Pseudo-Steady-State Productivity Formula for a Partially Penetrating Vertical Well in a Box-Shaped Reservoir
For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long-producing times. Taking a partially penetrating well as a uniform line sink in three dimensional space, by the orthogonal decomposition of Dirac function and using Green's function to three-dimensional Laplace equation with homogeneous Neumann boundary condition, this paper presents step-by-step derivations of a pseudo-steady-state productivity formula for a partially penetrating vertical well arbitrarily located in a closed anisotropic box-shaped drainage volume. A formula for calculating pseudo skin factor due to partial penetration is derived in detailed steps. A convenient expression is presented for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle, and for arbitrary position of the well within the rectangle.
Well productivity is one of primary concerns in oil field development and provides the basis for oil field development strategy. To determine the economical feasibility of drilling a well, the engineers need reliable methods to estimate its expected productivity. Well productivity is often evaluated using the productivity index, which is defined as the production rate per unit pressure drawdown. Petroleum engineers often relate the productivity evaluation to the long-time performance behavior of a well, that is, the behavior during pseudo-steady-state or steady-state flow.
For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long producing times. In these reservoirs, also called volumetric reservoirs, there can be no flow across the impermeable outer boundary, such as a sealing fault, and fluid production must come from the expansion and pressure decline of the reservoir. This condition of no flow boundary is also encountered in a well that is offset on four sides.
Flow enters the pseudo-steady-state regime when the pressure transient reaches all boundaries after drawdown for a sufficiently long-time. During this period, the rate of pressure decline is almost identical at all points in the reservoir and wellbore. Therefore, the difference between the average reservoir pressure and pressure in the wellbore approaches a constant with respect to time. Pseudo-steady-state productivity index is defined as the production rate divided by the difference of average reservoir pressure and wellbore pressure, hence the productivity index is basically constant [1, 2].
In many oil reservoirs the producing wells are completed as partially penetrating wells. If a vertical well partially penetrates the formation, the streamlines converge and the area for flow decreases in the vicinity of the wellbore, which results in added resistance, that is, a pseudoskin factor. Only semianalytical and semi-empirical expressions are available in the literature to calculate pseudoskin factor due to partial penetration.
Rarely do wells drain ideally shaped drainage areas. Even if they are assigned regular geographic drainage areas, they become distorted after production commences, either because of the presence of natural boundaries or because of lopsided production rates in adjoining wells. The drainage area is then shaped by the assigned production share of a particular well. An oil reservoir often has irregular shape, but a rectangular shape is often used to approximate an irregular shape by petroleum engineers, so it is important to study well performance in a rectangular or box-shaped reservoir [1, 2].
2. Literature Review
The pseudo-steady-state productivity formula of a fully penetrating vertical well which is located at the center of a closed isotropic circular reservoir is [3, page 63]
where is average reservoir pressure in the circular drainage area, is flowing wellbore pressure, is permeability, is payzone thickness, is oil viscosity, is oil formation volume factor, is radius of circular drainage area, is wellbore radius, and is the factor which allows the use of field units and practical units, and it can be found in [3, page 52, Table ].
Formula (2.1) is only applicable for a fully penetrating vertical well at the center of a circular drainage area with impermeable outer boundary.
If a vertical well is partially penetrate the formation, the streamlines converge and the area for flow decreases in the region around the wellbore, and this added resistance is included by introducing the pseudoskin factor, . Thus, (2.1) may be rewritten to include the pseudoskin factor due to partial penetration as [4, page 92]:
Assume that the well-drilled length is equal to the well producing length, (i.e., perforated interval,) , and define partial penetration factor :
Pseudoskin factor formula given by Brons and Marting is 
Pseudoskin factor formula given by Papatzacos is 
where has the same meaning as in (2.5), and
and is the distance from the top of the reservoir to the top of the open interval.
Pseudoskin factor formula given by Bervaldier is 
By solving-three-dimensional Laplace equation with homogeneous Dirichlet boundary condition, Lu et al. presented formulas to calculate in steady state .
To account for irregular drainage shapes or asymmetrical positioning of a well within its drainage area, a series of shape factors was developed by Dietz . Formula (2.1) can be generalized for any shape into the following formula:
where is shape factor, and is drainage area.
Dietz evaluated shape factor for various geometries, in particular, for rectangles of various aspect ratios with single well in various locations. He obtained his results graphically, from the straight line portion of various pressure build-up curves. Earlougher et al.  carried out summations of exponential integrals to obtain dimensionless pressure drops at various points within a square drainage area and then used superposition of various square shapes to obtain pressure drops for rectangular shapes. The linear portions of the pressure drop curves so obtained, corresponding to pseudo-steady-state, were then used to obtain shape factors for various rectangles.
The methods used by Dietz and Earlougher et al. are limited to rectangles whose sides are integral ratios, and the well must be located at some special positions within the rectangle.
Lu and Tiab presented formulas to calculate productivity index and pseudoskin factor in pseudo-steady-state for a partially penetrating vertical well in a box-shaped reservoir, they also presented a convenient expression for calculating the shape factor of an isotropic rectangle reservoir [1, 2]. But in [1, 2], they did not provide detail derivation steps of their formulas.
