Effects of Reservoir Boundary Conditions, Drainage Shape, and Well Location on Productivity of a Vertical Well

. This paper gives a review of steady state and pseudosteady state productivity equations for an unfractured fully penetrating vertical well in a permeability anisotropic reservoir. This paper also studies the effects of drainage area, reservoir boundary conditions, drainage shape, and well location on productivity. The production performances of an unfractured vertical well in a circular reservoir, a sector fault reservoir and a rectangular reservoir are studied and compared. Mechanical skin factor is included in the productivity equations. This paper examines the steady state and pseudosteady state production performance of oil wells with constant ﬂ ow rates in different drainage shapes, a library of productivity equations is introduced, several combinations of closed and/or constant pressure boundary conditions are considered at lateral reservoir boundaries. The equations introduced in this paper can be used to determine the economical feasibility of a drilling an unfractured fully penetrating vertical well. It is concluded that, drainage area and reservoir boundary conditions have signi ﬁ cant effects on productivity of a well, and productivity is a weak function of drainage shape and well location.


Introduction
Well productivity is one of primary concerns in field development and provides the basis for field development strategy. To determine the economical feasibility of drilling a well, petroleum engineers need reliable methods to estimate its expected productivity. Petroleum engineers often relate the productivity evaluation to the long-time performance behavior of a well, that is, the behavior during pseudosteady state or steady state flow [1].
Substituting Darcy's equation into the equation of continuity, the steady state productivity equation for an unfractured fully penetrating vertical well in a permeability isotropic circular reservoir with constant pressure outer boundary is obtained below [2]: where k is reservoir permeability; h is payzone thickness; μ is oil viscosity; P e , P w are pressures at drainage outer boundary and wellbore, respectively; R e , R w are radii of drainage area and wellbore, respectively; Q w is well flow rate; F D is the unit conversion factor [3]. In oil field units, F D ¼ 0:001127 [4]. The pseudosteady state productivity equation for an unfractured fully penetrating vertical well in a permeability isotropic circular reservoir with closed outer boundary is given by [5]: where P a is the average reservoir pressure throughout the circular drainage area.
To account for irregular drainage shapes or asymmetrical positioning of a well within its drainage area, the following productivity equation was proposed by Dietz [6]: where C A is shape factor and A is the drainage area.
Dietz [6] evaluated shape factor for rectangles with single well in various locations, but the shape factors obtained by Dietz are only applicable to rectangular shapes whose sides are integral ratios. Hagoort [7] presented an algorithm to calculate productivity of a well in a rectangle with arbitrary aspect ratio.
Well productivity is often evaluated using the productivity index, which is defined as the production rate per unit pressure drawdown [8]. Hagoort [9] presented an analytical formula for the stabilized productivity index of an arbitrary well in an arbitrary closed, naturally fractured reservoir that can be modeled as a double-porosity reservoir. Hagoort [10] also presented an algorithm to calculate the productivity index of a well with a vertical, infinite-conductivity fracture in a closed rectangular reservoir for a wide range of fracture lengths and reservoir aspect ratios. Friehauf et al. [11] developed a model to calculate the productivity index of a finiteconductivity fractured well, including the effect of fracture-face damage caused by fluid leakoff.
The production performance of multiple wells system has received attention in the last two decades.
Valko et al. [12] presented pseudosteady state productivity index for multiple wells producing from a closed rectangular reservoir. Umnuayponwiwat et al. [13] presented equations of inflow performance of multiple vertical and horizontal wells in closed systems. Marhaendrajana and Blasingame [14] presented a solution and associated analysis methodology to evaluate single well performance behavior in a multiple wells reservoir system.
Lu [1] presented steady state and pseudosteady state productivity equations for an off-center unfractured partially penetrating vertical well in a circular reservoir and a rectangular reservoir, but the mechanical skin factor due to formation damage or stimulation is not included in his equations.
The term vertical well in this paper is defined as a wellbore penetrating nearly vertically into a nearly horizontal, nonhydraulically fractured payzone [15].
A review of steady state and pseudosteady state productivity equations for an unfractured fully penetrating vertical well in a permeability anisotropic reservoir is given in this paper. The effects of drainage area, reservoir boundary conditions, drainage shape, and well location on productivity are also studied. The production performances of an unfractured vertical well in a circular reservoir, a sector fault reservoir, and a rectangular reservoir are compared. Several combinations of closed and/or constant pressure boundary conditions are considered at lateral reservoir boundaries, and mechanical skin factor is included in the productivity equations.
In this paper, all parameters of reservoir, wellbore, and fluid properties in the following equations are in oil field units as shown in Table 1. The unit conversion factor F D in the following equations is equal to 0.001127, F D ¼ 0:001127.

