝑡𝜙𝜇𝐶𝑡𝐿2.(3.7)

The dimensionless wellbore radius is [8]

𝑅𝑤𝐷=𝐾𝑧/𝐾𝑥𝐾𝑦1/6𝐾𝑥/𝐾𝑦1/4+𝐾𝑦/𝐾𝑥1/4𝑅𝑤.2𝐿(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 𝑄𝑤, and there holds [8]

𝑄𝑞=𝑤𝐿𝑝𝐷=𝑄𝑤𝐿𝐷.(3.9)

Dimensionless pressures are defined by

𝑃𝐷=𝐾𝑎𝐿𝑃𝑖𝑃,𝑃𝜇𝑞𝐵(3.10)𝑤𝐷=𝐾𝑎𝐿𝑃𝑖𝑃𝑤.𝜇𝑞𝐵(3.11)

Then (3.1) becomes

𝜕𝑃𝐷𝜕𝑡𝐷𝜕2𝑃𝐷𝜕𝑥2𝐷+𝜕2𝑃𝐷𝜕𝑦2𝐷+𝜕2𝑃𝐷𝜕𝑧2𝐷𝑥=𝛿𝐷𝑥𝐷𝛿𝑦𝐷𝑦𝐷𝛿𝑧𝐷𝑧𝐷,(3.12) in the dimensionless box-shaped drainage volume

Ω𝐷=0,𝑎𝐷×0,𝑏𝐷×0,𝐻𝐷,(3.13) with boundary condition

𝜕𝑃𝐷𝜕𝑁𝐷||||Γ𝐷=0,(3.14) and initial condition

𝑃𝐷||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𝜕𝜕𝑡2𝑃𝜕𝑥2+𝜕2𝑃𝜕𝑦2+𝜕2𝑃𝜕𝑧2=𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧.(4.1)

Rewrite (3.14) below

𝜕𝑃|||𝜕𝑁Γ=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 {𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)} [11, 12]:

𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)=1𝑎𝑏𝐻𝑑𝑙𝑑𝑚𝑑𝑛cos𝑙𝜋𝑥𝑎cos𝑚𝜋𝑦𝑏cos𝑛𝜋𝑧𝐻,(4.4) where 𝑙,𝑚,𝑛 are nonnegative numbers, and

𝑑𝑙=11if𝑙=0,2if𝑙>0,(4.5) 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]):

𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧=𝑙,𝑚,𝑛=0𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧.(4.6)

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

𝑙,𝑚,𝑛=0𝐹𝑙𝑚𝑛(𝑥,𝑦,𝑧)=𝑙=0𝑚=0𝑛=0𝐹𝑙𝑚𝑛(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>0𝐹𝑙𝑚𝑛(𝑥,𝑦,𝑧)=𝑙0𝑚0𝑛0𝐹𝑙𝑚𝑛(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

𝑃𝑡,𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=𝑙,𝑚,𝑛=0𝑒𝑙𝑚𝑛(𝑡)𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧),(4.9) 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

𝑙,𝑚,𝑛=0𝜕𝑒𝑙𝑚𝑛(𝑡)𝑔𝜕𝑡𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑒𝑙𝑚𝑛(𝑔𝑡)Δ𝑙𝑚𝑛(=𝑥,𝑦,𝑧)𝑙,𝑚,𝑛=0𝜕𝑒𝑙𝑚𝑛(𝑡)𝜕𝑡+𝑒𝑙𝑚𝑛(𝑡)𝜆𝑙𝑚𝑛𝑔𝑙𝑚𝑛=(𝑥,𝑦,𝑧)𝑙,𝑚,𝑛=0𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧),(4.10) where Δ is the three-dimensional Laplace operator

𝜕Δ=2𝜕𝑥2+𝜕2𝜕𝑦2+𝜕2𝜕𝑧2,𝜆(4.11)𝑙𝑚𝑛=𝑙𝜋𝑎2+𝑚𝜋𝑏2+𝑛𝜋𝐻2.(4.12)

From (4.3) and (4.9),

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

𝜕𝑒𝑙𝑚𝑛(𝑡)𝜕𝑡+𝜆𝑙𝑚𝑛𝑒𝑙𝑚𝑛(𝑡)=𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧,(4.14) because 𝜆000=0, from (4.14),

