Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 907206 | 35 pages | https://doi.org/10.1155/2010/907206

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

Academic Editor: Alexander P. Seyranian
Received08 Sep 2009
Accepted09 Feb 2010
Published08 Apr 2010

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]

𝑄𝑀=𝐹𝐷𝑃2πœ‹πΎπ»π‘Žβˆ’π‘ƒπ‘€ξ€Έ/(πœ‡π΅)𝑅ln𝑒/π‘…π‘€ξ€Έβˆ’3/4,(2.1) where π‘ƒπ‘Ž is average reservoir pressure in the circular drainage area, 𝑃𝑀 is flowing wellbore pressure, 𝐾 is permeability, 𝐻 is payzone thickness, πœ‡ is oil viscosity, 𝐡 is oil formation volume factor, 𝑅𝑒 is radius of circular drainage area, 𝑅𝑀 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]:

𝑄𝑀=𝐹𝐷𝑃2πœ‹πΎπ»π‘Žβˆ’π‘ƒπ‘€ξ€Έ/(πœ‡π΅)𝑅ln𝑒/π‘…π‘€ξ€Έβˆ’3/4+𝑆ps.(2.2)

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 πœ‚:

πΏπœ‚=𝑝𝐻=𝐿𝐻.(2.3)

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

𝑆ps=ξ‚΅1πœ‚ξ‚Άξ€Ίξ€·β„Žβˆ’1lnπ·ξ€Έξ€»βˆ’πΊ(πœ‚),(2.4) where

β„Žπ·=ξ‚΅π»π‘…π‘€πΎξ‚Άξ‚΅β„ŽπΎπ‘£ξ‚Ά1/2,𝐺(2.5)(πœ‚)=2.948βˆ’7.363πœ‚+11.45πœ‚2βˆ’4.675πœ‚3.(2.6)

Pseudoskin factor formula given by Papatzacos is [6]

𝑆ps=ξ‚΅1πœ‚ξ‚Άξ‚΅βˆ’1lnπœ‹β„Žπ·2ξ‚Ά+ξ‚΅1πœ‚ξ‚Άξƒ¬ξ‚΅πœ‚lnΞ¨2+πœ‚ξ‚Άξ‚΅1βˆ’1Ξ¨2ξ‚Άβˆ’11/2ξƒ­,(2.7) where β„Žπ· has the same meaning as in (2.5), and

Ξ¨1=π»β„Ž1+0.25𝐿𝑝,Ξ¨2=π»β„Ž1+0.75𝐿𝑝,(2.8) and β„Ž1 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=ξ‚΅1πœ‚ξ‚Άξƒ¬ξ€·πΏβˆ’1ln𝑝/𝑅𝑀1βˆ’π‘…π‘€/πΏπ‘ξ€Έξƒ­βˆ’1.(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:

𝑄𝑀=𝐹𝐷𝑃2πœ‹πΎπ»π‘Žβˆ’π‘ƒπ‘€ξ€Έ/(πœ‡π΅)(𝐢1/2)ln2.2458𝐴/𝐴𝑅2𝑀,(2.10) where 𝐢𝐴 is shape factor, and 𝐴 is drainage area.

Dietz evaluated shape factor 𝐢𝐴 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 (π‘₯ direction) π‘Ž, and width (𝑦 direction) 𝑏. The well is parallel to the 𝑧 direction with a length 𝐿≀𝐻, and we assume 𝑏β‰₯π‘Ž.

The following assumptions are made.

(1)The reservoir is homogeneous, anisotropic, and has constant 𝐾π‘₯,𝐾𝑦,𝐾𝑧 permeabilities, thickness 𝐻, and porosity πœ™. All the boundaries of the box-shaped drainage volume are sealed.(2)The reservoir pressure is initially constant. At time 𝑑=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 𝑅𝑀, represented in the model by a uniform line sink.(4)A single phase fluid, of small and constant compressibility 𝐢𝑓, constant viscosity πœ‡, and formation volume factor 𝐡, flows from the reservoir to the well at a constant rate 𝑄𝑀. 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 (π‘₯ξ…ž,π‘¦ξ…ž,0) and (π‘₯ξ…ž,π‘¦ξ…ž,𝐿). We suppose the point (π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž) is on the well line, and its point convergence intensity is π‘ž.

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 (π‘₯,𝑦,𝑧) caused by the point (π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž), we have to obtain the basic solution of the following Laplace equation:

𝐾π‘₯πœ•2π‘ƒπœ•π‘₯2+πΎπ‘¦πœ•2π‘ƒπœ•π‘¦2+πΎπ‘§πœ•2π‘ƒπœ•π‘§2=πœ™πœ‡πΆπ‘‘πœ•π‘ƒξ€·πœ•π‘‘+πœ‡π‘žπ΅π›Ώπ‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ,(3.1) in the box-shaped drainage volume:

Ξ©=(0,π‘Ž)Γ—(0,𝑏)Γ—(0,𝐻),(3.2) and we always assume

𝑏β‰₯π‘Žβ‰«π»,(3.3) and 𝛿(π‘₯βˆ’π‘₯ξ…ž), 𝛿(π‘¦βˆ’π‘¦ξ…ž), 𝛿(π‘§βˆ’π‘§ξ…ž) are Dirac functions.

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

πœ•π‘ƒ|||πœ•π‘Ξ“=0,(3.4) where πœ•π‘ƒ/πœ•π‘|Ξ“ is the exterior normal derivative of pressure on the surface of box-shaped drainage volume Ξ“=πœ•Ξ©.

The reservoir pressure is initially constant

𝑃||𝑑=0=𝑃𝑖.(3.5)

Define average permeability:

πΎπ‘Ž=𝐾π‘₯𝐾𝑦𝐾𝑧1/3.(3.6)

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

π‘₯𝐷=ξ‚€π‘₯πΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘₯ξ‚Ά1/2,𝑦𝐷=ξ‚€π‘¦πΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘¦ξ‚Ά1/2,𝑧𝐷=ξ‚€π‘§πΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘§ξ‚Ά1/2,π‘Žπ·=ξ‚€π‘ŽπΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘₯ξ‚Ά1/2,𝑏𝐷=ξ‚€π‘πΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘¦ξ‚Ά1/2,𝐿𝐷=ξ‚΅πΎπ‘ŽπΎπ‘§ξ‚Ά1/2,𝐻𝐷=ξ‚€π»πΏξ‚ξ‚΅πΎπ‘ŽπΎπ‘§ξ‚Ά1/2,𝑑𝐷=πΎπ‘Žπ‘‘πœ™πœ‡πΆπ‘‘πΏ2.(3.7)

The dimensionless wellbore radius is [8]

𝑅𝑀𝐷=𝐾𝑧/√𝐾π‘₯𝐾𝑦1/6𝐾π‘₯/𝐾𝑦1/4+𝐾𝑦/𝐾π‘₯ξ€Έ1/4𝑅𝑀.2𝐿(3.8)

