Productivity Formulas for a Partially Penetrating Vertical Well in a Circular Cylinder Drainage Volume

Abstract

Taking a partially penetrating vertical well as a uniform line sink in three-dimensional space, by developing necessary mathematical analysis, this paper presents steady state productivity formulas for an off-center partially penetrating vertical well in a circular cylinder drainage volume with constant pressure at outer boundary. This paper also gives formulas for calculating the pseudo-skin factor due to partial penetration. If top and bottom reservoir boundaries are impermeable, the radius of the cylindrical system and off-center distance appears in the productivity formulas. If the reservoir has a gas cap or bottom water, the effects of the radius and off-center distance on productivity can be ignored. It is concluded that, for a partially penetrating vertical well, different productivity equations should be used under different reservoir boundary conditions.

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.

The productivity index expresses an intuitive feeling, that is, once the well production is stabilized, the ratio of production rate to some pressure difference between the reservoir and the well must depend on the geometry of the reservoir/well system only. Indeed, a long time ago, petroleum engineers observed that in a bounded reservoir or a reservoir with strong water drive, the productivity index of a well stabilizes in a long time asymptote.

When an oil reservoir is bounded with a constant pressure boundary (such as a gas cap or an aquifer), flow reaches the steady-state regime after the pressure transient reaches the constant pressure boundary. Rate and pressure become constant with time at all points in the reservoir and wellbore once steady-state flow is established. Therefore, the productivity index during steady-state flow is a constant.

Strictly speaking, steady-state flow can occur only if the flow across the drainage boundary is equal to the flow across the wellbore wall at well radius, and the fluid properties remain constant throughout the oil reservoir. These conditions may never be met in an oil reservoir; however, in oil reservoirs produced by a strong water drive, whereby the water influx rate at reservoir outer boundary equals the well producing rate, the pressure change with time is so slight that it is practically undetectable. In such cases, the assumption of steady-state is acceptable.

In many oil 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. If a vertical well partially penetrates the formation, there is an added resistance to flow in the vicinity of the wellbore. The streamlines converge and the area for flow decreases, which results in added resistance.

The problem of fluid flow into wells with partial penetration has received much attention in the past, the exact solution of the partial penetration problem presents great analytical problems. Brons and Marting [4], Papatzacos [5], Basinev [3] developed solutions to the two dimensional diffusivity equation, which included flow of fluid in the vertical direction. They only obtained semianalytical and semiempirical expressions to calculate the added resistance due to partial penetration.

The primary goal of this study is to present new steady-state productivity formulas for a partially penetrating vertical well in a circular cylinder drainage reservoir with constant pressure at outer boundary. Analytical solutions are derived by making the assumption of uniform fluid withdrawal along the portion of the wellbore open to flow. The producing portion of a partially penetrating vertical well is modeled as a uniform line sink. This paper also gives new expressions for calculating the added resistance due to partial penetration, by solving the three-dimensional Laplace equation.

2. Literature Review

Putting Darcy's equation into the equation of continuity, the productivity formula of a fully penetrating vertical well in a homogeneous, isotropic permeability reservoir is obtained [1, page 52]: 𝑄_𝑤=𝐹_𝐷2𝜋𝐾𝐻(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)ln(𝑅_𝑒/𝑅_𝑤), (2.1) where 𝑃_𝑒 is outer boundary pressure, 𝑃_𝑤 is flowing wellbore pressure, 𝐾 is permeability, 𝐻 is payzone thickness, 𝜇 is oil viscosity, 𝐵 is oil formation volume factor, 𝑅_𝑒 is drainage radius, 𝑅_𝑤 is wellbore radius, and 𝐹_𝐷 is the factor which allows the use of field units, and it can be found in a Table 1 at page 52 of [1].

Formula (2.1) is applicable for a fully penetrating vertical well in a circular drainage area with constant pressure outer boundary.

If a vertical well partially penetrates the formation, there is an added resistance to flow which is limited to the region around the wellbore, this added resistance is included by introducing the pseudo-skin factor, 𝑆_ps. Thus, Formula (2.1) may be rewritten to include the pseudo-skin factor due to partial penetration as [2, page 92] 𝑄_𝑤=𝐹_𝐷2𝜋𝐾𝐻(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)ln(𝑅_𝑒/𝑅_𝑤)+𝑆_ps. (2.2)

Define partial penetration factor 𝜂: 𝐿𝜂=^𝑝𝐻, (2.3) where 𝐿_𝑝 is the producing well length, that is, perforated interval.

Several authors obtained semianalytical and semiempirical expressions for evaluating pseudo-skin factor due to partial penetration.

Bervaldier's pseudo-skin factor formula [3]: 𝑆_ps1=(𝜂−1)[ln(𝐿_𝑝/𝑅_𝑤)(1−𝑅_𝑤/𝐿_𝑝)−1]. (2.4)

Brons and Marting's pseudo-skin factor formula [4] is as follows: 𝑆_ps1=(𝜂−1)[ln(ℎ_𝐷)−𝐺(𝜂)], (2.5) where ℎ_𝐷𝐻=(2𝑅_𝑤𝐾)(^ℎ𝐾_𝑣)^1/2,𝐺(𝜂)=2.948−7.363𝜂+11.45𝜂²−4.675𝜂³. (2.6)

Papatzacos's pseudo-skin factor formula [5] is as follows: 𝑆_ps1=(𝜂−1)ln(𝜋ℎ_𝐷21)+(𝜂𝜂)ln[(𝐼2+𝜂)(¹−1𝐼₂)−1^1/2], (2.7) where 𝐼₁=𝐻ℎ₁+0.25𝐿_𝑝,𝐼₂=𝐻ℎ₁+0.75𝐿_𝑝, (2.8) and ℎ₁ is the distance from the top of the reservoir to the top of the open interval.

It must be pointed out that the aforementioned formulas are only applicable to a reservoir with both impermeable top and bottom boundaries.

3. 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 𝑅_𝑒.

Figure 1 

Partially penetrating vertical well model.

The following assumptions are made.

(1)The reservoir has constant 𝐾_𝑥,𝐾_𝑦,𝐾_𝑧 permeabilities, thickness 𝐻, porosity 𝜙. During production, the partially penetrating vertical well has a circular cylinder drainage volume with height 𝐻 and radius 𝑅_𝑒. The well is located at 𝑅₀ away from the axis of symmetry of the cylindrical body.(2)At time 𝑡=0, pressure is uniformly distributed in the reservoir, equal to the initial pressure 𝑃_𝑖. If the reservoir has constant pressure boundaries (edge water, gas cap, bottom water), the pressure is equal to the initial value at such boundaries during production.(3)The production occurs through a partially penetrating vertical well of radius 𝑅_𝑤, represented in the model by a uniform line sink, the drilled well length is 𝐿, the producing well length is 𝐿_𝑝.(4)A single phase fluid, of small and constant compressibility 𝐶_𝑓, constant viscosity 𝜇, and formation volume factor 𝐵, flows from the reservoir to the well. Fluid properties are independent of pressure. Gravity forces are neglected.(5)There is no water encroachment and no water/gas coning. Edge water, gas cap, and bottom water are taken as constant pressure boundaries, multiphase flow effects are ignored.(6)Any additional pressure drops caused by formation damage, stimulation, or perforation are ignored, we only consider pseudo-skin factor due to partial penetration.

The porous media domain is: Ω={(𝑥,𝑦,𝑧)∣𝑥²+𝑦²<𝑅²_𝑒,0<𝑧<𝐻}, (3.1) where 𝑅_𝑒 is cylinder radius, Ω 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 hold 𝐿_𝑝=𝐿₂−𝐿₁,𝐿_𝑝≤𝐿≤𝐻. (3.2)

We assume 𝐾_𝑥=𝐾_𝑦=𝐾_ℎ,𝐾_𝑧=𝐾_𝑣, (3.3) and define average permeability: 𝐾_𝑎=(𝐾_𝑥𝐾_𝑦𝐾_𝑧)^1/3=𝐾_ℎ^2/3𝐾_𝑣^1/3. (3.4)

The reservoir initial pressure is a constant: 𝑃|_𝑡=0=𝑃_𝑖. (3.5)

The pressure at constant pressure boundaries (edge water, gas cap, bottom water) is assumed to be equal to the reservoir initial pressure during production: 𝑃_𝑒=𝑃_𝑖. (3.6)

Suppose point (𝑅₀,0,𝑧^′) is in the producing portion, and its point convergence intensity is 𝑞, in order to obtain the pressure at point (𝑥,𝑦,𝑧) caused by the point (𝑅₀,0,𝑧^′), according to mass conservation law and Darcy's law, we have to obtain the basic solution of the diffusivity equation in Ω [6, 7]: 𝐾_ℎ𝜕²𝑃𝜕𝑥²+𝐾_ℎ𝜕²𝑃𝜕𝑦²+𝐾_𝑣𝜕²𝑃𝜕𝑧²=𝜇𝑞𝐵𝛿(𝑥−𝑅₀)𝛿(𝑦)𝛿(𝑧−𝑧^′),inΩ, (3.7) where 𝐶_𝑡 is total compressibility coefficient of porous media, 𝛿(𝑥−𝑅₀),𝛿(𝑦),𝛿(𝑧−𝑧^′) are Dirac functions.

