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.


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.
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,

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 P a is average reservoir pressure in the circular drainage area, 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 radius of circular drainage area, R w is wellbore radius, and F D is the factor which allows the use of field units and practical SI units, and it can be found in 3, page 52, Table 5.1 .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, S ps .Thus, 2.1 may be rewritten to include the pseudoskin factor due to partial penetration as 4, page 92 : 2πKH P a − P w / μB ln R e /R w − 3/4 S ps .

2.2
S ps can be calculated by semianalytical and semiempirical expressions presented by Brons, Marting, Papatzacos, and Bervaldier 5-7 .Assume that the well-drilled length is equal to the well producing length, i.e., perforated interval, L p L, and define partial penetration factor η: Pseudoskin factor formula given by Brons and Marting is 5 where G η 2.948 − 7.363η 11.45η 2 − 4.675η 3 .

2.6
Pseudoskin factor formula given by Papatzacos is 6 where h D has the same meaning as in 2.5 , and

2.8
and h 1 is the distance from the top of the reservoir to the top of the open interval.Pseudoskin factor formula given by Bervaldier is 7

2.9
It must be pointed out that the well location in the reservoir has no effect on S ps calculated by 2.4 , 2.7 , and 2.9 .
By solving-three-dimensional Laplace equation with homogeneous Dirichlet boundary condition, Lu et al. presented formulas to calculate S ps in steady state 8 .
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 9 .Formula 2.1 can be generalized for any shape into the following formula: where C A is shape factor, and A is drainage area.Dietz evaluated shape factor C A 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. 10 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 pseudosteady-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 8 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.

Partially Penetrating Vertical Well Model
Figure 1 is a schematic of a partially penetrating well.A partially penetrating vertical well of length L drains a box-shaped reservoir with height H, length x direction a, and width y direction b.The well is parallel to the z direction with a length L ≤ H, and we assume b ≥ a.
The following assumptions are made.
1 The reservoir is homogeneous, anisotropic, and has constant K x , K y , K z permeabilities, thickness H, and porosity φ.All the boundaries of the box-shaped drainage volume are sealed.3 The production occurs through a partially penetrating vertical well of radius R w , represented in the model by a uniform line sink.
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 at a constant rate Q w .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 x , y , 0 and x , y , L .We suppose the point x , y , z is on the well line, and its point convergence intensity is q.
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 8 .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 x, y, z caused by the point x , y , z , we have to obtain the basic solution of the following Laplace equation: All the boundaries of the box-shaped drainage volume are sealed, that is, where ∂P/∂N| Γ is the exterior normal derivative of pressure on the surface of box-shaped drainage volume Γ ∂Ω.
The reservoir pressure is initially constant Define average permeability: In order to simplify 3.1 , we take the following dimensionless transforms:

3.7
The dimensionless wellbore radius is 8 Assume that q is the point convergence intensity at the point sink x , y , z , the partially penetrating well is a uniform line sink, the total productivity of the well is Q w , and there holds 8

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 D. Consequently, 3.12 is expressed as We want to solve 4.1 under the boundary condition 4.2 and initial condition 4.3 , and to obtain point sink solution when the time t is so long that the pseudo-steady-state is reached.
If the boundary condition is 4.2 , there exists the following complete normalized orthogonal system {g lmn x, y, z } 11, 12 : where l, m, n are nonnegative numbers, and In order to simplify the following derivations, we define the following notation: which means in any function F x, y, z , the subscripts l, m, n of any variable must count from 0 to infinite.And define which means in any function F x, y, z , the subscripts l, m, n of any variable must be no less than zero, and at least one of the three subscripts l, m, n must be positive to guarantee l m n > 0. And the upper limit of the subscripts l, m, n is infinite.Let exp −λ lmn t g lmn x , y , z g lmn x, y, z λ lmn .

4.17
Define 18  where g 000 means g lmn when l m n 0. Thus, f x, y, z can be decomposed as 13, 14 : f, g lmn x , y , z g lmn x, y, z l m n>0 δ x − x δ y − y δ z − z , g lmn x , y , z g lmn x, y, z l m n>0 g lmn x , y , z g lmn x, y, z .

Mathematical Problems in Engineering 13
The drainage volume is V abH.

4.38
Recall 4.28 , the average pressure throughout the reservoir is

4.39
The wellbore pressure at point x w , y w , z w is where Ψ w is the value of Ψ at wellbore point x w , y w , z w .Combining 4.39 and 4.40 gives which implies P a,v − P w is independent of time.

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 D.
The producing portion of the partially penetrating well is between point x , y , 0 and point x , y , L , recall 4.4 and 4.19 , in order to obtain uniform line sink solution, we integrate Ψ with respect to z from 0 to L, then then it is easy to prove

5.7
Recall 5.1 and 5.2 , and use 5.3 -75 , J x, y, z; x , y , z ; l, m, n can be decomposed as

Mathematical Problems in Engineering 15
Define the following notations: and the average value of J at wellbore can be written as

5.17
Mathematical Problems in Engineering Recall 5.4 and 5.9 , J z is for the case l m 0, n > 0, and at wellbore of the offcenter well,

5.18
The average value of J z w along the well length is

5.19
where we have used 5.17 .For a fully penetrating well, L H, then J z,a w 0. 5.20 Recall 5.5 and 5.10 , J yz is for the case l 0, m > 0, n ≥ 0, and at wellbore of the off-center well,

5.21
where we use the following formulas 16, page 47 :

Mathematical Problems in Engineering
The average value of J yz w along the well length is

5.24
where we use the following formulas 16, page 47 : and we may simplify 5.24 further J yz,a w 2bL aH

5.27
For a fully penetrating well, L H, then

5.40
So, in 5.37 , M 100 is sufficient to reach engineering accuracy.Recall 5.6 and 5.11 , J xyz is for the case l > 0, m ≥ 0, n ≥ 0, and at wellbore of the off-center well,

5.42
The average value of J xyz w along the well length is

5.43
where we use 5.22 and 5.25 .

