Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 267964 | 33 pages | https://doi.org/10.1155/2009/267964

Pressure Drop Equations for a Partially Penetrating Vertical Well in a Circular Cylinder Drainage Volume

Academic Editor: Saad A. Ragab
Received08 Oct 2008
Accepted02 Jan 2009
Published19 Mar 2009

Abstract

Taking a partially penetrating vertical well as a uniform line sink in three-dimensional space, by developing necessary mathematical analysis, this paper presents unsteady-state pressure drop equations for an off-center partially penetrating vertical well in a circular cylinder drainage volume with constant pressure at outer boundary. First, the point sink solution to the diffusivity equation is derived, then using superposition principle, pressure drop equations for a uniform line sink model are obtained. This paper also gives an equation to calculate pseudoskin factor due to partial penetration. The proposed equations provide fast analytical tools to evaluate the performance of a vertical well which is located arbitrarily in a circular cylinder drainage volume. It is concluded that the well off-center distance has significant effect on well pressure drop behavior, but it does not have any effect on pseudoskin factor due to partial penetration. Because the outer boundary is at constant pressure, when producing time is sufficiently long, steady-state is definitely reached. When well producing length is equal to payzone thickness, the pressure drop equations for a fully penetrating well are obtained.

1. Introduction

For both fully and partially penetrating vertical wells, steady-state and unsteady-state pressure-transient testings are useful tools for evaluating in situ reservoir and wellbore parameters that describe the production characteristics of a well. The use of transient well testing for determining reservoir parameters and well productivity has become common, in the past years, analytic solutions have been presented for the pressure behavior of partially penetrating vertical wells.

The problem of fluid flow into wells with partial penetration has received much attention in the past years in petroleum engineering [1–7].

In many oil and gas reservoirs the producing wells are completed as partially penetrating wells; that is, only a portion of the pay zone is perforated. This may be done for a variety of reasons, but the most common one is to prevent or delay the unwanted fluids into the wellbore. The exact solution of the partial penetration problem presents great analytical problems because the boundary conditions that the solutions of the partial differential equations must satisfy are mixed; that is, on one of the boundaries the pressure is specified on one portion and the flux on the other. This difficult occurs at the wellbore, for the flux over the nonproductive section of the well is zero, the potential over the perforated interval must be constant.

This problem may be overcome in the case of constant rate production by making the assumption that the flux into the well is uniform over the entire perforated interval, so that on the wellbore the flux is specified over the total formation thickness. This approximation naturally leads to an error in the solution since the potential (pressure) will not be uniform over the perforated interval, but it has been shown that this occurrence is not too significant.

Many different techniques have been used for solving the partial penetration problem, namely, finite difference method [2], Fourier, Hankel and Laplace transforms [3–5], Green's functions [6]. The analytical expressions and the numerical results obtained for reservoir pressures by different methods were essentially identical, however, there are some differences between the values of wellbore pressures computed from numerical models and those obtained from analytical solutions [7].

The primary goal of this study is to present unsteady state pressure drop equations for an off-center partially penetrating vertical well in a circular cylinder drainage volume. Analytical solutions are derived by making the assumption of uniform fluid withdrawal along the portion of the wellbore open to flow. Taking the producing portion of a partially penetrating well as a uniform line sink, using principle of potential superposition, pressure drop equations for a partially penetrating well are obtained.

2. Partially Penetrating Vertical Well Model

Figure 1 is a schematic of an off-center partially penetrating vertical well. A partially penetrating well of drilled length 𝐿 drains a circular cylinder porous volume with height 𝐻 and radius 𝑅𝑒.

The following assumptions are made.

(1)The porous media volume is circular cylinder which has constant 𝐾π‘₯,𝐾𝑦,𝐾𝑧 permeabilities, thickness 𝐻, porosity πœ™. And the porous volume is bounded by top and bottom impermeable boundaries.(2)The pressure is initially constant in the cylindrical body, during production the pressure remains constant and equal to the initial pressure 𝑃𝑖 at the lateral surface.(3)The production occurs through a partially penetrating vertical well of radius 𝑅𝑀, represented in the model by a uniform line sink which is located at 𝑅0 away from the axis of symmetry of the cylindrical body. The drilled well length is 𝐿, the producing well length is πΏπ‘π‘Ÿ.(4)A single-phase fluid, of small and constant compressibility 𝐢𝑓, constant viscosity πœ‡, and formation volume factor 𝐡, flows from the porous media to the well. Fluids properties are independent of pressure. Gravity forces are neglected.

The porous media domain isξ€½Ξ©=(π‘₯,𝑦,𝑧)∣π‘₯2+𝑦2<𝑅2𝑒,0<𝑧<𝐻,(2.1)where 𝑅𝑒 is cylinder radius, Ξ© is the cylindrical body.

Located at 𝑅0 away from the center of the cylindrical body, the coordinates of the top and bottom points of the well line are (𝑅0,0,0) and (𝑅0,0,𝐿), respectively, while point (𝑅0,0,𝐿1) and point (𝑅0,0,𝐿2) are the beginning point and end point of the producing portion of the well, respectively. The well is a uniform line sink between (𝑅0,0,𝐿1) and (𝑅0,0,𝐿2), and there holdsπΏπ‘π‘Ÿ=𝐿2βˆ’πΏ1,πΏπ‘π‘Ÿβ‰€πΏβ‰€π».(2.2)

We assume𝐾π‘₯=𝐾𝑦=πΎβ„Ž,𝐾𝑧=𝐾𝑣(2.3)and define average permeability πΎπ‘Ž=(𝐾π‘₯𝐾𝑦𝐾𝑧)1/3=πΎβ„Ž2/3𝐾𝑣1/3.(2.4)

Suppose point (𝑅0,0,π‘§ξ…ž) is on the producing portion, and its point convergence intensity is π‘ž, in order to obtain the pressure at point (π‘₯,𝑦,𝑧) caused by the point (𝑅0,0,π‘§ξ…ž), according to mass conservation law and Darcy's law, we have to obtain the basic solution of the diffusivity equation in Ξ© [8]:πΎβ„Žπœ•2π‘ƒπœ•π‘₯2+πΎβ„Žπœ•2π‘ƒπœ•π‘¦2+πΎπ‘£πœ•2π‘ƒπœ•π‘§2=πœ™πœ‡πΆπ‘‘πœ•π‘ƒπœ•π‘‘+πœ‡π‘žπ΅π›Ώ(π‘₯βˆ’π‘…0)𝛿(𝑦)𝛿(π‘§βˆ’π‘§ξ…ž),inΞ©,(2.5)where 𝐢𝑑 is total compressibility coefficient of porous media, 𝛿(π‘₯βˆ’π‘…0), 𝛿(𝑦), 𝛿(π‘§βˆ’π‘§β€²) are Dirac functions.

The initial condition is𝑃(𝑑,π‘₯,𝑦,𝑧)|𝑑=0=𝑃𝑖,inΞ©.(2.6)

The lateral boundary condition is𝑃(𝑑,π‘₯,𝑦,𝑧)=𝑃𝑖,onΞ“,(2.7)where Ξ“ is the cylindrical lateral surface: ξ€½Ξ“=(π‘₯,𝑦,𝑧)∣π‘₯2+𝑦2=𝑅2𝑒,0<𝑧<𝐻.(2.8)

The porous media domain is bounded by top and bottom impermeable boundaries, soπœ•π‘ƒ|||πœ•π‘§π‘§=0=0;πœ•π‘ƒ|||πœ•π‘§π‘§=𝐻=0.(2.9)

In order to simplify the above equations, we take the following dimensionless transforms:π‘₯𝐷=2π‘₯𝐿,𝑦𝐷=2𝑦𝐿,𝑧𝐷=ξ‚€2π‘§πΏπΎξ‚ξ‚€β„ŽπΎπ‘£ξ‚1/2,𝐿(2.10)𝐷𝐾=2β„ŽπΎπ‘£ξ‚1/2,𝐻𝐷=ξ‚€2π»πΏπΎξ‚ξ‚€β„ŽπΎπ‘£ξ‚1/2,𝐿(2.11)π‘π‘Ÿπ·=𝐿2π·βˆ’πΏ1𝐷=2(𝐿2βˆ’πΏ1)πΏπΎξ‚„ξ‚€β„ŽπΎπ‘£ξ‚1/2,𝑅(2.12)0𝐷=2𝑅0𝐿,𝑅𝑒𝐷=2𝑅𝑒𝐿,𝑅𝑀𝐷=2𝑅𝑀𝐿,𝑑(2.13)𝐷=4πΎβ„Žπ‘‘πœ™πœ‡πΆπ‘‘πΏ2.(2.14)