Reservoir Boundary Conditions and Drainage Shape
In this paper, the following assumptions are made: (1) The top and bottom boundaries of a three-dimensional reservoir are impermeable, and the reservoir is permeability anisotropic, the production occurs through an unfractured fully penetrating vertical well with radius R w , thus the three-dimensional reservoir is reduced to a two-dimensional reservoir. (2) Before production, the 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, and bottom water), the pressure P e is equal to the initial value P i at such boundaries during production. (3) 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. (4) There is no water encroachment or water/gas coning.
Edge water, gas cap, and bottom water are taken as constant pressure boundaries, multiphase flow effects are ignored.
In this paper, mechanical skin factor due to formation damage or stimulation is included in the productivity equations for an unfractured fully penetrating vertical well in a permeability anisotropic reservoir.
For an unfractured fully penetrating vertical well in an isotropic reservoir, if we refer to the additional pressure drop in the skin zone as ΔP s , then [16] k r is the radial permeability. In a rectangular reservoir, we always assume k r ¼ ffiffiffiffiffiffiffiffi ffi k x k y p . Figure 1 is a schematic of an off-center unfractured fully penetrating vertical well in a circular reservoir with radius R e and the off-center distance is R 0 . The following two cases of different lateral boundary conditions are considered for a circular reservoir: Case 1: Constant pressure lateral boundary Case 2: Impermeable lateral boundary ∂P ∂r Figure 2 is a schematic of an unfractured fully penetrating vertical well in a sector fault reservoir with radius R e , the well is on the bisector line, the well location angle θ w ¼ Φ=2, and sector angle Φ ¼ π=n; where n is an integer number, and R 0 is off-vertex distance of the well.
The two sides of the angle are impermeable, that is, the sector reservoir is with two sealing faults where ∂P=∂Nj OA;OB are the exterior normal derivatives of pressure on the two sides of angle of the sector area. Case 1: Constant pressure outer boundary If the outer boundary is with edge water, during production the pressure at the outer boundary is always equal to initial reservoir pressure P i .
Case 2: Outer boundary is impermeable ∂P ∂r Figure 3 is a schematic of an off-center unfractured fully penetrating vertical well in a rectangular reservoir with length (x direction) X e and width (y direction) Y e . The original point of the rectangular coordinate system is the lower left corner point of the rectangular area, the well is located at the point (X w , Y w ). And the following five cases of different lateral boundary conditions are applicable to a rectangular reservoir [17].
Case 1: the rectangular reservoir is surrounded by a strong edge water drive, such that the pressures at the four lateral boundaries are assumed constant and equal to the reservoir initial pressure during production   Journal of GeoEnergy 3 Case 2: Only two opposite lateral boundaries are at constant pressure. The other two opposite lateral boundaries are considered as no-flow (impermeable) boundaries Case 3: Only one lateral boundary is at constant pressure, while the other three lateral boundaries are considered as noflow (impermeable) boundaries Case 4: Only one lateral boundary is a no-flow (impermeable) boundary, while the other three are at constant pressure Case 5: Only two adjacent lateral boundaries are at constant pressure, while the other two adjacent lateral boundaries are no-flow (impermeable) boundaries The above five different lateral boundary conditions in Case 1 through Case 5 are shown in Figure 4(a) through Figure 4(e), respectively.

Productivity Equations in Steady State
This section presents the steady state productivity equations for an unfractured fully penetrating vertical well located in a circular reservoir, a sector reservoir and a rectangular reservoir, and the mechanical skin factor is included.

