#### 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]: where is outer boundary pressure, is flowing wellbore pressure, is permeability, is payzone thickness, is oil viscosity, is oil formation volume factor, is drainage radius, is wellbore radius, and 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, . Thus, Formula (2.1) may be rewritten to include the pseudo-skin factor due to partial penetration as [2, page 92]

Define partial penetration factor : where 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]:

Brons and Marting's pseudo-skin factor formula [4] is as follows: where

Papatzacos's pseudo-skin factor formula [5] is as follows: where and 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 drains a circular cylinder porous volume with height and radius .

The following assumptions are made.

(1)The reservoir has constant permeabilities, thickness , porosity . During production, the partially penetrating vertical well has a circular cylinder drainage volume with height and radius . The well is located at away from the axis of symmetry of the cylindrical body.(2)At time , pressure is uniformly distributed in the reservoir, equal to the initial pressure . 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 , represented in the model by a uniform line sink, 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 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: where 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 hold

We assume and define average permeability:

The reservoir initial pressure is a constant:

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

Suppose point is in 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 [6, 7]: where is total compressibility coefficient of porous media, are Dirac functions.

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

The dimensionless wellbore radius is [8]

Assume is the point convergence intensity at the point sink , the partially penetrating well is a uniform line sink, the total productivity of the well is , and there holds

Define the dimensionless pressures:

Then (3.7) becomes [6, 7] where

If point and point are with distances and , respectively, from the axis of symmetry of the cylindrical body, then the dimensionless off-center distances are

There holds where

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

If then

Moreover if then

Recall (3.19), there holds since there holds (3.21), and according to the aforementioned calculations in (3.22), (3.23), (3.24), and (3.25), we obtain

Because thus

Combining (3.27) and (3.29), we have

#### 4. Boundary Conditions

In this paper, we always assume constant pressure lateral boundary: on cylindrical lateral surface:

Recall (3.15), the dimensionless form of constant pressure lateral boundary condition is on

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 and are both impermeable (e.g., the reservoir does not have gas cap drive or bottom water drive), then (ii)If the circular cylinder drainage volume is with impermeable boundary at , constant pressure boundary at , (e.g., the reservoir has gas cap drive), then (iii)If the circular cylinder drainage volume is with impermeable boundary at , constant pressure boundary at (e.g., the reservoir has bottom water drive), then (iv)If the circular cylinder drainage volume is with top and bottom constant pressure boundaries, that is, the boundaries at and are both constant pressure boundaries (e.g., the reservoir has both gas cap drive and bottom water drive), then#### 5. Point Sink Solutions

For convenience, we use dimensionless variables given by (3.8) through (3.13), but we drop the subscript . 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: where

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: where .

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 is 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

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

If the observation point is on the wellbore, then

Define

##### 5.1. Impermeable Upper and Lower Boundaries

If upper and lower boundaries are impermeable, obviously for such impermeable boundary conditions, we have where Let and substituting (5.12) into (5.1) and compare the coefficients of , we obtain in circular area , and on circumference and is two-dimensional Laplace operator,

*Case 1. *If , then

Using Green's function of Laplace problem in a circular domain, we obtain [9â€“11]

*Case 2. *If , then satisfies (5.13). Since satisfies the equations:

So is a basic solution of (5.13), where

Define
thus
then satisfies homogeneous equation
and 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 [9â€“11]:
where are undetermined coefficients.

Because is continuously bounded within , but , so there holds

There hold [7, 12]
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 .

Also there hold [13, page 979]

Let (note that ) putting (5.26) into (5.27), and using (5.28), we have the following Cosine Fourier expansions of [13, page 952]:

Note that on , comparing coefficients of Cosine Fourier expansions of in (5.29) and (5.24), we obtain

Define and recall (5.24), then we have where

There hold [13, page 919]

Note that in Formula (5.8) is in dimensionless form, recall Formulas (3.9), (3.12) and (3.18), for dimensionless , there hold thus There holds Combining Formulas (3.18), (5.38), and (5.39), we obtain where

There holds where we use Formula (3.26),

If , there holds [13, page 916] thus for , where we use and if , then

Putting Formula (5.47) into Formula (5.42), we obtain

Note that then we obtain where we use [13, page 919]

Note that and we have thus Formula (5.52) can be simplified as follows: where we use Formulas (5.45) and (3.26)

Combining Formulas (5.40), (5.43), and (5.55), we prove

Combining Formulas (5.4), (5.12), (5.17), (5.20), (5.22), (5.29), and (5.57), the point convergence pressure of point is

##### 5.2. Constant Pressure Upper or Lower Boundaries

If the reservoir is with gas cap and impermeable bottom boundary, then and assume the outer boundary is at constant pressure on cylindrical surface .

Define then under the boundary condition of (5.59), we have

Let where satisfies in , and on .

Let where and has the same meaning as in Formula (5.6).

Thus satisfies homogeneous equation: in , and on .

Using polar coordinates, we have then and point convergence pressure of point is:

If the reservoir is with bottom water and impermeable top boundary, then and recall Formula (5.60), the outer boundary is at constant pressure.

Define then under the boundary condition of (5.73), we have

Let where satisfies in , and on .

Let where thus satisfies in , and on .

Using polar coordinates, we have then and point convergence pressure of point is

If the reservoir is with gas cap and bottom water, then and recall Formula (5.60), the outer boundary is at constant pressure.

Define where has the same meaning as in Formula (5.8).

Under the boundary condition of (5.86), we have

Let where satisfies in , and on .

Let where thus satisfies in , and on .

Using polar coordinates, we have then and point convergence pressure of point is

#### 6. Uniform Line Sink Solutions

For convenience, we drop the subscript for dimensionless variables. In order to calculate the pressure at wellbore, assume the observation point is on the wellbore, and recall Formula (5.7), and

The partially penetrating vertical well is taken as a uniform line sink. Recall Formula (5.9), integrate at both sides of Formula (5.58) from to , if the upper and lower boundaries are impermeable, pressure at wellbore point is

Recall Formula (5.34), there hold if and if .

Combining to Formulas (3.9), (3.12), (3.13), and (5.8), we obtain so

According to Formula (6.6), we have