Assume that π‘ž is the point convergence intensity at the point sink (π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž), the partially penetrating well is a uniform line sink, the total productivity of the well is 𝑄𝑀, and there holds [8]

π‘„π‘ž=𝑀𝐿𝑝𝐷=𝑄𝑀𝐿𝐷.(3.9)

Dimensionless pressures are defined by

𝑃𝐷=πΎπ‘ŽπΏξ€·π‘ƒπ‘–ξ€Έβˆ’π‘ƒ,π‘ƒπœ‡π‘žπ΅(3.10)𝑀𝐷=πΎπ‘ŽπΏξ€·π‘ƒπ‘–βˆ’π‘ƒπ‘€ξ€Έ.πœ‡π‘žπ΅(3.11)

Then (3.1) becomes

πœ•π‘ƒπ·πœ•π‘‘π·βˆ’ξƒ©πœ•2π‘ƒπ·πœ•π‘₯2𝐷+πœ•2π‘ƒπ·πœ•π‘¦2𝐷+πœ•2π‘ƒπ·πœ•π‘§2𝐷ξƒͺξ€·π‘₯=π›Ώπ·βˆ’π‘₯ξ…žπ·ξ€Έπ›Ώξ€·π‘¦π·βˆ’π‘¦ξ…žπ·ξ€Έπ›Ώξ€·π‘§π·βˆ’π‘§ξ…žπ·ξ€Έ,(3.12) in the dimensionless box-shaped drainage volume

Ω𝐷=ξ€·0,π‘Žπ·ξ€ΈΓ—ξ€·0,𝑏𝐷×0,𝐻𝐷,(3.13) with boundary condition

πœ•π‘ƒπ·πœ•π‘π·||||Γ𝐷=0,(3.14) and initial condition

𝑃𝐷||𝑑𝐷=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

πœ•π‘ƒβˆ’ξ‚΅πœ•πœ•π‘‘2π‘ƒπœ•π‘₯2+πœ•2π‘ƒπœ•π‘¦2+πœ•2π‘ƒπœ•π‘§2ξ‚Άξ€·=𝛿π‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ.(4.1)

Rewrite (3.14) below

πœ•π‘ƒ|||πœ•π‘Ξ“=0,(4.2) and (3.15) becomes

𝑃||𝑑=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 𝑑 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 {π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)} [11, 12]:

π‘”π‘™π‘šπ‘›ξƒŽ(π‘₯,𝑦,𝑧)=1π‘Žπ‘π»π‘‘π‘™π‘‘π‘šπ‘‘π‘›ξ‚€cosπ‘™πœ‹π‘₯π‘Žξ‚ξ‚€cosπ‘šπœ‹π‘¦π‘ξ‚ξ‚€cosπ‘›πœ‹π‘§π»ξ‚,(4.4) where 𝑙,π‘š,𝑛 are nonnegative numbers, and

𝑑𝑙=⎧βŽͺ⎨βŽͺ⎩11if𝑙=0,2if𝑙>0,(4.5) and π‘‘π‘š,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]):

𝛿π‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ=βˆžξ“π‘™,π‘š,𝑛=0π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ.(4.6)

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

βˆžξ“π‘™,π‘š,𝑛=0πΉπ‘™π‘šπ‘›(π‘₯,𝑦,𝑧)=βˆžξ“βˆžπ‘™=0ξ“βˆžπ‘š=0𝑛=0πΉπ‘™π‘šπ‘›(π‘₯,𝑦,𝑧),(4.7) which means in any function 𝐹(π‘₯,𝑦,𝑧), the subscripts 𝑙,π‘š,𝑛 of any variable must count from 0 to infinite.

And define

𝑙+π‘š+𝑛>0πΉπ‘™π‘šπ‘›(π‘₯,𝑦,𝑧)=βˆžξ“βˆžπ‘™β‰₯0ξ“βˆžπ‘šβ‰₯0𝑛β‰₯0πΉπ‘™π‘šπ‘›(π‘₯,𝑦,𝑧)(𝑙+π‘š+𝑛>0),(4.8) which means in any function 𝐹(π‘₯,𝑦,𝑧), the subscripts 𝑙,π‘š,𝑛 of any variable must be no less than zero, and at least one of the three subscripts 𝑙,π‘š,𝑛 must be positive to guarantee 𝑙+π‘š+𝑛>0. And the upper limit of the subscripts 𝑙,π‘š,𝑛 is infinite.

Let

𝑃𝑑,π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=βˆžξ“π‘™,π‘š,𝑛=0π‘’π‘™π‘šπ‘›(𝑑)π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧),(4.9) where π‘’π‘™π‘šπ‘›(𝑑) 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

βˆžξ“π‘™,π‘š,𝑛=0ξ‚»πœ•π‘’π‘™π‘šπ‘›(𝑑)π‘”πœ•π‘‘π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)βˆ’π‘’π‘™π‘šπ‘›(𝑔𝑑)Ξ”π‘™π‘šπ‘›(ξ€»ξ‚Ό=π‘₯,𝑦,𝑧)βˆžξ“π‘™,π‘š,𝑛=0ξ‚»πœ•π‘’π‘™π‘šπ‘›(𝑑)πœ•π‘‘+π‘’π‘™π‘šπ‘›(𝑑)πœ†π‘™π‘šπ‘›ξ‚Όπ‘”π‘™π‘šπ‘›=(π‘₯,𝑦,𝑧)βˆžξ“π‘™,π‘š,𝑛=0π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧),(4.10) where Ξ” is the three-dimensional Laplace operator

πœ•Ξ”=2πœ•π‘₯2+πœ•2πœ•π‘¦2+πœ•2πœ•π‘§2,πœ†(4.11)π‘™π‘šπ‘›=ξ‚€π‘™πœ‹π‘Žξ‚2+ξ‚€π‘šπœ‹π‘ξ‚2+ξ‚€π‘›πœ‹π»ξ‚2.(4.12)

From (4.3) and (4.9),

π‘’π‘™π‘šπ‘›(0)=0,(4.13) compare the coefficients of π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧) at both sides of (4.10), we obtain

πœ•π‘’π‘™π‘šπ‘›(𝑑)πœ•π‘‘+πœ†π‘™π‘šπ‘›π‘’π‘™π‘šπ‘›(𝑑)=π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ,(4.14) because πœ†000=0, from (4.14),

𝑒000(𝑑)=𝑔000ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘‘=π‘‘βˆš.π‘Žπ‘π»(4.15)

When πœ†π‘™π‘šπ‘›β‰ 0(𝑙+π‘š+𝑛>0), solve (4.14),

π‘’π‘™π‘šπ‘›ξ€Ίξ€·(𝑑)=1βˆ’expβˆ’πœ†π‘™π‘šπ‘›π‘‘π‘”ξ€Έξ€»π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπœ†π‘™π‘šπ‘›.(4.16)