𝑒000(𝑡)=𝑔000𝑥,𝑦,𝑧𝑡=𝑡.𝑎𝑏𝐻(4.15)

When 𝜆𝑙𝑚𝑛0(𝑙+𝑚+𝑛>0), solve (4.14),

𝑒𝑙𝑚𝑛(𝑡)=1exp𝜆𝑙𝑚𝑛𝑡𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝜆𝑙𝑚𝑛.(4.16)

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

𝑃𝑡,𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=𝑙,𝑚,𝑛=0𝑒𝑙𝑚𝑛(𝑡)𝑔𝑙𝑚𝑛(=𝑡𝑥,𝑦,𝑧)𝑔𝑎𝑏𝐻000+(𝑥,𝑦,𝑧)𝑙+𝑚+𝑛>01exp𝜆𝑙𝑚𝑛𝑡𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆𝑙𝑚𝑛=𝑡+𝑎𝑏𝐻𝑙+𝑚+𝑛>0𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆𝑙𝑚𝑛𝑙+𝑚+𝑛>0exp𝜆𝑙𝑚𝑛𝑡𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆𝑙𝑚𝑛.(4.17)

Define

𝐼1=𝑡,𝐼𝑎𝑏𝐻(4.18)2=Ψ𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=𝑙+𝑚+𝑛>0𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆𝑙𝑚𝑛,𝐼(4.19)3=𝑙+𝑚+𝑛>0exp𝜆𝑙𝑚𝑛𝑡𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝜆𝑙𝑚𝑛,(4.20) then

𝑃𝑡,𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=𝐼1+𝐼2𝐼3.(4.21)

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

Ψ𝑎,𝑣=1𝑉Ω=1Ψ(𝑥,𝑦,𝑧)𝑑𝑉𝑉𝑎0𝑏0𝐻0Ψ𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=1𝑑𝑥𝑑𝑦𝑑𝑧𝑉𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝜆𝑙𝑚𝑛𝑎0𝑏0𝐻0𝑙+𝑚+𝑛>0𝑔𝑙𝑚𝑛(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

𝑎0cos𝑙𝜋𝑥𝑎𝑑𝑥=0.(4.24) So,

𝑎0𝑏0𝐻0𝑙+𝑚+𝑛>0𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧)𝑑𝑥𝑑𝑦𝑑𝑧=0,(4.25) consequently,

Ψ𝑎,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,

𝐼30,(4.27) then

𝑃𝑡,𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=𝑡𝑎𝑏𝐻+Ψ𝑥,𝑦,𝑧;𝑥,𝑦,𝑧.(4.28)

Substituting (4.28) into (4.1), we have

1𝑎𝑏𝐻ΔΨ=𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧.(4.29)

Define

1𝑓(𝑥,𝑦,𝑧)=ΔΨ=𝑎𝑏𝐻+𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧,(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 𝑔𝑙𝑚𝑛 when 𝑙=𝑚=𝑛=0.

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

𝑓(𝑥,𝑦,𝑧)=𝑙,𝑚,𝑛=0𝑓,𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(=𝑥,𝑦,𝑧)𝑙+𝑚+𝑛>0𝑓,𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛=(𝑥,𝑦,𝑧)𝑙+𝑚+𝑛>0𝛿𝑥𝑥𝛿𝑦𝑦𝛿𝑧𝑧,𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛=(𝑥,𝑦,𝑧)𝑙+𝑚+𝑛>0𝑔𝑙𝑚𝑛𝑥,𝑦,𝑧𝑔𝑙𝑚𝑛(𝑥,𝑦,𝑧).(4.37)

The drainage volume is

𝑉=𝑎𝑏𝐻.(4.38)

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

𝑃𝑎,𝑣=1𝑉Ω𝑡𝑃(𝑥,𝑦,𝑧)𝑑𝑥𝑑𝑦𝑑𝑧=𝑎𝑏𝐻+Ψ𝑎,𝑣.(4.39)

The wellbore pressure at point (𝑥𝑤,𝑦𝑤,𝑧𝑤) is

𝑃𝑤=𝑡𝑎𝑏𝐻+Ψ𝑤,(4.40) where Ψ𝑤 is the value of Ψ at wellbore point (𝑥𝑤,𝑦𝑤,𝑧𝑤).

