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]

๐‘„๐‘ค=๐น๐ท๎€ท๐‘ƒ2๐œ‹๐พ๐ป๐‘Žโˆ’๐‘ƒ๐‘ค๎€ธ/(๐œ‡๐ต)๎€ท๐‘…ln๐‘’/๐‘…๐‘ค๎€ธโˆ’3/4,(2.1) where ๐‘ƒ๐‘Ž is average reservoir pressure in the circular drainage area, ๐‘ƒ๐‘ค is flowing wellbore pressure, ๐พ is permeability, ๐ป is payzone thickness, ๐œ‡ is oil viscosity, ๐ต is oil formation volume factor, ๐‘…๐‘’ is radius of circular drainage area, ๐‘…๐‘ค is wellbore radius, and ๐น๐ท is the factor which allows the use of field units and practical ๐‘†๐ผ 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, ๐‘†ps. Thus, (2.1) may be rewritten to include the pseudoskin factor due to partial penetration as [4, page 92]:

๐‘„๐‘ค=๐น๐ท๎€ท๐‘ƒ2๐œ‹๐พ๐ป๐‘Žโˆ’๐‘ƒ๐‘ค๎€ธ/(๐œ‡๐ต)๎€ท๐‘…ln๐‘’/๐‘…๐‘ค๎€ธโˆ’3/4+๐‘†ps.(2.2)

๐‘†ps can be calculated by semianalytical and semiempirical expressions presented by Brons, Marting, Papatzacos, and Bervaldier [5โ€“7].

Assume that the well-drilled length is equal to the well producing length, (i.e., perforated interval,) ๐ฟ๐‘=๐ฟ, and define partial penetration factor ๐œ‚:

๐ฟ๐œ‚=๐‘๐ป=๐ฟ๐ป.(2.3)

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

๐‘†ps=๎‚ต1๐œ‚๎‚ถ๎€บ๎€ทโ„Žโˆ’1ln๐ท๎€ธ๎€ปโˆ’๐บ(๐œ‚),(2.4) where

โ„Ž๐ท=๎‚ต๐ป๐‘…๐‘ค๐พ๎‚ถ๎‚ตโ„Ž๐พ๐‘ฃ๎‚ถ1/2,๐บ(2.5)(๐œ‚)=2.948โˆ’7.363๐œ‚+11.45๐œ‚2โˆ’4.675๐œ‚3.(2.6)

Pseudoskin factor formula given by Papatzacos is [6]

๐‘†ps=๎‚ต1๐œ‚๎‚ถ๎‚ตโˆ’1ln๐œ‹โ„Ž๐ท2๎‚ถ+๎‚ต1๐œ‚๎‚ถ๎ƒฌ๎‚ต๐œ‚lnฮจ2+๐œ‚๎‚ถ๎‚ต1โˆ’1ฮจ2๎‚ถโˆ’11/2๎ƒญ,(2.7) where โ„Ž๐ท has the same meaning as in (2.5), and

ฮจ1=๐ปโ„Ž1+0.25๐ฟ๐‘,ฮจ2=๐ปโ„Ž1+0.75๐ฟ๐‘,(2.8) and โ„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]

๐‘†ps=๎‚ต1๐œ‚๎‚ถ๎ƒฌ๎€ท๐ฟโˆ’1ln๐‘/๐‘…๐‘ค๎€ธ๎€ท1โˆ’๐‘…๐‘ค/๐ฟ๐‘๎€ธ๎ƒญโˆ’1.(2.9)

It must be pointed out that the well location in the reservoir has no effect on ๐‘†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 ๐‘†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:

๐‘„๐‘ค=๐น๐ท๎€ท๐‘ƒ2๐œ‹๐พ๐ป๐‘Žโˆ’๐‘ƒ๐‘ค๎€ธ/(๐œ‡๐ต)(๎€บ๎€ท๐ถ1/2)ln2.2458๐ด/๐ด๐‘…2๐‘ค๎€ธ๎€ป,(2.10) where ๐ถ๐ด is shape factor, and ๐ด is drainage area.

Dietz evaluated shape factor ๐ถ๐ด 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 ๐ฟ drains a box-shaped reservoir with height ๐ป, length (๐‘ฅ direction) ๐‘Ž, and width (๐‘ฆ direction) ๐‘. The well is parallel to the ๐‘ง direction with a length ๐ฟโ‰ค๐ป, and we assume ๐‘โ‰ฅ๐‘Ž.

The following assumptions are made.

(1)The reservoir is homogeneous, anisotropic, and has constant ๐พ๐‘ฅ,๐พ๐‘ฆ,๐พ๐‘ง permeabilities, thickness ๐ป, and porosity ๐œ™. All the boundaries of the box-shaped drainage volume are sealed.(2)The reservoir pressure is initially constant. At time ๐‘ก=0, pressure is uniformly distributed in the reservoir, equal to the initial pressure ๐‘ƒ๐‘–.(3)The production occurs through a partially penetrating vertical well of radius ๐‘…๐‘ค, represented in the model by a uniform line sink.(4)A single phase fluid, of small and constant compressibility ๐ถ๐‘“, constant viscosity ๐œ‡, and formation volume factor ๐ต, flows from the reservoir to the well at a constant rate ๐‘„๐‘ค. 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 (๐‘ฅ๎…ž,๐‘ฆ๎…ž,0) and (๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐ฟ). We suppose the point (๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž) is on the well line, and its point convergence intensity is ๐‘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 (๐‘ฅ,๐‘ฆ,๐‘ง) caused by the point (๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž), we have to obtain the basic solution of the following Laplace equation:

๐พ๐‘ฅ๐œ•2๐‘ƒ๐œ•๐‘ฅ2+๐พ๐‘ฆ๐œ•2๐‘ƒ๐œ•๐‘ฆ2+๐พ๐‘ง๐œ•2๐‘ƒ๐œ•๐‘ง2=๐œ™๐œ‡๐ถ๐‘ก๐œ•๐‘ƒ๎€ท๐œ•๐‘ก+๐œ‡๐‘ž๐ต๐›ฟ๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ,(3.1) in the box-shaped drainage volume:

ฮฉ=(0,๐‘Ž)ร—(0,๐‘)ร—(0,๐ป),(3.2) and we always assume

๐‘โ‰ฅ๐‘Žโ‰ซ๐ป,(3.3) and ๐›ฟ(๐‘ฅโˆ’๐‘ฅ๎…ž), ๐›ฟ(๐‘ฆโˆ’๐‘ฆ๎…ž), ๐›ฟ(๐‘งโˆ’๐‘ง๎…ž) are Dirac functions.

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

๐œ•๐‘ƒ|||๐œ•๐‘ฮ“=0,(3.4) where ๐œ•๐‘ƒ/๐œ•๐‘|ฮ“ is the exterior normal derivative of pressure on the surface of box-shaped drainage volume ฮ“=๐œ•ฮฉ.