In order to simplify the equations, we take the following dimensionless transforms: 𝑥_𝐷𝑥=(𝐿𝐾)(^𝑎𝐾_ℎ)^1/2,𝑦_𝐷𝑦=(𝐿𝐾)(^𝑎𝐾_ℎ)^1/2,𝑧_𝐷𝑧=(𝐿𝐾)(^𝑎𝐾_𝑣)^1/2,𝐿_𝐷𝐾=(^𝑎𝐾_𝑣)^1/2,𝐻_𝐷𝐻=(𝐿𝐾)(^𝑎𝐾_𝑣)^1/2𝐿_1𝐷,𝐿=(¹𝐿𝐾)(^𝑎𝐾_𝑣)^1/2,𝐿_2𝐷𝐿=(²𝐿𝐾)(^𝑎𝐾_𝑣)^1/2𝐿_𝑝𝐷,=𝐿_2𝐷−𝐿_1𝐷𝐿=(^𝑝𝐿𝐾)(^𝑎𝐾_𝑣)^1/2𝑅_𝑒𝐷,𝑅=(^𝑒𝐿𝐾)(^𝑎𝐾_ℎ)^1/2,𝑅_0𝐷𝑅=(⁰𝐿𝐾)(^𝑎𝐾_ℎ)^1/2. (3.8)(3.9)(3.10)(3.11)(3.12)

The dimensionless wellbore radius is [8] 𝑅_𝑤𝐷𝐾≈[(^ℎ𝐾_𝑣)^1/4𝐾+(^ℎ𝐾_𝑣)^−1/4𝑅](^𝑤2𝐿). (3.13)

Assume 𝑞 is the point convergence intensity at the point sink (𝑅₀,0,𝑧^′), the partially penetrating well is a uniform line sink, the total productivity of the well is 𝑄, and there holds 𝑄𝑞=𝐿_𝑝𝐷. (3.14)

Define the dimensionless pressures: 𝑃_𝐷=𝐾_𝑎𝐿(𝑃_𝑒−𝑃)𝜇𝑞𝐵,𝑃_𝑤𝐷=𝐾_𝑎𝐿(𝑃_𝑒−𝑃_𝑤).𝜇𝑞𝐵 (3.15)

Then (3.7) becomes [6, 7] 𝜕²𝑃_𝐷𝜕𝑥_𝐷²+𝜕²𝑃_𝐷𝜕𝑦_𝐷²+𝜕²𝑃_𝐷𝜕𝑧_𝐷²=−𝛿(𝑥_𝐷−𝑅_0𝐷)𝛿(𝑦_𝐷)𝛿(𝑧_𝐷−𝑧^′_𝐷),inΩ_𝐷, (3.16) where Ω_𝐷={(𝑥_𝐷,𝑦_𝐷,𝑧_𝐷)∣𝑥²_𝐷+𝑦²_𝐷<𝑅²_𝑒𝐷,0<𝑧_𝐷<𝐻_𝐷}. (3.17)

If point 𝐫₀ and point 𝐫 are with distances 𝜌₀ and 𝜌, respectively, from the axis of symmetry of the cylindrical body, then the dimensionless off-center distances are 𝜌_0𝐷𝜌=(⁰𝐿𝐾)(^𝑎𝐾_ℎ)^1/2,𝜌_𝐷𝜌=(𝐿𝐾)(^𝑎𝐾_ℎ)^1/2. (3.18)

There holds (𝜋𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷−√𝜌_0𝐷𝜌_𝐷𝐾)=(^𝑣𝐾_ℎ)^1/2(𝜋𝐿2𝐻)(4𝑅_𝑒𝐿−2𝜌₀𝐿−2𝜌𝐿−2√𝜌₀𝜌𝐿)𝐾=(^𝑣𝐾_ℎ)^1/2(𝜋𝑅_𝑒𝐻𝜌)(2−⁰𝑅_𝑒−𝜌𝑅_𝑒−√𝜌₀𝜌𝑅_𝑒)𝐾=(^𝑣𝐾_ℎ)^1/2(𝜋𝑅_𝑒𝐻)(2−𝜗₀−𝜗−𝜗₀𝜗), (3.19) where 𝜗₀=𝜌₀𝑅_𝑒𝜌,𝜗=𝑅_𝑒. (3.20)

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 assume that 𝜗₀≤0.6,𝜗≤0.6 (3.21)

If 𝜗₀𝐾=𝜗=0.6,^𝑣𝐾_ℎ𝑅=0.25,^𝑒𝐻=15, (3.22) then (𝐾_𝑣𝐾_ℎ)^1/2(𝜋𝑅_𝑒𝐻)(2−𝜗₀−𝜗−𝜗₀𝜗)=0.25^1/2×(𝜋×15)×(2.0−0.6−0.6−√0.6×0.6)=4.7124,exp(−4.7124)=8.983×10⁻³. (3.23)

Moreover if 𝜗₀𝐾=𝜗=0.5,^𝑣𝐾_ℎ𝑅=0.5,^𝑒𝐻=10, (3.24) then (𝐾_𝑣𝐾_ℎ)^1/2(𝜋𝑅_𝑒𝐻)(2−𝜗₀−𝜗−𝜗₀𝜗)=0.5^1/2×(𝜋×10)×(2.0−0.5−0.5−√0.5×0.5)=11.107,exp(−11.107)=1.501×10⁻⁵. (3.25)

Recall (3.19), there holds 𝜋exp[−(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷−√𝜌_0𝐷𝜌_𝐷𝐾)]=(^𝑣𝐾_ℎ)^1/2(𝜋𝑅_𝑒𝐻)(2−𝜗₀−𝜗−𝜗₀𝜗), (3.26) since there holds (3.21), and according to the aforementioned calculations in (3.22), (3.23), (3.24), and (3.25), we obtain 𝜋exp[−(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷−√𝜌_0𝐷𝜌_𝐷)]≈0. (3.27)

Because 𝜋0<(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷−√𝜌_0𝐷𝜌_𝐷𝜋)<(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷), (3.28) thus 𝜋exp[−(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷−√𝜌_0𝐷𝜌_𝐷𝜋)]>exp[−(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷)]. (3.29)

Combining (3.27) and (3.29), we have 𝜋exp[−(𝐻_𝐷)(2𝑅_𝑒𝐷−𝜌_0𝐷−𝜌_𝐷)]≈0. (3.30)

4. Boundary Conditions

In this paper, we always assume constant pressure lateral boundary: 𝑃(𝑥,𝑦,𝑧)=𝑃_𝑒=𝑃_𝑖, (4.1) on cylindrical lateral surface: Γ={(𝑥,𝑦,𝑧)∣𝑥²+𝑦²=𝑅²_𝑒,0<𝑧<𝐻}. (4.2)

Recall (3.15), the dimensionless form of constant pressure lateral boundary condition is 𝑃_𝐷(𝑥_𝐷,𝑦_𝐷,𝑧_𝐷)=0, (4.3) on Γ_𝐷={(𝑥_𝐷,𝑦_𝐷,𝑧_𝐷)∣𝑥²_𝐷+𝑦²_𝐷=𝑅²_𝑒𝐷,0<𝑧_𝐷<𝐻_𝐷}. (4.4)

Also we have the following dimensionless equations for top and bottom boundary conditions:

(i)If the circular cylinder drainage volume is with top and bottom impermeable boundaries, that is, the boundaries at 𝑧=0 and 𝑧=𝐻 are both impermeable (e.g., the reservoir does not have gas cap drive or bottom water drive), then 𝜕𝑃_𝐷𝜕𝑧_𝐷|_𝑧𝐷=0=0;𝜕𝑃_𝐷𝜕𝑧_𝐷|_{𝑧𝐷=𝐻𝐷}=0. (4.5)(ii)If the circular cylinder drainage volume is with impermeable boundary at 𝑧=𝐻, constant pressure boundary at 𝑧=0, (e.g., the reservoir has gas cap drive), then 𝑃_𝐷|_𝑧𝐷=0=0;𝜕𝑃_𝐷𝜕𝑧_𝐷|_{𝑧𝐷=𝐻𝐷}=0. (4.6)(iii)If the circular cylinder drainage volume is with impermeable boundary at 𝑧=0, constant pressure boundary at 𝑧=𝐻 (e.g., the reservoir has bottom water drive), then 𝑃_𝐷|_{𝑧𝐷=𝐻𝐷}=0;𝜕𝑃_𝐷𝜕𝑧_𝐷|_𝑧𝐷=0=0. (4.7)(iv)If the circular cylinder drainage volume is with top and bottom constant pressure boundaries, that is, the boundaries at 𝑧=0 and 𝑧=𝐻 are both constant pressure boundaries (e.g., the reservoir has both gas cap drive and bottom water drive), then 𝑃_𝐷|_𝑧𝐷=0=0;𝑃_𝐷|_{𝑧𝐷=𝐻𝐷}=0. (4.8)

5. Point Sink Solutions

For convenience, we use dimensionless variables given by (3.8) through (3.13), but we drop the subscript 𝐷. In order to obtain the dimensionless pressure of a point sink in a circular cylinder reservoir, we need to solve a dimensionless Laplace equation in dimensionless space: 𝜕²𝑃𝜕𝑥²+𝜕²𝑃𝜕𝑦²+𝜕²𝑃𝜕𝑧²=−𝛿(𝑥−𝑥^′)𝛿(𝑦)𝛿(𝑧−𝑧^′),inΩ, (5.1) where Ω={(𝑥,𝑦,𝑧)∣𝑥²+𝑦²<𝑅²_𝑒,0<𝑧<𝐻}. (5.2)

The following initial reservoir condition and lateral reservoir boundary condition will be used to obtain point sink pressure in a circular cylinder reservoir with constant pressure outer boundary: 𝑃(𝑡,𝑥,𝑦,𝑧)|_𝑡=0=0,inΩ,𝑃(𝑡,𝑥,𝑦,𝑧)=0,onΓ, (5.3) where Γ={(𝑥,𝑦,𝑧)∣𝑥²+𝑦²=𝑅²_𝑒,0<𝑧<𝐻}.

The problem under consideration is that of fluid flow toward a point sink from an off-center position within a circular of radius 𝑅_𝑒. 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 𝜃. The inverse point of the point sink 𝐫₀ with respect to the circle is point 𝐫_∗. Point 𝐫_∗ is with a distance 𝜌_∗ 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 that 𝜌_∗=𝑅²_𝑒𝜌₀. (5.4)

Figure 2 

Geometric representation of a circular system.

Assume 𝑅^′ is the distance between point 𝐫 and point 𝐫₀, then 𝑅^′=𝜌²+𝜌²₀−2𝜌𝜌₀cos𝜃. (5.5)

If the observation point 𝐫 is on the drainage circle, 𝜌=𝑅_𝑒, then 𝑅^′=𝑅²_𝑒+𝜌²₀−2𝑅_𝑒𝜌₀cos𝜃,𝑅_𝑒>𝜌₀>0. (5.6)

If the observation point 𝐫 is on the wellbore, then 𝑅^′=𝑅_𝑤. (5.7)

Define 𝜆_𝑛=𝑛𝜋𝐻. (5.8)

5.1. Impermeable Upper and Lower Boundaries

If upper and lower boundaries are impermeable, 𝜕𝑃|𝜕𝑧_𝑧=0=0,𝜕𝑃|𝜕𝑧_𝑧=𝐻=0, (5.9) obviously for such impermeable boundary conditions, we have 𝛿(𝑧−𝑧^′∑)=^∞𝑛=0cos(𝜆_𝑛𝑧^′)cos(𝜆_𝑛𝑧)(𝐻𝑑_𝑛), (5.10) where 𝑑_𝑛=11,if𝑛=0,2,if𝑛>0. (5.11) Let 𝑃(𝑥,𝑦,𝑧)=^∞_𝑛=0𝜑_𝑛(𝑥,𝑦)cos(𝜆_𝑛𝑧), (5.12) and substituting (5.12) into (5.1) and compare the coefficients of cos(𝜆_𝑛𝑧), we obtain Δ𝜑_𝑛−𝜆²_𝑛𝜑_𝑛=−cos(𝜆_𝑛𝑧^′)𝛿(𝑥−𝑥^′)𝛿(𝑦)(𝐻𝑑_𝑛), (5.13) in circular area Ω₁={(𝑥,𝑦)∣𝑥²+𝑦²<𝑅²_𝑒}, and 𝜑_𝑛=0, (5.14) on circumference Γ₁={(𝑥,𝑦)∣𝑥²+𝑦²=𝑅²_𝑒}, and Δ is two-dimensional Laplace operator, Δ𝜑_𝑛=𝜕²𝜑_𝑛𝜕𝑥²+𝜕²𝜑_𝑛𝜕𝑦². (5.15)

Case 1. If 𝑛=0, then Δ𝜑₀=−𝛿(𝑥−𝑥^′)𝛿(𝑦)𝐻,inΩ₁,𝜑₀=0,onΓ₁. (5.16)
Using Green's function of Laplace problem in a circular domain, we obtain [9–11] 𝜑₀(𝑥,𝑦;𝑥^′1,0)=(2𝜋𝐻){ln[|𝜌−𝜌_∗||𝜌−𝜌₀||𝜌₀|𝑅_𝑒]}. (5.17)

Case 2. If 𝑛>0, then 𝜑_𝑛 satisfies (5.13). Since [−1/(2𝜋)]𝐾₀(𝜆_𝑛𝑅^′) satisfies the equations: Δ𝑢−𝜆²_𝑛𝑢=𝛿(𝑥−𝑥^′)𝛿(𝑦),𝑢=0,onΓ₁. (5.18)
So [−𝛼_𝑛/(2𝜋)]𝐾₀(𝜆_𝑛𝑅^′) is a basic solution of (5.13), where 𝛼_𝑛1=(−𝐻𝑑_𝑛)cos(𝑛𝜋𝑧^′𝐻2)=(−𝐻)cos(𝜆_𝑛𝑧^′𝛽),_𝑛=𝛼_𝑛(2𝜋)=(−22𝜋𝐻)cos(𝑛𝜋𝑧^′𝐻)=(−1𝜋𝐻)cos(𝜆_𝑛𝑧^′). (5.19)(5.20)
Define 𝜓_𝑛=𝜑_𝑛+𝛽_𝑛𝐾₀(𝜆_𝑛𝑅^′), (5.21) thus 𝜑_𝑛=𝜓_𝑛−𝛽_𝑛𝐾₀(𝜆_𝑛𝑅^′), (5.22) then 𝜓_𝑛 satisfies homogeneous equation Δ𝜓_𝑛−𝜆²_𝑛𝜓_𝑛=0,inΩ₁,𝜓_𝑛=𝛽_𝑛𝐾₀(𝜆_𝑛𝑅^′),onΓ₁, (5.23) and 𝑅^′ has the same meaning as in (5.6).
Under polar coordinates representation of Laplace operator and by using methods of separation of variables, we obtain a general solution [9–11]: 𝜓_𝑛=[𝐴_0𝑛𝐼₀(𝜆_𝑛𝜌)+𝐵_0𝑛𝐾₀(𝜆_𝑛𝜌)][𝑎_0𝑛𝜃+𝑏_0𝑛]+^∞_𝑚=1[𝐴_𝑚𝑛𝐼_𝑚(𝜆_𝑛𝜌)+𝐵_𝑚𝑛𝐾_𝑚(𝜆_𝑛𝜌)]×[𝑎_𝑚𝑛cos(𝑚𝜃)+𝑏_𝑚𝑛sin(𝑚𝜃)], (5.24) where 𝐴_𝑖𝑛,𝐵_𝑖𝑛,𝑎_𝑖𝑛,𝑏_𝑖𝑛,𝑖=0,1,… are undetermined coefficients.
Because 𝜓_𝑛 is continuously bounded within Ω₁, but 𝐾_𝑖(0)=∞, so there holds 𝐵_𝑖𝑛=0,𝑖=0,1,…. (5.25)
There hold [7, 12] 𝐾_𝜐(𝑧)=(𝜋𝑖2)𝑒^{𝜐𝜋𝑖/2}𝐻_𝜐⁽¹⁾𝐼(𝑧𝑖),_𝜐(𝑧)=𝑒^{−𝜐𝜋𝑖/2}𝐽_𝜐(𝑧𝑖), (5.26) where 𝐾_𝜐(𝑧) is modified Bessel function of second kind and order 𝜐,𝐼_𝜐(𝑧) is modified Bessel function of first kind and order 𝜐,𝐽_𝜐(𝑧) is Bessel function of first kind and order 𝜐,𝐻_𝜐⁽¹⁾(𝑧) is Hankel function of first kind and order 𝜐, and 𝑖=√−1.