Combining (4.39) and (4.40) gives

𝑃𝑎,𝑣𝑃𝑤=Ψ𝑎,𝑣Ψ𝑤,(4.41) which implies 𝑃𝑎,𝑣𝑃𝑤 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

𝐽𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=;𝑙,𝑚,𝑛𝐿0Ψ𝑥,𝑦,𝑧;𝑥,𝑦,𝑧𝑑𝑧=𝑙+𝑚+𝑛>0𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥,𝑦,𝑧,;𝑙,𝑚,𝑛(5.1) where

𝑙+𝑚+𝑛>0𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=;𝑙,𝑚,𝑛𝑙+𝑚+𝑛>01𝑎𝑏𝐻𝑑𝑙𝑑𝑚𝑑𝑛𝜆𝑙𝑚𝑛cos𝑙𝜋𝑥𝑎×cos𝑚𝜋𝑦𝑏cos𝑛𝜋𝑧𝐻×cos𝑚𝜋𝑦𝑏cos𝑙𝜋𝑥𝑎𝐿0cos𝑛𝜋𝑧𝐻𝑑𝑧=𝑙+𝑚+𝑛>01𝑎𝑏𝐻𝑑𝑙𝑑𝑚𝑑𝑛𝜆𝑙𝑚𝑛cos𝑙𝜋𝑥𝑎cos𝑚𝜋𝑦𝑏cos𝑛𝜋𝑧𝐻×𝐻𝜋𝑛cos𝑙𝜋𝑥𝑎cos𝑚𝜋𝑦𝑏sin𝑛𝜋𝐿𝐻if𝑙0,𝐿cos𝑚𝜋𝑦𝑏cos𝑙𝜋𝑥𝑎if𝑙=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,𝑛0},(5.5)3={(𝑙,𝑚,𝑛)𝑙>0,𝑚0,𝑛0},(5.6) then it is easy to prove

𝐶=𝐶1𝐶2𝐶3,𝐶1𝐶2=,𝐶2𝐶3=,𝐶3𝐶1=.(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>0𝑙𝑚𝑛𝑥,𝑦,𝑧;𝑥,𝑦,𝑧=;𝑙,𝑚,𝑛𝑛=100𝑛+𝑚=1𝑛=00𝑚𝑛+𝑙=1𝑚=0𝑛=0𝑙𝑚𝑛.(5.8)

Define the following notations:

𝐽𝑧=𝑛=100𝑛,𝐽(5.9)𝑦𝑧=𝑚=1𝑛=00𝑚𝑛𝐽,(5.10)𝑥𝑦𝑧=𝑙=1𝑚=0𝑛=0𝑙𝑚𝑛,(5.11) so

𝐽=𝐽𝑧+𝐽𝑦𝑧+𝐽𝑥𝑦𝑧,(5.12) and the average value of 𝐽 at wellbore can be written as

𝐽𝑎,𝑤=𝐽𝑧,𝑎𝑤+𝐽𝑦𝑧,𝑎𝑤+𝐽𝑥𝑦𝑧,𝑎𝑤.(5.13)

Rearrange (4.12) and obtain

𝜆𝑙𝑚𝑛=𝑙𝜋𝑎2+𝑚𝜋𝑏2+𝑛𝜋𝐻2=𝜋𝐻2𝑛2+𝜇2𝑙𝑚,(5.14) where

𝜇2𝑙𝑚=𝑙𝐻𝑎2+𝑚𝐻𝑏2=𝐻𝑏2𝑚2+𝑙𝑏𝑎2,𝜇𝑙0=𝑙𝐻𝑎,𝜆𝑙𝑚0=𝜋𝐻2𝜇2𝑙𝑚,𝜆0𝑚𝑛=𝑚𝜋𝑏2+𝑛𝜋𝐻2=𝜋𝐻2𝑛2+𝑚𝐻𝑏2,𝜆00𝑛=𝑛2𝜋2𝐻2.(5.15)