The reservoir pressure is initially constant

๐‘ƒ||๐‘ก=0=๐‘ƒ๐‘–.(3.5)

Define average permeability:

๐พ๐‘Ž=๎€ท๐พ๐‘ฅ๐พ๐‘ฆ๐พ๐‘ง๎€ธ1/3.(3.6)

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

๐‘ฅ๐ท=๎‚€๐‘ฅ๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ฅ๎‚ถ1/2,๐‘ฆ๐ท=๎‚€๐‘ฆ๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ฆ๎‚ถ1/2,๐‘ง๐ท=๎‚€๐‘ง๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ง๎‚ถ1/2,๐‘Ž๐ท=๎‚€๐‘Ž๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ฅ๎‚ถ1/2,๐‘๐ท=๎‚€๐‘๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ฆ๎‚ถ1/2,๐ฟ๐ท=๎‚ต๐พ๐‘Ž๐พ๐‘ง๎‚ถ1/2,๐ป๐ท=๎‚€๐ป๐ฟ๎‚๎‚ต๐พ๐‘Ž๐พ๐‘ง๎‚ถ1/2,๐‘ก๐ท=๐พ๐‘Ž๐‘ก๐œ™๐œ‡๐ถ๐‘ก๐ฟ2.(3.7)

The dimensionless wellbore radius is [8]

๐‘…๐‘ค๐ท=๎€ท๐พ๐‘ง/โˆš๐พ๐‘ฅ๐พ๐‘ฆ๎€ธ1/6๎‚ƒ๎€ท๐พ๐‘ฅ/๐พ๐‘ฆ๎€ธ1/4+๎€ท๐พ๐‘ฆ/๐พ๐‘ฅ๎€ธ1/4๎‚„๐‘…๐‘ค.2๐ฟ(3.8)

Assume that ๐‘ž is the point convergence intensity at the point sink (๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž), the partially penetrating well is a uniform line sink, the total productivity of the well is ๐‘„๐‘ค, and there holds [8]

๐‘„๐‘ž=๐‘ค๐ฟ๐‘๐ท=๐‘„๐‘ค๐ฟ๐ท.(3.9)

Dimensionless pressures are defined by

๐‘ƒ๐ท=๐พ๐‘Ž๐ฟ๎€ท๐‘ƒ๐‘–๎€ธโˆ’๐‘ƒ,๐‘ƒ๐œ‡๐‘ž๐ต(3.10)๐‘ค๐ท=๐พ๐‘Ž๐ฟ๎€ท๐‘ƒ๐‘–โˆ’๐‘ƒ๐‘ค๎€ธ.๐œ‡๐‘ž๐ต(3.11)

Then (3.1) becomes

๐œ•๐‘ƒ๐ท๐œ•๐‘ก๐ทโˆ’๎ƒฉ๐œ•2๐‘ƒ๐ท๐œ•๐‘ฅ2๐ท+๐œ•2๐‘ƒ๐ท๐œ•๐‘ฆ2๐ท+๐œ•2๐‘ƒ๐ท๐œ•๐‘ง2๐ท๎ƒช๎€ท๐‘ฅ=๐›ฟ๐ทโˆ’๐‘ฅ๎…ž๐ท๎€ธ๐›ฟ๎€ท๐‘ฆ๐ทโˆ’๐‘ฆ๎…ž๐ท๎€ธ๐›ฟ๎€ท๐‘ง๐ทโˆ’๐‘ง๎…ž๐ท๎€ธ,(3.12) in the dimensionless box-shaped drainage volume

ฮฉ๐ท=๎€ท0,๐‘Ž๐ท๎€ธร—๎€ท0,๐‘๐ท๎€ธร—๎€ท0,๐ป๐ท๎€ธ,(3.13) with boundary condition

๐œ•๐‘ƒ๐ท๐œ•๐‘๐ท||||ฮ“๐ท=0,(3.14) and initial condition

๐‘ƒ๐ท||๐‘ก๐ท=0=0.(3.15)

4. Point Sink Solution

For convenience in the following reference, we use dimensionless transforms given by (3.7)โ€“(3.10), every variable, drainage domain, initial and boundary conditions should be taken as dimensionless, but we drop the subscript ๐ท.

Consequently, (3.12) is expressed as

๐œ•๐‘ƒโˆ’๎‚ต๐œ•๐œ•๐‘ก2๐‘ƒ๐œ•๐‘ฅ2+๐œ•2๐‘ƒ๐œ•๐‘ฆ2+๐œ•2๐‘ƒ๐œ•๐‘ง2๎‚ถ๎€ท=๐›ฟ๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ.(4.1)

Rewrite (3.14) below

๐œ•๐‘ƒ|||๐œ•๐‘ฮ“=0,(4.2) and (3.15) becomes

๐‘ƒ||๐‘ก=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 ๐‘ก 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 {๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)} [11, 12]:

๐‘”๐‘™๐‘š๐‘›๎ƒŽ(๐‘ฅ,๐‘ฆ,๐‘ง)=1๐‘Ž๐‘๐ป๐‘‘๐‘™๐‘‘๐‘š๐‘‘๐‘›๎‚€cos๐‘™๐œ‹๐‘ฅ๐‘Ž๎‚๎‚€cos๐‘š๐œ‹๐‘ฆ๐‘๎‚๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚,(4.4) where ๐‘™,๐‘š,๐‘› are nonnegative numbers, and

๐‘‘๐‘™=โŽงโŽชโŽจโŽชโŽฉ11if๐‘™=0,2if๐‘™>0,(4.5) and ๐‘‘๐‘š,๐‘‘๐‘› 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]):

๐›ฟ๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ=โˆž๎“๐‘™,๐‘š,๐‘›=0๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ.(4.6)

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

โˆž๎“๐‘™,๐‘š,๐‘›=0๐น๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)=โˆž๎“โˆž๐‘™=0๎“โˆž๐‘š=0๎“๐‘›=0๐น๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง),(4.7) which means in any function ๐น(๐‘ฅ,๐‘ฆ,๐‘ง), the subscripts ๐‘™,๐‘š,๐‘› of any variable must count from 0 to infinite.

And define

๎“๐‘™+๐‘š+๐‘›>0๐น๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)=โˆž๎“โˆž๐‘™โ‰ฅ0๎“โˆž๐‘šโ‰ฅ0๎“๐‘›โ‰ฅ0๐น๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)(๐‘™+๐‘š+๐‘›>0),(4.8) which means in any function ๐น(๐‘ฅ,๐‘ฆ,๐‘ง), the subscripts ๐‘™,๐‘š,๐‘› of any variable must be no less than zero, and at least one of the three subscripts ๐‘™,๐‘š,๐‘› must be positive to guarantee ๐‘™+๐‘š+๐‘›>0. And the upper limit of the subscripts ๐‘™,๐‘š,๐‘› is infinite.

Let

๐‘ƒ๎€ท๐‘ก,๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=โˆž๎“๐‘™,๐‘š,๐‘›=0๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง),(4.9) where ๐‘’๐‘™๐‘š๐‘›(๐‘ก) 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

โˆž๎“๐‘™,๐‘š,๐‘›=0๎‚ป๐œ•๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐‘”๐œ•๐‘ก๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)โˆ’๐‘’๐‘™๐‘š๐‘›(๎€บ๐‘”๐‘ก)ฮ”๐‘™๐‘š๐‘›(๎€ป๎‚ผ=๐‘ฅ,๐‘ฆ,๐‘ง)โˆž๎“๐‘™,๐‘š,๐‘›=0๎‚ป๐œ•๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐œ•๐‘ก+๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐œ†๐‘™๐‘š๐‘›๎‚ผ๐‘”๐‘™๐‘š๐‘›=(๐‘ฅ,๐‘ฆ,๐‘ง)โˆž๎“๐‘™,๐‘š,๐‘›=0๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง),(4.10) where ฮ” is the three-dimensional Laplace operator

๐œ•ฮ”=2๐œ•๐‘ฅ2+๐œ•2๐œ•๐‘ฆ2+๐œ•2๐œ•๐‘ง2,๐œ†(4.11)๐‘™๐‘š๐‘›=๎‚€๐‘™๐œ‹๐‘Ž๎‚2+๎‚€๐‘š๐œ‹๐‘๎‚2+๎‚€๐‘›๐œ‹๐ป๎‚2.(4.12)

From (4.3) and (4.9),

๐‘’๐‘™๐‘š๐‘›(0)=0,(4.13) compare the coefficients of ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง) at both sides of (4.10), we obtain

๐œ•๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐œ•๐‘ก+๐œ†๐‘™๐‘š๐‘›๐‘’๐‘™๐‘š๐‘›(๐‘ก)=๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ,(4.14) because ๐œ†000=0, from (4.14),

๐‘’000(๐‘ก)=๐‘”000๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘ก=๐‘กโˆš.๐‘Ž๐‘๐ป(4.15)

When ๐œ†๐‘™๐‘š๐‘›โ‰ 0(๐‘™+๐‘š+๐‘›>0), solve (4.14),

๐‘’๐‘™๐‘š๐‘›๎€บ๎€ท(๐‘ก)=1โˆ’expโˆ’๐œ†๐‘™๐‘š๐‘›๐‘ก๐‘”๎€ธ๎€ป๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐œ†๐‘™๐‘š๐‘›.(4.16)