Also there hold [13, page 979] 𝐻₀⁽¹⁾(𝜎𝑅^′)=𝐽₀(𝜎𝜌₀)𝐻₀⁽¹⁾(𝜎𝑅_𝑒)+2^∞_𝑚=1𝐽_𝑚(𝜎𝜌₀)𝐻_𝑚⁽¹⁾(𝜎𝑅_𝑒𝐾)cos(𝑚𝜃),₀(𝜆_𝑛𝑅^′)=(𝜋𝑖2)𝐻₀⁽¹⁾(𝑖𝜆_𝑛𝑅^′). (5.27)(5.28)

Let 𝜎=𝑖𝜆_𝑛, (note that 𝑖²=−1) putting (5.26) into (5.27), and using (5.28), we have the following Cosine Fourier expansions of 𝐾₀(𝜆_𝑛𝑅^′) [13, page 952]: 𝐾₀(𝜆_𝑛𝑅^′)=(𝜋𝑖2)[𝐽₀(𝑖𝜆_𝑛𝜌₀)𝐻₀⁽¹⁾(𝑖𝜆_𝑛𝑅_𝑒)+2^∞_𝑚=1𝐽_𝑚(𝑖𝜆_𝑛𝜌₀)𝐻_𝑚⁽¹⁾(𝑖𝜆_𝑛𝑅_𝑒)cos(𝑚𝜃)]=𝐽₀(𝑖𝜆_𝑛𝜌₀)𝐾₀(𝜆_𝑛𝑅_𝑒)+2^∞_𝑚=1𝑒^{−𝑚𝜋𝑖/2}𝐽_𝑚(𝑖𝜆_𝑛𝜌₀)𝐾_𝑚(𝜆_𝑛𝑅_𝑒)cos(𝑚𝜃)=𝐼₀(𝜆_𝑛𝜌₀)𝐾₀(𝜆_𝑛𝑅_𝑒)+2^∞_𝑚=1𝐼_𝑚(𝜆_𝑛𝜌₀)𝐾_𝑚(𝜆_𝑛𝑅_𝑒)cos(𝑚𝜃). (5.29)

Note that 𝜓_𝑛=𝛽_𝑛𝐾₀(𝜆_𝑛𝑅^′) on Γ₁, comparing coefficients of Cosine Fourier expansions of 𝐾₀(𝜆_𝑛𝑅^′) in (5.29) and (5.24), we obtain 𝑎_0𝑛=0,𝑏_0𝑛=1,𝑏_𝑖𝑛=0,𝑖=1,2,…. (5.30)

Define 𝑌_𝑚𝑛=𝑎_𝑚𝑛𝐴_𝑚𝑛,𝑛=0,1,2,…, (5.31) and recall (5.24), then we have 𝜓_𝑛=^∞_𝑚=0𝑌_𝑚𝑛𝐼_𝑚(𝜆_𝑛𝜌)cos(𝑚𝜃),𝑛=0,1,2,…, (5.32) where 𝑌_0𝑛=𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝜌₀)𝐼₀(𝜆_𝑛𝑅_𝑒),𝑌_𝑚𝑛=2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝑅_𝑒). (5.33)

There hold [13, page 919] 𝐼_𝑚(𝑥)≃exp(𝑥)(2𝜋𝑥)^1/2,𝐾_𝑚(𝑥)≃[𝜋/(2𝑥)]^1/2exp(𝑥),𝑥≫1,∀𝑚≥0. (5.34)

Note that 𝐻 in Formula (5.8) is in dimensionless form, recall Formulas (3.9), (3.12) and (3.18), for dimensionless 𝐻,𝑅_𝑒,𝜌₀,𝜌, there hold 𝜆_𝑛𝑅_𝑒≫1,𝜆_𝑛𝜌₀≫1,𝜆_𝑛𝜌≫1, (5.35) thus 𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)≈𝜋exp(−2𝜆_𝑛𝑅_𝑒𝐼),_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)≈[exp(𝜆_𝑛𝜌₀)(2𝜋𝜆_𝑛𝜌₀)^1/2,exp(𝜆_𝑛𝜌)(2𝜋𝜆_𝑛𝜌)^1/2]=exp[𝜆_𝑛(𝜌+𝜌₀)](2𝜋𝜆_𝑛)(𝜌𝜌₀)^1/2,𝑌_𝑚𝑛𝐼_𝑚(𝜆_𝑛𝜌)=2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)≈(2𝛽_𝑛)[𝜋exp(−2𝜆_𝑛𝑅_𝑒)]{exp[𝜆_𝑛(𝜌+𝜌₀)](2𝜋𝜆_𝑛)(𝜌𝜌₀)^1/2}=(2𝛽_𝑛𝜋)[(2𝜋𝜆_𝑛)(𝜌𝜌₀)^1/2]exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀𝛽−𝜌)]=[^𝑛𝜆_𝑛(𝜌𝜌₀)^1/2]exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]. (5.36)(5.37)(5.38) There holds |𝜓_𝑛|=^∞_𝑚=0|𝑌_𝑚𝑛𝐼_𝑚(𝜆_𝑛<𝜌)cos(𝑚𝜃)|_∞𝑚=0|𝑌_𝑚𝑛𝐼_𝑚(𝜆_𝑛=𝜌)|_∞𝑚=0|2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)|. (5.39) Combining Formulas (3.18), (5.38), and (5.39), we obtain ^∞_𝑛=1|𝜓_𝑛cos(𝑛𝜋𝑧𝐻≤)|_∞𝑛=1|𝜓_𝑛=_{∞∞𝑛=1𝑚=0}||2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)≤_∞𝑛=1|[|2𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝜌₀)𝐼₀(𝜆_𝑛𝜌)𝐼₀(𝜆_𝑛𝑅_𝑒)|+^∞_𝑚=1|2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)=|]_∞𝑛=1|2𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝜌₀)𝐼₀(𝜆_𝑛𝜌)𝐼₀(𝜆_𝑛𝑅_𝑒)|+^∞∞_𝑛=1𝑚=1|2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)|=Ξ₁+Ξ₂, (5.40)(1)(2)(3)(4)(5) where Ξ₁=^∞_𝑛=1|2𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝜌₀)𝐼₀(𝜆_𝑛𝜌)𝐼₀(𝜆_𝑛𝑅_𝑒)Ξ|,₂=^∞∞_𝑛=1𝑚=1|2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)|. (5.41)(5.42)

There holds Ξ₁=^∞_𝑛=1|2𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝜌₀)𝐼₀(𝜆_𝑛𝜌)𝐼₀(𝜆_𝑛𝑅_𝑒)|≈^∞_𝑛=1[|𝛽_𝑛|𝜆_𝑛(𝜌𝜌₀)^1/2]exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀=−𝜌)]_∞𝑛=1[1𝑛𝜋²(𝜌𝜌₀)^1/2]exp[−(𝑛𝜋𝐻)(2𝑅_𝑒−𝜌₀<−𝜌)]_∞𝑛=1[1𝜋²(𝜌𝜌₀)^1/2]exp[−(𝑛𝜋𝐻)(2𝑅_𝑒−𝜌₀1−𝜌)]≈[𝜋²(𝜌𝜌₀)^1/2]{exp[−(𝜋/𝐻)(2𝑅_𝑒−𝜌₀−𝜌)]1−exp[−(𝜋/𝐻)(2𝑅_𝑒−𝜌₀}−𝜌)]≈0, (5.43) where we use Formula (3.26), 𝜋exp[−(𝐻)(2𝑅_𝑒−𝜌₀−𝜌)]≈0,𝑥+𝑥²+𝑥³+𝑥⁴+𝑥⁵𝑥+⋯=1−𝑥,0<𝑥<1. (5.44)(5.45)

If 𝑚>−1/2, there holds [13, page 916] 𝐼_𝑚(𝑧)=[(𝑧/2)^𝑚Γ(𝑚+1/2)Γ(1/2)]¹₋₁(1−𝑡²)^𝑚−1/2cosh(𝑧𝑡)𝑑𝑡, (5.46) thus for 𝑚≥1, 𝐼_𝑚(𝜆_𝑛𝜌)≤[(𝜆_𝑛𝜌/2)^𝑚]Γ(𝑚+1/2)Γ(1/2)¹₋₁cosh(𝜆_𝑛𝜌𝑡)𝑑𝑡=[2(𝜆_𝑛𝜌/2)^𝑚(𝜆_𝑛𝜌)Γ(𝑚+1/2)Γ(1/2)]sinh(𝜆_𝑛𝜌)=[(𝜆_𝑛𝜌/2)^𝑚−1Γ(𝑚+1/2)Γ(1/2)]sinh(𝜆_𝑛𝜌)<[(𝜆_𝑛𝜌/2)^𝑚−12Γ(𝑚+1/2)Γ(1/2)]exp(𝜆_𝑛𝜌), (5.47) where we use ¹₋₁cosh(𝜆_𝑛𝜌𝑡)𝑑𝑡=2sinh(𝜆_𝑛𝜌)𝜆_𝑛𝜌,sinh(𝜆_𝑛𝜌)<exp(𝜆_𝑛𝜌)2, (5.48) and if −1<𝑡<1,𝑚≥1, then (1−𝑡²)^𝑚−1/2≤1. (5.49)