There hold [16, page 47]

𝑛=1sin(𝑛𝑥)𝑛3=𝜋2𝑥6𝜋𝑥24+𝑥3(120𝑥2𝜋),(5.16)𝑛=11cos(𝑛𝑥)𝑛4=𝜋2𝑥212𝜋𝑥3+𝑥12448(0𝑥2𝜋).(5.17)

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

𝑦=𝑦0,𝑥0,𝑥=𝑥+𝑅𝑤,0𝑧=𝑧𝐽𝐿,𝑧𝑤=𝑛=11𝑎𝑏𝐻𝑑𝑛𝜆00𝑛cos𝑛𝜋𝑧𝐻𝐿0cos𝑛𝜋𝑧𝐻𝑑𝑧=2𝑎𝑏𝐻𝑛=1𝐻2𝜋2𝑛2cos𝑛𝜋𝑧𝐻𝐻𝑛𝜋sin𝑛𝜋𝐿𝐻=2𝐻2𝑎𝑏𝜋3𝑛=11𝑛3sin𝑛𝜋𝐿𝐻cos𝑛𝜋𝑧𝐻.(5.18)

The average value of (𝐽𝑧)𝑤 along the well length is

𝐽𝑧,𝑎𝑤=1𝐿𝐿0𝐽𝑧=1𝑑𝑧𝐿𝑛=12𝐻2𝜋3𝑎𝑏𝑛3sin𝑛𝜋𝐿𝐻𝐿0cos𝑛𝜋𝑧𝐻=𝑑𝑧𝑛=12𝐻2𝜋3𝑎𝑏𝐿𝑛3sin𝑛𝜋𝐿𝐻𝐻𝑛𝜋sin𝑛𝜋𝐿𝐻=𝑛=12𝐻3𝜋4𝑎𝑏𝐿𝑛4sin2𝑛𝜋𝐿𝐻=𝐻3𝜋4𝑎𝑏𝐿𝑛=11𝑛41cos2𝑛𝜋𝐿𝐻=𝐻3𝜋4𝑎𝑏𝐿2𝜋𝐿𝐻2𝜋2𝜋12122𝜋𝐿𝐻+1482𝜋𝐿𝐻2=4𝐻𝐿1𝑎𝑏𝐿12+𝐿6𝐻212𝐻2=2𝐻𝐿13𝑎𝑏2𝐿𝐻+𝐿22𝐻2,(5.19) where we have used (5.17).

For a fully penetrating well, 𝐿=𝐻, then

𝐽𝑧,𝑎𝑤=0.(5.20)

Recall (5.5) and (5.10), 𝐽𝑦𝑧 is for the case 𝑙=0,𝑚>0,𝑛0, and at wellbore of the off-center well,

𝑦=𝑦0,𝑥0,𝑥=𝑥+𝑅𝑤,0𝑧=𝑧𝐽𝐿,𝑦𝑧𝑤=1𝑎𝑏𝐻𝑚=1𝑛=01𝑑𝑚𝑑𝑛𝜆0𝑚𝑛cos2𝑚𝜋𝑦𝑏cos𝑛𝜋𝑧𝐻𝐿0cos𝑛𝜋𝑧𝐻𝑑𝑧=2𝑎𝑏𝐻𝑚=1𝑛=0cos2𝑚𝜋𝑦/𝑏cos(𝑛𝜋𝑧/𝐻)𝜋2𝑑𝑛(𝑛/𝐻)2+(𝑚/𝑏)2𝐿0cos𝑛𝜋𝑧𝐻𝑑𝑧=2𝑎𝑏𝐻𝑚=1𝑛=12(𝐻/𝑛𝜋)cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝜋2(𝑛/𝐻)2+(𝑚/𝑏)2+cos2𝑚𝜋𝑦𝑏𝑏2𝐿𝜋2𝑚2=2𝐻3𝜋3𝑎𝑏𝐻𝑚=1𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝑛𝑛2+(𝑚𝐻/𝑏)2+cos2𝑚𝜋𝑦𝑏𝑏2𝐿𝜋𝐻3𝑚2=2𝐻2𝜋3𝑎𝑏𝜋𝐿𝑏2𝐻3𝑚=11𝑚2cos2𝑚𝜋𝑦𝑏+2𝐻2𝜋3𝑎𝑏𝑚=1𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝑛𝑛2+(𝑚𝐻/𝑏)2=2𝑏𝐿𝜋2𝑎𝐻𝑚=11𝑚2cos2𝑚𝜋𝑦𝑏+2𝐻2𝜋3𝑎𝑏𝑚=1𝑛=12cos(𝑛𝜋𝑧/𝐻)sin(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝑛𝑛2+(𝑚𝐻/𝑏)2,(5.21) where we use the following formulas [16, page 47]:

𝑚=11𝑚2𝜋cos(𝑚𝑥)=26𝜋𝑥2+𝑥24(0𝑥2𝜋),(5.22)𝑚=11𝑚2cos2𝜋(𝑚𝑥)=26𝜋𝑥2+𝑥22(0𝑥𝜋).(5.23)

The average value of (𝐽𝑦𝑧)𝑤 along the well length is

𝐽𝑦𝑧,𝑎𝑤=1𝐿𝐿0𝐽𝑦𝑧=𝑑𝑧2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻2𝑎𝑏𝐿𝜋3𝑚=1𝑛=12sin(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝑛𝑛2+(𝑚𝐻/𝑏)2𝐿0cos𝑛𝜋𝑧𝐻=𝑑𝑧2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻2𝑎𝑏𝐿𝜋3𝑚=1𝑛=12𝐻sin2(𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝜋𝑛2𝑛2+(𝑚𝐻/𝑏)2=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻3𝑎𝑏𝐿𝜋4𝑚=1𝑛=1[]1cos(2𝑛𝜋𝐿/𝐻)cos2𝑚𝜋𝑦/𝑏𝑛2𝑛2+(𝑚𝐻/𝑏)2=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻3𝑎𝑏𝐿𝜋4𝑚=1𝑛=1𝑏𝑚𝐻2cos2𝑚𝜋𝑦𝑏×1cos2𝑛𝜋𝐿𝐻,1𝑛21𝑛2+(𝑚𝐻/𝑏)2=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+𝐻32𝑎𝑏𝐿𝜋4𝑚=1𝑏𝑚𝐻2cos2𝑚𝜋𝑦𝑏×𝑛=11𝑛2cos(2𝑛𝜋𝐿/𝐻)𝑛21𝑛2+(𝑚𝐻/𝑏)2+cos(2𝑛𝜋𝐿/𝐻)𝑛2+(𝑚𝐻/𝑏)2=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻3𝑎𝑏𝐿𝜋4𝑚=1𝑏𝑚𝐻2cos2𝑚𝜋𝑦𝑏×𝜋26𝜋26𝜋22𝜋𝐿𝐻+142𝜋𝐿𝐻2𝑏𝜋2𝑚𝐻coth𝑚𝐻𝜋𝑏12𝑏𝑚𝐻2+𝑏𝜋[(]2𝑚𝐻cosh𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)1sinh(𝑚𝐻𝜋/𝑏)2𝑏𝑚𝐻2,(5.24) where we use the following formulas [16, page 47]:

𝑛=1cos(𝑛𝑥)𝑛2+𝛽2=𝜋[]2𝛽cosh𝛽(𝜋𝑥)1sinh(𝛽𝜋)2𝛽2(0𝑥2𝜋),(5.25)𝑛=11𝑛2+𝛽2=𝜋12𝛽coth(𝛽𝜋)2𝛽2(0𝑥2𝜋),(5.26) and we may simplify (5.24) further