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

𝑃𝑑,π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=βˆžξ“π‘™,π‘š,𝑛=0π‘’π‘™π‘šπ‘›(𝑑)π‘”π‘™π‘šπ‘›(=𝑑π‘₯,𝑦,𝑧)√ξƒͺπ‘”π‘Žπ‘π»000+(π‘₯,𝑦,𝑧)𝑙+π‘š+𝑛>0ξ€Ίξ€·1βˆ’expβˆ’πœ†π‘™π‘šπ‘›π‘‘π‘”ξ€Έξ€»π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)πœ†π‘™π‘šπ‘›=𝑑+ξ“π‘Žπ‘π»π‘™+π‘š+𝑛>0π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)πœ†π‘™π‘šπ‘›βˆ’ξ“π‘™+π‘š+𝑛>0ξ€·expβˆ’πœ†π‘™π‘šπ‘›π‘‘ξ€Έπ‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)πœ†π‘™π‘šπ‘›.(4.17)

Define

𝐼1=𝑑,πΌπ‘Žπ‘π»(4.18)2ξ€·=Ξ¨π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=𝑙+π‘š+𝑛>0π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)πœ†π‘™π‘šπ‘›,𝐼(4.19)3=𝑙+π‘š+𝑛>0ξ€·expβˆ’πœ†π‘™π‘šπ‘›π‘‘ξ€Έπ‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)πœ†π‘™π‘šπ‘›,(4.20) then

𝑃𝑑,π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=𝐼1+𝐼2βˆ’πΌ3.(4.21)

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

Ξ¨π‘Ž,𝑣=ξ‚€1π‘‰ξ‚ξ€œΞ©=ξ‚€1Ξ¨(π‘₯,𝑦,𝑧)π‘‘π‘‰π‘‰ξ‚ξ€œπ‘Ž0ξ€œπ‘0ξ€œπ»0Ξ¨ξ€·π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=ξ‚€1𝑑π‘₯π‘‘π‘¦π‘‘π‘§π‘‰ξ‚ξƒ©π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπœ†π‘™π‘šπ‘›ξƒͺξ€œπ‘Ž0ξ€œπ‘0ξ€œπ»0𝑙+π‘š+𝑛>0π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)𝑑π‘₯𝑑𝑦𝑑𝑧.(4.22)

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

𝑙>0,(4.23) then

ξ€œπ‘Ž0ξ‚€cosπ‘™πœ‹π‘₯π‘Žξ‚π‘‘π‘₯=0.(4.24) So,

ξ€œπ‘Ž0ξ€œπ‘0ξ€œπ»0𝑙+π‘š+𝑛>0π‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧)𝑑π‘₯𝑑𝑦𝑑𝑧=0,(4.25) consequently,

Ξ¨π‘Ž,𝑣=0.(4.26)

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

𝐼3β‰ˆ0,(4.27) then

𝑃𝑑,π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=π‘‘ξ€·π‘Žπ‘π»+Ξ¨π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ.(4.28)

Substituting (4.28) into (4.1), we have

1ξ€·π‘Žπ‘π»βˆ’Ξ”Ξ¨=𝛿π‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ.(4.29)

Define

ξ‚€1𝑓(π‘₯,𝑦,𝑧)=βˆ’Ξ”Ξ¨=βˆ’ξ‚ξ€·π‘Žπ‘π»+𝛿π‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ,(4.30) note that Ξ¨ is equal to 𝐼2 in (4.19), and

πœ•Ξ¨πœ•π‘=0,onΞ“.(4.31)

From Green's Formula [15],

ξ€œ0=Ξ“πœ•Ξ¨ξ€œπœ•π‘π‘‘π‘†=Ξ©ξ€œΞ”Ξ¨π‘‘π‘‰=βˆ’Ξ©π‘“(π‘₯,𝑦,𝑧)𝑑𝑉,(4.32) that is,

ξ€œΞ©π‘“(π‘₯,𝑦,𝑧)𝑑𝑉=0,(4.33) where 𝑉 is volume of drainage domain Ξ©.

Define the following notation of internal product of functions 𝑓(π‘₯,𝑦,𝑧) and 𝑔(π‘₯,𝑦,𝑧):

ξ€œβŒŠπ‘“(π‘₯,𝑦,𝑧),𝑔(π‘₯,𝑦,𝑧)βŒ‹=Ξ©ξ€œπ‘“(π‘₯,𝑦,𝑧)𝑔(π‘₯,𝑦,𝑧)𝑑π‘₯𝑑𝑦𝑑𝑧=Ω𝑓(π‘₯,𝑦,𝑧)𝑔(π‘₯,𝑦,𝑧)𝑑𝑉,(4.34) where βŒŠπ‘“,π‘”βŒ‹ means the internal product of functions 𝑓 and 𝑔.

From (4.33), we know that the internal product of 𝑓(π‘₯,𝑦,𝑧) and constant number 1 is zero

βŒŠπ‘“(π‘₯,𝑦,𝑧),1βŒ‹=0,(4.35) and it is easy to prove

βŒŠπ‘“(π‘₯,𝑦,𝑧),𝑔000βŒ‹=0,(4.36) where 𝑔000 means π‘”π‘™π‘šπ‘› when 𝑙=π‘š=𝑛=0.

Thus, 𝑓(π‘₯,𝑦,𝑧) can be decomposed as [13, 14]:

𝑓(π‘₯,𝑦,𝑧)=βˆžξ“π‘™,π‘š,𝑛=0𝑓,π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žπ‘”ξ€Έξžπ‘™π‘šπ‘›(=π‘₯,𝑦,𝑧)𝑙+π‘š+𝑛>0𝑓,π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žπ‘”ξ€Έξžπ‘™π‘šπ‘›=(π‘₯,𝑦,𝑧)𝑙+π‘š+𝑛>0𝛿π‘₯βˆ’π‘₯ξ…žξ€Έπ›Ώξ€·π‘¦βˆ’π‘¦ξ…žξ€Έπ›Ώξ€·π‘§βˆ’π‘§ξ…žξ€Έ,π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žπ‘”ξ€Έξžπ‘™π‘šπ‘›=(π‘₯,𝑦,𝑧)𝑙+π‘š+𝑛>0π‘”π‘™π‘šπ‘›ξ€·π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘”π‘™π‘šπ‘›(π‘₯,𝑦,𝑧).(4.37)

The drainage volume is

𝑉=π‘Žπ‘π».(4.38)

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

π‘ƒπ‘Ž,𝑣=ξ‚€1π‘‰ξ‚ξ€œΞ©π‘‘π‘ƒ(π‘₯,𝑦,𝑧)𝑑π‘₯𝑑𝑦𝑑𝑧=π‘Žπ‘π»+Ξ¨π‘Ž,𝑣.(4.39)

The wellbore pressure at point (π‘₯𝑀,𝑦𝑀,𝑧𝑀) is

𝑃𝑀=π‘‘π‘Žπ‘π»+Ψ𝑀,(4.40) where Ψ𝑀 is the value of Ξ¨ at wellbore point (π‘₯𝑀,𝑦𝑀,𝑧𝑀).