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

๐‘ƒ๎€ท๐‘ก,๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=โˆž๎“๐‘™,๐‘š,๐‘›=0๐‘’๐‘™๐‘š๐‘›(๐‘ก)๐‘”๐‘™๐‘š๐‘›(=๎ƒฉ๐‘ก๐‘ฅ,๐‘ฆ,๐‘ง)โˆš๎ƒช๐‘”๐‘Ž๐‘๐ป000+๎“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘™+๐‘š+๐‘›>0๎€บ๎€ท1โˆ’expโˆ’๐œ†๐‘™๐‘š๐‘›๐‘ก๐‘”๎€ธ๎€ป๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐œ†๐‘™๐‘š๐‘›=๐‘ก+๎“๐‘Ž๐‘๐ป๐‘™+๐‘š+๐‘›>0๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐œ†๐‘™๐‘š๐‘›โˆ’๎“๐‘™+๐‘š+๐‘›>0๎€ทexpโˆ’๐œ†๐‘™๐‘š๐‘›๐‘ก๎€ธ๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐œ†๐‘™๐‘š๐‘›.(4.17)

Define

๐ผ1=๐‘ก,๐ผ๐‘Ž๐‘๐ป(4.18)2๎€ท=ฮจ๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๎“๐‘™+๐‘š+๐‘›>0๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐œ†๐‘™๐‘š๐‘›,๐ผ(4.19)3=๎“๐‘™+๐‘š+๐‘›>0๎€ทexpโˆ’๐œ†๐‘™๐‘š๐‘›๐‘ก๎€ธ๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐œ†๐‘™๐‘š๐‘›,(4.20) then

๐‘ƒ๎€ท๐‘ก,๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๐ผ1+๐ผ2โˆ’๐ผ3.(4.21)

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

ฮจ๐‘Ž,๐‘ฃ=๎‚€1๐‘‰๎‚๎€œฮฉ=๎‚€1ฮจ(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘‰๐‘‰๎‚๎€œ๐‘Ž0๎€œ๐‘0๎€œ๐ป0ฮจ๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๎‚€1๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘‘๐‘ง๐‘‰๎‚๎ƒฉ๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐œ†๐‘™๐‘š๐‘›๎ƒช๎€œ๐‘Ž0๎€œ๐‘0๎€œ๐ป0๎“๐‘™+๐‘š+๐‘›>0๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘‘๐‘ง.(4.22)

Note that ๐‘™+๐‘š+๐‘›>0 implies that at least one of ๐‘™,๐‘š,๐‘› must be greater than 0, without losing generality, we may assume

๐‘™>0,(4.23) then

๎€œ๐‘Ž0๎‚€cos๐‘™๐œ‹๐‘ฅ๐‘Ž๎‚๐‘‘๐‘ฅ=0.(4.24) So,

๎€œ๐‘Ž0๎€œ๐‘0๎€œ๐ป0๎“๐‘™+๐‘š+๐‘›>0๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘‘๐‘ง=0,(4.25) consequently,

ฮจ๐‘Ž,๐‘ฃ=0.(4.26)

If time ๐‘ก is sufficiently long, pseudo-steady-state is reached, ๐ผ3 decreases by exponential law, ๐ผ3 will vanish, that is,

๐ผ3โ‰ˆ0,(4.27) then

๐‘ƒ๎€ท๐‘ก,๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๐‘ก๎€ท๐‘Ž๐‘๐ป+ฮจ๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ.(4.28)

Substituting (4.28) into (4.1), we have

1๎€ท๐‘Ž๐‘๐ปโˆ’ฮ”ฮจ=๐›ฟ๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ.(4.29)

Define

๎‚€1๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)=โˆ’ฮ”ฮจ=โˆ’๎‚๎€ท๐‘Ž๐‘๐ป+๐›ฟ๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ,(4.30) note that ฮจ is equal to ๐ผ2 in (4.19), and

๐œ•ฮจ๐œ•๐‘=0,onฮ“.(4.31)

From Green's Formula [15],

๎€œ0=ฮ“๐œ•ฮจ๎€œ๐œ•๐‘๐‘‘๐‘†=ฮฉ๎€œฮ”ฮจ๐‘‘๐‘‰=โˆ’ฮฉ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘‰,(4.32) that is,

๎€œฮฉ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘‰=0,(4.33) where ๐‘‰ is volume of drainage domain ฮฉ.

Define the following notation of internal product of functions ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง) and ๐‘”(๐‘ฅ,๐‘ฆ,๐‘ง):

๎€œโŒŠ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง),๐‘”(๐‘ฅ,๐‘ฆ,๐‘ง)โŒ‹=ฮฉ๎€œ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘”(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘‘๐‘ง=ฮฉ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘”(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘‰,(4.34) where โŒŠ๐‘“,๐‘”โŒ‹ means the internal product of functions ๐‘“ and ๐‘”.

From (4.33), we know that the internal product of ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง) and constant number 1 is zero

โŒŠ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง),1โŒ‹=0,(4.35) and it is easy to prove

โŒŠ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง),๐‘”000โŒ‹=0,(4.36) where ๐‘”000 means ๐‘”๐‘™๐‘š๐‘› when ๐‘™=๐‘š=๐‘›=0.

Thus, ๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง) can be decomposed as [13, 14]:

๐‘“(๐‘ฅ,๐‘ฆ,๐‘ง)=โˆž๎“๐‘™,๐‘š,๐‘›=0๎๐‘“,๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๐‘”๎€ธ๎ž๐‘™๐‘š๐‘›(=๎“๐‘ฅ,๐‘ฆ,๐‘ง)๐‘™+๐‘š+๐‘›>0๎๐‘“,๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๐‘”๎€ธ๎ž๐‘™๐‘š๐‘›=๎“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘™+๐‘š+๐‘›>0๎๐›ฟ๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ๐›ฟ๎€ท๐‘ฆโˆ’๐‘ฆ๎…ž๎€ธ๐›ฟ๎€ท๐‘งโˆ’๐‘ง๎…ž๎€ธ,๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๐‘”๎€ธ๎ž๐‘™๐‘š๐‘›=๎“(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘™+๐‘š+๐‘›>0๐‘”๐‘™๐‘š๐‘›๎€ท๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘”๐‘™๐‘š๐‘›(๐‘ฅ,๐‘ฆ,๐‘ง).(4.37)

The drainage volume is

๐‘‰=๐‘Ž๐‘๐ป.(4.38)

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

๐‘ƒ๐‘Ž,๐‘ฃ=๎‚€1๐‘‰๎‚๎€œฮฉ๐‘ก๐‘ƒ(๐‘ฅ,๐‘ฆ,๐‘ง)๐‘‘๐‘ฅ๐‘‘๐‘ฆ๐‘‘๐‘ง=๐‘Ž๐‘๐ป+ฮจ๐‘Ž,๐‘ฃ.(4.39)

The wellbore pressure at point (๐‘ฅ๐‘ค,๐‘ฆ๐‘ค,๐‘ง๐‘ค) is

๐‘ƒ๐‘ค=๐‘ก๐‘Ž๐‘๐ป+ฮจ๐‘ค,(4.40) where ฮจ๐‘ค is the value of ฮจ at wellbore point (๐‘ฅ๐‘ค,๐‘ฆ๐‘ค,๐‘ง๐‘ค).

Combining (4.39) and (4.40) gives

๐‘ƒ๐‘Ž,๐‘ฃโˆ’๐‘ƒ๐‘ค=ฮจ๐‘Ž,๐‘ฃโˆ’ฮจ๐‘ค,(4.41) which implies ๐‘ƒ๐‘Ž,๐‘ฃโˆ’๐‘ƒ๐‘ค 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 ๐ท.