𝐽𝑦𝑧,𝑎𝑤=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻3𝑎𝑏𝐿𝜋4𝑚=1cos2𝑚𝜋𝑦𝑏𝑏𝑚𝐻2×𝜋2𝐿𝐻𝜋2𝐿2𝐻2+𝑏𝜋[]2𝑚𝐻cosh(𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)sinh(𝑚𝐻𝜋/𝑏)coth𝑚𝐻𝜋𝑏=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝐻3𝑎𝑏𝐿𝜋4𝑚=1cos2𝑚𝜋𝑦𝑏1𝑚2×𝜋2𝐿𝑏2𝐻3𝜋2𝐿2𝑏2𝐻4+𝑏3𝜋2𝑚𝐻3×[]cosh(𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)sinh(𝑚𝐻𝜋/𝑏)coth𝑚𝐻𝜋𝑏=2𝑏𝐿1𝑎𝐻6𝑦+𝑦2𝑏22𝑏2+2𝑏𝑎𝜋2𝐿1𝐻𝜋26𝜋2𝑦+𝜋2𝑏2𝑦22𝑏2+𝑏2𝑎𝐿𝜋3𝑚=1cos2𝑚𝜋𝑦𝑏1𝑚3×[]cosh(𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)sinh(𝑚𝐻𝜋/𝑏)coth𝑚𝐻𝜋𝑏=2𝑏𝑎16𝑦+𝑦2𝑏22𝑏2+𝑏2𝑎𝐿𝜋3𝑚=1cos2𝑚𝜋𝑦/𝑏𝑚3[]cosh(𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)sinh(𝑚𝐻𝜋/𝑏)coth𝑚𝐻𝜋𝑏.(5.27)

For a fully penetrating well, 𝐿=𝐻, then

𝐽𝑦𝑧,𝑎𝑤=2𝑏𝑎16𝑦+𝑦2𝑏22𝑏2.(5.28)

Define

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

𝑓[][][],(𝑥)=𝛼cosh(𝛼𝑥)sinh𝛼(1𝑥)𝛼cosh𝛼(1𝑥)sinh(𝛼𝑥)=𝛼sinh𝛼(12𝑥)(5.30) consequently,

𝑓12=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

𝐿𝑥=𝐻,(5.33) and the producing length 𝐿 is a variable, define

[]𝐹(𝐿)=cosh𝛽𝜋(12𝐿/𝐻)cosh(𝛽𝜋)=[][]sinh(𝛽𝜋)2sinh𝛽𝜋(1𝐿/𝐻)sinh𝛽𝜋𝐿/(𝐻),sinh(𝛽𝜋)(5.34) thus when 𝐿=𝐻/2,|𝐹(𝐿)| reaches maximum value,

||||𝐹(𝐿)max=|||𝐹𝐻2|||=2sinh2(𝛽𝜋/2)=sinh(𝛽𝜋)2sinh2(𝛽𝜋/2)=2sinh(𝛽𝜋/2)cosh(𝛽𝜋/2)sinh(𝛽𝜋/2)cosh(𝛽𝜋/2)<1,(5.35) so 𝐹(𝐿) is a bounded function, let

𝛽=𝑚𝐻𝑏,(5.36) then

𝐽𝑦𝑧,𝑎𝑤=2𝑏𝑎16𝑦+𝑦2𝑏22𝑏2+𝑏2𝑎𝐿𝜋3×𝑚=1cos2𝑚𝜋𝑦/𝑏𝑚3[]cosh(𝑚𝐻𝜋/𝑏)(12𝐿/𝐻)sinh(𝑚𝐻𝜋/𝑏)coth𝑚𝐻𝜋𝑏=2𝑏𝑎16𝑦+𝑦2𝑏22𝑏2+𝑏2𝑎𝐿𝜋3×𝑚=1cos2𝑚𝜋𝑦/𝑏𝑚3[]2sinh(𝑚𝐻𝜋/𝑏)(1𝐿/𝐻)sinh(𝑚𝐿𝜋/𝑏)sinh(𝑚𝐻𝜋/𝑏)2𝑏𝑎16𝑦+𝑦2𝑏22𝑏2+𝑏2𝑎𝐿𝜋3×𝑀𝑚=1cos2𝑚𝜋𝑦/𝑏𝑚3[]2sinh(𝑚𝐻𝜋/𝑏)(1𝐿/𝐻)sinh(𝑚𝐿𝜋/𝑏).sinh(𝑚𝐻𝜋/𝑏)(5.37)