Assuming π‘ž is the point convergence intensity at the point sink (𝑅0,0,𝑧′), the partially penetrating well is a uniform line sink, the total flow rate of the well is 𝑄, and there holdsπ‘„π‘ž=πΏπ‘π‘Ÿπ·.(2.15)

Define dimensionless pressures 𝑃𝐷=4πœ‹πΏ(πΎβ„ŽπΎπ‘£)1/2(π‘ƒπ‘–βˆ’π‘ƒ),𝑃(πœ‡π‘žπ΅)(2.16)𝑀𝐷=4πœ‹πΏ(πΎβ„ŽπΎπ‘£)1/2(π‘ƒπ‘–βˆ’π‘ƒπ‘€).(πœ‡π‘žπ΅)(2.17)

Note that if 𝑐 is a positive constant, there holds [9]𝛿(𝑐π‘₯)=𝛿(π‘₯)𝑐,(2.18)consequently, (2.5) becomes [8, 9]πœ•2π‘ƒπ·πœ•π‘₯𝐷2+πœ•2π‘ƒπ·πœ•π‘¦π·2+πœ•2π‘ƒπ·πœ•π‘§π·2=πœ•π‘ƒπ·πœ•π‘‘π·βˆ’8πœ‹π›Ώ(π‘₯π·βˆ’π‘…0𝐷)𝛿(𝑦𝐷)𝛿(π‘§π·βˆ’π‘§ξ…žπ·),inΩ𝐷,(2.19)whereΩ𝐷=ξ€½(π‘₯𝐷,𝑦𝐷,𝑧𝐷)∣π‘₯2𝐷+𝑦2𝐷<𝑅2𝑒𝐷,0<𝑧𝐷<𝐻𝐷.(2.20)

If point 𝐫0 and point 𝐫 are with distances 𝜌0 and 𝜌, respectively, from the circular center, then the dimensionless off-center distances are𝜌0𝐷=2𝜌0𝐿,𝜌𝐷=2𝜌𝐿,(2.21)

There holdsξ‚€πœ‹π»π·ξ‚ξ€·2π‘…π‘’π·βˆ’πœŒ0π·βˆ’πœŒπ·βˆ’βˆšπœŒ0π·πœŒπ·ξ€Έ=ξ‚€πΎπ‘£πΎβ„Žξ‚1/2ξ‚€πœ‹πΏ2𝐻4π‘…π‘’πΏβˆ’2𝜌0πΏβˆ’2πœŒπΏβˆ’2√𝜌0πœŒπΏξ‚=ξ‚€πΎπ‘£πΎβ„Žξ‚1/2ξ‚€πœ‹π‘…π‘’π»πœŒξ‚ξ‚€2βˆ’0π‘…π‘’βˆ’πœŒπ‘…π‘’βˆ’βˆšπœŒ0πœŒπ‘…π‘’ξ‚=ξ‚€πΎπ‘£πΎβ„Žξ‚1/2ξ‚€πœ‹π‘…π‘’π»ξ‚ξ‚€2βˆ’πœ—0βˆšβˆ’πœ—βˆ’πœ—0πœ—ξ‚,(2.22)whereπœ—0=𝜌0π‘…π‘’πœŒ,πœ—=𝑅𝑒.(2.23)

Since the reservoir is with constant pressure outer boundary (edge water), in order to delay water encroachment, a producing well must keep a sufficient distance from the outer boundary. Thus in this paper, it is reasonable to assumeπœ—0≀0.6,πœ—β‰€0.6.(2.24)

Ifπœ—0𝐾=πœ—=0.6,π‘£πΎβ„Žπ‘…=0.25,𝑒𝐻=15(2.25)thenξ‚€πΎπ‘£πΎβ„Žξ‚1/2ξ‚€πœ‹π‘…π‘’π»ξ‚ξ€·2βˆ’πœ—0βˆšβˆ’πœ—βˆ’πœ—0πœ—ξ€Έ=0.251/2βˆšΓ—(πœ‹Γ—15)Γ—(2.0βˆ’0.6βˆ’0.6βˆ’0.6Γ—0.6),exp(βˆ’4.7124)=8.983Γ—10βˆ’3;(2.26) and ifπœ—0𝐾=πœ—=0.5,π‘£πΎβ„Žπ‘…=0.5,𝑒𝐻=10,(2.27)thenξ‚€πΎπ‘£πΎβ„Žξ‚1/2ξ‚€πœ‹π‘…π‘’π»ξ‚ξ€·2βˆ’πœ—0βˆšβˆ’πœ—βˆ’πœ—0πœ—ξ€Έ=0.51/2βˆšΓ—(πœ‹Γ—10)Γ—(2.0βˆ’0.5βˆ’0.5βˆ’0.5Γ—0.5)=11.107,exp(βˆ’11.107)=1.501Γ—10βˆ’5.(2.28)

Recall (2.22), according to the above calculations, without losing generality, there holdsξ‚ƒβˆ’ξ‚€πœ‹exp𝐻𝐷2π‘…π‘’π·βˆ’πœŒ0π·βˆ’πœŒπ·βˆ’βˆšπœŒ0π·πœŒπ·ξ€Έξ‚„β‰ˆ0.(2.29)

In the same manner, we haveξ‚ƒβˆ’ξ‚€πœ‹exp𝐻𝐷2π‘…π‘’π·βˆ’πœŒ0π·βˆ’πœŒπ·ξ€Έξ‚„β‰ˆ0.(2.30)

3. Point Sink Solution

For convenience in the following reference, we use dimensionless transforms given by (2.10) through (2.17), every variable, domain, initial and boundary conditions below should be taken as dimensionless, but we drop the subscript 𝐷.

Thus, if the point sink is at (π‘₯β€²,0,𝑧′), (2.19) can be written asπœ•π‘ƒπœ•π‘‘βˆ’Ξ”π‘ƒ=8πœ‹π›Ώ(π‘₯βˆ’π‘₯ξ…ž)𝛿(𝑦)𝛿(π‘§βˆ’π‘§ξ…ž),inΞ©,(3.1)whereξ€½Ξ©=(π‘₯,𝑦,𝑧)∣π‘₯2+𝑦2<𝑅2𝑒,πœ•,0<𝑧<𝐻Δ𝑃=2π‘ƒπœ•π‘₯2+πœ•2π‘ƒπœ•π‘¦2+πœ•2π‘ƒπœ•π‘§2.(3.2)

The equation of initial condition is changed to𝑃(𝑑,π‘₯,𝑦,𝑧)|𝑑=0=0,inΞ©.(3.3)

The equation of lateral boundary condition is changed to𝑃(𝑑,π‘₯,𝑦,𝑧)=0,onΞ“,(3.4)whereξ€½Ξ“=(π‘₯,𝑦,𝑧)∣π‘₯2+𝑦2=𝑅2𝑒,0<𝑧<𝐻.(3.5)

The problem under consideration is that of fluid flow toward a point sink from an off-center position within a circular of radius 𝑅𝑒. We want to determine the pressure change at an observation point with a distance 𝜌 from the center of circle.