The producing portion of the partially penetrating well is between point (๐‘ฅ๎…ž,๐‘ฆ๎…ž,0) and point (๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐ฟ), recall (4.4) and (4.19), in order to obtain uniform line sink solution, we integrate ฮจ with respect to ๐‘ง๎…ž from 0 to ๐ฟ, then

๐ฝ๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๎€œ;๐‘™,๐‘š,๐‘›๐ฟ0ฮจ๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ๐‘‘๐‘ง๎…ž=๎“๐‘™+๐‘š+๐‘›>0โ„‘๐‘™๐‘š๐‘›๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ,;๐‘™,๐‘š,๐‘›(5.1) where

๎“๐‘™+๐‘š+๐‘›>0โ„‘๐‘™๐‘š๐‘›๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=๎“;๐‘™,๐‘š,๐‘›๐‘™+๐‘š+๐‘›>0๎‚ต1๐‘Ž๐‘๐ป๐‘‘๐‘™๐‘‘๐‘š๐‘‘๐‘›๐œ†๐‘™๐‘š๐‘›๎‚ถ๎‚€cos๐‘™๐œ‹๐‘ฅ๐‘Ž๎‚๎‚€ร—cos๐‘š๐œ‹๐‘ฆ๐‘๎‚๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚๎‚ตร—cos๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎€œ๐ฟ0๎‚ตcos๐‘›๐œ‹๐‘ง๎…ž๐ป๎‚ถ๐‘‘๐‘ง๎…ž=๎“๐‘™+๐‘š+๐‘›>0๎‚ต1๐‘Ž๐‘๐ป๐‘‘๐‘™๐‘‘๐‘š๐‘‘๐‘›๐œ†๐‘™๐‘š๐‘›๎‚ถ๎‚€cos๐‘™๐œ‹๐‘ฅ๐‘Ž๎‚๎‚€cos๐‘š๐œ‹๐‘ฆ๐‘๎‚๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚ร—โŽงโŽชโŽจโŽชโŽฉ๎‚€๐ป๎‚๎‚ต๐œ‹๐‘›cos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎‚ตcos๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚€sin๐‘›๐œ‹๐ฟ๐ป๎‚๎‚ตif๐‘™โ‰ 0,๐ฟcos๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถif๐‘™=0.(5.2)

Define

๐ถ๐ถ={(๐‘™,๐‘š,๐‘›)โˆถ๐‘™+๐‘š+๐‘›>0},(5.3)1๐ถ={(๐‘™,๐‘š,๐‘›)โˆถ๐‘™=๐‘š=0,๐‘›>0},(5.4)2๐ถ={(๐‘™,๐‘š,๐‘›)โˆถ๐‘™=0,๐‘š>0,๐‘›โ‰ฅ0},(5.5)3={(๐‘™,๐‘š,๐‘›)โˆถ๐‘™>0,๐‘šโ‰ฅ0,๐‘›โ‰ฅ0},(5.6) then it is easy to prove

๐ถ=๐ถ1โˆช๐ถ2โˆช๐ถ3,๐ถ1โˆฉ๐ถ2=โˆ…,๐ถ2โˆฉ๐ถ3=โˆ…,๐ถ3โˆฉ๐ถ1=โˆ….(5.7)

Recall (5.1) and (5.2), and use (5.3)โ€“(75), ๐ฝ(๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž;๐‘™,๐‘š,๐‘›) can be decomposed as

๎“๐ฝ=๐‘™+๐‘š+๐‘›>0โ„‘๐‘™๐‘š๐‘›๎€ท๐‘ฅ,๐‘ฆ,๐‘ง;๐‘ฅ๎…ž,๐‘ฆ๎…ž,๐‘ง๎…ž๎€ธ=;๐‘™,๐‘š,๐‘›โˆž๎“๐‘›=1โ„‘00๐‘›+โˆž๎“โˆž๐‘š=1๎“๐‘›=0โ„‘0๐‘š๐‘›+โˆž๎“โˆž๐‘™=1๎“โˆž๐‘š=0๎“๐‘›=0โ„‘๐‘™๐‘š๐‘›.(5.8)

Define the following notations:

๐ฝ๐‘ง=โˆž๎“๐‘›=1โ„‘00๐‘›,๐ฝ(5.9)๐‘ฆ๐‘ง=โˆž๎“โˆž๐‘š=1๎“๐‘›=0โ„‘0๐‘š๐‘›๐ฝ,(5.10)๐‘ฅ๐‘ฆ๐‘ง=โˆž๎“โˆž๐‘™=1๎“โˆž๐‘š=0๎“๐‘›=0โ„‘๐‘™๐‘š๐‘›,(5.11) so

๐ฝ=๐ฝ๐‘ง+๐ฝ๐‘ฆ๐‘ง+๐ฝ๐‘ฅ๐‘ฆ๐‘ง,(5.12) and the average value of ๐ฝ at wellbore can be written as

๐ฝ๐‘Ž,๐‘ค=๎€ท๐ฝ๐‘ง,๐‘Ž๎€ธ๐‘ค+๎€ท๐ฝ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค+๎€ท๐ฝ๐‘ฅ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค.(5.13)

Rearrange (4.12) and obtain

๐œ†๐‘™๐‘š๐‘›=๎‚€๐‘™๐œ‹๐‘Ž๎‚2+๎‚€๐‘š๐œ‹๐‘๎‚2+๎‚€๐‘›๐œ‹๐ป๎‚2=๎‚€๐œ‹๐ป๎‚2๎€ท๐‘›2+๐œ‡2๐‘™๐‘š๎€ธ,(5.14) where

๐œ‡2๐‘™๐‘š=๎‚€๐‘™๐ป๐‘Ž๎‚2+๎‚€๐‘š๐ป๐‘๎‚2=๎‚€๐ป๐‘๎‚2๎‚ธ๐‘š2+๎‚€๐‘™๐‘๐‘Ž๎‚2๎‚น,๐œ‡๐‘™0=๐‘™๐ป๐‘Ž,๐œ†๐‘™๐‘š0=๎‚€๐œ‹๐ป๎‚2๐œ‡2๐‘™๐‘š,๐œ†0๐‘š๐‘›=๎‚€๐‘š๐œ‹๐‘๎‚2+๎‚€๐‘›๐œ‹๐ป๎‚2=๎‚€๐œ‹๐ป๎‚2๎‚ธ๐‘›2+๎‚€๐‘š๐ป๐‘๎‚2๎‚น,๐œ†00๐‘›=๐‘›2๐œ‹2๐ป2.(5.15)

There hold [16, page 47]

โˆž๎“๐‘›=1sin(๐‘›๐‘ฅ)๐‘›3=๐œ‹2๐‘ฅ6โˆ’๐œ‹๐‘ฅ24+๐‘ฅ3(120โ‰ค๐‘ฅโ‰ค2๐œ‹),(5.16)โˆž๎“๐‘›=11โˆ’cos(๐‘›๐‘ฅ)๐‘›4=๐œ‹2๐‘ฅ2โˆ’12๐œ‹๐‘ฅ3+๐‘ฅ12448(0โ‰ค๐‘ฅโ‰ค2๐œ‹).(5.17)

Recall (5.4) and (5.9), ๐ฝ๐‘ง is for the case ๐‘™=๐‘š=0,๐‘›>0, and at wellbore of the off-center well,

๐‘ฆ=๐‘ฆ๎…žโ‰ 0,๐‘ฅ๎…žโ‰ 0,๐‘ฅ=๐‘ฅ๎…ž+๐‘…๐‘ค,0โ‰ค๐‘ง=๐‘ง๎…ž๎€ท๐ฝโ‰ค๐ฟ,๐‘ง๎€ธ๐‘ค=โˆž๎“๐‘›=1๎‚ต1๐‘Ž๐‘๐ป๐‘‘๐‘›๐œ†00๐‘›๎‚ถ๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚๎€œ๐ฟ0๎‚ตcos๐‘›๐œ‹๐‘ง๎…ž๐ป๎‚ถ๐‘‘๐‘ง๎…ž=๎‚€2๎‚๐‘Ž๐‘๐ปโˆž๎“๐‘›=1๎‚ต๐ป2๐œ‹2๐‘›2๎‚ถ๎‚€cos๐‘›๐œ‹๐‘ง๐ป๐ป๎‚๎‚€๎‚๎‚€๐‘›๐œ‹sin๐‘›๐œ‹๐ฟ๐ป๎‚=๎‚ต2๐ป2๐‘Ž๐‘๐œ‹3๎‚ถโˆž๎“๐‘›=1๎‚€1๐‘›3๎‚๎‚€sin๐‘›๐œ‹๐ฟ๐ป๎‚๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚.(5.18)

