#### 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 drains a circular cylinder porous volume with height and radius .

The following assumptions are made.

(1)The porous media volume is circular cylinder which has constant permeabilities, thickness , porosity . 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 at the lateral surface.(3)The production occurs through a partially penetrating vertical well of radius , represented in the model by a uniform line sink which is located at away from the axis of symmetry of the cylindrical body. The drilled well length is , the producing well length is .(4)A single-phase fluid, of small and constant compressibility , constant viscosity , and formation volume factor , flows from the porous media to the well. Fluids properties are independent of pressure. Gravity forces are neglected.

The porous media domain iswhere is cylinder radius, is the cylindrical body.

Located at away from the center of the cylindrical body, the coordinates of the top and bottom points of the well line are () and (), respectively, while point () and point () are the beginning point and end point of the producing portion of the well, respectively. The well is a uniform line sink between () and (), and there holds

We assumeand define average permeability

Suppose point is on the producing portion, and its point convergence intensity is , in order to obtain the pressure at point caused by the point , according to mass conservation law and Darcy's law, we have to obtain the basic solution of the diffusivity equation in [8]:where is total compressibility coefficient of porous media, , , are Dirac functions.

The initial condition is

The lateral boundary condition iswhere is the cylindrical lateral surface:

The porous media domain is bounded by top and bottom impermeable boundaries, so

In order to simplify the above equations, we take the following dimensionless transforms:

Assuming is the point convergence intensity at the point sink , the partially penetrating well is a uniform line sink, the total flow rate of the well is , and there holds

Define dimensionless pressures

Note that if is a positive constant, there holds [9]consequently, (2.5) becomes [8, 9]where

If point and point are with distances and , respectively, from the circular center, then the dimensionless off-center distances are

There holdswhere

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

Ifthen and ifthen

Recall (2.22), according to the above calculations, without losing generality, there holds

In the same manner, we have

#### 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 .

Thus, if the point sink is at , (2.19) can be written aswhere

The equation of initial condition is changed to

The equation of lateral boundary condition is changed towhere

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

Figure 2 is a geometric representation of the system. In Figure 2, the point sink and the observation point , are with distances and , respectively, from the circular center; and the two points are separated at the center by an angle . The inverse point of the point sink with respect to the circle is point . Point with a distance from the center, and 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

Assume is the distance between point and point , then [9, 10]

If the observation point is on the drainage circle, , then

If the observation point is on the wellbore, then

Recall (2.9), obviously for impermeable upper and lower boundary conditions, there holds [9, 10]where

Letand substitute (3.12) into (3.1) and compare the coefficients of , we obtainin circular , andon circumference andwhere

Taking the Laplace transform at the both sides of (3.13), thenwhereand is Laplace transform variable.

Define

*Case 1. *If ,
thenwhere

*Case 2. *If ,
then satisfies (3.17).

Define

Recall (3.8), and is a basic solution of (3.17), since ,
we haveso letwherethusand satisfies homogeneous equation: 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]:where are undetermined coefficients.

Because is continuously bounded within , but , there holds

There hold [9, 10]where is modified Bessel function of second kind and order is modified Bessel function of first kind and order is Bessel function of first kind and order is Hankel function of first kind and order , and .

And there hold (see [14, page 979])

Let (note that ), substituting (3.31) into (3.32) and using (3.33), we have the following Cosine Fourier expansions of (see [14, page 952]):

So, we obtain

Note that on , and comparing coefficients of Cosine Fourier expansions of in (3.35) and (3.29), we obtain

Defineand recall (3.29), then we havewhere

In the appendix, we can prove

Thus we only consider the case , in (3.38) and (3.40), let , we havewherewhere

And there holdswhere is Inverse Laplace transform operator.

Since are simple poles of meromorphic function , if using partial fraction expansion of meromorphic function, there holds [15]where are residues at poles respectively, and is the th root of equation

From (3.48), we havewhere

According to the convolution theorem [12], from (3.44), there holdswhere

Recall (3.38), (3.44), and (3.51), there holds

Using Laplace asymptotic integration (see [16, page 221]), when is sufficiently large, thentherefore,

There holds [9]

Using (3.48) and (3.56), and note thatso (3.46) can be written aswhere denotes , thus from (3.48), we obtaintherefore,

is defined as real part operator, for example, means real part of , There holdsand define is a complex number.

Note that , recall (3.53), define

In (3.60), letand definethus we obtainand there holds [9, 10]thus if we recall (3.26) and definethenand has the same meaning as in (3.7).