Combining (4.39) and (4.40) gives

π‘ƒπ‘Ž,π‘£βˆ’π‘ƒπ‘€=Ξ¨π‘Ž,π‘£βˆ’Ξ¨π‘€,(4.41) which implies π‘ƒπ‘Ž,π‘£βˆ’π‘ƒπ‘€ 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 (π‘₯ξ…ž,π‘¦ξ…ž,0) and point (π‘₯ξ…ž,π‘¦ξ…ž,𝐿), recall (4.4) and (4.19), in order to obtain uniform line sink solution, we integrate Ξ¨ with respect to π‘§ξ…ž from 0 to 𝐿, then

𝐽π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=ξ€œ;𝑙,π‘š,𝑛𝐿0Ξ¨ξ€·π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έπ‘‘π‘§ξ…ž=𝑙+π‘š+𝑛>0β„‘π‘™π‘šπ‘›ξ€·π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ,;𝑙,π‘š,𝑛(5.1) where

𝑙+π‘š+𝑛>0β„‘π‘™π‘šπ‘›ξ€·π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=;𝑙,π‘š,𝑛𝑙+π‘š+𝑛>0ξ‚΅1π‘Žπ‘π»π‘‘π‘™π‘‘π‘šπ‘‘π‘›πœ†π‘™π‘šπ‘›ξ‚Άξ‚€cosπ‘™πœ‹π‘₯π‘Žξ‚ξ‚€Γ—cosπ‘šπœ‹π‘¦π‘ξ‚ξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξ‚΅Γ—cosπ‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξ€œπΏ0ξ‚΅cosπ‘›πœ‹π‘§ξ…žπ»ξ‚Άπ‘‘π‘§ξ…ž=𝑙+π‘š+𝑛>0ξ‚΅1π‘Žπ‘π»π‘‘π‘™π‘‘π‘šπ‘‘π‘›πœ†π‘™π‘šπ‘›ξ‚Άξ‚€cosπ‘™πœ‹π‘₯π‘Žξ‚ξ‚€cosπ‘šπœ‹π‘¦π‘ξ‚ξ‚€cosπ‘›πœ‹π‘§π»ξ‚Γ—βŽ§βŽͺ⎨βŽͺβŽ©ξ‚€π»ξ‚ξ‚΅πœ‹π‘›cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξ‚΅cosπ‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚€sinπ‘›πœ‹πΏπ»ξ‚ξ‚΅if𝑙≠0,𝐿cosπ‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άif𝑙=0.(5.2)

Define

𝐢𝐢={(𝑙,π‘š,𝑛)βˆΆπ‘™+π‘š+𝑛>0},(5.3)1𝐢={(𝑙,π‘š,𝑛)βˆΆπ‘™=π‘š=0,𝑛>0},(5.4)2𝐢={(𝑙,π‘š,𝑛)βˆΆπ‘™=0,π‘š>0,𝑛β‰₯0},(5.5)3={(𝑙,π‘š,𝑛)βˆΆπ‘™>0,π‘šβ‰₯0,𝑛β‰₯0},(5.6) then it is easy to prove

𝐢=𝐢1βˆͺ𝐢2βˆͺ𝐢3,𝐢1∩𝐢2=βˆ…,𝐢2∩𝐢3=βˆ…,𝐢3∩𝐢1=βˆ….(5.7)

Recall (5.1) and (5.2), and use (5.3)–(75), 𝐽(π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž;𝑙,π‘š,𝑛) can be decomposed as

𝐽=𝑙+π‘š+𝑛>0β„‘π‘™π‘šπ‘›ξ€·π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…žξ€Έ=;𝑙,π‘š,π‘›βˆžξ“π‘›=1β„‘00𝑛+βˆžξ“βˆžπ‘š=1𝑛=0β„‘0π‘šπ‘›+βˆžξ“βˆžπ‘™=1ξ“βˆžπ‘š=0𝑛=0β„‘π‘™π‘šπ‘›.(5.8)

Define the following notations:

𝐽𝑧=βˆžξ“π‘›=1β„‘00𝑛,𝐽(5.9)𝑦𝑧=βˆžξ“βˆžπ‘š=1𝑛=0β„‘0π‘šπ‘›π½,(5.10)π‘₯𝑦𝑧=βˆžξ“βˆžπ‘™=1ξ“βˆžπ‘š=0𝑛=0β„‘π‘™π‘šπ‘›,(5.11) so

𝐽=𝐽𝑧+𝐽𝑦𝑧+𝐽π‘₯𝑦𝑧,(5.12) and the average value of 𝐽 at wellbore can be written as

π½π‘Ž,𝑀=𝐽𝑧,π‘Žξ€Έπ‘€+𝐽𝑦𝑧,π‘Žξ€Έπ‘€+𝐽π‘₯𝑦𝑧,π‘Žξ€Έπ‘€.(5.13)

Rearrange (4.12) and obtain

πœ†π‘™π‘šπ‘›=ξ‚€π‘™πœ‹π‘Žξ‚2+ξ‚€π‘šπœ‹π‘ξ‚2+ξ‚€π‘›πœ‹π»ξ‚2=ξ‚€πœ‹π»ξ‚2𝑛2+πœ‡2π‘™π‘šξ€Έ,(5.14) where

πœ‡2π‘™π‘š=ξ‚€π‘™π»π‘Žξ‚2+ξ‚€π‘šπ»π‘ξ‚2=𝐻𝑏2ξ‚Έπ‘š2+ξ‚€π‘™π‘π‘Žξ‚2ξ‚Ή,πœ‡π‘™0=π‘™π»π‘Ž,πœ†π‘™π‘š0=ξ‚€πœ‹π»ξ‚2πœ‡2π‘™π‘š,πœ†0π‘šπ‘›=ξ‚€π‘šπœ‹π‘ξ‚2+ξ‚€π‘›πœ‹π»ξ‚2=ξ‚€πœ‹π»ξ‚2𝑛2+ξ‚€π‘šπ»π‘ξ‚2ξ‚Ή,πœ†00𝑛=𝑛2πœ‹2𝐻2.(5.15)

There hold [16, page 47]

βˆžξ“π‘›=1sin(𝑛π‘₯)𝑛3=πœ‹2π‘₯6βˆ’πœ‹π‘₯24+π‘₯3(120≀π‘₯≀2πœ‹),(5.16)βˆžξ“π‘›=11βˆ’cos(𝑛π‘₯)𝑛4=πœ‹2π‘₯2βˆ’12πœ‹π‘₯3+π‘₯12448(0≀π‘₯≀2πœ‹).(5.17)