The average value of (๐ฝ๐‘ง)๐‘ค along the well length is

๎€ท๐ฝ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€1๐ฟ๎‚๎€œ๐ฟ0๐ฝ๐‘ง=๎‚€1๐‘‘๐‘ง๐ฟ๎‚โˆž๎“๐‘›=1๎‚ต2๐ป2๐œ‹3๐‘Ž๐‘๐‘›3๎‚ถ๎‚€sin๐‘›๐œ‹๐ฟ๐ป๎‚๎€œ๐ฟ0๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚=๐‘‘๐‘งโˆž๎“๐‘›=1๎‚ต2๐ป2๐œ‹3๐‘Ž๐‘๐ฟ๐‘›3๎‚ถ๎‚€sin๐‘›๐œ‹๐ฟ๐ป๐ป๎‚๎‚ƒ๎‚€๎‚๎‚€๐‘›๐œ‹sin๐‘›๐œ‹๐ฟ๐ป=๎‚๎‚„โˆž๎“๐‘›=1๎‚ต2๐ป3๐œ‹4๐‘Ž๐‘๐ฟ๐‘›4๎‚ถsin2๎‚€๐‘›๐œ‹๐ฟ๐ป๎‚=๎‚ต๐ป3๐œ‹4๎‚ถ๐‘Ž๐‘๐ฟโˆž๎“๐‘›=1๎‚€1๐‘›4๎‚€๎‚๎‚ƒ1โˆ’cos2๐‘›๐œ‹๐ฟ๐ป=๎‚ต๐ป๎‚๎‚„3๐œ‹4๎‚ถ๎‚€๐‘Ž๐‘๐ฟ2๐œ‹๐ฟ๐ป๎‚2๎‚ธ๐œ‹2โˆ’๐œ‹12๎‚€122๐œ‹๐ฟ๐ป๎‚+1๎‚€482๐œ‹๐ฟ๐ป๎‚2๎‚น=๎‚€4๐ป๐ฟ๎‚๎‚ต1๐‘Ž๐‘โˆ’๐ฟ12+๐ฟ6๐ป212๐ป2๎‚ถ=๎‚€2๐ป๐ฟ๎‚๎‚ต13๐‘Ž๐‘2โˆ’๐ฟ๐ป+๐ฟ22๐ป2๎‚ถ,(5.19) where we have used (5.17).

For a fully penetrating well, ๐ฟ=๐ป, then

๎€ท๐ฝ๐‘ง,๐‘Ž๎€ธ๐‘ค=0.(5.20)

Recall (5.5) and (5.10), ๐ฝ๐‘ฆ๐‘ง is for the case ๐‘™=0,๐‘š>0,๐‘›โ‰ฅ0, and at wellbore of the off-center well,

๐‘ฆ=๐‘ฆ๎…žโ‰ 0,๐‘ฅ๎…žโ‰ 0,๐‘ฅ=๐‘ฅ๎…ž+๐‘…๐‘ค,0โ‰ค๐‘ง=๐‘ง๎…ž๎€ท๐ฝโ‰ค๐ฟ,๐‘ฆ๐‘ง๎€ธ๐‘ค=๎‚€1๎‚๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘š=1๎“๐‘›=0๎‚ต1๐‘‘๐‘š๐‘‘๐‘›๐œ†0๐‘š๐‘›๎‚ถcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚๎€œ๐ฟ0๎‚ตcos๐‘›๐œ‹๐‘ง๎…ž๐ป๎‚ถ๐‘‘๐‘ง๎…ž=๎‚€2๎‚๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘š=1๎“๐‘›=0๎ƒฏcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘cos(๐‘›๐œ‹๐‘ง/๐ป)๐œ‹2๐‘‘๐‘›๎€บ(๐‘›/๐ป)2+(๐‘š/๐‘)2๎€ป๎€œ๐ฟ0๎‚ตcos๐‘›๐œ‹๐‘ง๎…ž๐ป๎‚ถ๐‘‘๐‘ง๎…ž๎ƒฐ=๎‚€2๎‚๐‘Ž๐‘๐ปโˆž๎“๐‘š=1๎ƒฏโˆž๎“๐‘›=12(๐ป/๐‘›๐œ‹)cos(๐‘›๐œ‹๐‘ง/๐ป)sin(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐œ‹2๎€บ(๐‘›/๐ป)2+(๐‘š/๐‘)2๎€ป+cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๐‘๎‚ถ๎‚ต2๐ฟ๐œ‹2๐‘š2๎‚ถ๎ƒฐ=๎‚ต2๐ป3๐œ‹3๎‚ถ๐‘Ž๐‘๐ปโˆž๎“๐‘š=1๎ƒฏโˆž๎“๐‘›=12cos(๐‘›๐œ‹๐‘ง/๐ป)sin(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘›๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป+cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๐‘๎‚ถ๎‚ต2๐ฟ๐œ‹๐ป3๐‘š2๎‚ถ๎ƒฐ=๎‚ต2๐ป2๐œ‹3๐‘Ž๐‘๎‚ถ๎‚ต๐œ‹๐ฟ๐‘2๐ป3๎‚ถโˆž๎“๐‘š=1๎‚€1๐‘š2๎‚cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ+๎‚ต2๐ป2๐œ‹3๎‚ถ๐‘Ž๐‘โˆž๎“โˆž๐‘š=1๎“๐‘›=1๎ƒฏ2cos(๐‘›๐œ‹๐‘ง/๐ป)sin(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘›๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป๎ƒฐ=๎‚€2๐‘๐ฟ๐œ‹2๎‚๐‘Ž๐ปโˆž๎“๐‘š=1๎‚€1๐‘š2๎‚cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ+๎‚ต2๐ป2๐œ‹3๎‚ถ๐‘Ž๐‘โˆž๎“โˆž๐‘š=1๎“๐‘›=1๎ƒฏ2cos(๐‘›๐œ‹๐‘ง/๐ป)sin(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘›๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป๎ƒฐ,(5.21) where we use the following formulas [16, page 47]:

โˆž๎“๐‘š=1๎‚€1๐‘š2๎‚๐œ‹cos(๐‘š๐‘ฅ)=26โˆ’๐œ‹๐‘ฅ2+๐‘ฅ24(0โ‰ค๐‘ฅโ‰ค2๐œ‹),(5.22)โˆž๎“๐‘š=1๎‚€1๐‘š2๎‚cos2๐œ‹(๐‘š๐‘ฅ)=26โˆ’๐œ‹๐‘ฅ2+๐‘ฅ22(0โ‰ค๐‘ฅโ‰ค๐œ‹).(5.23)