The primary goal of this paper is to present step-by-step derivations of the pseudo-steady-state productivity formula and pseudoskin factor formula for a partially penetrating vertical well in an anisotropic box-shaped reservoir, which were given in [1, 2]. A similar procedure in  is given in this paper, point sink solution is first derived by the orthogonal decomposition of Dirac function and Green's function to Laplace equation with homogeneous Neumann boundary condition, then using the principle of superposition, point sink solution is integrated along the well length, uniform line sink solution is obtained, and rearrange the resulting solution, pseudo-steady-state productivity formula and shape factor formula are obtained. A convenient expression is derived for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle and for arbitrary position of the well within the rectangle.
3. Partially Penetrating Vertical Well Model
Figure 1 is a schematic of a partially penetrating well. A partially penetrating vertical well of length drains a box-shaped reservoir with height , length ( direction) , and width ( direction) . The well is parallel to the direction with a length , and we assume
The following assumptions are made.(1)The reservoir is homogeneous, anisotropic, and has constant permeabilities, thickness , and porosity . All the boundaries of the box-shaped drainage volume are sealed.(2)The reservoir pressure is initially constant. At time , pressure is uniformly distributed in the reservoir, equal to the initial pressure .(3)The production occurs through a partially penetrating vertical well of radius , represented in the model by a uniform line sink.(4)A single phase fluid, of small and constant compressibility , constant viscosity , and formation volume factor , flows from the reservoir to the well at a constant rate . Fluids properties are independent of pressure.(5)No gravity effect is considered. Any additional pressure drops caused by formation damage, stimulation, or perforation are ignored, we only consider pseudoskin factor due to partial penetration.
The partially penetrating vertical well is taken as a uniform line sink in three dimensional space. The coordinates of the two end points of the uniform link sink are () and (). We suppose the point is on the well line, and its point convergence intensity is .
By the orthogonal decomposition of Dirac function and using Green's function to Laplace equation with homogeneous Dirichlet boundary condition, Lu et al. obtained point sink solution and uniform line sink solution to steady-state productivity equation of a partially penetrating vertical well in a circular cylinder reservoir . For a box-shaped reservoir and a circular cylinder reservoir, the Laplace equation of a point sink is the same, in order to obtain the pressure at point caused by the point , we have to obtain the basic solution of the following Laplace equation:
in the box-shaped drainage volume:
and we always assume
and , , are Dirac functions.
All the boundaries of the box-shaped drainage volume are sealed, that is,
where is the exterior normal derivative of pressure on the surface of box-shaped drainage volume .
The reservoir pressure is initially constant
Define average permeability:
In order to simplify (3.1), we take the following dimensionless transforms:
The dimensionless wellbore radius is 
Assume that 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 
Dimensionless pressures are defined by
Then (3.1) becomes
in the dimensionless box-shaped drainage volume
with boundary condition
and initial condition
4. Point Sink Solution
For convenience in the following reference, we use dimensionless transforms given by (3.7)–(3.10), every variable, drainage domain, initial and boundary conditions should be taken as dimensionless, but we drop the subscript .
Consequently, (3.12) is expressed as
Rewrite (3.14) below
and (3.15) becomes
where are nonnegative numbers, and
and have similar definitions.
In order to simplify the following derivations, we define the following notation:
which means in any function , the subscripts of any variable must count from to infinite.
which means in any function , the subscripts of any variable must be no less than zero, and at least one of the three subscripts must be positive to guarantee . And the upper limit of the subscripts is infinite.
where are undetermined coefficients.
where is the three-dimensional Laplace operator
compare the coefficients of at both sides of (4.10), we obtain
because from (4.14),
When solve (4.14),
Recall (4.19), the average value of throughout of the total volume of the box-shaped reservoir is
Note that implies that at least one of must be greater than , without losing generality, we may assume
If time is sufficiently long, pseudo-steady-state is reached, decreases by exponential law, will vanish, that is,
note that is equal to in (4.19), and
From Green's Formula ,
where is volume of drainage domain .
Define the following notation of internal product of functions and :
where means the internal product of functions and .
From (4.33), we know that the internal product of and constant number is zero
and it is easy to prove
where means when .
The drainage volume is
Recall (4.28), the average pressure throughout the reservoir is
The wellbore pressure at point is
where is the value of at wellbore point .
which implies is independent of time.
5. Uniform Line Sink Solution
For convenience, in the following reference, every variable, drainage domain, initial and boundary conditions should be taken as dimensionless, but we drop the subscript .
The producing portion of the partially penetrating well is between point and point , recall (4.4) and (4.19), in order to obtain uniform line sink solution, we integrate with respect to from to , then
then it is easy to prove
Define the following notations:
and the average value of at wellbore can be written as
Rearrange (4.12) and obtain
There hold [16, page 47]
The average value of along the well length is
where we have used (5.17).
For a fully penetrating well, , then
where we use the following formulas [16, page 47]:
The average value of along the well length is
where we use the following formulas [16, page 47]:
and we may simplify (5.24) further
For a fully penetrating well, , then
since the derivative of is
When and ,
When reaches maximum value, let
and the producing length is a variable, define
thus when reaches maximum value,
so is a bounded function, let
where is function:
So, in (5.37), is sufficient to reach engineering accuracy.
Let recast (5.26), we obtain