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

𝑦=π‘¦ξ…žβ‰ 0,π‘₯ξ…žβ‰ 0,π‘₯=π‘₯ξ…ž+𝑅𝑀,0≀𝑧=π‘§ξ…žξ€·π½β‰€πΏ,𝑧𝑀=βˆžξ“π‘›=1ξ‚΅1π‘Žπ‘π»π‘‘π‘›πœ†00𝑛cosπ‘›πœ‹π‘§π»ξ‚ξ€œπΏ0ξ‚΅cosπ‘›πœ‹π‘§ξ…žπ»ξ‚Άπ‘‘π‘§ξ…ž=ξ‚€2ξ‚π‘Žπ‘π»βˆžξ“π‘›=1𝐻2πœ‹2𝑛2ξ‚Άξ‚€cosπ‘›πœ‹π‘§π»π»ξ‚ξ‚€ξ‚ξ‚€π‘›πœ‹sinπ‘›πœ‹πΏπ»ξ‚=ξ‚΅2𝐻2π‘Žπ‘πœ‹3ξ‚Άβˆžξ“π‘›=1ξ‚€1𝑛3sinπ‘›πœ‹πΏπ»ξ‚ξ‚€cosπ‘›πœ‹π‘§π»ξ‚.(5.18)

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

𝐽𝑧,π‘Žξ€Έπ‘€=ξ‚€1πΏξ‚ξ€œπΏ0𝐽𝑧=ξ‚€1π‘‘π‘§πΏξ‚βˆžξ“π‘›=1ξ‚΅2𝐻2πœ‹3π‘Žπ‘π‘›3ξ‚Άξ‚€sinπ‘›πœ‹πΏπ»ξ‚ξ€œπΏ0ξ‚€cosπ‘›πœ‹π‘§π»ξ‚=π‘‘π‘§βˆžξ“π‘›=1ξ‚΅2𝐻2πœ‹3π‘Žπ‘πΏπ‘›3ξ‚Άξ‚€sinπ‘›πœ‹πΏπ»π»ξ‚ξ‚ƒξ‚€ξ‚ξ‚€π‘›πœ‹sinπ‘›πœ‹πΏπ»=ξ‚ξ‚„βˆžξ“π‘›=1ξ‚΅2𝐻3πœ‹4π‘Žπ‘πΏπ‘›4ξ‚Άsin2ξ‚€π‘›πœ‹πΏπ»ξ‚=𝐻3πœ‹4ξ‚Άπ‘Žπ‘πΏβˆžξ“π‘›=1ξ‚€1𝑛41βˆ’cos2π‘›πœ‹πΏπ»=𝐻3πœ‹4ξ‚Άξ‚€π‘Žπ‘πΏ2πœ‹πΏπ»ξ‚2ξ‚Έπœ‹2βˆ’πœ‹12ξ‚€122πœ‹πΏπ»ξ‚+1ξ‚€482πœ‹πΏπ»ξ‚2ξ‚Ή=ξ‚€4𝐻𝐿1π‘Žπ‘βˆ’πΏ12+𝐿6𝐻212𝐻2ξ‚Ά=ξ‚€2𝐻𝐿13π‘Žπ‘2βˆ’πΏπ»+𝐿22𝐻2ξ‚Ά,(5.19) where we have used (5.17).

For a fully penetrating well, 𝐿=𝐻, then

𝐽𝑧,π‘Žξ€Έπ‘€=0.(5.20)

Recall (5.5) and (5.10), 𝐽𝑦𝑧 is for the case 𝑙=0,π‘š>0,𝑛β‰₯0, and at wellbore of the off-center well,

𝑦=π‘¦ξ…žβ‰ 0,π‘₯ξ…žβ‰ 0,π‘₯=π‘₯ξ…ž+𝑅𝑀,0≀𝑧=π‘§ξ…žξ€·π½β‰€πΏ,𝑦𝑧𝑀=ξ‚€1ξ‚π‘Žπ‘π»βˆžξ“βˆžπ‘š=1𝑛=0ξ‚΅1π‘‘π‘šπ‘‘π‘›πœ†0π‘šπ‘›ξ‚Άcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξ€œπΏ0ξ‚΅cosπ‘›πœ‹π‘§ξ…žπ»ξ‚Άπ‘‘π‘§ξ…ž=ξ‚€2ξ‚π‘Žπ‘π»βˆžξ“βˆžπ‘š=1𝑛=0ξƒ―cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏cos(π‘›πœ‹π‘§/𝐻)πœ‹2𝑑𝑛(𝑛/𝐻)2+(π‘š/𝑏)2ξ€»ξ€œπΏ0ξ‚΅cosπ‘›πœ‹π‘§ξ…žπ»ξ‚Άπ‘‘π‘§ξ…žξƒ°=ξ‚€2ξ‚π‘Žπ‘π»βˆžξ“π‘š=1ξƒ―βˆžξ“π‘›=12(𝐻/π‘›πœ‹)cos(π‘›πœ‹π‘§/𝐻)sin(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘πœ‹2ξ€Ί(𝑛/𝐻)2+(π‘š/𝑏)2ξ€»+cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘π‘ξ‚Άξ‚΅2πΏπœ‹2π‘š2ξ‚Άξƒ°=ξ‚΅2𝐻3πœ‹3ξ‚Άπ‘Žπ‘π»βˆžξ“π‘š=1ξƒ―βˆžξ“π‘›=12cos(π‘›πœ‹π‘§/𝐻)sin(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏𝑛𝑛2+(π‘šπ»/𝑏)2ξ€»+cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘π‘ξ‚Άξ‚΅2πΏπœ‹π»3π‘š2ξ‚Άξƒ°=ξ‚΅2𝐻2πœ‹3π‘Žπ‘ξ‚Άξ‚΅πœ‹πΏπ‘2𝐻3ξ‚Άβˆžξ“π‘š=1ξ‚€1π‘š2cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Ά+ξ‚΅2𝐻2πœ‹3ξ‚Άπ‘Žπ‘βˆžξ“βˆžπ‘š=1𝑛=1ξƒ―2cos(π‘›πœ‹π‘§/𝐻)sin(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏𝑛𝑛2+(π‘šπ»/𝑏)2ξ€»ξƒ°=ξ‚€2π‘πΏπœ‹2ξ‚π‘Žπ»βˆžξ“π‘š=1ξ‚€1π‘š2cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Ά+ξ‚΅2𝐻2πœ‹3ξ‚Άπ‘Žπ‘βˆžξ“βˆžπ‘š=1𝑛=1ξƒ―2cos(π‘›πœ‹π‘§/𝐻)sin(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏𝑛𝑛2+(π‘šπ»/𝑏)2ξ€»ξƒ°,(5.21) where we use the following formulas [16, page 47]:

βˆžξ“π‘š=1ξ‚€1π‘š2ξ‚πœ‹cos(π‘šπ‘₯)=26βˆ’πœ‹π‘₯2+π‘₯24(0≀π‘₯≀2πœ‹),(5.22)βˆžξ“π‘š=1ξ‚€1π‘š2cos2πœ‹(π‘šπ‘₯)=26βˆ’πœ‹π‘₯2+π‘₯22(0≀π‘₯β‰€πœ‹).(5.23)

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

𝐽𝑦𝑧,π‘Žξ€Έπ‘€=ξ‚€1πΏξ‚ξ€œπΏ0𝐽𝑦𝑧=𝑑𝑧2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻2π‘Žπ‘πΏπœ‹3ξ‚Άβˆžξ“βˆžπ‘š=1𝑛=1ξƒ―2sin(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏𝑛𝑛2+(π‘šπ»/𝑏)2ξ€»ξ€œπΏ0ξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξƒ°=𝑑𝑧2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻2π‘Žπ‘πΏπœ‹3ξ‚Άβˆžξ“βˆžπ‘š=1𝑛=1ξƒ―2𝐻sin2(π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘πœ‹π‘›2𝑛2+(π‘šπ»/𝑏)2ξ€»ξƒ°=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻3π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“βˆžπ‘š=1𝑛=1ξƒ―[]1βˆ’cos(2π‘›πœ‹πΏ/𝐻)cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/𝑏𝑛2𝑛2+(π‘šπ»/𝑏)2ξ€»ξƒ°=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻3π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“βˆžπ‘š=1𝑛=1ξ‚€π‘ξ‚π‘šπ»2cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξ‚Έξ‚€1βˆ’cos2π‘›πœ‹πΏπ»ξ‚,1𝑛2βˆ’1𝑛2+(π‘šπ»/𝑏)2ξ‚Ή=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2𝑏2ξ…ž2𝑏2ξ‚Ά+𝐻32π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“π‘š=1ξ‚€π‘ξ‚π‘šπ»2cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—βˆžξ“π‘›=1ξ‚Έ1𝑛2βˆ’cos(2π‘›πœ‹πΏ/𝐻)𝑛2βˆ’1𝑛2+(π‘šπ»/𝑏)2+cos(2π‘›πœ‹πΏ/𝐻)𝑛2+(π‘šπ»/𝑏)2ξ‚Ή=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻3π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“π‘š=1ξ‚€π‘ξ‚π‘šπ»2cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξ‚»πœ‹26βˆ’ξ‚Έπœ‹26βˆ’πœ‹2ξ‚€2πœ‹πΏπ»ξ‚+14ξ‚€2πœ‹πΏπ»ξ‚2ξ‚Ήβˆ’ξ‚Έξ‚€π‘πœ‹ξ‚ξ‚€2π‘šπ»cothπ‘šπ»πœ‹π‘ξ‚βˆ’12ξ‚€π‘ξ‚π‘šπ»2ξ‚Ή+ξ‚Έξ‚€π‘πœ‹ξ‚[(]2π‘šπ»coshπ‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)βˆ’1sinh(π‘šπ»πœ‹/𝑏)2ξ‚€π‘ξ‚π‘šπ»2,ξ‚Ήξ‚Ό(5.24) where we use the following formulas [16, page 47]:

βˆžξ“π‘›=1cos(𝑛π‘₯)𝑛2+𝛽2=ξ‚΅πœ‹[]2𝛽cosh𝛽(πœ‹βˆ’π‘₯)ξ‚Όβˆ’1sinh(π›½πœ‹)2𝛽2(0≀π‘₯≀2πœ‹),(5.25)βˆžξ“π‘›=11𝑛2+𝛽2=ξ‚΅πœ‹ξ‚Ά12𝛽coth(π›½πœ‹)βˆ’2𝛽2(0≀π‘₯≀2πœ‹),(5.26) and we may simplify (5.24) further

𝐽𝑦𝑧,π‘Žξ€Έπ‘€=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻3π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“π‘š=1cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚€π‘ξ‚π‘šπ»2Γ—ξ‚»πœ‹2πΏπ»βˆ’πœ‹2𝐿2𝐻2+ξ‚€π‘πœ‹ξ‚ξ‚»[]2π‘šπ»cosh(π‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)ξ‚€sinh(π‘šπ»πœ‹/𝑏)βˆ’cothπ‘šπ»πœ‹π‘ξ‚=ξ‚€ξ‚Όξ‚Ό2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚΅2𝐻3π‘Žπ‘πΏπœ‹4ξ‚Άβˆžξ“π‘š=1cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚€1π‘š2ξ‚Γ—βŽ§βŽͺ⎨βŽͺβŽ©πœ‹2𝐿𝑏2𝐻3βˆ’πœ‹2𝐿2𝑏2𝐻4+𝑏3πœ‹2π‘šπ»3ξ‚ΆΓ—ξ‚»[]cosh(π‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)ξ‚€sinh(π‘šπ»πœ‹/𝑏)βˆ’cothπ‘šπ»πœ‹π‘ξ‚ξ‚Όξƒ°=ξ‚€2𝑏𝐿1π‘Žπ»6βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά+ξ‚€2π‘π‘Žπœ‹2𝐿1βˆ’π»ξ‚ξƒ©πœ‹26βˆ’πœ‹2π‘¦ξ…ž+πœ‹2𝑏2𝑦2β€²2𝑏2ξƒͺ+𝑏2π‘ŽπΏπœ‹3ξ‚Άβˆžξ“π‘š=1cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚Άξ‚€1π‘š3×[]cosh(π‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)ξ‚€sinh(π‘šπ»πœ‹/𝑏)βˆ’cothπ‘šπ»πœ‹π‘ξ‚ξ‚Ό=ξ‚€2π‘π‘Žξ‚ξ‚΅16βˆ’π‘¦ξ…ž+𝑦2𝑏2ξ…ž2𝑏2ξ‚Ά+𝑏2π‘ŽπΏπœ‹3ξ‚Άβˆžξ“π‘š=1cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘š3ξƒ­ξ‚»[]cosh(π‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)ξ‚€sinh(π‘šπ»πœ‹/𝑏)βˆ’cothπ‘šπ»πœ‹π‘ξ‚ξ‚Ό.(5.27)

For a fully penetrating well, 𝐿=𝐻, then

𝐽𝑦𝑧,π‘Žξ€Έπ‘€=ξ‚€2π‘π‘Žξ‚ξ‚΅16βˆ’π‘¦ξ…ž+𝑦2π‘ξ…ž22𝑏2ξ‚Ά.(5.28)