Figure 2 is a geometric representation of the system. In Figure 2, the point sink 𝐫0 and the observation point 𝐫, are with distances 𝜌0 and 𝜌, respectively, from the circular center; and the two points are separated at the center by an angle πœƒ. The inverse point of the point sink 𝐫0 with respect to the circle is point π«βˆ—. Point π«βˆ— with a distance πœŒβˆ— from the center, and 𝜌1 from the observation point. The inverse point is the point outside the circle, on the extension of the line connecting the center and the point sink, and such thatπœŒβˆ—=𝑅2π‘’πœŒ0.(3.6)

Assume 𝑅′ is the distance between point 𝐫 and point 𝐫0, then [9, 10]π‘…ξ…ž=ξ”πœŒ2+𝜌20βˆ’2𝜌𝜌0cosπœƒ.(3.7)

If the observation point 𝐫 is on the drainage circle, 𝜌=𝑅𝑒, thenπ‘…ξ…ž=𝑅2𝑒+𝜌20βˆ’2π‘…π‘’πœŒ0cosπœƒ,𝑅𝑒>𝜌0>0.(3.8)

If the observation point 𝐫 is on the wellbore, thenπ‘…ξ…ž=𝑅𝑀.(3.9)

Recall (2.9), obviously for impermeable upper and lower boundary conditions, there holds [9, 10]𝛿(π‘§βˆ’π‘§ξ…ž)=βˆžξ“π‘˜=0ξ‚€cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚cosπ‘˜πœ‹π‘§π»/(π»π‘‘π‘˜),(3.10)whereπ‘‘π‘˜=⎧βŽͺ⎨βŽͺ⎩11,ifπ‘˜=0,2,ifπ‘˜>0.(3.11)

Let𝑃(𝑑;π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž)=βˆžξ“π‘˜=0πœ‘π‘˜ξ‚€(𝑑,π‘₯,𝑦)cosπ‘˜πœ‹π‘§π»ξ‚,(3.12)and substitute (3.12) into (3.1) and compare the coefficients of cos(π‘˜πœ‹π‘§/𝐻), we obtainπœ•πœ‘π‘˜πœ•π‘‘+πœ†2π‘˜πœ‘π‘˜βˆ’ξ‚€πœ•2πœ‘π‘˜πœ•π‘₯2+πœ•2πœ‘π‘˜πœ•π‘¦2=8πœ‹cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚π›Ώ(π‘₯βˆ’π‘₯ξ…ž)𝛿(𝑦)/(π»π‘‘π‘˜)(3.13)in circular Ξ©1={(π‘₯,𝑦)∣π‘₯2+𝑦2<𝑅2𝑒}, andπœ‘π‘˜=0,(3.14)on circumference Ξ“1={(π‘₯,𝑦)∣π‘₯2+𝑦2=𝑅2𝑒}, andπœ‘π‘˜|𝑑=0=0,(3.15)whereπœ†π‘˜=π‘˜πœ‹π».(3.16)

Taking the Laplace transform at the both sides of (3.13), thenξ‚€πœ•2ξ‚Šπœ‘π‘˜πœ•π‘₯2+πœ•2ξ‚Šπœ‘π‘˜πœ•π‘¦2ξ‚βˆ’ξ€·π‘ +πœ†2π‘˜ξ€Έξ‚Šπœ‘π‘˜=π›Όπ‘˜π›Ώ(π‘₯βˆ’π‘₯ξ…ž)𝛿(𝑦)𝑠,inΞ©1,(3.17)ξ‚Šπœ‘π‘˜=0,onΞ“1,(3.18)whereπ›Όπ‘˜=ξ‚€βˆ’8πœ‹π»π‘‘π‘˜ξ‚ξ‚€cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚,(3.19)and 𝑠 is Laplace transform variable.

Defineπ›½π‘˜=ξ‚€1𝛼2πœ‹π‘˜=ξ‚€βˆ’4π»π‘‘π‘˜ξ‚ξ‚€cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚.(3.20)

Case 1. If π‘˜=0, thenπœ•2ξ‚Šπœ“0πœ•π‘₯2+πœ•2ξ‚Šπœ“0πœ•π‘¦2βˆ’π‘ ξ‚Šπœ“0=𝛼0𝛿(π‘₯βˆ’π‘₯ξ…ž)𝛿(𝑦)𝑠,inΞ©1,(3.21)where𝛼0=ξ‚€βˆ’8πœ‹π»ξ‚,ξ‚Šπœ“0=0,onΞ“1.(3.22)

Case 2. If π‘˜>0, then ξ‚Šπœ‘π‘˜ satisfies (3.17).
Defineπœπ‘˜=ξ”πœ†2π‘˜+𝑠.(3.23)
Recall (3.8), and [βˆ’π›½π‘˜/𝑠]𝐾0(πœπ‘˜π‘…β€²) is a basic solution of (3.17), since π‘˜>0, we haveπ›Όπ‘˜=ξ‚€βˆ’16πœ‹π»ξ‚ξ‚€cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚,π›½π‘˜=ξ‚€βˆ’8𝐻cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚,(3.24)so letξ‚Šπœ“π‘˜=ξ‚Šπœ‘π‘˜+ξ‚Šπœ‡π‘˜,(3.25)whereξ‚Šπœ‡π‘˜=π›½π‘˜πΎ0[πœπ‘˜π‘…ξ…ž]𝑠,(3.26)thusξ‚Šπœ‘π‘˜=ξ‚Šπœ“π‘˜βˆ’ξ‚Šπœ‡π‘˜,(3.27)and ξ‚Šπœ“π‘˜ satisfies homogeneous equation:πœ•2ξ‚Šπœ“π‘˜πœ•π‘₯2+πœ•2ξ‚Šπœ“π‘˜πœ•π‘¦2βˆ’ξ€·π‘ +πœ†2π‘˜ξ€Έξ‚Šπœ“π‘˜=0,inΞ©1,ξ‚Šπœ“π‘˜=π›½π‘˜πΎ0𝑠+πœ†2π‘˜π‘…ξ…žξ€Έπ‘ ,onΞ“1,(3.28)𝑅′ has the same meaning as in (3.8).

Under polar coordinates representation of Laplace operator and by using methods of separation of variables, we obtain a general solution [11–13]:ξ‚Šπœ“π‘˜(𝑠,π‘₯,𝑦;𝑠,π‘₯ξ…žξ€Ίπ΄,0)=0π‘˜πΌ0(πœπ‘˜πœŒ)+𝐡0π‘˜πΎ0(πœπ‘˜π‘ŽπœŒ)ξ€»ξ€Ί0π‘˜πœƒ+𝑏0π‘˜ξ€»+βˆžξ“π‘š=1ξ€Ίπ΄π‘šπ‘˜πΌπ‘š(πœπ‘˜πœŒ)+π΅π‘šπ‘˜πΎπ‘š(πœπ‘˜π‘ŽπœŒ)ξ€»ξ€Ίπ‘šπ‘˜cos(π‘šπœƒ)+π‘π‘šπ‘˜ξ€»,sin(π‘šπœƒ)(3.29)where π΄π‘–π‘˜,π΅π‘–π‘˜,π‘Žπ‘–π‘˜,π‘π‘–π‘˜,𝑖=0,1,2,…, are undetermined coefficients.

Because ξ‚Šπœ“π‘˜(𝑠,π‘₯,𝑦;𝑠,π‘₯β€²,0) is continuously bounded within Ξ©1, but 𝐾𝑖(0)=∞, there holdsπ΅π‘–π‘˜=0,𝑖=0,1,2,….(3.30)

There hold [9, 10]πΎπœξ‚€(𝑧)=πœ‹π‘–2ξ‚π‘’πœπœ‹π‘–/2𝐻𝜐(1)𝐼(𝑧𝑖),𝜐(𝑧)=π‘’βˆ’πœπœ‹π‘–/2𝐽𝜐(𝑧𝑖),(3.31)where 𝐾𝜐(𝑧) is modified Bessel function of second kind and order 𝜐,𝐼𝜐(𝑧) is modified Bessel function of first kind and order 𝜐,𝐽𝜐(𝑧) is Bessel function of first kind and order 𝜐,𝐻𝜐(1)(𝑧) is Hankel function of first kind and order 𝜐, and βˆšπ‘–=βˆ’1.