Circular Reservoir.
If an off-center unfractured fully penetrating vertical well is located in a permeability isotropic circular reservoir which has a constant pressure outer boundary, then the steady state productivity equation is [1]   Journal of GeoEnergy where R 0 is off-center distance of the well, S m is mechanical skin factor due to formation damage or stimulation, and F D is the unit conversion factor. In oil field units, F D ¼ 0:001127. If the well is in a permeability anisotropic reservoir, then

Impermeable boundary
The definitions of R eD ; R oD ; R wD are given in Appendix A. For a well located at the center, the off-center distance R 0 = 0, then Equation (16) reduces to If S m = 0, Equation (18) is equivalent to Equation (1), which is the well-known steady state productivity equation for an unfractured fully penetrating vertical well located at the center of a permeability isotropic circular reservoir in the literature [2].

Sector Fault Reservoir.
If an unfractured fully penetrating vertical well is located in a permeability anisotropic sector fault reservoir with outer boundaries indicated in Equations (8) and (9), the well is on the bisector line, the well location angle and n is an integer number, we have Γ is sector shape function in steady state, the definitions of R oD ; R wD are given in Appendix A. If permeability is isotropic, k r ¼ k; consequently, then For n = 1, 2, 3, 4, 5, 6, 8, 9, 10, and Γ can be found in Table 2.

Rectangular Reservoir.
If an off-center unfractured fully penetrating vertical well is located in a permeability isotropic rectangular reservoir which has at least one constant pressure outer boundary, we have the following five cases: Case 1: The lateral boundary condition is defined by Equation (11), then [1] where ð25Þ For a fully penetrating vertical well at the center of an isotropic square reservoir, we have ; then Equation (23) can be approximated by the following expression where

Journal of GeoEnergy
Equation (28) is the steady state productivity equation for a fully penetrating vertical well located at the center of a permeability isotropic square reservoir surrounded by a strong aquifer.

Case 3:
The lateral boundary condition is defined by Equation (13), then where Case 4: The lateral boundary condition is defined by Equation (14), then where Journal of GeoEnergy Case 5: The lateral boundary condition is defined by Equation (15), then where

Journal of GeoEnergy
If the reservoir is permeability anisotropic in the above five cases, then permeability k in Equations (23), (28), (32), (33), (38), and (43) should be replaced by ffiffiffiffiffiffiffiffi ffi k x k y p ; and X w ; Y w ; X e ; Y e ; R w in Equations (11)

Productivity Equations in Pseudosteady State
If all reservoir boundaries are impermeable, and the producing time is sufficiently long, then the pseudosteady state can be reached [3], and the wellbore pressure must decline at the same rate as the average reservoir pressure. This section presents pseudosteady state productivity equations for an unfractured fully penetrating vertical well located in a permeability anisotropic circular reservoir, a sector fault reservoir, and a rectangular reservoir, and the mechanical skin factor is included.

Circular Reservoir.
The productivity of an off-center unfractured fully penetrating vertical well in a permeability anisotropic closed circular reservoir in pseudosteady state can be calculated by [1] where and P a is the average reservoir pressure throughout the circular drainage area, the definitions of R eD ; R oD ; R wD are given in Appendix A.
If permeability is isotropic, k r ¼ k; consequently, and For a well located at the center, the off-center distance R 0 = 0, then Equation (55) reduces to If S m = 0, Equation (56) is equivalent to Equation (2), which is the well-known pseudosteady state productivity equation for an unfractured fully penetrating vertical well located at the center of a permeability isotropic closed circular reservoir in the literature [5]. (8) and (10), the well is on the bisector line, the well location angle θ w ¼ Φ=2 and sector angle Φ ¼ 2π=m; we have

Sector Fault Reservoir. If an unfractured fully penetrating vertical well is located in a permeability anisotropic closed sector fault reservoir with outer boundaries indicated in Equations
Journal of GeoEnergy Λ is sector shape function in pseudosteady state, the definitions of R eD ; R oD ; R wD are given in Appendix A.
If permeability is isotropic, k r ¼ k; consequently, and For m = 2, 3, 4, and 6, Λ can be found in Table 3.