Mathematical Problems in Engineering
Let x 0, recast 5.26 , we obtain

5.47
where we use the following formula 16, page 46 : and the following simplifications:

5.49
For a fully penetrating well, L H, 5.47 is simplified as Recall 5.13 , then

5.51
Recall 4.28 and 4.40 , the average wellbore pressure along the uniform line sink is P a,w t abH J a,w , 5.52 then 4.41 becomes P a,v − P a,w J a,v − J a,w , 5.53 which implies P a,v − P a,w is independent of time.

Productivity Formula and Shape Factor Formula
Note that 5.53 is in dimensionless form, that is, P a,vD − P a,wD J a,vD − J a,wD .
6.17 Formula 6.6 is recommended to calculate productivity index in pseudo-steady-state, because it does not require the shape factor, and it is applicable to an off-center partially penetrating vertical well in pseudo-steady-state arbitrarily located in an anisotropic boxshaped reservoir.
So, the step-by-step derivations of pseudo-steady-state productivity formula and shape factor formula which were published in 1, 2 have been given in the above sections.

Examples and Discussions
The following examples are given to calculate well productivity index, pseudoskin factor due to partial penetration, and shape factor.

Example One
Use 6.6 to calculate productivity index of a partially penetrating vertical well in pseudosteady-state in a closed box-shaped anisotropic reservoir.The wellbore, reservoir, and fluid properties data practical SI units are given in Table 1. where

Example Two
Using the formulas given by Brons and Marting, Papatzacos, Bervaldier, calculate pseudoskin factor of the well in Example One.
Solution.If we use Brons and Marting's pseudoskin factor formula, then Formulas 2.4 , 2.7 , and 2.9 cannot account for the effect of well location inside a finite drainage volume on S ps .But 6.8 is applicable to a well arbitrarily located in a boxshaped reservoir, S ps is a function of well location parameters X w and Y w , and S ps is also a function of reservoir size parameters X e and Y e .This is the reason why significant differences exist between S ps calculated by 6.8 and S ps calculated by 2.4 , 2.7 , and 2.9 .

Example Three
A fully penetrating vertical well is located at the center of an isotropic rectangular reservoir with X e /Y e 4, calculate the shape factor and compare with the corresponding shape factors given by Dietz and Earlougher et al.

Solution.
Since the well is located at the center of the rectangular reservoir with X e /Y e 4, use 6. 7.17 The corresponding shape factor given by Dietz 9 is C A 5.38, and C A 5.3790 given by Earlougher et al. 10 .Thus, there does not exist significant difference between the shape factor values calculated by our proposed formula and given by Dietz and Earlougher et al., which indicates that our proposed formula is reliable and reasonable accurate.
More examples are given in 1, 2 to calculate productivity index and pseudoskin factor due to partial penetration by using the proposed formulas, the values of shape factors obtained by the methods of Dietz, Earlougher, and the proposed shape factor formula are compared.The proposed formulas are shown to be reliable and reasonable accurate by the examples in 1, 2 , because the proposed equations are derived by solving analytically the involved three-dimensional Laplace equation, they are a fast analytical tool to evaluate well performance in pseudo-steady-state.

Summary and Conclusions
The summary and conclusions of this paper are given below.
1 A pseudo-steady-state productivity formula for an off-center partially penetrating vertical well in a closed box-shaped reservoir is presented.
2 A formula for calculating pseudoskin factor due to partial penetration is presented; the pseudoskin factor of a vertical well in a box-shaped reservoir is a function of well location and reservoir size.

4 . 5 and
d m , d n have similar definitions.According to the complete normalized orthogonal systems of the Laplace equation's basic solution, Dirac function has the following expression for homogeneous Neumann boundary condition 13, 14 :

I
J x, y, z; x , y , z ; l, m, n L 0 Ψ x, y, z; x , y , z dz l m n>0 I lmn x, y, z; x , y , z ; l, m, n , lmn x, y, z; x , y , z ; l, m, n l m n>0
Substituting 4.9 into left-hand side of 4.1 , and substituting 4.6 into right-hand side of 4.1 , we obtain n 0 e lmn t g lmn x, y, z , 4.9where e lmn t are undetermined coefficients.comparethe coefficients of g lmn x, y, z at both sides of 4.10 , we obtain lmn / 0 l m n > 0 , solve 4.14 , e lmn t 1 − exp −λ lmn t g lmn x , y , z
We use 6.6 , F D 86.4 for practical SI units, the productivity index the production rate per unit pressure drawdown in pseudo-steady-state of the given well is 16 ,

3
The proposed formulas are reliable and reasonable accurate, because the proposed formulas are derived by the orthogonal decomposition of Dirac function and Green's function to Laplace equation with homogeneous Neumann boundary condition, they are a fast analytical tool to evaluate well performance in pseudosteady-state.