Putting Formula (5.47) into Formula (5.42), we obtain Ξ₂=^∞∞_𝑛=1𝑚=1|2𝛽_𝑛𝐾_𝑚(𝜆_𝑛𝑅_𝑒)𝐼_𝑚(𝜆_𝑛𝜌₀)𝐼_𝑚(𝜆_𝑛𝜌)𝐼_𝑚(𝜆_𝑛𝑅_𝑒)|<^∞∞_𝑛=1𝑚=1(2𝜋|𝛽_𝑛|)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)][(𝜆_𝑛𝜌/2)^𝑚−12Γ(𝑚+1/2)Γ(1/2)][(𝜆_𝑛𝜌₀/2)^𝑚−1]=2Γ(𝑚+1/2)Γ(1/2)_{∞∞𝑛=1𝑚=1}(2𝐻)(𝜆²_𝑛𝜌𝜌₀/4)^𝑚−1[2Γ(𝑚+1/2)Γ(1/2)]²exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀=−𝜌)]_∞𝑛=1(2𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]^∞_𝑚=1(𝜆²_𝑛𝜌𝜌₀/4)^𝑚−1[2Γ(𝑚+1/2)Γ(1/2)]². (5.50)

Note that √Γ(𝑚+1/2)=1×3×5×⋯×(2𝑚−1)𝜋2^𝑚>1×2×6×⋯×(2𝑚−2)√𝜋2^𝑚=2^𝑚−1(𝑚−1)!√𝜋2^𝑚=(𝑚−1)!√𝜋2, (5.51) then we obtain Ξ₂<^∞_𝑛=1(2𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]^∞_𝑚=1(𝜆²_𝑛𝜌𝜌₀/4)^𝑚−1[2Γ(𝑚+1/2)Γ(1/2)]²<^∞_𝑛=1(2𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]^∞_𝑚=1(𝜆²_𝑛𝜌𝜌₀/4)^𝑚−1[(𝑚−1)!𝜋]²=^∞_𝑛=1(2𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]^∞_𝑘=0(𝜆_𝑛√𝜌𝜌₀/2)^2𝑘(𝑘!𝜋)²=^∞_𝑛=1(2𝜋²𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]𝐼₀(𝜆_𝑛√𝜌𝜌₀), (5.52) where we use [13, page 919] 𝐼₀(𝑧)=^∞_𝑘=0(𝑧/2)^2𝑘(𝑘!)²1,Γ(2)=√𝜋. (5.53)

Note that 𝜆_𝑛√𝜌𝜌₀≫1, and we have 𝐼₀(𝜆_𝑛√𝜌𝜌₀)≈exp(𝜆_𝑛√𝜌𝜌₀)(2𝜋𝜆_𝑛√𝜌𝜌₀)^1/2, (5.54) thus Formula (5.52) can be simplified as follows: Ξ₂<^∞_𝑛=1(2𝜋²𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)]𝐼₀(𝜆_𝑛√𝜌𝜌₀)≈^∞_𝑛=1(2𝜋²𝐻)exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌)][exp(𝜆_𝑛√𝜌𝜌₀)(2𝜋𝜆_𝑛√𝜌𝜌₀)^1/2]<^∞_𝑛=1[2𝜋²𝐻(2𝜋)^1/2(𝜌𝜌₀)^1/4]exp[−𝜆_𝑛(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀)]𝜆_𝑛^1/2=^∞_𝑛=1[2𝜋³(2𝑛𝐻)^1/2(𝜌𝜌₀)^1/4]exp[−(𝑛𝜋𝐻)(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀<)]_∞𝑛=1[2𝜋³(2𝐻)^1/2(𝜌𝜌₀)^1/4]exp[−(𝑛𝜋𝐻)(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀2)]=[𝜋³(2𝐻)^1/2(𝜌𝜌₀)^1/4]{exp[−(𝜋/𝐻)(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀)]1−exp[−(𝜋/𝐻)(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀})]≈0, (5.55) where we use Formulas (5.45) and (3.26) 𝜋exp[−(𝐻)(2𝑅_𝑒−𝜌₀−𝜌−√𝜌𝜌₀)]≈0. (5.56)

Combining Formulas (5.40), (5.43), and (5.55), we prove ^∞_𝑛=1𝜓_𝑛cos(𝑛𝜋𝑧𝐻)≈0. (5.57)

Combining Formulas (5.4), (5.12), (5.17), (5.20), (5.22), (5.29), and (5.57), the point convergence pressure of point (𝑥^′,0,𝑧^′) is 𝑃(𝑥,𝑦,𝑧;𝑥^′,0,𝑧^′1)=(2𝜋𝐻){ln[|𝑅²_𝑒−𝜌𝜌₀||𝜌−𝜌₀|𝑅_𝑒+]}_∞𝑛=1[𝛽_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅^′)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝛽_𝑛𝐾₀(𝜆_𝑛𝑅^′)]cos(𝜆_𝑛𝑧)=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅^′)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝐾₀(𝜆_𝑛𝑅^′)]cos(𝜆_𝑛𝑧)cos(𝜆_𝑛𝑧^′)1+(2𝜋𝐻){ln[|𝑅²_𝑒−𝜌𝜌₀||𝜌−𝜌₀|𝑅_𝑒]}. (5.58)

5.2. Constant Pressure Upper or Lower Boundaries

If the reservoir is with gas cap and impermeable bottom boundary, then 𝑃|_𝑧=0=0,𝜕𝑃|𝜕𝑧_𝑧=𝐻=0, (5.59) and assume the outer boundary is at constant pressure 𝑃=0, (5.60) on cylindrical surface Γ={(𝑥,𝑦)∣𝑥²+𝑦²=𝑅²_𝑒,0<𝑧<𝐻}.

Define 𝜔_𝑛=[(2𝑛−1)𝜋],𝑔(2𝐻)_𝑛(𝑧)=2𝐻sin(𝜔_𝑛𝑧),(𝑛=1,2,3,…), (5.61) then under the boundary condition of (5.59), we have 𝛿(𝑧−𝑧^′)=^∞_𝑛=1𝑔_𝑛(𝑧)𝑔_𝑛(𝑧^′). (5.62)

Let 𝑃(𝑥,𝑦,𝑧)=^∞_𝑛=1𝜑_𝑛(𝑥,𝑦)sin(𝜔_𝑛𝑧), (5.63) where 𝜑_𝑛 satisfies Δ𝜑_𝑛−𝜔²_𝑛𝜑_𝑛2=(−𝐻)𝛿(𝑥−𝑥^′)𝛿(𝑦)sin(𝜔_𝑛𝑧^′), (5.64) in Ω₁, and 𝜑_𝑛=0, (5.65) on Γ₁.

Let 𝜓_𝑛=𝜑_𝑛+𝜁_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.66) where 𝜁_𝑛=(−1𝜋𝐻)sin(𝜔_𝑛𝑧^′), (5.67) and 𝑅^′ has the same meaning as in Formula (5.6).

Thus 𝜓_𝑛 satisfies homogeneous equation: Δ𝜓_𝑛−𝜔²_𝑛𝜓_𝑛=0, (5.68) in Ω₁, and 𝜓_𝑛=𝜁_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.69) on Γ₁.

Using polar coordinates, we have 𝜓_𝑛=𝜁_𝑛𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒), (5.70) then 𝜑_𝑛=𝜓_𝑛−𝜁_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.71) and point convergence pressure of point (𝑥^′,0,𝑧^′) is: 𝑃(𝑥,𝑦,𝑧;𝑥^′,0,𝑧^′)=^∞_𝑛=1[𝜁_𝑛𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝜁_𝑛𝐾₀(𝜔_𝑛𝑅^′)]sin(𝜔_𝑛𝑧)=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝐾₀(𝜔_𝑛𝑅^′)]sin(𝜔_𝑛𝑧)sin(𝜔_𝑛𝑧^′). (5.72)

If the reservoir is with bottom water and impermeable top boundary, then 𝑃|_𝑧=𝐻=0,𝜕𝑃|𝜕𝑧_𝑧=0=0, (5.73) and recall Formula (5.60), the outer boundary is at constant pressure.