Rectangular Reservoir.
If an off-center unfractured fully penetrating vertical well is located inside a permeability anisotropic closed rectangular reservoir, then the well productivity in pseudosteady state can be calculated by [1] where P a is average reservoir pressure throughout the rectangular drainage area, and The definitions of X wD ; Y wD ; X eD ; Y eD ; R wD are given in Appendix A.
If permeability is isotropic, k x ¼ k y ¼ k; consequently, and If a well is located at the center of a closed rectangular reservoir, then Equation (63) reduces to If a well is at the center of a closed square reservoir, X e ¼ Y e ; then Equation (66) can be further simplified to Equation (67) is the pseudosteady state productivity equation for an unfractured fully penetrating vertical well located at the center of a permeability isotropic closed square reservoir.
Recall Equation (3), Dietz [6] developed the following equation to calculate the productivity of a vertical well in a permeability isotropic closed rectangular reservoir, where C A is shape factor, which is used to account for asymmetrical positioning of a well within a rectangular reservoir.

Application and Analysis
All the above equations proposed by the authors of this paper are only applicable to an unfractured fully penetrating vertical well in a permeability anisotropic reservoir. The equations for partially penetrating vertical wells, horizontal wells, or hydraulically fractured wells will be available in our future work. The performance of oil reservoirs is highly affected by many parameters, such as formation petrophysical properties, fluid properties, and reservoir and wellbore configuration. The effects of these parameters on reservoir performance are represented by productivity index (PI). PI is one of the major parameters in reservoir-management/development plans. It is defined as the surface production rate per unit pressure drawdown, Journal of GeoEnergy In steady state, ΔP ¼ P i − P w is constant with time; in pseudosteady state, ΔP ¼ P a − P w is also constant with time. Thus, in steady state and pseudosteady state, unless the conditions progressively deteriorate because of formation damage and skin factor, PI is constant.

Effects of Drainage Shape and Well Location
Example 1. Figure 5 shows a square reservoir X e ¼ Y e ð Þand its inscribed circular reservoir. A fully penetrating vertical well is located at X w ; ð Y w Þ in the square reservoir. Assume Y w ¼ Y e =2 ¼ X e =2 and X w is variable. Calculate steady state productivity indexes if the square reservoir is with constant pressure outer boundary. And calculate the productivity indexes of the well which is located in the inscribed circular reservoir with constant pressure outer boundary. Reservoir and fluid properties data in field units are given in Table 4. Mechanical skin factor S m ¼ 0. (8) and (23), and note that

Solution. We use Equations
Steady state productivity indexes of the well are given in Table 5.
Example 2. If the outer boundaries are impermeable in Figure 5, calculate the productivity indexes in pseudosteady state. Reservoir and fluid properties data are the same as those given in Table 4.
Tables 5 and 6 indicate that in steady state and in pseudosteady state, the productivity index is a weak function of drainage shape and well location, neither drainage shape nor well location has significant effects on the production performance of a well.
Steady state is dominated by a constant pressure outer boundary flow regime, which implies that the same volume of fluid is being moved at the wellbore and at the outer boundary. Table 5 indicates that in steady state, for a given well, when the off-center distance increases (when R 0 increases and X w decreases) the productivity index also increases. Because    the off-center well is near the constant pressure outer boundary, under the same pressure drop, the fluid moves through a short distance into the off-center well, consequently the productivity index is higher [8]. Pseudosteady state is dominated by a closed outer boundary flow regime, the produced fluid is evenly distributed in the reservoir. There is a zero flow rate at the closed outer boundary and maximum flow rate at the wellbore. Table 6 indicates that in pseudosteady state, for a given well, when the offcenter distance increases, (when R 0 increases and Xw decreases), the productivity index decreases. Because in pseudosteady state, no driving force is from the closed outer boundary, for a well located at the center of a closed reservoir, all flowlines toward the wellbore are radial or parallel to each other, no curved flowlines; but for an off-center well, the flowlines toward the wellbore are curved, more energy is dissipated under the same pressure drop, and consequently the productivity index is smaller [18].