The average value of (๐ฝ๐‘ฆ๐‘ง)๐‘ค along the well length is

๎€ท๐ฝ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€1๐ฟ๎‚๎€œ๐ฟ0๐ฝ๐‘ฆ๐‘ง=๎‚€๐‘‘๐‘ง2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป2๐‘Ž๐‘๐ฟ๐œ‹3๎‚ถโˆž๎“โˆž๐‘š=1๎“๐‘›=1๎ƒฏ2sin(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘›๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป๎€œ๐ฟ0๎‚€cos๐‘›๐œ‹๐‘ง๐ป๎‚๎ƒฐ=๎‚€๐‘‘๐‘ง2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป2๐‘Ž๐‘๐ฟ๐œ‹3๎‚ถโˆž๎“โˆž๐‘š=1๎“๐‘›=1๎ƒฏ2๐ปsin2(๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐œ‹๐‘›2๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป๎ƒฐ=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป3๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“โˆž๐‘š=1๎“๐‘›=1๎ƒฏ[]1โˆ’cos(2๐‘›๐œ‹๐ฟ/๐ป)cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘›2๎€บ๐‘›2+(๐‘š๐ป/๐‘)2๎€ป๎ƒฐ=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป3๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“โˆž๐‘š=1๎“๐‘›=1๎‚€๐‘๎‚๐‘š๐ป2cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎‚ธ๎‚€1โˆ’cos2๐‘›๐œ‹๐ฟ๐ป๎‚,1๐‘›2โˆ’1๐‘›2+(๐‘š๐ป/๐‘)2๎‚น=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘2๎…ž2๐‘2๎‚ถ+๎‚ต๐ป32๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“๐‘š=1๎‚€๐‘๎‚๐‘š๐ป2cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—โˆž๎“๐‘›=1๎‚ธ1๐‘›2โˆ’cos(2๐‘›๐œ‹๐ฟ/๐ป)๐‘›2โˆ’1๐‘›2+(๐‘š๐ป/๐‘)2+cos(2๐‘›๐œ‹๐ฟ/๐ป)๐‘›2+(๐‘š๐ป/๐‘)2๎‚น=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป3๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“๐‘š=1๎‚€๐‘๎‚๐‘š๐ป2cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎‚ป๐œ‹26โˆ’๎‚ธ๐œ‹26โˆ’๐œ‹2๎‚€2๐œ‹๐ฟ๐ป๎‚+14๎‚€2๐œ‹๐ฟ๐ป๎‚2๎‚นโˆ’๎‚ธ๎‚€๐‘๐œ‹๎‚๎‚€2๐‘š๐ปcoth๐‘š๐ป๐œ‹๐‘๎‚โˆ’12๎‚€๐‘๎‚๐‘š๐ป2๎‚น+๎‚ธ๎‚€๐‘๐œ‹๎‚[(]2๐‘š๐ปcosh๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)โˆ’1sinh(๐‘š๐ป๐œ‹/๐‘)2๎‚€๐‘๎‚๐‘š๐ป2,๎‚น๎‚ผ(5.24) where we use the following formulas [16, page 47]:

โˆž๎“๐‘›=1cos(๐‘›๐‘ฅ)๐‘›2+๐›ฝ2=๎‚ต๐œ‹[]2๐›ฝ๎‚ถ๎‚ปcosh๐›ฝ(๐œ‹โˆ’๐‘ฅ)๎‚ผโˆ’1sinh(๐›ฝ๐œ‹)2๐›ฝ2(0โ‰ค๐‘ฅโ‰ค2๐œ‹),(5.25)โˆž๎“๐‘›=11๐‘›2+๐›ฝ2=๎‚ต๐œ‹๎‚ถ12๐›ฝcoth(๐›ฝ๐œ‹)โˆ’2๐›ฝ2(0โ‰ค๐‘ฅโ‰ค2๐œ‹),(5.26) and we may simplify (5.24) further

๎€ท๐ฝ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป3๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“๐‘š=1cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚€๐‘๎‚๐‘š๐ป2ร—๎‚ป๐œ‹2๐ฟ๐ปโˆ’๐œ‹2๐ฟ2๐ป2+๎‚€๐‘๐œ‹๎‚๎‚ป[]2๐‘š๐ปcosh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)โˆ’coth๐‘š๐ป๐œ‹๐‘๎‚=๎‚€๎‚ผ๎‚ผ2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚ต2๐ป3๐‘Ž๐‘๐ฟ๐œ‹4๎‚ถโˆž๎“๐‘š=1cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚€1๐‘š2๎‚ร—โŽงโŽชโŽจโŽชโŽฉ๐œ‹2๐ฟ๐‘2๐ป3โˆ’๐œ‹2๐ฟ2๐‘2๐ป4+๎‚ต๐‘3๐œ‹2๐‘š๐ป3๎‚ถร—๎‚ป[]cosh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)โˆ’coth๐‘š๐ป๐œ‹๐‘๎‚๎‚ผ๎ƒฐ=๎‚€2๐‘๐ฟ๎‚๎‚ต1๐‘Ž๐ป6โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ+๎‚€2๐‘๐‘Ž๐œ‹2๐ฟ๎‚๎‚€1โˆ’๐ป๎‚๎ƒฉ๐œ‹26โˆ’๐œ‹2๐‘ฆ๎…ž+๐œ‹2๐‘2๐‘ฆ2โ€ฒ2๐‘2๎ƒช+๎‚ต๐‘2๐‘Ž๐ฟ๐œ‹3๎‚ถโˆž๎“๐‘š=1cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถ๎‚€1๐‘š3๎‚ร—๎‚ป[]cosh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)โˆ’coth๐‘š๐ป๐œ‹๐‘๎‚๎‚ผ=๎‚€2๐‘๐‘Ž๎‚๎‚ต16โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘2๎…ž2๐‘2๎‚ถ+๎‚ต๐‘2๐‘Ž๐ฟ๐œ‹3๎‚ถโˆž๎“๐‘š=1๎ƒฌcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘š3๎ƒญ๎‚ป[]cosh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)โˆ’coth๐‘š๐ป๐œ‹๐‘๎‚๎‚ผ.(5.27)

For a fully penetrating well, ๐ฟ=๐ป, then

๎€ท๐ฝ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€2๐‘๐‘Ž๎‚๎‚ต16โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘๎…ž22๐‘2๎‚ถ.(5.28)

Define

[]๐‘“(๐‘ฅ)=sinh๐›ผ(1โˆ’๐‘ฅ)sinh(๐›ผ๐‘ฅ),(5.29) since the derivative of ๐‘“(๐‘ฅ) is

๐‘“๎…ž[][][],(๐‘ฅ)=๐›ผcosh(๐›ผ๐‘ฅ)sinh๐›ผ(1โˆ’๐‘ฅ)โˆ’๐›ผcosh๐›ผ(1โˆ’๐‘ฅ)sinh(๐›ผ๐‘ฅ)=๐›ผsinh๐›ผ(1โˆ’2๐‘ฅ)(5.30) consequently,

๐‘“๎…ž๎‚€12๎‚=0.(5.31)

When ๐‘ฅ=0 and ๐‘ฅ=1,

๐‘“(0)=๐‘“(1)=0.(5.32)

When ๐‘ฅ=1/2,๐‘“(๐‘ฅ) reaches maximum value, let

๐ฟ๐‘ฅ=๐ป,(5.33) and the producing length ๐ฟ is a variable, define

[]๐น(๐ฟ)=cosh๐›ฝ๐œ‹(1โˆ’2๐ฟ/๐ป)โˆ’cosh(๐›ฝ๐œ‹)=[][]sinh(๐›ฝ๐œ‹)โˆ’2sinh๐›ฝ๐œ‹(1โˆ’๐ฟ/๐ป)sinh๐›ฝ๐œ‹๐ฟ/(๐ป),sinh(๐›ฝ๐œ‹)(5.34) thus when ๐ฟ=๐ป/2,|๐น(๐ฟ)| reaches maximum value,

||||๐น(๐ฟ)max=|||๐น๎‚€๐ป2๎‚|||=2sinh2(๐›ฝ๐œ‹/2)=sinh(๐›ฝ๐œ‹)2sinh2(๐›ฝ๐œ‹/2)=2sinh(๐›ฝ๐œ‹/2)cosh(๐›ฝ๐œ‹/2)sinh(๐›ฝ๐œ‹/2)cosh(๐›ฝ๐œ‹/2)<1,(5.35) so ๐น(๐ฟ) is a bounded function, let