In the above equations, is exponential integral function,

Recall (3.27), there holds

Combining (3.12), (3.26), (3.27), and (3.41), we obtain

Equation (3.73) is the pressure distribution equation of an off-center point sink in the cylindrical body. If the point sink and the observation point are not on a radius of the drainage circle, , recall (3.7), 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, then

In Figure 2, if the point sink and the observation point are on a radius, then

#### 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 .

The well line sink is located along the line . If the observation point is on the wellbore, , note that , and there hold thenand recall (3.64), then

Recall (3.66), thenand recall (3.70), then

Define

In order to calculate the pressure at the wellbore, using principle of potential superposition, integrating at both sides of (3.72) from to , then

Recall (3.26), and note that we haveand definebecause when is very small, (time is sufficiently long), there holdsso when time is sufficiently long,

Recall (3.12), (3.24), (3.27), and (3.41), when time is sufficiently long, define

Therefore, the wellbore pressure at point is

Considering the bottom point of the well line sink, then , thus , in this case, (4.12) reduces towherewhere is the integer part of

For it holds the following estimate:

So, (4.14) reduces to

Combining (4.7), (4.13), and (4.17), pressure at the bottom point of the producing portion is

In order to obtain average wellbore pressure, recall (4.12) and (4.17), integrate both sides of (4.13) with respect to from to , then divided by , average wellbore pressure is obtained:where we use

#### 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 iswhere is the integer part of .

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 denotes pseudo-skin factor due to partial penetration.

If , the drilled well length is equal to formation thickness, for a fully penetrating well, (5.1) reduces to

If the well is located at the center of the cylindrical body, then there holds

Thus, (5.1) reduces towhere has the same meaning as in (5.2).

If the well is a fully penetrating well in an infinite reservoir, , there holds

Thus, (5.6) reduces to

Substitute (2.12) and (2.15) into (2.17), then simplify and rearrange the resulting equation, we obtainwhere is total flow rate of the well, and 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 is a function of and is not a function of well off-center distance or drainage radius .

For an isotropic reservoir, (5.2) reduces toand (5.3) reduces to is the integer part of .

If we definethen (6.1) can be written as

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

*Solution**Case 1. *Figure 3 shows the trend of with when ,
it can be found that is a weak decreasing function of .

*Case 2.*Figure 4 shows the trend of with when , it can be found that is an increasing function of . When is a constant, we may assume is a constant, then is also a constant; when increases, also increases, thus the well producing length decreases, and pseudo-skin factor due to partial penetration increases.

*Case 3.*Figure 5 shows the trend of with when , it can be found that is a decreasing function of . When is a constant, we may assume is a constant, then is also a constant; when increases, also increases, thus the well producing length increases, and pseudo-skin factor due to partial penetration decreases.

*Example 6.2. *A fully penetrating off-center vertical well,
ifcompare the wellbore pressure
responses when .

*Solution*

Equation (5.4) is used to calculate ,
the results are shown in Figure 6.

Figure 6 shows that at early times, the well is in
infinite acting period. When producing time is long, the influence from outer
boundary appears. Because the outer boundary is at constant pressure, when the
producing time is sufficiently long, steady state will be reached, becomes a constant.

At a given time ,
if drainage radius is a constant, when well off-center distance increases, decreases, which indicates the effect from
constant pressure outer boundary is more pronounced.

*Example 6.3. *A fully penetrating off-center vertical
well, ifcompare the wellbore pressure
responses when .

*Solution*

Equation (5.4) is used to calculate ,
the results are shown in Figure 7.

Figure 7 shows that at a given time ,
if well off-center distance is a constant, when drainage radius increases, also increases, which indicates the effect from
constant pressure outer boundary is more pronounced.

#### 7. Conclusions

The following conclusions are reached.

(1)The proposed equations provide fast analytical tools to evaluate the performance of a vertical well which is located arbitrarily in a circular drainage volume with constant pressure outer boundary.(2)The well off-center distance has significant effect on well pressure drop behavior, but it does not have any effect on pseudo-skin factor due to partial penetration.(3)Because the outer boundary is at constant pressure, when producing time is sufficiently long, steady-state is definitely reached.(4)At a given time in a given drainage volume, if the well off-center distance increases, the pressure drop at wellbore decreases.(5)When well producing length is equal to payzone thickness, the pressure drop equations for a fully penetrating well are obtained.

#### Appendix

In this appendix, we want to prove (3.41).

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

There hold [14]

Sinceand note that is in dimensionless form in the above equation, recall (2.11), (2.13) and (2.21), for dimensionless , there holdthus, we obtain

There holds