Effects of Drainage Area
Example 3. A fully penetrating vertical well is located on the bisector line of a sector fault reservoir.
The sector radius R e ¼ 1000 ft, the off-vertex distance R 0 ¼ 500 ft, wellbore radius R w ¼ 0:5 ft, other reservoir, and fluid properties data are the same as those given in Table 4. Mechanical skin factor S m ¼ 0. Calculate the steady state and pseudosteady state productivity index when the sector angle Φ ¼ π; π=2; π=3: 5.2.1. Solution. The equations in Table 2 and 3 are used in Equations (22) and (60), the productivity indexes are given in Table 7. Table 7 indicates that in steady state and in pseudosteady state, the productivity index is a strong function of drainage area. Table 7, when the sector angle Φ decreases, steady state productivity index decreases. Because when Φ decreases, the drainage area also decreases, the length of the constant pressure outer boundary decreases, the driving force from the outer boundary decreases, and consequently the productivity index decreases. Table 7 also indicates that in pseudosteady state, when the sector angle Φ decreases, the productivity index increases. No driving force is on the closed outer boundary, the produced fluid is evenly distributed in the reservoir. When Φ decreases, the drainage area also decreases, the fluid moves through a shorter distance into the wellbore, less energy is dissipated under the same pressure drop, and consequently the productivity index increases.

Effects of Reservoir Boundary Conditions
Example 4. A fully penetrating vertical well is located at the center of a square reservoir. Reservoir and fluid properties data are the same as those given in Table 4. Mechanical skin factor S m ¼ 0. Calculate the steady state productivity index for each case in for the five cases in Figure 4, by using Equations (23), (32), (33), (38), and (43), we can obtain the productivity indexes shown in Table 8. Table 8 indicates that reservoir boundary conditions have significant effects on the production performance of a well.
As shown in Table 8, the productivity index of Case 1 is the biggest in the five cases as shown in Figure 4, and the productivity index of Case 3 is the smallest. Because all outer boundaries are at constant pressure for Case 1, the driving forces are from four boundaries, consequently the productivity index is the biggest. For Case 3, only one constant pressure outer boundary, the driving force is only from one outer boundary, thus the productivity index is the smallest in the five cases. Three constant pressure outer boundaries for Case 4, thus the productivity index is the second biggest. For Case 2 and Case 5, only two outer boundaries are at constant pressure, thus their productivity indexes are with intermediate values.

The Comparisons of Lu's Method and Dietz's Method Example 5.
A fully penetrating vertical well is located at the center of a closed rectangular reservoir. Reservoir width Y e = 2000 ft, other reservoir data and fluid properties data are the same as those given in Table 4. Mechanical skin factor S m ¼ 0. If aspect ratio X e =Y e ¼ 1; 2; 4; 5; 4:5, calculate the pseudosteady state productivity index by Lu's method and Dietz's method.

Solution.
When X e =Y e ¼ 1; we use Equation (67) to calculate the productivity index, then For a well located at the center of a closed square reservoir, shape factor C A is equal to 30.9 (Dietz, 1965), Equation (68) When aspect ratio X e =Y e ¼ 4; 5, the calculation results are shown in Table 9, we can find that little difference between the results in each case obtained by the two methods.
Dietz's method can not be used to calculate PI when X e =Y e ¼ 4:5.
It must be pointed out the shape factors obtained by Dietz are only applicable to rectangular shapes whose sides are integral ratios, that is X e =Y e ¼ 1; 2; 4; 5, but our proposed equations are applicable to a rectangular reservoir with arbitrary aspect ratio, shape factors are not required.

Summary
The disadvantage of steady state and pseudosteady state productivity equations in the literature is that those equations are only applicable to permeability isotropic reservoirs. The advantage of the productivity equations given by the authors of this paper is that the proposed equations are applicable to permeability anisotropic reservoir and can be used to study the effects of drainage area, reservoir boundary conditions, drainage shape, and well location on steady state and pseudosteady state productivity.
Another advantage of the productivity equations given by the authors is that the proposed equations are applicable to a well arbitrarily located in a circular reservoir and a rectangular reservoir. A summary of productivity equations for a permeability isotropic reservoir is given in Table 10.
The disadvantage of the productivity equation and the shape factors obtained by Dietz [6] is that Dietz's equation and shape factors are only applicable to rectangular shapes whose sides are integral ratios. The advantage of the equations given by the authors of this paper is that the proposed equations are applicable to a rectangular reservoir with arbitrary aspect ratio. The equations proposed by the authors can calculate the productivity of a well directly, shape factors are not required, thus shape factors for a rectangular reservoir is out of date.