๐›ฝ=๐‘š๐ป๐‘,(5.36) then

๎€ท๐ฝ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€2๐‘๐‘Ž๎‚๎‚ต16โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘2๎…ž2๐‘2๎‚ถ+๎‚ต๐‘2๐‘Ž๐ฟ๐œ‹3๎‚ถร—โˆž๎“๐‘š=1๎ƒฌcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘š3๎ƒญ๎‚ป[]cosh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’2๐ฟ/๐ป)๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)โˆ’coth๐‘š๐ป๐œ‹๐‘๎‚๎‚ผ=๎‚€2๐‘๐‘Ž๎‚๎‚ต16โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘2๎…ž2๐‘2๎‚ถ+๎‚ต๐‘2๐‘Ž๐ฟ๐œ‹3๎‚ถร—โˆž๎“๐‘š=1๎ƒฌcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘š3๎ƒญ๎‚ป[]โˆ’2sinh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’๐ฟ/๐ป)sinh(๐‘š๐ฟ๐œ‹/๐‘)๎‚ผโ‰ˆ๎‚€sinh(๐‘š๐ป๐œ‹/๐‘)2๐‘๐‘Ž๎‚๎‚ต16โˆ’๐‘ฆ๎…ž+๐‘ฆ2๐‘2๎…ž2๐‘2๎‚ถ+๎‚ต๐‘2๐‘Ž๐ฟ๐œ‹3๎‚ถร—๐‘€๎“๐‘š=1๎ƒฌcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘š3๎ƒญ๎‚ป[]โˆ’2sinh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’๐ฟ/๐ป)sinh(๐‘š๐ฟ๐œ‹/๐‘)๎‚ผ.sinh(๐‘š๐ป๐œ‹/๐‘)(5.37)

Since 0<๐ฟ/๐ป<1, from (5.34) and (5.35), there holds

โˆž๎“๐‘š=101|||||๎ƒฌcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘š3๎ƒญ๎‚ป[]โˆ’2sinh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’๐ฟ/๐ป)sinh(๐‘š๐ฟ๐œ‹/๐‘)๎‚ผ|||||โ‰คsinh(๐‘š๐ป๐œ‹/๐‘)โˆž๎“๐‘š=1011๐‘š3=๐œ(3)โˆ’100๎“๐‘š=11๐‘š3=4.9502ร—10โˆ’5,(5.38) where ๐œ(3) is ๐‘…๐‘–๐‘’๐‘š๐‘Ž๐‘›๐‘›-๐œ function:

๐œ(3)=โˆž๎“๐‘š=11๐‘š3=1.202057,(5.39) thus

โˆž๎“๐‘š=1๎‚€1๐‘š3๎‚๎‚ป[]2sinh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’๐ฟ/๐ป)sinh(๐‘š๐ฟ๐œ‹/๐‘)๎‚ผโ‰ˆsinh(๐‘š๐ป๐œ‹/๐‘)100๎“๐‘š=1๎‚€1๐‘š3๎‚๎‚ป[]2sinh(๐‘š๐ป๐œ‹/๐‘)(1โˆ’๐ฟ/๐ป)sinh(๐‘š๐ฟ๐œ‹/๐‘)๎‚ผ.sinh(๐‘š๐ป๐œ‹/๐‘)(5.40)

So, in (5.37), ๐‘€=100 is sufficient to reach engineering accuracy.

Recall (5.6) and (5.11), ๐ฝ๐‘ฅ๐‘ฆ๐‘ง is for the case ๐‘™>0,๐‘šโ‰ฅ0,๐‘›โ‰ฅ0, and at wellbore of the off-center well,

๐‘ฆ=๐‘ฆ๎…žโ‰ 0,๐‘ฅ๎…žโ‰ 0,๐‘ฅ=๐‘ฅ๎…ž+๐‘…๐‘ค,0โ‰ค๐‘ง=๐‘ง๎…žโ‰ค๐ฟ,(5.41) then

๎€ท๐ฝ๐‘ฅ๐‘ฆ๐‘ง๎€ธ๐‘ค=๎‚€1๎‚ร—๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘™=1๎“โˆž๐‘š=0๎“๐‘›=0๎€ท๎ƒฏ๎ƒฌcos(๐‘›๐œ‹๐‘ง/๐ป)cos๐‘™๐œ‹๐‘ฅ๎…ž๎€ธ๎€ท๐‘ฅ/๐‘Žcos๎€บ๎€ท๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ป๎€ธ๎€ธ/๐‘Žcos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘™๐‘‘๐‘š๐‘‘๐‘›๐œ†๐‘™๐‘š๐‘›๎ƒญร—๎€œ๐ฟ0๎‚ตcos๐‘›๐œ‹๐‘ง๎…ž๐ป๎‚ถ๐‘‘๐‘ง๎…ž๎‚ผ=๎‚€1๎‚๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘™=1๎“๐‘š=0๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=1๎‚ธ4(๐ป/๐‘›๐œ‹)sin(๐‘›๐œ‹๐ฟ/๐ป)cos(๐‘›๐œ‹๐‘ง/๐ป)๐‘‘๐‘š๐œ†๐‘™๐‘š๐‘›๎‚น+2๐ฟ๐‘‘๐‘š๐œ†๐‘™๐‘š0๎ƒฐ.(5.42)

The average value of (๐ฝ๐‘ฅ๐‘ฆ๐‘ง)๐‘ค along the well length is ๎€ท๐ฝ๐‘ฅ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚€1๎‚๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘™=1๎“๐‘š=0๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=1๎ƒฌ4โˆซ(๐ป/๐‘›๐œ‹)sin(๐‘›๐œ‹๐ฟ/๐ป)๐ฟ0cos(๐‘›๐œ‹๐‘ง/๐ป)๐‘‘๐‘ง๐‘‘๐‘š๐œ†๐‘™๐‘š๐‘›๐ฟ๎ƒญ+2๐ฟ๐‘‘๐‘š๐œ†๐‘™๐‘š0๎ƒฐ=๎‚€1๎‚๐‘Ž๐‘๐ปโˆž๎“โˆž๐‘™=1๎“๐‘š=0๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=1๎‚ธ4(๐ป/๐‘›๐œ‹)2sin2(๐‘›๐œ‹๐ฟ/๐ป)๐‘‘๐‘š๐œ†๐‘™๐‘š๐‘›๐ฟ๎‚น+2๐ฟ๐‘‘๐‘š๐œ†๐‘™๐‘š0๎ƒฐ=๎‚ต๐ป4๐‘Ž๐‘๐ป๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญโˆž๎“๐‘š=0cos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=12[]1โˆ’cos(2๐‘›๐œ‹๐ฟ/๐ป)๐‘‘๐‘š๐‘›2๎€ท๐‘›2+๐œ‡2๐‘™๐‘š๎€ธ๐ฟ+2๐œ‹2๐ฟ๐‘‘๐‘š๐ป2๐œ‡2๐‘™๐‘š๎ƒฐ=๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญ๎ƒฏโˆž๎“๐‘š=0๎‚ต2๐‘‘๐‘š๎‚ถcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=1๎ƒฉ1๐œ‡2๐‘™๐‘š๐ฟ๎ƒช๎‚ƒ๎‚€1โˆ’cos2๐‘›๐œ‹๐ฟ๐ปร—๎ƒฉ1๎‚๎‚„๐‘›2โˆ’1๐‘›2+๐œ‡2๐‘™๐‘š๎ƒช+๐œ‹2๐ฟ๐ป2๐œ‡2๐‘™๐‘š๎ƒฐ=๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญ๎ƒฏโˆž๎“๐‘š=0๎ƒฉ2๐‘‘๐‘š๐œ‡2๐‘™๐‘š๐ฟ๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏโˆž๎“๐‘›=1๎ƒฌ1๐‘›2โˆ’cos(2๐‘›๐œ‹๐ฟ/๐ป)๐‘›2โˆ’1๐‘›2+๐œ‡2๐‘™๐‘š+cos(2๐‘›๐œ‹๐ฟ/๐ป)๐‘›2+๐œ‡2๐‘™๐‘š+๎‚ต๐œ‹๎ƒญ๎ƒฐ2๐ฟ2๐ป2๎‚ถ๎ƒฐ=๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญโˆž๎“๐‘š=0๎ƒฉ2๐‘‘๐‘š๐œ‡2๐‘™๐‘š๐ฟ๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎‚ป๐œ‹26โˆ’๎‚ธ๐œ‹26โˆ’๎‚€๐œ‹2๎‚๎‚€2๐œ‹๐ฟ๐ป๎‚+๎‚€14๎‚๎‚€2๐œ‹๐ฟ๐ป๎‚2๎‚นโˆ’๎ƒฌ๎‚ต๐œ‹2๐œ‡๐‘™๐‘š๎‚ถ๎€ท๐œ‡coth๐‘™๐‘š๐œ‹๎€ธโˆ’๎ƒฉ12๐œ‡2๐‘™๐‘š+๎ƒฌ๎‚ต๐œ‹๎ƒช๎ƒญ2๐œ‡๐‘™๐‘š๎‚ถ๎€บ๐œ‡cosh๐‘™๐‘š๎€ป๐œ‹(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธโˆ’12๐œ‡2๐‘™๐‘š+๎‚ต๐œ‹๎ƒญ๎ƒฐ2๐ฟ2๐ป2๎‚ถ๎ƒฐ,(5.43) where we use (5.22) and (5.25).