And there hold (see [14, page 979])𝐻0(1)(πœŽπ‘…ξ…ž)=𝐽0(𝜎𝜌0)𝐻0(1)(πœŽπ‘…π‘’)+2βˆžξ“π‘š=1π½π‘š(𝜎𝜌0)π»π‘š(1)(πœŽπ‘…π‘’πΎ)cos(π‘šπœƒ),(3.32)0(πœπ‘˜π‘…ξ…žξ‚€)=πœ‹π‘–2𝐻0(1)ξ€·π‘–πœπ‘˜π‘…ξ…žξ€Έ.(3.33)

Let 𝜎=π‘–πœπ‘˜, (note that 𝑖2=βˆ’1), substituting (3.31) into (3.32) and using (3.33), we have the following Cosine Fourier expansions of 𝐾0(πœπ‘˜π‘…β€²) (see [14, page 952]):𝐾0ξ€·πœπ‘˜π‘…ξ…žξ€Έ=ξ‚€πœ‹π‘–2𝐽0(π‘–πœπ‘˜πœŒ0)𝐻0(1)(π‘–πœπ‘˜π‘…π‘’)+2βˆžξ“π‘š=1π½π‘š(π‘–πœπ‘˜πœŒ0)π»π‘š(1)(π‘–πœπ‘˜π‘…π‘’ξ‚Ή)cos(π‘šπœƒ)=𝐽0(π‘–πœπ‘˜πœŒ0)𝐾0(πœπ‘˜π‘…π‘’)+2βˆžξ“π‘š=1π‘’βˆ’π‘šπœ‹π‘–/2π½π‘š(π‘–πœπ‘˜πœŒ0)πΎπ‘š(πœπ‘˜π‘…π‘’)cos(π‘šπœƒ)=𝐼0(πœπ‘˜πœŒ0)𝐾0(πœπ‘˜π‘…π‘’)+2βˆžξ“π‘š=1πΌπ‘š(πœπ‘˜πœŒ0)πΎπ‘š(πœπ‘˜π‘…π‘’)cos(π‘šπœƒ).(3.34)

So, we obtainπ›½π‘˜πΎ0(πœπ‘˜π‘…ξ…ž)𝑠=π›½π‘˜ξ€ΊπΌ0(πœπ‘˜πœŒ0)𝐾0(πœπ‘˜π‘…π‘’βˆ‘)+2βˆžπ‘š=1πΌπ‘š(πœπ‘˜πœŒ0)πΎπ‘š(πœπ‘˜π‘…π‘’ξ€»)cos(π‘šπœƒ)𝑠.(3.35)

Note that ξ‚Šπœ“π‘˜=π›½π‘˜πΎ0(πœπ‘˜π‘…β€²)/𝑠 on Ξ“1, and comparing coefficients of Cosine Fourier expansions of 𝐾0(πœπ‘˜π‘…β€²)/𝑠 in (3.35) and (3.29), we obtainπ‘Ž0π‘˜=0,𝑏0π‘˜=1,π‘π‘–π‘˜=0,𝑖=1,2,….(3.36)

Defineπ‘Œπ‘šπ‘˜=π‘Žπ‘šπ‘˜π΄π‘šπ‘˜,π‘˜=0,1,2,…(3.37)and recall (3.29), then we haveξ‚Šπœ“π‘˜(𝑠,π‘₯,𝑦;𝑠,π‘₯ξ…ž,0)=βˆžξ“π‘š=0π‘Œπ‘šπ‘˜πΌπ‘š(πœπ‘˜πœŒ)cos(π‘šπœƒ),π‘˜=0,1,2,…,(3.38)whereπ‘Œ0π‘˜=π›½π‘˜πΎ0(πœπ‘˜π‘…π‘’)𝐼0(πœπ‘˜πœŒ0)𝑠𝐼0(πœπ‘˜π‘…π‘’),π‘Œ(3.39)π‘šπ‘˜=2π›½π‘˜πΎπ‘š(πœπ‘˜π‘…π‘’)πΌπ‘š(πœπ‘˜πœŒ0)π‘ πΌπ‘š(πœπ‘˜π‘…π‘’).(3.40)

In the appendix, we can proveβˆžξ“π‘˜=1|||ξ‚Šπœ“π‘˜ξ‚€cosπ‘˜πœ‹π‘§π»ξ‚|||β‰ˆ0.(3.41)

Thus we only consider the case π‘˜=0, in (3.38) and (3.40), let π‘˜=0, we haveξ‚Šπœ“0(𝑠,π‘₯,𝑦;𝑠,π‘₯ξ…ž,0)=βˆžξ“π‘š=0π‘Œπ‘š0πΌπ‘š(𝜁0𝜌)cos(π‘šπœƒ),(3.42)where𝜁0=βˆšπ‘Œπ‘ ,(3.43)π‘š0πΌπ‘š(𝜁0π›½πœŒ)=0πΎπ‘šξ€·βˆšπ‘ π‘…π‘’ξ€ΈπΌπ‘šξ€·βˆšπ‘ πœŒ0ξ€ΈπΌπ‘šξ€·βˆšξ€Έπ‘ πœŒπ‘ πΌπ‘šξ€·βˆšπ‘ π‘…π‘’ξ€Έ=𝑓1π‘š(𝑠)×𝑓2π‘š(𝑠),(3.44)where𝑓1π‘šξ‚€π›½(𝑠)=0π‘…π‘šπ‘’21+π‘š21+π‘šπ‘ π‘š/2πΎπ‘šξ€·βˆšπ‘ π‘…π‘’ξ€Έπ‘…π‘šπ‘’ξ‚„π‘“,π‘š=0,1,2,…,(3.45)2π‘šπΌ(𝑠)=π‘šξ€·βˆšπ‘ πœŒ0ξ€ΈπΌπ‘šξ€·βˆšξ€Έπ‘ πœŒπ‘ π‘š/2+1πΌπ‘šξ€·βˆšπ‘ π‘…π‘’ξ€Έ,π‘š=0,1,2,….(3.46)

And there holdsβ„’βˆ’1𝑓1π‘šξ€Ύ=𝛽(𝑠)0π‘…π‘šπ‘’21+π‘šξ€·ξ‚ξ‚ƒexpβˆ’(𝑅2𝑒/4𝑑)π‘‘π‘š+1ξ‚„,π‘š=0,1,2,…,(3.47)where β„’βˆ’1 is Inverse Laplace transform operator.

Since 𝑠=0,𝑠=βˆ’π›Ύπ‘šπ‘› are simple poles of meromorphic function 𝑓2π‘š(𝑠), if using partial fraction expansion of meromorphic function, there holds [15]𝑓2π‘šπ΅(𝑠)=π‘š0𝑠+βˆžξ“π‘›=1π΅π‘šπ‘›π‘ +π›Ύπ‘šπ‘›,(3.48)where π΅π‘š0,π΅π‘šπ‘› are residues at poles 𝑠=0,𝑠=βˆ’π›Ύπ‘šπ‘›, respectively, andπ΅π‘š0=(𝜌0𝜌)π‘š2π‘šπ‘š!π‘…π‘šπ‘’,π›Ύπ‘šπ‘›=πœ€2π‘šπ‘›π‘…2𝑒,(3.49)πœ€π‘šπ‘› is the 𝑛th root of equation π½π‘š(π‘₯)=0.

From (3.48), we haveβ„’βˆ’1𝑓2π‘šξ€Ύ(𝑠)=π΅π‘š0+βˆžξ“π‘›=1π΅π‘šπ‘›exp(βˆ’π›Ύπ‘šπ‘›π‘‘)=βˆžξ“π‘›=0π΅π‘šπ‘›exp(βˆ’π›Ύπ‘šπ‘›π‘‘),(3.50)where π›Ύπ‘š0=0.

According to the convolution theorem [12], from (3.44), there holdsβ„’βˆ’1𝑓1π‘š(𝑠)×𝑓2π‘šξ€Ύ=𝛽(𝑠)0π‘…π‘šπ‘’21+π‘šξ‚ξ‚»βˆžξ“π‘›=0π΅π‘šπ‘›ξ€œπ‘‘0ξ‚Έexp(βˆ’(𝑅2𝑒/4𝜏))πœπ‘š+1ξ‚Ήξ€Ίexpβˆ’π›Ύπ‘šπ‘›ξ€»ξ‚Ό(π‘‘βˆ’πœ)π‘‘πœ=πΆπ‘š0+π·π‘š,(3.51)whereπΆπ‘š0=𝛽0π‘…π‘šπ‘’π΅π‘š021+π‘šξ‚ξ€œπ‘‘0ξ‚ƒξ€·βˆ’ξ€·π‘…exp2𝑒/4πœξ€Έξ€Έπœπ‘š+1ξ‚„=ξ‚ƒπ›½π‘‘πœ0(𝜌0𝜌)π‘šπ‘š!Γ—21+2π‘šξ‚„ξ€œπ‘‘0ξ‚ƒξ€·βˆ’ξ€·π‘…exp2𝑒/4πœξ€Έξ€Έπœπ‘š+1ξ‚„π·π‘‘πœ,π‘š=𝛽0π‘…π‘šπ‘’21+π‘šξ‚ξ‚»βˆžξ“π‘›=1π΅π‘šπ‘›ξ€œπ‘‘0ξ‚ƒξ€·βˆ’ξ€·π‘…exp2𝑒/4πœξ€Έξ€Έπœπ‘š+1ξ‚„ξ€Ίexpβˆ’π›Ύπ‘šπ‘›ξ€»ξ‚Ό.(π‘‘βˆ’πœ)π‘‘πœ(3.52)