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

𝑚=101|||||cos2𝑚𝜋𝑦/𝑏𝑚3[]2sinh(𝑚𝐻𝜋/𝑏)(1𝐿/𝐻)sinh(𝑚𝐿𝜋/𝑏)|||||sinh(𝑚𝐻𝜋/𝑏)𝑚=1011𝑚3=𝜁(3)100𝑚=11𝑚3=4.9502×105,(5.38) where 𝜁(3) is 𝑅𝑖𝑒𝑚𝑎𝑛𝑛-𝜁 function:

𝜁(3)=𝑚=11𝑚3=1.202057,(5.39) thus

𝑚=11𝑚3[]2sinh(𝑚𝐻𝜋/𝑏)(1𝐿/𝐻)sinh(𝑚𝐿𝜋/𝑏)sinh(𝑚𝐻𝜋/𝑏)100𝑚=11𝑚3[]2sinh(𝑚𝐻𝜋/𝑏)(1𝐿/𝐻)sinh(𝑚𝐿𝜋/𝑏).sinh(𝑚𝐻𝜋/𝑏)(5.40)

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,𝑚0,𝑛0, and at wellbore of the off-center well,

𝑦=𝑦0,𝑥0,𝑥=𝑥+𝑅𝑤,0𝑧=𝑧𝐿,(5.41) then

𝐽𝑥𝑦𝑧𝑤=1×𝑎𝑏𝐻𝑙=1𝑚=0𝑛=0cos(𝑛𝜋𝑧/𝐻)cos𝑙𝜋𝑥𝑥/𝑎cos𝑙𝜋+𝑅𝑤/𝑎cos2𝑚𝜋𝑦/𝑏𝑑𝑙𝑑𝑚𝑑𝑛𝜆𝑙𝑚𝑛×𝐿0cos𝑛𝜋𝑧𝐻𝑑𝑧=1𝑎𝑏𝐻𝑙=1𝑚=0cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎cos2𝑚𝜋𝑦𝑏×𝑛=14(𝐻/𝑛𝜋)sin(𝑛𝜋𝐿/𝐻)cos(𝑛𝜋𝑧/𝐻)𝑑𝑚𝜆𝑙𝑚𝑛+2𝐿𝑑𝑚𝜆𝑙𝑚0.(5.42)

The average value of (𝐽𝑥𝑦𝑧)𝑤 along the well length is 𝐽𝑥𝑦𝑧,𝑎𝑤=1𝑎𝑏𝐻𝑙=1𝑚=0cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎cos2𝑚𝜋𝑦𝑏×𝑛=14(𝐻/𝑛𝜋)sin(𝑛𝜋𝐿/𝐻)𝐿0cos(𝑛𝜋𝑧/𝐻)𝑑𝑧𝑑𝑚𝜆𝑙𝑚𝑛𝐿+2𝐿𝑑𝑚𝜆𝑙𝑚0=1𝑎𝑏𝐻𝑙=1𝑚=0cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎cos2𝑚𝜋𝑦𝑏×𝑛=14(𝐻/𝑛𝜋)2sin2(𝑛𝜋𝐿/𝐻)𝑑𝑚𝜆𝑙𝑚𝑛𝐿+2𝐿𝑑𝑚𝜆𝑙𝑚0=𝐻4𝑎𝑏𝐻𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=0cos2𝑚𝜋𝑦𝑏×𝑛=12[]1cos(2𝑛𝜋𝐿/𝐻)𝑑𝑚𝑛2𝑛2+𝜇2𝑙𝑚𝐿+2𝜋2𝐿𝑑𝑚𝐻2𝜇2𝑙𝑚=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=02𝑑𝑚cos2𝑚𝜋𝑦𝑏×𝑛=11𝜇2𝑙𝑚𝐿1cos2𝑛𝜋𝐿𝐻×1𝑛21𝑛2+𝜇2𝑙𝑚+𝜋2𝐿𝐻2𝜇2𝑙𝑚=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=02𝑑𝑚𝜇2𝑙𝑚𝐿cos2𝑚𝜋𝑦𝑏×𝑛=11𝑛2cos(2𝑛𝜋𝐿/𝐻)𝑛21𝑛2+𝜇2𝑙𝑚+cos(2𝑛𝜋𝐿/𝐻)𝑛2+𝜇2𝑙𝑚+𝜋2𝐿2𝐻2=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=02𝑑𝑚𝜇2𝑙𝑚𝐿cos2𝑚𝜋𝑦𝑏×𝜋26𝜋26𝜋22𝜋𝐿𝐻+142𝜋𝐿𝐻2𝜋2𝜇𝑙𝑚𝜇coth𝑙𝑚𝜋12𝜇2𝑙𝑚+𝜋2𝜇𝑙𝑚𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋12𝜇2𝑙𝑚+𝜋2𝐿2𝐻2,(5.43) where we use (5.22) and (5.25).