Define ℎ_𝑛(𝑧)=2𝐻cos(𝜔_𝑛𝑧),(𝑛=1,2,3,…), (5.74) then under the boundary condition of (5.73), we have 𝛿(𝑧−𝑧^′)=^∞_𝑛=1ℎ_𝑛(𝑧)ℎ_𝑛(𝑧^′). (5.75)

Let 𝑃(𝑥,𝑦,𝑧)=^∞_𝑛=1𝜑_𝑛(𝑥,𝑦)cos(𝜔_𝑛𝑧), (5.76) where 𝜑_𝑛 satisfies Δ𝜑_𝑛−𝜔²_𝑛𝜑_𝑛2=(−𝐻)𝛿(𝑥−𝑥^′)𝛿(𝑦)cos(𝜔_𝑛𝑧^′), (5.77) in Ω₁, and 𝜑_𝑛=0, (5.78) on Γ₁.

Let 𝜓_𝑛=𝜑_𝑛+𝜂_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.79) where 𝜂_𝑛=(−1𝜋𝐻)cos(𝜔_𝑛𝑧^′), (5.80) thus 𝜓_𝑛 satisfies Δ𝜓_𝑛−𝜔²_𝑛𝜓_𝑛=0, (5.81) in Ω₁, and 𝜓_𝑛=𝜂_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.82) on Γ₁.

Using polar coordinates, we have 𝜓_𝑛=𝜂_𝑛𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒), (5.83) then 𝜑_𝑛=𝜓_𝑛−𝜂_𝑛𝐾₀(𝜔_𝑛𝑅^′), (5.84) and point convergence pressure of point (𝑥^′,0,𝑧^′) is 𝑃(𝑥,𝑦,𝑧;𝑥^′,0,𝑧^′)=^∞_𝑛=1[𝜂_𝑛𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝜂_𝑛𝐾₀(𝜔_𝑛𝑅^′)]cos(𝜔_𝑛𝑧)=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅^′)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝐾₀(𝜔_𝑛𝑅^′)]cos(𝜔_𝑛𝑧)cos(𝜔_𝑛𝑧^′). (5.85)

If the reservoir is with gas cap and bottom water, then 𝑃|_𝑧=0=0,𝑃|_𝑧=𝐻=0, (5.86) and recall Formula (5.60), the outer boundary is at constant pressure.

Define 𝑓_𝑛(𝑧)=2𝐻sin(𝜆_𝑛𝑧),(𝑛=1,2,3,…), (5.87) where 𝜆_𝑛 has the same meaning as in Formula (5.8).

Under the boundary condition of (5.86), we have 𝛿(𝑧−𝑧^′)=^∞_𝑛=1𝑓_𝑛(𝑧)𝑓_𝑛(𝑧^′). (5.88)

Let 𝑃(𝑥,𝑦,𝑧)=^∞_𝑛=1𝜑_𝑛(𝑥,𝑦)sin(𝜆_𝑛𝑧), (5.89) where 𝜑_𝑛 satisfies Δ𝜑_𝑛−𝜆²_𝑛𝜑_𝑛2=(−𝐻)𝛿(𝑥−𝑥^′)𝛿(𝑦)sin(𝜆_𝑛𝑧^′), (5.90) in Ω₁, and 𝜑_𝑛=0, (5.91) on Γ₁.

Let 𝜓_𝑛=𝜑_𝑛+𝜄_𝑛𝐾₀(𝜆_𝑛𝑅^′), (5.92) where 𝜄_𝑛=(−1𝜋𝐻)sin(𝜆_𝑛𝑧^′), (5.93) thus 𝜓_𝑛 satisfies Δ𝜓_𝑛−𝜆²_𝑛𝜓_𝑛=0, (5.94) in Ω₁, and 𝜓_𝑛=𝜄_𝑛𝐾₀(𝜆_𝑛𝑅^′), (5.95) on Γ₁.

Using polar coordinates, we have 𝜓_𝑛=𝜄_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅^′)𝐼₀(𝜆_𝑛𝑅_𝑒), (5.96) then 𝜑_𝑛=𝜓_𝑛−𝜄_𝑛𝐾₀(𝜆_𝑛𝑅^′), (5.97) and point convergence pressure of point (𝑥^′,0,𝑧^′) is 𝑃(𝑥,𝑦,𝑧;𝑥^′,0,𝑧^′)=^∞_𝑛=1[𝜄_𝑛𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅^′)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝜄_𝑛𝐾₀(𝜆_𝑛𝑅^′)]sin(𝜆_𝑛𝑧)=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅^′)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝐾₀(𝜆_𝑛𝑅^′)]sin(𝜆_𝑛𝑧)sin(𝜆_𝑛𝑧^′). (5.98)

6. Uniform Line Sink Solutions

For convenience, we drop the subscript 𝐷 for dimensionless variables. In order to calculate the pressure at wellbore, assume the observation point 𝐫 is on the wellbore, and recall Formula (5.7), 𝑅^′=𝑅_𝑤 and |𝜌−𝜌₀|=𝑅_𝑤,𝜌𝜌₀=(𝜌₀+𝑅_𝑤)𝜌₀≈𝜌²₀=𝑅²₀. (6.1)

The partially penetrating vertical well is taken as a uniform line sink. Recall Formula (5.9), integrate 𝑧^′ at both sides of Formula (5.58) from 𝐿₁ to 𝐿₂, if the upper and lower boundaries are impermeable, pressure at wellbore point (𝑅_𝑤,𝑧) is 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′𝐿=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)+(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝐾₀(𝜆_𝑛𝑅_𝑤)]×cos(𝑛𝜋𝑧𝐻)^𝐿2_𝐿1cos(𝑛𝜋𝑧^′𝐻)𝑑𝑧^′. (6.2)

Recall Formula (5.34), there hold 𝐾₀(𝑥)≈𝜋𝑒2𝑥^−𝑥,𝐾₀(𝑥)𝐼₀(𝑥)≈𝜋𝑒^−2𝑥, (6.3) if 𝑥>1, and 𝐼₀(𝑥)≈1.0, (6.4) if 0<𝑥<0.5.

Combining to Formulas (3.9), (3.12), (3.13), and (5.8), we obtain 𝜆_𝑛𝑅_𝑒>1,𝜆_𝑛𝑅_𝑤<0.5, (6.5) so 𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑒)≈𝜋𝑒^{−2𝜆𝑛𝑅𝑒},𝐼₀(𝜆_𝑛𝑅_𝑤)≈1.0. (6.6)

According to Formula (6.6), we have |(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)]cos(𝑛𝜋𝑧𝐻)^𝐿2_𝐿1cos(𝑛𝜋𝑧^′𝐻)𝑑𝑧^′|=|(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)𝐻](𝑛𝜋)cos(𝑛𝜋𝑧𝐻)[sin(𝑛𝜋𝐿₂𝐻)−sin(𝑛𝜋𝐿₁𝐻)]|≤|(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)𝐻](1𝑛𝜋)|=(𝜋²)^∞_𝑛=11𝑛|𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)|1≤(𝜋²)^∞_𝑛=1|𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)|1≈(𝜋²)^∞_𝑛=1𝜋𝑒^{−2𝜆𝑛𝑅𝑒}=𝑒^{−2𝜋𝑅𝑒/𝐻}𝜋(1−𝑒^{−2𝜋𝑅𝑒/𝐻})=1𝜋(𝑒^{2𝜋𝑅𝑒/𝐻}−1)≈0, (6.7) because the value of 𝑅_𝑒/𝐻 is very big.