Recall (3.38), (3.44), and (3.51), there holdsπœ“0(𝑑,π‘₯,𝑦;π‘₯ξ…ž,0)=β„’βˆ’1ξ€½ξ‚Šπœ“0(𝑠,π‘₯,𝑦;π‘₯ξ…žξ€Ύ=,0)βˆžξ“π‘š=0β„’βˆ’1𝑓1π‘š(𝑠)×𝑓2π‘šξ€Ύ=(𝑠)cos(π‘šπœƒ)βˆžξ“π‘š=0ξ€·πΆπ‘š0+π·π‘šξ€Έcos(π‘šπœƒ).(3.53)

Using Laplace asymptotic integration (see [16, page 221]), when π›Ύπ‘šπ‘› is sufficiently large, thenξ€œπ‘‘0ξ‚ƒξ€·βˆ’ξ€·π‘…exp2𝑒/4πœξ€Έξ€Έπœπ‘š+1ξ‚„ξ€Ίexpβˆ’π›Ύπ‘šπ‘›ξ€»ξ€·βˆ’ξ€·π‘…(π‘‘βˆ’πœ)π‘‘πœβ‰ˆexp2𝑒/4πœξ€Έξ€Έπ›Ύπ‘šπ‘›π‘‘π‘š+1,(3.54)therefore,π·π‘š=𝛽0π‘…π‘šπ‘’ξ€·βˆ’ξ€·π‘…exp2𝑒/4𝑑21+π‘šπ‘‘π‘š+1ξ‚„βˆžξ“π‘›=1π΅π‘šπ‘›π›Ύπ‘šπ‘›.(3.55)

There holds [9]πΌπ‘š(π‘₯)=βˆžξ“π‘˜=01ξ‚€π‘₯π‘˜!(π‘š+π‘˜)!2ξ‚π‘š+2π‘˜=1π‘₯π‘š!2ξ‚π‘šξ‚„ξ‚†1+π‘š!ξ‚€π‘₯(π‘š+1)!22+ξ‚ƒπ‘š!π‘₯2!(π‘š+2)!ξ‚„ξ‚€24.+β‹―(3.56)

Using (3.48) and (3.56), and note that11+π‘₯=1βˆ’π‘₯+π‘₯2βˆ’π‘₯3+𝑂(π‘₯3),(3.57)so (3.46) can be written as𝑓2π‘š=ξ€½ξ€·βˆš(𝑠)(1/π‘š!)π‘ πœŒ0ξ€Έ/2π‘šξ€Ίξ€·βˆš1+π’œπ‘ πœŒ0ξ€Έ/22ξ€·βˆš+β‹―ξ€»ξ€Ύξ€½(1/π‘š!)ξ€Έπ‘ πœŒ/2π‘šξ€Ίξ€·βˆš1+π’œξ€Έπ‘ πœŒ/22+⋯𝑠1+π‘š/2ξ€·βˆš{(1/π‘š!)𝑠𝑅𝑒/2π‘šξ€·βˆš[1+(1/(π‘š+1))𝑠𝑅𝑒/22=𝐡+β‹―]}π‘š0π‘ ξ‚ƒπœŒ1+20π‘ πœŒ4(π‘š+1)+⋯1+2𝑠𝑅4(π‘š+1)+⋯1βˆ’2𝑒𝑠=𝐡4(π‘š+1)+β‹―π‘š0𝑠+ξ‚€1πœŒπ‘š!0𝜌2π‘…π‘’ξ‚π‘šξ‚ƒ1ξ€·πœŒ4(π‘š+1)20+𝜌2βˆ’π‘…2𝑒,+𝑂(𝑠)(3.58)where π’œ denotes (π‘š!/(π‘š+1)!), thus from (3.48), we obtainβˆžξ“π‘›=1π΅π‘šπ‘›π›Ύπ‘šπ‘›=lim𝑠→0𝑓2π‘šπ΅(𝑠)βˆ’π‘š0𝑠=1𝜌4(π‘š+1)!ξ‚„ξ‚€0𝜌2π‘…π‘’ξ‚π‘šξ€·πœŒ20+𝜌2βˆ’π‘…2𝑒,(3.59)therefore,π·π‘š=𝛽0π‘…π‘šπ‘’ξ€·βˆ’ξ€·π‘…exp2𝑒/4𝑑21+π‘šπ‘‘π‘š+1ξ€·πœŒξ‚„ξ‚ƒ20+𝜌2βˆ’π‘…2π‘’ξ€ΈπœŒ4(π‘š+1)!ξ‚„ξ‚€0𝜌2π‘…π‘’ξ‚π‘š=𝛽0ξ€·βˆ’ξ€·π‘…exp2𝑒/4π‘‘ξ€Έξ€Έξ‚„ξ€·πœŒ8𝑑(π‘š+1)!20+𝜌2βˆ’π‘…2π‘’ξ€Έξ‚€πœŒ0πœŒξ‚4π‘‘π‘š.(3.60)

𝑅𝑃 is defined as real part operator, for example, 𝑅𝑃(π‘’π‘–π‘šπœƒ) means real part of π‘’π‘–π‘šπœƒ,𝑅𝑃(π‘’π‘–π‘šπœƒ)=cos(π‘šπœƒ).(3.61) There holdsξ‚€πœŒexp0πœŒπ‘’π‘–πœƒξ‚=4πœβˆžξ“π‘š=0ξ‚€1πœŒπ‘š!0πœŒπ‘’π‘–πœƒξ‚4πœπ‘š,(3.62)and defineπ‘…πœ‚=2𝑒4βˆ’ξ‚€πœŒ0πœŒπ‘’π‘–πœƒ4,(3.63)πœ‚ is a complex number.

Note that 𝛽0=βˆ’4/𝐻, recall (3.53), defineΞ›1=βˆžξ“π‘š=0πΆπ‘š0=𝛽cos(π‘šπœƒ)02ξ‚ξ‚»ξ€œπ‘‘0exp(βˆ’(𝑅2𝑒/4𝜏))πœξ‚„βˆžξ“π‘š=0cos(π‘šπœƒ)ξ‚€πœŒπ‘š!0πœŒξ‚4πœπ‘šξ‚„ξ‚Ό=ξ‚€π›½π‘‘πœ02ξ‚ξ‚»ξ€œΓ—π‘…π‘ƒπ‘‘0exp(βˆ’(𝑅2𝑒/4𝜏))πœξ‚„ξ‚€πœŒexp0πœŒπ‘’π‘–πœƒξ‚ξ‚Ό=𝛽4πœπ‘‘πœ02ξ‚ξ‚»ξ€œΓ—π‘…π‘ƒπ‘‘0exp(βˆ’(πœ‚/𝜏))πœξ‚„ξ‚Όξ‚€π›½π‘‘πœ=βˆ’02ξ‚ξ‚†ξ‚€βˆ’πœ‚Γ—π‘…π‘ƒπΈπ‘–π‘‘=ξ‚€2ξ‚ξ‚‡π»ξ‚ξ‚†ξ‚€βˆ’πœ‚Γ—π‘…π‘ƒπΈπ‘–π‘‘.(3.64)

In (3.60), letπœŒπœ’=0𝜌4𝑑,(3.65)and defineΞ›2=βˆžξ“π‘š=0π·π‘š=𝛽cos(π‘šπœƒ)0exp(βˆ’(𝑅2𝑒/4𝑑))ξ‚„ξ€·πœŒ8𝑑20+𝜌2βˆ’π‘…2π‘’ξ€Έβˆžξ“π‘š=0cos(π‘šπœƒ)ξ‚€πœŒ(π‘š+1)!0πœŒξ‚4π‘‘π‘šξ‚„=𝛽0exp(βˆ’(𝑅2𝑒/4𝑑))ξ‚„ξ€·πœŒ8𝑑20+𝜌2βˆ’π‘…2π‘’ξ€Έβˆžξ“π‘š=01π‘…π‘ƒξ‚€πœŒ(π‘š+1)!0πœŒπ‘’π‘–πœƒξ‚4π‘‘π‘šξ‚‡ξ‚ƒ=βˆ’exp(βˆ’(𝑅2𝑒/4𝑑))ξ‚„ξ€·πœŒ2𝑑𝐻20+𝜌2βˆ’π‘…2𝑒1Γ—π‘…π‘ƒξ‚†ξ‚€πœ’π‘’π‘–πœƒξ‚ξ€Ίξ€·expπœ’π‘’π‘–πœƒξ€Έξ€»ξ‚‡,βˆ’1(3.66)thus we obtainπœ“0=β„’βˆ’1ξ€½ξ‚Šπœ“0ξ€Ύ=Ξ›1+Ξ›2,(3.67)and there holds [9, 10]β„’βˆ’1𝐾0√(π‘Žπ‘ )𝑠1=βˆ’2ξ‚€βˆ’π‘ŽπΈπ‘–24𝑑,(3.68)thus if we recall (3.26) and defineΞ›3=βˆ’πœ‡0=βˆ’β„’βˆ’1ξ€½ξ‚Šπœ‡0ξ€Ύ,(3.69)thenΞ›3=βˆ’β„’βˆ’1𝛽0𝐾0(βˆšπ‘ π‘…ξ…ž)𝑠2=βˆ’π»ξ‚ξ‚€βˆ’π‘…πΈπ‘–ξ…ž24𝑑(3.70)and 𝑅′ has the same meaning as in (3.7).