Define

[]𝑓(π‘₯)=sinh𝛼(1βˆ’π‘₯)sinh(𝛼π‘₯),(5.29) since the derivative of 𝑓(π‘₯) is

π‘“ξ…ž[][][],(π‘₯)=𝛼cosh(𝛼π‘₯)sinh𝛼(1βˆ’π‘₯)βˆ’π›Όcosh𝛼(1βˆ’π‘₯)sinh(𝛼π‘₯)=𝛼sinh𝛼(1βˆ’2π‘₯)(5.30) consequently,

π‘“ξ…žξ‚€12=0.(5.31)

When π‘₯=0 and π‘₯=1,

𝑓(0)=𝑓(1)=0.(5.32)

When π‘₯=1/2,𝑓(π‘₯) reaches maximum value, let

𝐿π‘₯=𝐻,(5.33) and the producing length 𝐿 is a variable, define

[]𝐹(𝐿)=coshπ›½πœ‹(1βˆ’2𝐿/𝐻)βˆ’cosh(π›½πœ‹)=[][]sinh(π›½πœ‹)βˆ’2sinhπ›½πœ‹(1βˆ’πΏ/𝐻)sinhπ›½πœ‹πΏ/(𝐻),sinh(π›½πœ‹)(5.34) thus when 𝐿=𝐻/2,|𝐹(𝐿)| reaches maximum value,

||||𝐹(𝐿)max=|||𝐹𝐻2|||=2sinh2(π›½πœ‹/2)=sinh(π›½πœ‹)2sinh2(π›½πœ‹/2)=2sinh(π›½πœ‹/2)cosh(π›½πœ‹/2)sinh(π›½πœ‹/2)cosh(π›½πœ‹/2)<1,(5.35) so 𝐹(𝐿) is a bounded function, let

𝛽=π‘šπ»π‘,(5.36) then

𝐽𝑦𝑧,π‘Žξ€Έπ‘€=ξ‚€2π‘π‘Žξ‚ξ‚΅16βˆ’π‘¦ξ…ž+𝑦2𝑏2ξ…ž2𝑏2ξ‚Ά+𝑏2π‘ŽπΏπœ‹3ξ‚ΆΓ—βˆžξ“π‘š=1cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘š3ξƒ­ξ‚»[]cosh(π‘šπ»πœ‹/𝑏)(1βˆ’2𝐿/𝐻)ξ‚€sinh(π‘šπ»πœ‹/𝑏)βˆ’cothπ‘šπ»πœ‹π‘ξ‚ξ‚Ό=ξ‚€2π‘π‘Žξ‚ξ‚΅16βˆ’π‘¦ξ…ž+𝑦2𝑏2ξ…ž2𝑏2ξ‚Ά+𝑏2π‘ŽπΏπœ‹3ξ‚ΆΓ—βˆžξ“π‘š=1cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘š3ξƒ­ξ‚»[]βˆ’2sinh(π‘šπ»πœ‹/𝑏)(1βˆ’πΏ/𝐻)sinh(π‘šπΏπœ‹/𝑏)ξ‚Όβ‰ˆξ‚€sinh(π‘šπ»πœ‹/𝑏)2π‘π‘Žξ‚ξ‚΅16βˆ’π‘¦ξ…ž+𝑦2𝑏2ξ…ž2𝑏2ξ‚Ά+𝑏2π‘ŽπΏπœ‹3ξ‚ΆΓ—π‘€ξ“π‘š=1cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘š3ξƒ­ξ‚»[]βˆ’2sinh(π‘šπ»πœ‹/𝑏)(1βˆ’πΏ/𝐻)sinh(π‘šπΏπœ‹/𝑏)ξ‚Ό.sinh(π‘šπ»πœ‹/𝑏)(5.37)

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

βˆžξ“π‘š=101|||||cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘š3ξƒ­ξ‚»[]βˆ’2sinh(π‘šπ»πœ‹/𝑏)(1βˆ’πΏ/𝐻)sinh(π‘šπΏπœ‹/𝑏)ξ‚Ό|||||≀sinh(π‘šπ»πœ‹/𝑏)βˆžξ“π‘š=1011π‘š3=𝜁(3)βˆ’100ξ“π‘š=11π‘š3=4.9502Γ—10βˆ’5,(5.38) where 𝜁(3) is π‘…π‘–π‘’π‘šπ‘Žπ‘›π‘›-𝜁 function:

𝜁(3)=βˆžξ“π‘š=11π‘š3=1.202057,(5.39) thus

βˆžξ“π‘š=1ξ‚€1π‘š3[]2sinh(π‘šπ»πœ‹/𝑏)(1βˆ’πΏ/𝐻)sinh(π‘šπΏπœ‹/𝑏)ξ‚Όβ‰ˆsinh(π‘šπ»πœ‹/𝑏)100ξ“π‘š=1ξ‚€1π‘š3[]2sinh(π‘šπ»πœ‹/𝑏)(1βˆ’πΏ/𝐻)sinh(π‘šπΏπœ‹/𝑏)ξ‚Ό.sinh(π‘šπ»πœ‹/𝑏)(5.40)

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

Recall (5.6) and (5.11), 𝐽π‘₯𝑦𝑧 is for the case 𝑙>0,π‘šβ‰₯0,𝑛β‰₯0, and at wellbore of the off-center well,

𝑦=π‘¦ξ…žβ‰ 0,π‘₯ξ…žβ‰ 0,π‘₯=π‘₯ξ…ž+𝑅𝑀,0≀𝑧=π‘§ξ…žβ‰€πΏ,(5.41) then

𝐽π‘₯𝑦𝑧𝑀=ξ‚€1ξ‚Γ—π‘Žπ‘π»βˆžξ“βˆžπ‘™=1ξ“βˆžπ‘š=0𝑛=0cos(π‘›πœ‹π‘§/𝐻)cosπ‘™πœ‹π‘₯ξ…žξ€Έξ€·π‘₯/π‘Žcosξ€Ίξ€·π‘™πœ‹ξ…ž+𝑅𝑀/π‘Žcos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘‘π‘™π‘‘π‘šπ‘‘π‘›πœ†π‘™π‘šπ‘›ξƒ­Γ—ξ€œπΏ0ξ‚΅cosπ‘›πœ‹π‘§ξ…žπ»ξ‚Άπ‘‘π‘§ξ…žξ‚Ό=ξ‚€1ξ‚π‘Žπ‘π»βˆžξ“βˆžπ‘™=1ξ“π‘š=0ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=1ξ‚Έ4(𝐻/π‘›πœ‹)sin(π‘›πœ‹πΏ/𝐻)cos(π‘›πœ‹π‘§/𝐻)π‘‘π‘šπœ†π‘™π‘šπ‘›ξ‚Ή+2πΏπ‘‘π‘šπœ†π‘™π‘š0ξƒ°.(5.42)