Combining Formulas (6.2) and (6.7) yields 𝑃(𝑅_𝑤𝐿,𝑧)=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)1+()𝜋𝐻^∞_𝑛=1𝐾₀(𝜆_𝑛𝑅_𝑤)cos(𝑛𝜋𝑧𝐻)^𝐿2_𝐿1cos(𝑛𝜋𝑧^′𝐻)𝑑𝑧^′𝐿=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)+^∞_𝑛=1(1𝑛𝜋²)𝐾₀(𝜆_𝑛𝑅_𝑤)cos(𝑛𝜋𝑧𝐻)[sin(𝑛𝜋𝐿₂𝐻)−sin(𝑛𝜋𝐿₁𝐻)], (6.8) where we use ^𝐿2_𝐿1cos(𝑛𝜋𝑧^′𝐻)𝑑𝑧^′𝐻=(𝑛𝜋)[sin(𝑛𝜋𝐿₂𝐻)−sin(𝑛𝜋𝐿₁𝐻)]. (6.9)

In order to obtain the average wellbore pressure, integrate both sides of Formula (6.2) with respect to 𝑧 from 𝐿₁ to 𝐿₂, then divided by 𝐿_𝑝, we obtain 𝑃_𝑎(𝑅_𝑤1)=𝐿_𝑝^𝐿2_𝐿1𝑃(𝑅_𝑤𝐿,𝑧)𝑑𝑧=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)+^∞_𝑛=1(1𝑛𝜋²)𝐾₀(𝜆_𝑛𝑅_𝑤)[sin(𝑛𝜋𝐿₂𝐻)−sin(𝑛𝜋𝐿₁𝐻1)][𝐿_𝑝^𝐿2_𝐿1cos(𝑛𝜋𝑧𝐻𝐿)𝑑𝑧]=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)𝐻+(𝜋³𝐿_𝑝)^∞_𝑛=1(1𝑛²)𝐾₀(𝜆_𝑛𝑅_𝑤)[sin(𝑛𝜋𝐿₂𝐻)−sin(𝑛𝜋𝐿₁𝐻)]²𝐿=(^𝑝𝑅2𝜋𝐻)ln(^2𝑒−𝑅²₀𝑅_𝑒𝑅_𝑤)+(4𝐻𝜋³𝐿_𝑝)^∞_𝑛=1(1𝑛²)𝐾₀(𝜆_𝑛𝑅_𝑤)sin²(𝑛𝜋𝐿_𝑝2𝐻)cos²[𝑛𝜋(𝐿₂+𝐿₁)2𝐻]. (6.10)

Recall Formula (5.59), let 𝑅^′=𝑅_𝑤, integrate 𝑧^′ at both sides of Formula (5.72) from 𝐿₁ to 𝐿₂, if the reservoir is with gas cap and impermeable bottom boundary, pressure at wellbore point (𝑅_𝑤,𝑧) is 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅_𝑤)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝐾₀(𝜔_𝑛𝑅_𝑤)]Ξ₁, (6.11) where Ξ₁=sin(𝜔_𝑛𝑧)^𝐿2_𝐿1sin(𝜔_𝑛𝑧^′)𝑑𝑧^′=sin[(2𝑛−1)𝜋𝑧2𝐻][(−2𝐻)(2𝑛−1)𝜋]{cos[(2𝑛−1)𝜋𝐿₂2𝐻]−cos[(2𝑛−1)𝜋𝐿₁2𝐻]}. (6.12)

With a similar procedure in Formula (6.7), there holds |(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅_𝑤)𝐼₀(𝜔_𝑛𝑅_𝑒)]sin[(2𝑛−1)𝜋𝑧2𝐻][(−2𝐻)(2𝑛−1)𝜋]|×|{cos[(2𝑛−1)𝜋𝐿₂2𝐻]−cos[(2𝑛−1)𝜋𝐿₁2𝐻]}|≈0, (6.13) thus 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′1=()𝜋𝐻^∞_𝑛=1𝐾₀(𝜔_𝑛𝑅_𝑤)sin[(2𝑛−1)𝜋𝑧2𝐻][−2𝐻](2𝑛−1)𝜋×{cos[(2𝑛−1)𝜋𝐿₂2𝐻]−cos[(2𝑛−1)𝜋𝐿₁=2𝐻]}_∞𝑛=1[2(2𝑛−1)𝜋²]𝐾₀(𝜔_𝑛𝑅_𝑤)sin[(2𝑛−1)𝜋𝑧]2𝐻×{cos[(2𝑛−1)𝜋𝐿₁2𝐻]−cos[(2𝑛−1)𝜋𝐿₂2𝐻]}. (6.14)

In order to obtain the average wellbore pressure, integrate both sides of Formula (6.14) with respect to 𝑧 from 𝐿₁ to 𝐿₂, then divided by 𝐿_𝑝, we obtain 𝑃_𝑎(𝑅_𝑤1)=𝐿_𝑝^𝐿2_𝐿1𝑃(𝑅_𝑤=1,𝑧)𝑑𝑧𝐿_𝑝^∞_𝑛=1[2(2𝑛−1)𝜋²]𝐾₀(𝜔_𝑛𝑅_𝑤){cos[(2𝑛−1)𝜋𝐿₁2𝐻]−cos[(2𝑛−1)𝜋𝐿₂2𝐻]}×^𝐿2_𝐿1sin[(2𝑛−1)𝜋𝑧2𝐻]𝑑𝑧=(4𝐻𝜋³𝐿_𝑝)^∞_𝑛=11(2𝑛−1)²𝐾₀(𝜔_𝑛𝑅_𝑤){cos[(2𝑛−1)𝜋𝐿₂2𝐻]−cos[(2𝑛−1)𝜋𝐿₁2𝐻]}²=(16𝐻𝜋³𝐿_𝑝)^∞_𝑛=11(2𝑛−1)²𝐾₀(𝜔_𝑛𝑅_𝑤)sin²[(2𝑛−1)𝜋𝐿_𝑝4𝐻]sin²[(2𝑛−1)𝜋(𝐿₂+𝐿₁)4𝐻]. (6.15)

Recall Formula (5.73), let 𝑅^′=𝑅_𝑤, integrate 𝑧^′ at both sides of Formula (5.85) from 𝐿₁ to 𝐿₂, if the reservoir is with bottom water and impermeable top boundary, pressure at wellbore point (𝑅_𝑤,𝑧) is 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅_𝑤)𝐼₀(𝜔_𝑛𝑅_𝑒)−𝐾₀(𝜔_𝑛𝑅_𝑤)]Ξ₂, (6.16) where Ξ₂=cos(𝜔_𝑛𝑧)^𝐿2_𝐿1cos(𝜔_𝑛𝑧^′)𝑑𝑧^′=cos[(2𝑛−1)𝜋𝑧2𝐻][(2𝐻)(2𝑛−1)𝜋]{sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁2𝐻]}. (6.17)

With a similar procedure in Formula (6.7), there hold |(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜔_𝑛𝑅_𝑒)𝐼₀(𝜔_𝑛𝑅_𝑤)𝐼₀(𝜔_𝑛𝑅_𝑒)]cos[(2𝑛−1)𝜋𝑧2𝐻][2𝐻(2𝑛−1)𝜋]|×|{sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁2𝐻]}|≈0,𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′1=()𝜋𝐻^∞_𝑛=1𝐾₀(𝜔_𝑛𝑅_𝑤)cos[(2𝑛−1)𝜋𝑧2𝐻][2𝐻](2𝑛−1)𝜋×{sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁=2𝐻]}_∞𝑛=1[2(2𝑛−1)𝜋²]𝐾₀(𝜔_𝑛𝑅_𝑤)cos[(2𝑛−1)𝜋𝑧]2𝐻×{sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁2𝐻]}. (6.18)(6.19)

In order to obtain the average wellbore pressure, integrate both sides of Formula (6.19) with respect to 𝑧 from 𝐿₁ to 𝐿₂, then divided by 𝐿_𝑝, we obtain 𝑃_𝑎(𝑅_𝑤1)=𝐿_𝑝^𝐿2_𝐿1𝑃(𝑅_𝑤=1,𝑧)𝑑𝑧𝐿_𝑝^∞_𝑛=1[2(2𝑛−1)𝜋²]𝐾₀(𝜔_𝑛𝑅_𝑤){sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁2𝐻]}×^𝐿2_𝐿1cos[(2𝑛−1)𝜋𝑧2𝐻]𝑑𝑧=(4𝐻𝜋³𝐿_𝑝)^∞_𝑛=11(2𝑛−1)²𝐾₀(𝜔_𝑛𝑅_𝑤){sin[(2𝑛−1)𝜋𝐿₂2𝐻]−sin[(2𝑛−1)𝜋𝐿₁2𝐻]}²=(16𝐻𝜋³𝐿_𝑝)^∞_𝑛=11(2𝑛−1)²𝐾₀(𝜔_𝑛𝑅_𝑤)sin²[(2𝑛−1)𝜋𝐿_𝑝4𝐻]cos²[(2𝑛−1)𝜋(𝐿₂+𝐿₁)4𝐻]. (6.20)

Recall Formula (5.86), let 𝑅^′=𝑅_𝑤, integrate 𝑧^′ at both sides of Formula (5.98) from 𝐿₁ to 𝐿₂, if the reservoir is with bottom water and gas cap, pressure at wellbore point (𝑅_𝑤,𝑧) is 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′=(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)−𝐾₀(𝜆_𝑛𝑅_𝑤)]Ξ₃, (6.21) where Ξ₃=sin(𝜆_𝑛𝑧)^𝐿2_𝐿1sin(𝜆_𝑛𝑧^′)𝑑𝑧^′=sin(𝑛𝜋𝑧𝐻)(−𝐻𝑛𝜋)[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻)]. (6.22)