In the above equations, 𝐸𝑖(βˆ’π‘₯) is exponential integral function,ξ€œπΈπ‘–(βˆ’π‘₯)=βˆ’π‘₯βˆ’βˆžexp(𝑒)𝑒𝑑𝑒,(0<π‘₯<∞).(3.71)

Recall (3.27), there holdsπœ‘0(𝑑,π‘₯,𝑦;π‘₯ξ…ž,0)=πœ“0βˆ’πœ‡0=Ξ›1+Ξ›2+Ξ›3.(3.72)

Combining (3.12), (3.26), (3.27), and (3.41), we obtain𝑃(𝑑;π‘₯,𝑦,𝑧;π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž)=πœ‘0+β„’βˆ’1ξ‚»βˆžξ“π‘˜=1ξ‚Šπœ‘π‘˜ξ‚€cosπ‘˜πœ‹π‘§π»ξ‚ξ‚Ό=πœ‘0+β„’βˆ’1ξ‚»βˆžξ“π‘˜=1(ξ‚Šπœ“π‘˜βˆ’ξ‚Šπœ‡π‘˜ξ‚€)cosπ‘˜πœ‹π‘§π»ξ‚ξ‚Ό=πœ‘0βˆ’βˆžξ“π‘˜=1ξ‚€cosπ‘˜πœ‹π‘§π»ξ‚β„’βˆ’1ξ‚†π›½π‘˜πΎ0𝑠+πœ†2π‘˜π‘…ξ…žξ€Έπ‘ ξ‚‡.(3.73)

Equation (3.73) is the pressure distribution equation of an off-center point sink in the cylindrical body. If the point sink 𝐫0 and the observation point 𝐫 are not on a radius of the drainage circle, πœƒβ‰ 0, recall (3.7), 𝑅′ cannot be simplified, we cannot obtain exact inverse Laplace transform of (3.73), but if necessary, we may obtain numerical inverse Laplace transform results.

If the point sink is at the center of the drainage circle, then𝜌0𝑅=0,πœ‚=2𝑒4,πœ‘0=ξ‚€2π»ξ‚€βˆ’π‘…ξ‚ξ‚ƒπΈπ‘–2π‘’ξ‚ξ‚€βˆ’πœŒ4π‘‘βˆ’πΈπ‘–2+4𝑑exp(βˆ’(𝑅2𝑒/4𝑑))𝑅2𝑑𝐻2π‘’βˆ’πœŒ2ξ€Έ.(3.74)

In Figure 2, if the point sink 𝐫0 and the observation point 𝐫 are on a radius, thenπ‘…πœƒ=0,πœ‚=2𝑒4βˆ’ξ‚€πœŒ0𝜌4,πœ‘0=ξ‚€2π»ξ‚ƒβˆ’ξ‚€π‘…ξ‚ξ‚†πΈπ‘–2π‘’βˆ’πœŒπœŒ0ξ‚ƒβˆ’4π‘‘ξ‚ξ‚„βˆ’πΈπ‘–(πœŒβˆ’πœŒ0)2ξ‚„βˆ’ξ‚€πœŒ4𝑑20+𝜌2βˆ’π‘…2π‘’πœŒπœŒ0exp𝜌𝜌0βˆ’π‘…2π‘’ξ‚ξ‚€βˆ’π‘…4π‘‘βˆ’exp2𝑒.4𝑑(3.75)

4. Uniform Line Sink Solution

Although the off-center partially penetrating vertical well is represented in the model by a line sink, we only concern in the pressures at the wellbore face.

For convenience, in the following reference, every variable below is dimensionless but we drop the subscript 𝐷.

The well line sink is located along the line {(π‘₯β€²,0,𝑧)∢𝐿1≀𝑧≀𝐿2}. If the observation point 𝐫 is on the wellbore, 𝑅′=𝑅𝑀, note that 𝑅0≫𝑅𝑀, and there holdπœƒ=0,𝜌=𝜌0+𝑅𝑀=𝑅0+𝑅𝑀,𝜌0=𝑅0,πœŒβˆ’πœŒ0=𝑅𝑀,𝜌0πœŒβ‰ˆπ‘…20,𝜌+𝜌0β‰ˆ2𝜌0=2𝑅0,(4.1) thenπ‘…πœ‚=2𝑒4βˆ’π‘…204,πœ’=𝜌𝜌0β‰ˆπ‘…4𝑑20,4𝑑(4.2)and recall (3.64), thenΞ›1(𝑑;𝑅0𝛽,0)=βˆ’02ξ‚ξ‚ƒβˆ’ξ‚€π‘…πΈπ‘–2π‘’βˆ’π‘…204𝑑.(4.3)

Recall (3.66), thenΞ›2(𝑑;𝑅0ξ‚ƒξ€·βˆ’ξ€·π‘…,0)=βˆ’2exp2𝑒/4𝑑𝐻𝑅20ξ‚„ξ€·2𝑅20βˆ’π‘…2𝑒𝑅exp204π‘‘βˆ’1,(4.4)and recall (3.70), thenΞ›3(𝑑;𝑅0𝛽,0)=02ξ‚ξ‚ƒβˆ’πΈπ‘–(πœŒβˆ’πœŒ0)2ξ‚„=𝛽4𝑑02ξ‚ξ‚€βˆ’π‘…πΈπ‘–2𝑀4𝑑.(4.5)

DefineΞ“1=ξ€œπΏ2𝐿1Ξ›1(𝑑;𝑅0,0)π‘‘π‘§ξ…ž=ξ€œπΏ2𝐿1ξ‚ƒβˆ’ξ‚€π›½02ξ‚ƒβˆ’ξ‚€π‘…ξ‚ξ‚„πΈπ‘–2π‘’βˆ’π‘…204π‘‘ξ‚ξ‚„π‘‘π‘§ξ…žξ‚€π›½=βˆ’02(𝐿2βˆ’πΏ1ξ‚ƒβˆ’ξ‚€π‘…)𝐸𝑖2π‘’βˆ’π‘…20=ξ‚€4𝑑2πΏπ‘π‘Ÿπ»ξ‚ξ‚ƒβˆ’ξ‚€π‘…πΈπ‘–2π‘’βˆ’π‘…20,Ξ“4𝑑2=ξ€œπΏ2𝐿1Ξ›2(𝑑;𝑅0,0)π‘‘π‘§ξ…ž=ξ€œπΏ2𝐿1βˆ’ξ‚ƒ2exp(βˆ’(𝑅2𝑒/4𝑑))𝐻𝑅20ξ‚„ξ€·2𝑅20βˆ’π‘…2𝑒𝑅exp204π‘‘βˆ’1π‘‘π‘§ξ…žξ‚ƒ=βˆ’2πΏπ‘ξ€·βˆ’ξ€·π‘…exp2𝑒/4𝑑𝐻𝑅20ξ‚„ξ€·2𝑅20βˆ’π‘…2𝑒𝑅exp204π‘‘βˆ’1=βˆ’2πΏπ‘π‘Ÿπ»π‘…202𝑅20βˆ’π‘…2𝑒𝑅exp20βˆ’π‘…2π‘’ξ‚ξ‚€βˆ’π‘…4π‘‘βˆ’exp2𝑒,Ξ“4𝑑3=ξ€œπΏ2𝐿1Ξ›3(𝑑;𝑅0,0)π‘‘π‘§ξ…ž=𝛽02ξ‚ξ€œπΏ2𝐿1ξ‚€βˆ’π‘…πΈπ‘–2𝑀4π‘‘π‘‘π‘§ξ…žξ‚€=βˆ’2πΏπ‘π‘Ÿπ»ξ‚ξ‚€βˆ’π‘…πΈπ‘–2𝑀.4𝑑(4.6)