Let 𝑥=0, recast (5.26), we obtain

1𝛽2+2𝑛=11𝑛2+𝛽2=𝜋𝛽coth(𝛽𝜋),𝑛=01𝑛2+𝛽2𝑑𝑛=𝜋𝛽coth(𝛽𝜋).(5.44) So,

𝐽𝑥𝑦𝑧,𝑎𝑤=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=02𝑑𝑚𝜇2𝑙𝑚𝐿cos2𝑚𝜋𝑦𝑏×𝜋2𝐿𝐻𝜋2𝐿2𝐻2+𝜋2𝐿2𝐻2+𝜋2𝜇𝑙𝑚𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋𝜇coth𝑙𝑚𝜋=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎×𝑚=0𝜋𝑑𝑚𝜇3𝑙𝑚𝐿cos2𝑚𝜋𝑦𝑏×𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋𝜇coth𝑙𝑚𝜋+2𝐿𝜋2𝐿𝐻𝑚=0cos2𝑚𝜋𝑦/𝑏𝑑𝑚𝜇2𝑙𝑚=𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎×𝑚=0𝜋𝑑𝑚𝜇3𝑙𝑚𝐿cos2𝑚𝜋𝑦𝑏×𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋𝜇coth𝑙𝑚𝜋+𝑎𝑏𝜋3𝐻3𝑙coth𝑙𝑏𝜋𝑎+𝜋2𝐻𝑚=0cos2𝑚𝜋𝑦/𝑏𝑑𝑚𝜇2𝑙𝑚=𝐻3𝑎𝑏𝐿𝜋3𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎×𝑚=01𝑑𝑚𝜇3𝑙𝑚cos2𝑚𝜋𝑦𝑏×𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋𝜇coth𝑙𝑚𝜋+𝐻3𝑎𝑏𝜋4𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎,𝑎𝑏𝜋3𝐻3𝑙coth𝑙𝑏𝜋𝑎+𝜋2𝐻𝑚=0cos2𝑚𝜋𝑦/𝑏𝑑𝑚𝜇2𝑙𝑚.(5.45)

Since

𝜋2𝐻𝑚=0cos2𝑚𝜋𝑦/𝑏𝑑𝑚𝜇2𝑙𝑚=𝑏2𝐻3𝑚=0cos2𝑚𝜋𝑦/𝑏𝑑𝑚𝑚2+(𝑏𝑙/𝑎)2=𝑏2𝐻3𝜋𝑎𝑏𝑙cosh𝜋𝑏𝑙12𝑦/𝑏/𝑎𝑏sinh(𝜋𝑏𝑙/𝑎)𝐻3𝜋𝑎𝑙exp2𝜋𝑦𝑙𝑎,𝐻3𝑎𝑏𝜋4|||||𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑏𝐻3𝜋𝑎𝑙exp2𝜋𝑦𝑙𝑎|||||1𝜋3ln1exp2𝜋𝑦/𝑎0,(5.46) thus

𝐽𝑥𝑦𝑧,𝑎𝑤𝐻3𝑎𝑏𝐿𝜋3𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝑚=01𝑑𝑚𝜇3𝑙𝑚cos2𝑚𝜋𝑦𝑏×𝜇cosh𝑙𝑚𝜋(12𝐿/𝐻)𝜇sinh𝑙𝑚𝜋𝜇coth𝑙𝑚𝜋+1𝜋𝑙=11𝑙coth𝑙𝑏𝜋𝑎cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤𝑎𝐻3𝑎𝑏𝐿𝜋3𝑙=1cos𝑙𝜋𝑥𝑎𝑥cos𝑙𝜋+𝑅𝑤<