With a similar procedure in Formula (6.7), there holds |(−1)𝜋𝐻^∞_𝑛=1[𝐾₀(𝜆_𝑛𝑅_𝑒)𝐼₀(𝜆_𝑛𝑅_𝑤)𝐼₀(𝜆_𝑛𝑅_𝑒)]sin(𝑛𝜋𝑧𝐻)(−𝐻𝑛𝜋)|×|[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻)]|≈0, (6.23) thus 𝑃(𝑅_𝑤,𝑧)=^𝐿2_𝐿1𝑃(𝑅_𝑤,𝑧;𝑅_𝑤,𝑧^′)𝑑𝑧^′1=()𝜋𝐻^∞_𝑛=1𝐾₀(𝜆_𝑛𝑅_𝑤)sin(𝑛𝜋𝑧𝐻)(−𝐻𝑛𝜋)[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻=)]_∞𝑛=1(−1𝑛𝜋²)𝐾₀(𝜆_𝑛𝑅_𝑤)sin(𝑛𝜋𝑧𝐻)[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻)]. (6.24)

In order to obtain the average pressure, integrate both sides of Formula (6.24) with respect to 𝑧 from 𝐿₁ to 𝐿₂, then divided by 𝐿_𝑝, we obtain 𝑃_𝑎(𝑅_𝑤1)=𝐿_𝑝^𝐿2_𝐿1𝑃(𝑅_𝑤=1,𝑧)𝑑𝑧𝐿_𝑝^∞_𝑛=1(−1𝑛𝜋²)𝐾₀(𝜆_𝑛𝑅_𝑤)[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻)]^𝐿2_𝐿1sin(𝑛𝜋𝑧𝐻𝐻)𝑑𝑧=(𝜋³𝐿_𝑝)^∞_𝑛=1(1𝑛²)𝐾₀(𝜆_𝑛𝑅_𝑤)[cos(𝑛𝜋𝐿₂𝐻)−cos(𝑛𝜋𝐿₁𝐻)]²=(4𝐻𝜋³𝐿_𝑝)^∞_𝑛=1(1𝑛²)𝐾₀(𝜆_𝑛𝑅_𝑤)sin²(𝑛𝜋𝐿_𝑝2𝐻)sin²[𝑛𝜋(𝐿₂+𝐿₁)2𝐻]. (6.25)

7. Partially Penetrating Vertical Well Productivity

Note that Formula (6.10) is in dimensionless form, substitute Formulas (3.14) and (3.15) into Formula (6.10), then rearrange and simplify the resulting formula, the productivity formula for a partially penetrating vertical well is obtained: 𝑄_𝑤=𝐹_𝐷2𝜋𝐾_ℎ𝐻(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)ln(𝑅_𝑒𝐷/𝑅_𝑤𝐷)+𝑆_ps, (7.1) where 𝑃_𝑤 is average wellbore pressure, 𝑆_ps is pseudo-skin factor due to partial penetration: 𝑆_ps=(8𝐻²_𝐷𝜋²𝐿²_𝑝𝐷)^𝑁_𝑛=1(1𝑛²)𝐾₀[(𝑛𝜋𝐻_𝐷)𝑅_𝑤𝐷]sin²(𝑛𝜋𝐿_𝑝𝐷2𝐻_𝐷)cos²[𝑛𝜋(𝐿_2𝐷+𝐿_1𝐷)2𝐻_𝐷]. (7.2)

However, (7.1) is only applicable to a circular cylinder drainage reservoir has both impermeable top and bottom boundaries, and has constant pressure lateral boundary.

If the well is fully penetrating, (𝐿_𝑝=𝐿=𝐻,) then 𝑆_ps=0, and if the reservoir is an isotropic permeability reservoir, that is, 𝐾_𝑎=𝐾_ℎ=𝐾_𝑣=𝐾, then Formula (7.1) reduces to Formula (2.1).

Substituting Formulas (3.14) and (3.15) into Formula (6.15), then rearranging and simplifing the resulting formula, we obtain the productivity formula for a partially penetrating vertical well in a reservoir with impermeable bottom boundary and constant pressure top boundary (gas cap): 𝑄_𝑤=𝐹_𝐷𝜋³𝐾_ℎ𝐿²_𝑝(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)16𝐻Θ₁, (7.3) where Θ₁=^𝑁_𝑛=11(2𝑛−1)²𝐾₀{[(2𝑛−1)𝜋2𝐻_𝐷]𝑅_𝑤𝐷}×sin²[(2𝑛−1)𝜋𝐿_𝑝𝐷4𝐻_𝐷]sin²[(2𝑛−1)𝜋(𝐿_2𝐷+𝐿_1𝐷)4𝐻_𝐷]. (7.4)

Substituting Formulas (3.14) and (3.15) into Formula (6.20), then rearranging and simplifing the resulting formula, we obtain the productivity formula for a partially penetrating vertical well in a reservoir with impermeable top boundary and constant pressure bottom boundary (bottom water): 𝑄_𝑤=𝐹_𝐷𝜋³𝐾_ℎ𝐿²_𝑝(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)16𝐻Θ₂, (7.5) where Θ₂=^𝑁_𝑛=11(2𝑛−1)²𝐾₀{[(2𝑛−1)𝜋2𝐻_𝐷]𝑅_𝑤𝐷}×sin²[(2𝑛−1)𝜋𝐿_𝑝𝐷4𝐻_𝐷]cos²[(2𝑛−1)𝜋(𝐿_2𝐷+𝐿_1𝐷)4𝐻_𝐷]. (7.6)

Substituting Formulas (3.14) and (3.15) into Formula (6.25), then rearranging and simplifing the resulting formula, we obtain the productivity formula for a partially penetrating vertical well in a reservoir with both bottom water and gas cap: 𝑄_𝑤=𝐹_𝐷𝜋³𝐾_ℎ𝐿²_𝑝(𝑃_𝑒−𝑃_𝑤)/(𝜇𝐵)4𝐻Θ₃, (7.7) where Θ₃=^𝑁_𝑛=11𝑛²𝐾₀[(𝑛𝜋𝐻_𝐷)𝑅_𝑤𝐷]sin²(𝑛𝜋𝐿_𝑝𝐷2𝐻_𝐷)sin²[𝑛𝜋(𝐿_2𝐷+𝐿_1𝐷)2𝐻_𝐷]. (7.8)

It must be pointed out that 𝑁=500 in the summation of the series in the aforementioned formulas is sufficient to reach engineering accuracy.

8. Conclusions and Discussions

Comparing Formulas (7.1), (7.3), (7.5), and (7.7), we can reach the the following conclusions.

If top and bottom reservoir boundaries are impermeable, the radius of the cylindrical system and off-center distance appears in the productivity formulas; if the reservoir has a gas cap or bottom water, the effects of the radius and off-center distance on productivity can be ignored.

Because the pay zone thickness 𝐻 is very small compared with the circular cylinder drainage radius 𝑅_𝑒, when the reservoir has gas cap or bottom water which provides the main drive mechanism, the lateral boundary has little influence on productivity, the well is producing as if the reservoir is infinite. The performance is the same for a centered well and an off-center well, even the reservoir is with an edge water boundary (constant pressure outer boundary). The effects of drainage radius and off-center distance on productivity are negligible, thus 𝑅_𝑒 and 𝑅₀ do not show up in Formulas (7.3), (7.5), and (7.7). However, (7.3), (7.5), and (7.7) are applicable to both centered and off-center wells.

If both top and bottom boundaries are impermeable, then 𝑅_𝑒 and 𝑅₀ play important roles in well productivity, as indicated by Formula (7.1).

Formula (7.2) for pseudo-skin factor is obtained by solving three-dimensional Laplace equation, thus it is more reliable than semianalytical and semiempirical expressions of pseudo-skin factor, that is, Formulas (2.4) and (2.7).

Field data will be provided in forthcoming papers in the near future, which can show that the proposed formulas are reliable and accurate, they are fast analytical tools to evaluate well performance.

In this paper, we always assume the lateral boundary is at constant pressure, so the proposed formulas are only applicable for steady-state, this study can be improved by considering:

(1)pseudo-steady-state productivity formulas for an oil reservoir with impermeable upper, lower, and lateral boundaries;(2)steady-state productivity formulas for an oil reservoir with impermeable lateral boundary, but constant pressure upper or lower boundary;(3)productivity formulas for a partially penetrating vertical well in a box-shaped reservoir, and the corresponding equation of pseudo-skin factor due to partial penetration;(4)non-Darcy flow effect on an oil well productivity.