Let ๐‘ฅ=0, recast (5.26), we obtain

1๐›ฝ2+2โˆž๎“๐‘›=11๐‘›2+๐›ฝ2=๎‚ต๐œ‹๐›ฝ๎‚ถcoth(๐›ฝ๐œ‹),โˆž๎“๐‘›=01๎€ท๐‘›2+๐›ฝ2๎€ธ๐‘‘๐‘›=๎‚ต๐œ‹๐›ฝ๎‚ถcoth(๐›ฝ๐œ‹).(5.44) So,

๎€ท๐ฝ๐‘ฅ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘ค=๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญโˆž๎“๐‘š=0๎ƒฉ2๐‘‘๐‘š๐œ‡2๐‘™๐‘š๐ฟ๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏ๎‚ต๐œ‹2๐ฟ๐ปโˆ’๐œ‹2๐ฟ2๐ป2+๐œ‹2๐ฟ2๐ป2๎‚ถ+๎‚ต๐œ‹2๐œ‡๐‘™๐‘š๎‚ถ๎ƒฌ๎€บ๐œ‡cosh๐‘™๐‘š๐œ‹๎€ป(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธ๎€ท๐œ‡โˆ’coth๐‘™๐‘š๐œ‹๎€ธ=๎‚ต๐ป๎ƒญ๎ƒฐ3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญร—๎ƒฏโˆž๎“๐‘š=0๎ƒฉ๐œ‹๐‘‘๐‘š๐œ‡3๐‘™๐‘š๐ฟ๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏ๎€บ๐œ‡cosh๐‘™๐‘š๎€ป๐œ‹(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธ๎€ท๐œ‡โˆ’coth๐‘™๐‘š๐œ‹๎€ธ๎ƒฐ+๎‚€2๐ฟ๎‚๎‚ต๐œ‹2๐ฟ๐ป๎‚ถโˆž๎“๐‘š=0cos2๎€ท๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘š๐œ‡2๐‘™๐‘š๎ƒฐ=๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญร—๎ƒฏโˆž๎“๐‘š=0๎ƒฉ๐œ‹๐‘‘๐‘š๐œ‡3๐‘™๐‘š๐ฟ๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏ๎€บ๐œ‡cosh๐‘™๐‘š๎€ป๐œ‹(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธ๎€ท๐œ‡โˆ’coth๐‘™๐‘š๐œ‹๎€ธ๎ƒฐ+๎‚ต๐‘Ž๐‘๐œ‹3๐ป3๐‘™๎‚ถ๎‚€coth๐‘™๐‘๐œ‹๐‘Ž๎‚+๎‚ต๐œ‹2๐ป๎‚ถโˆž๎“๐‘š=0๎€ทcos2๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘š๐œ‡2๐‘™๐‘š๎ƒฐ=๎‚ต๐ป3๐‘Ž๐‘๐ฟ๐œ‹3๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญร—๎ƒฏโˆž๎“๐‘š=0๎ƒฉ1๐‘‘๐‘š๐œ‡3๐‘™๐‘š๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏ๎€บ๐œ‡cosh๐‘™๐‘š๎€ป๐œ‹(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธ๎€ท๐œ‡โˆ’coth๐‘™๐‘š๐œ‹๎€ธ+๎‚ต๐ป๎ƒฐ๎ƒฐ3๐‘Ž๐‘๐œ‹4๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž,๎‚ต๐‘Ž๐‘๐œ‹3๐ป3๐‘™๎‚ถ๎‚€coth๐‘™๐‘๐œ‹๐‘Ž๎‚+๎‚ต๐œ‹2๐ป๎‚ถโˆž๎“๐‘š=0๎€ทcos2๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘š๐œ‡2๐‘™๐‘š๎ƒญ.(5.45)

Since

๎‚ต๐œ‹2๐ป๎‚ถโˆž๎“๐‘š=0๎ƒฌ๎€ทcos2๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘š๐œ‡2๐‘™๐‘š๎ƒญ=๎‚ต๐‘2๐ป3๎‚ถโˆž๎“๐‘š=0๎ƒฏ๎€ทcos2๐‘š๐œ‹๐‘ฆ๎…ž๎€ธ/๐‘๐‘‘๐‘š๎€บ๐‘š2+(๐‘๐‘™/๐‘Ž)2๎€ป๎ƒฐ=๎‚ต๐‘2๐ป3๎‚ถ๎‚€๐œ‹๐‘Ž๎‚๎ƒฏ๎€บ๎€ท๐‘๐‘™cosh๐œ‹๐‘๐‘™1โˆ’2๐‘ฆ๎…ž๎€ธ๎€ป/๐‘/๐‘Ž๎ƒฐโ‰ˆ๎‚ต๐‘sinh(๐œ‹๐‘๐‘™/๐‘Ž)๐ป3๎‚ถ๎‚€๐œ‹๐‘Ž๐‘™๎‚๎‚ตโˆ’exp2๐œ‹๐‘ฆ๎…ž๐‘™๐‘Ž๎‚ถ,๎‚ต๐ป3๐‘Ž๐‘๐œ‹4๎‚ถ|||||โˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญ๎‚ต๐‘๐ป3๎‚ถ๎‚€๐œ‹๐‘Ž๐‘™๎‚๎‚ตโˆ’exp2๐œ‹๐‘ฆ๎…ž๐‘™๐‘Ž๎‚ถ|||||โ‰ค๎‚€1๐œ‹3๎‚๎€บ๎€ทln1โˆ’expโˆ’2๐œ‹๐‘ฆ๎…ž/๐‘Ž๎€ธ๎€ปโ‰ˆ0,(5.46) thus

๎€ท๐ฝ๐‘ฅ๐‘ฆ๐‘ง,๐‘Ž๎€ธ๐‘คโ‰ˆ๎‚ต๐ป3๐‘Ž๐‘๐ฟ๐œ‹3๎‚ถโˆž๎“๐‘™=1๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos๐‘™๐œ‹๎…ž+๐‘…๐‘ค๎€ธ๐‘Ž๎ƒญโˆž๎“๐‘š=0๎ƒฉ1๐‘‘๐‘š๐œ‡3๐‘™๐‘š๎ƒชcos2๎‚ต๐‘š๐œ‹๐‘ฆ๎…ž๐‘๎‚ถร—๎ƒฏ๎€บ๐œ‡cosh๐‘™๐‘š๐œ‹๎€ป(1โˆ’2๐ฟ/๐ป)๎€ท๐œ‡sinh๐‘™๐‘š๐œ‹๎€ธ๎€ท๐œ‡โˆ’coth๐‘™๐‘š๐œ‹๎€ธ๎ƒฐ+๎‚€1๐œ‹๎‚โˆž๎“๐‘™=1๎‚€1๐‘™๎‚๎‚€coth๐‘™๐‘๐œ‹๐‘Ž๎‚๎‚ตcos๐‘™๐œ‹๐‘ฅ๎…ž๐‘Ž๎‚ถ๎ƒฌ๎€ท๐‘ฅcos