The average value of (𝐽π‘₯𝑦𝑧)𝑀 along the well length is 𝐽π‘₯𝑦𝑧,π‘Žξ€Έπ‘€=ξ‚€1ξ‚π‘Žπ‘π»βˆžξ“βˆžπ‘™=1ξ“π‘š=0ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=14∫(𝐻/π‘›πœ‹)sin(π‘›πœ‹πΏ/𝐻)𝐿0cos(π‘›πœ‹π‘§/𝐻)π‘‘π‘§π‘‘π‘šπœ†π‘™π‘šπ‘›πΏξƒ­+2πΏπ‘‘π‘šπœ†π‘™π‘š0ξƒ°=ξ‚€1ξ‚π‘Žπ‘π»βˆžξ“βˆžπ‘™=1ξ“π‘š=0ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=1ξ‚Έ4(𝐻/π‘›πœ‹)2sin2(π‘›πœ‹πΏ/𝐻)π‘‘π‘šπœ†π‘™π‘šπ‘›πΏξ‚Ή+2πΏπ‘‘π‘šπœ†π‘™π‘š0ξƒ°=𝐻4π‘Žπ‘π»πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­βˆžξ“π‘š=0cos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=12[]1βˆ’cos(2π‘›πœ‹πΏ/𝐻)π‘‘π‘šπ‘›2𝑛2+πœ‡2π‘™π‘šξ€ΈπΏ+2πœ‹2πΏπ‘‘π‘šπ»2πœ‡2π‘™π‘šξƒ°=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­ξƒ―βˆžξ“π‘š=0ξ‚΅2π‘‘π‘šξ‚Άcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=11πœ‡2π‘™π‘šπΏξƒͺ1βˆ’cos2π‘›πœ‹πΏπ»Γ—ξƒ©1𝑛2βˆ’1𝑛2+πœ‡2π‘™π‘šξƒͺ+πœ‹2𝐿𝐻2πœ‡2π‘™π‘šξƒ°=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­ξƒ―βˆžξ“π‘š=02π‘‘π‘šπœ‡2π‘™π‘šπΏξƒͺcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―βˆžξ“π‘›=11𝑛2βˆ’cos(2π‘›πœ‹πΏ/𝐻)𝑛2βˆ’1𝑛2+πœ‡2π‘™π‘š+cos(2π‘›πœ‹πΏ/𝐻)𝑛2+πœ‡2π‘™π‘š+ξ‚΅πœ‹ξƒ­ξƒ°2𝐿2𝐻2ξ‚Άξƒ°=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­βˆžξ“π‘š=02π‘‘π‘šπœ‡2π‘™π‘šπΏξƒͺcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξ‚»πœ‹26βˆ’ξ‚Έπœ‹26βˆ’ξ‚€πœ‹22πœ‹πΏπ»ξ‚+ξ‚€142πœ‹πΏπ»ξ‚2ξ‚Ήβˆ’ξƒ¬ξ‚΅πœ‹2πœ‡π‘™π‘šξ‚Άξ€·πœ‡cothπ‘™π‘šπœ‹ξ€Έβˆ’ξƒ©12πœ‡2π‘™π‘š+ξƒ¬ξ‚΅πœ‹ξƒͺξƒ­2πœ‡π‘™π‘šξ‚Άξ€Ίπœ‡coshπ‘™π‘šξ€»πœ‹(1βˆ’2𝐿/𝐻)ξ€·πœ‡sinhπ‘™π‘šπœ‹ξ€Έβˆ’12πœ‡2π‘™π‘š+ξ‚΅πœ‹ξƒ­ξƒ°2𝐿2𝐻2ξ‚Άξƒ°,(5.43) where we use (5.22) and (5.25).

Let π‘₯=0, recast (5.26), we obtain

1𝛽2+2βˆžξ“π‘›=11𝑛2+𝛽2=ξ‚΅πœ‹π›½ξ‚Άcoth(π›½πœ‹),βˆžξ“π‘›=01𝑛2+𝛽2𝑑𝑛=ξ‚΅πœ‹π›½ξ‚Άcoth(π›½πœ‹).(5.44) So,

𝐽π‘₯𝑦𝑧,π‘Žξ€Έπ‘€=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­βˆžξ“π‘š=02π‘‘π‘šπœ‡2π‘™π‘šπΏξƒͺcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―ξ‚΅πœ‹2πΏπ»βˆ’πœ‹2𝐿2𝐻2+πœ‹2𝐿2𝐻2ξ‚Ά+ξ‚΅πœ‹2πœ‡π‘™π‘šξ‚Άξƒ¬ξ€Ίπœ‡coshπ‘™π‘šπœ‹ξ€»(1βˆ’2𝐿/𝐻)ξ€·πœ‡sinhπ‘™π‘šπœ‹ξ€Έξ€·πœ‡βˆ’cothπ‘™π‘šπœ‹ξ€Έ=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­Γ—ξƒ―βˆžξ“π‘š=0ξƒ©πœ‹π‘‘π‘šπœ‡3π‘™π‘šπΏξƒͺcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―ξ€Ίπœ‡coshπ‘™π‘šξ€»πœ‹(1βˆ’2𝐿/𝐻)ξ€·πœ‡sinhπ‘™π‘šπœ‹ξ€Έξ€·πœ‡βˆ’cothπ‘™π‘šπœ‹ξ€Έξƒ°+ξ‚€2πΏξ‚ξ‚΅πœ‹2πΏπ»ξ‚Άβˆžξ“π‘š=0cos2ξ€·π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘‘π‘šπœ‡2π‘™π‘šξƒ°=𝐻3π‘Žπ‘πœ‹4ξ‚Άβˆžξ“π‘™=1ξ‚΅cosπ‘™πœ‹π‘₯ξ…žπ‘Žξ‚Άξƒ¬ξ€·π‘₯cosπ‘™πœ‹ξ…ž+π‘…π‘€ξ€Έπ‘Žξƒ­Γ—ξƒ―βˆžξ“π‘š=0ξƒ©πœ‹π‘‘π‘šπœ‡3π‘™π‘šπΏξƒͺcos2ξ‚΅π‘šπœ‹π‘¦ξ…žπ‘ξ‚ΆΓ—ξƒ―ξ€Ίπœ‡coshπ‘™π‘šξ€»πœ‹(1βˆ’2𝐿/𝐻)ξ€·πœ‡sinhπ‘™π‘šπœ‹ξ€Έξ€·πœ‡βˆ’cothπ‘™π‘šπœ‹ξ€Έξƒ°+ξ‚΅π‘Žπ‘πœ‹3𝐻3𝑙cothπ‘™π‘πœ‹π‘Žξ‚+ξ‚΅πœ‹2π»ξ‚Άβˆžξ“π‘š=0ξ€·cos2π‘šπœ‹π‘¦ξ…žξ€Έ/π‘π‘‘π‘šπœ‡2π‘™π‘šξƒ°=𝐻3π‘Žπ‘πΏπœ‹3ξ‚Άβˆžξ“π‘™=1ξ‚΅c