In order to calculate the pressure at the wellbore, using principle of potential superposition, integrating 𝑧′ at both sides of (3.72) from 𝐿1 to 𝐿2, thenΞ¨0ξ€œ(𝑑)=𝐿2𝐿1πœ‘0(𝑑;𝑅0,0)π‘‘π‘§ξ…ž=Ξ“1+Ξ“2+Ξ“3=ξ‚€2πΏπ‘π‘Ÿπ»ξ‚ξ‚ƒβˆ’ξ‚€π‘…πΈπ‘–2π‘’βˆ’π‘…20βˆ’ξ‚€4𝑑2πΏπ‘π‘Ÿπ»π‘…202𝑅20βˆ’π‘…2𝑒𝑅exp20βˆ’π‘…2π‘’ξ‚ξ‚€βˆ’π‘…4π‘‘βˆ’exp2π‘’βˆ’ξ‚€4𝑑2πΏπ‘π‘Ÿπ»ξ‚ξ‚€βˆ’π‘…πΈπ‘–2𝑀=ξ‚€4𝑑2πΏπ‘π‘Ÿπ»ξ‚ƒβˆ’ξ‚€π‘…ξ‚ξ‚†πΈπ‘–2π‘’βˆ’π‘…20ξ‚€βˆ’π‘…4π‘‘ξ‚ξ‚„βˆ’πΈπ‘–2π‘€ξ‚βˆ’ξ‚€14𝑑𝑅202𝑅20βˆ’π‘…2𝑒𝑅exp20βˆ’π‘…2π‘’ξ‚ξ‚€βˆ’π‘…4π‘‘βˆ’exp2𝑒.4𝑑(4.7)

Recall (3.26), and note that πœŒβˆ’πœŒ0=𝑅𝑀, we haveξπœ‡π‘˜=π›½π‘˜πΎ0𝑅𝑀𝑠+πœ†2π‘˜ξ€Έπ‘ ,(4.8)and defineξπœŽπ‘˜=ξ€œπΏ2𝐿1ξπœ‡π‘˜π‘‘π‘§ξ…ž=π›½π‘˜ξ‚€1π‘ ξ‚ξ€œπΏ2𝐿1𝐾0𝑅𝑀𝑠+πœ†2π‘˜ξ€Έπ‘‘π‘§ξ…žξ‚€8=βˆ’ξ‚πΎπ»π‘ 0𝑅𝑀𝑠+πœ†2π‘˜ξ€Έξ€œπΏ2𝐿1ξ‚€cosπ‘˜πœ‹π‘§ξ…žπ»ξ‚π‘‘π‘§ξ…žξ‚€8=βˆ’ξ‚πΎπ‘˜πœ‹π‘ 0𝑅𝑀𝑠+πœ†2π‘˜ξ€Έξ‚ƒξ‚€sinπ‘˜πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘˜πœ‹πΏ1𝐻,(4.9)because when 𝑠 is very small, (time 𝑑 is sufficiently long), there holds𝐾0𝑅𝑀𝑠+πœ†2π‘˜ξ€Έ=𝐾0ξ€·π‘…π‘€πœ†π‘˜ξ”1+𝑠/πœ†2π‘˜ξ€Έβ‰ˆπΎ0(π‘…π‘€πœ†π‘˜),(4.10)so when time is sufficiently long,πœŽπ‘˜=β„’βˆ’1ξ€½ξπœŽπ‘˜ξ€Ύξ‚€8=βˆ’ξ‚πΎπ‘˜πœ‹0(π‘…π‘€πœ†π‘˜)sinπ‘˜πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘˜πœ‹πΏ1𝐻.(4.11)

Recall (3.12), (3.24), (3.27), and (3.41), when time 𝑑 is sufficiently long, defineπ‘ˆ=βˆžξ“π‘˜=1πœŽπ‘˜ξ‚€cosπ‘˜πœ‹π‘§π»ξ‚=βˆžξ“π‘˜=1βˆ’ξ‚€8ξ‚πΎπ‘˜πœ‹0(π‘…π‘€πœ†π‘˜)sinπ‘˜πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘˜πœ‹πΏ1𝐻cosπ‘˜πœ‹π‘§π»ξ‚ξ‚€8=βˆ’πœ‹ξ‚βˆžξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sinπ‘›πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘›πœ‹πΏ1𝐻cosπ‘›πœ‹π‘§π»ξ‚.(4.12)

Therefore, the wellbore pressure at point (𝑅0+𝑅𝑀,𝑧) is𝑃(π‘…π‘€ξ€œ,𝑧)=𝐿2𝐿1𝑃(𝑅0+𝑅𝑀,0,𝑧,𝑑;𝑅0,0,π‘§ξ…ž)π‘‘π‘§ξ…žβ‰ˆΞ¨0(𝑑)βˆ’π‘ˆ.(4.13)

Considering the bottom point of the well line sink, then 𝑧=πΏπ‘π‘Ÿ,𝐿1=0, thus 𝐿2=πΏπ‘π‘Ÿ, in this case, (4.12) reduces toξ‚€8π‘ˆ=βˆ’πœ‹ξ‚βˆžξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sinπ‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚ξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξ‚€4=βˆ’πœ‹ξ‚βˆžξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚=𝐼1+𝐼2,(4.14)where𝐼1ξ‚€4=βˆ’πœ‹ξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚,𝐼2ξ‚€4=βˆ’πœ‹ξ‚βˆžξ“π‘›=𝑁+1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚,𝐻𝑁=4πœ‹π‘…π‘€ξ‚„,(4.15)where [𝐻/πœ‹π‘…π‘€] is the integer part of 𝐻/πœ‹π‘…π‘€.

For 𝐼2 it holds the following estimate:|𝐼2|||||=βˆžξ“π‘›=𝑁+1ξ‚€4πœ‹ξ‚πΎ0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚||||≀||||βˆžξ“π‘›=𝑁+1ξ‚€4πœ‹ξ‚πΎ0(π‘›πœ‹π‘…π‘€/𝐻)𝑛||||β‰€ξ€œβˆžπ‘ξ‚€4πœ‹ξ‚πΎ0(4π‘₯/𝑁)π‘₯=ξ‚€4𝑑π‘₯πœ‹ξ‚ξ€œβˆž4𝐾0(𝑦)𝑦𝑑𝑦=2.7Γ—10βˆ’3β‰ˆ0.(4.16)

So, (4.14) reduces toπ‘ˆβ‰ˆπΌ1ξ‚€4=βˆ’πœ‹ξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚.(4.17)

Combining (4.7), (4.13), and (4.17), pressure at the bottom point of the producing portion is𝑃(𝑅𝑀,πΏπ‘π‘Ÿ)=Ξ¨0ξ‚€4(𝑑)+πœ‹ξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sin2π‘›πœ‹πΏπ‘π‘Ÿπ»ξ‚.(4.18)

In order to obtain average wellbore pressure, recall (4.12) and (4.17), integrate both sides of (4.13) with respect to 𝑧 from 𝐿1 to 𝐿2, then divided by πΏπ‘π‘Ÿ, average wellbore pressure is obtained:π‘ƒπ‘Ž,𝑀=1πΏπ‘π‘Ÿξ€œπΏ2𝐿1𝑃(𝑅𝑀,𝑧)π‘‘π‘§β‰ˆΞ¨0ξ‚€8(𝑑)+πœ‹ξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛sinπ‘›πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘›πœ‹πΏ1𝐻1ξ‚Ήξ‚ΈπΏπ‘π‘Ÿξ€œπΏ2𝐿1ξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξ‚Ήπ‘‘π‘§=Ξ¨0ξ‚€(𝑑)+8π»πœ‹2πΏπ‘π‘Ÿξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛2ξ‚Έξ‚€sinπ‘›πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘›πœ‹πΏ1𝐻2=Ξ¨0ξ‚€(𝑑)+32π»πœ‹2πΏπ‘π‘Ÿξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛2sin2ξ‚€π‘›πœ‹πΏπ‘π‘Ÿξ‚2𝐻cos2ξ‚ƒπ‘›πœ‹(𝐿2+𝐿1)ξ‚„,2𝐻(4.19)where we use1πΏπ‘π‘Ÿξ€œπΏ2𝐿1ξ‚€cosπ‘›πœ‹π‘§π»ξ‚ξ‚€π»π‘‘π‘§=π‘›πœ‹πΏπ‘π‘Ÿξ‚€ξ‚ξ‚ƒsinπ‘›πœ‹πΏ2π»ξ‚ξ‚€βˆ’sinπ‘›πœ‹πΏ1𝐻.(4.20)

5. Dimensionless Wellbore Pressure Equations

Combining (4.7) and (4.19), the dimensionless average wellbore pressure of an off-center partially penetrating vertical well in a circular cylinder drainage volume is𝑃𝑀𝐷=ξ‚€2πΏπ‘π‘Ÿπ·π»π·ξ‚ƒβˆ’ξ‚€π‘…ξ‚ξ‚†πΈπ‘–2π‘’π·βˆ’π‘…20𝐷4π‘‘π·ξ‚€βˆ’π‘…ξ‚ξ‚„βˆ’πΈπ‘–2𝑀𝐷4π‘‘π·ξ‚βˆ’ξ‚€1𝑅20𝐷2𝑅20π·βˆ’π‘…2𝑒𝐷𝑅exp20π·βˆ’π‘…2𝑒𝐷4π‘‘π·ξ‚ξ‚€βˆ’π‘…βˆ’exp2𝑒𝐷4𝑑𝐷+𝑆𝑝,(5.1)where𝑆𝑝=ξ‚€32π»π·πœ‹2πΏπ‘π‘Ÿπ·ξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€π·/𝐻𝐷)𝑛2sin2ξ‚€π‘›πœ‹πΏπ‘π‘Ÿπ·2𝐻𝐷cos2ξ‚ƒπ‘›πœ‹(𝐿2𝐷+𝐿1𝐷)2𝐻𝐷,𝐻(5.2)𝑁=4π·πœ‹π‘…π‘€π·ξ‚„,(5.3)[𝐻𝐷/πœ‹π‘…π‘€π·] is the integer part of 𝐻𝐷/πœ‹π‘…π‘€π·.

Equation (5.1) is applicable to impermeable upper and lower boundaries and long after the time when pressure transient reaches the upper and lower boundaries. And 𝑆𝑝 denotes pseudo-skin factor due to partial penetration.

If πΏπ‘π‘Ÿ=𝐿=𝐻, the drilled well length is equal to formation thickness, for a fully penetrating well, 𝑆𝑝=0, (5.1) reduces toπ‘ƒπ‘€π·ξ‚†ξ‚ƒβˆ’ξ‚€π‘…=2𝐸𝑖2π‘’π·βˆ’π‘…20𝐷4π‘‘π·ξ‚€βˆ’π‘…ξ‚ξ‚„βˆ’πΈπ‘–2𝑀𝐷4π‘‘π·ξ‚βˆ’ξ‚€1𝑅20𝐷2𝑅20π·βˆ’π‘…2𝑒𝐷𝑅exp20π·βˆ’π‘…2𝑒𝐷4π‘‘π·ξ‚ξ‚€βˆ’π‘…βˆ’exp2𝑒𝐷4𝑑𝐷.(5.4)

If the well is located at the center of the cylindrical body, then π‘₯β€²=𝑅0=0, there holdslim𝑅0𝐷→02𝑅20π·βˆ’π‘…2𝑒𝐷𝑅20𝐷𝑅exp20π·βˆ’π‘…2𝑒𝐷4π‘‘π·ξ‚ξ‚€βˆ’π‘…βˆ’exp2𝑒𝐷4𝑑𝐷𝑅=βˆ’2𝑒𝐷4𝑑𝐷expβˆ’π‘…2𝑒𝐷4𝑑𝐷.(5.5)

Thus, (5.1) reduces to𝑃𝑀𝐷=ξ‚€2πΏπ‘π‘Ÿπ·π»π·ξ‚ƒβˆ’ξ‚€π‘…ξ‚ξ‚†πΈπ‘–2𝑒𝐷4π‘‘π·ξ‚€βˆ’π‘…ξ‚ξ‚„βˆ’πΈπ‘–2𝑀𝐷4𝑑𝐷+𝑅2𝑒𝐷4π‘‘π·ξ‚ξ‚ƒβˆ’ξ‚€π‘…exp2𝑒𝐷4𝑑𝐷+𝑆𝑝,(5.6)where 𝑆𝑝 has the same meaning as in (5.2).

If the well is a fully penetrating well in an infinite reservoir, 𝑅𝑒=∞, there holdsξ‚€βˆ’π‘…πΈπ‘–2𝑒𝐷4𝑑𝐷𝑅=0,2𝑒𝐷4π‘‘π·ξ‚ξ‚€βˆ’π‘…exp2𝑒𝐷4𝑑𝐷=0.(5.7)

Thus, (5.6) reduces toπ‘ƒπ‘€π·ξ‚€βˆ’π‘…=βˆ’2𝐸𝑖2𝑀𝐷4𝑑𝐷.(5.8)

Substitute (2.12) and (2.15) into (2.17), then simplify and rearrange the resulting equation, we obtainπ‘ƒπ‘–βˆ’π‘ƒπ‘€=ξ‚€πœ‡π‘„π΅8πœ‹πΎβ„ŽπΏπ‘π‘Ÿξ‚π‘ƒπ‘€π·,(5.9)where 𝑄 is total flow rate of the well, and 𝑃𝑀𝐷 can be calculated by (5.1), (5.4), (5.6), and (5.8) for different cases.

During production, the unsteady state pressure drop of an off-center partially penetrating vertical well in a circular cylinder drainage volume can be calculated by (5.9).

6. Examples and Discussions

Recall (5.2), pseudo-skin factor due to partial penetration 𝑆𝑝 is a function of 𝐿1,𝐿2 and 𝐻 is not a function of well off-center distance 𝑅0 or drainage radius 𝑅𝑒.

For an isotropic reservoir, (5.2) reduces to𝑆𝑝=ξ‚€32π»πœ‹2πΏπ‘π‘Ÿξ‚π‘ξ“π‘›=1𝐾0(π‘›πœ‹π‘…π‘€/𝐻)𝑛2sin2ξ‚€π‘›πœ‹πΏπ‘π‘Ÿξ‚2𝐻cos2ξ‚ƒπ‘›πœ‹(𝐿1+𝐿2)ξ‚„2𝐻,(6.1)and (5.3) reduces to𝐻𝑁=4πœ‹π‘…π‘€ξ‚„,(6.2)[𝐻/πœ‹π‘…π‘€] is the integer part of 𝐻/πœ‹π‘…π‘€.

If we define𝑓1=𝐿1𝐻,𝑓2=𝐿2𝐻,𝑓3=𝑅𝑀𝐻,(6.3)then (6.1) can be written as𝑆𝑝=32πœ‹2(𝑓2βˆ’π‘“1)𝑁𝑛=1𝐾0(π‘›πœ‹π‘“3)𝑛2sin2ξ‚ƒπ‘›πœ‹2(𝑓2βˆ’π‘“1)ξ‚„cos2ξ‚ƒπ‘›πœ‹2(𝑓2+𝑓1)ξ‚„.(6.4)

Example 6.1. Equation (6.4) shows that pseudo-skin factor 𝑆𝑝 is a function of the three parameters 𝑓1,𝑓2,𝑓3, fix two parameters, and generate plots that show the trend of 𝑆𝑝 with the third parameter.

Solution
Case 1. Figure 3 shows the trend of 𝑆𝑝 with 𝑓3 when 𝑓1=0.2,𝑓2=0.8, it can be found that 𝑆𝑝 is a weak decreasing function of 𝑓3.


Case 2. Figure 4 shows the trend of 𝑆𝑝 with 𝑓1 when 𝑓2=0.9,𝑓3=0.002, it can be found that 𝑆𝑝 is an increasing function of 𝑓1. When 𝑓2 is a constant, we may assume 𝐻 is a constant, then 𝐿2 is also a constant; when 𝑓1 increases, 𝐿1 also increases, thus the well producing length πΏπ‘π‘Ÿ=𝐿2βˆ’πΏ1 decreases, and pseudo-skin factor due to partial penetration increases.