A Comprehensive Solution for Partially Penetrating Wells with Various Reservoir Structures

1. Introduction

The pressure behavior of partially penetrating wells (PPWs) has been investigated extensively in the literature [1-5]. Many scholars have considered flow of the fluid in the reservoir toward the well as a radial flow with an additional pressure drop near the wellbore caused by the partial penetration [6-11]. This additional pressure drop is characterized by a positive geometrical skin that is called pseudoskin, due to partial completion. Several analytical and empirical dispositions have been developed to evaluate the pseudoskin and the time for which the assumption of radial flow becomes valid [9-13]. On the basis of the spherical flow, some researchers have provided methods for interpreting pressure transient well test data [14-17]. Many solutions have been developed to the two dimensional (2D) diffusivity equation, which included flow of the fluid in the vertical direction [14, 18-22].

On the basis of all the studies performed on PPWs, the presented solutions have some limitations. The most important restriction is the non-generality of these solutions which cannot be used for a wide range of parameters. In fact, most of the proposed methods developed for estimating the pseudoskin are not valid for all the time periods of the test, especially for the early time period. On the other hand, in most studies, wellbore storage and skin factor have not been included in the solutions. To obtain a comprehensive analytical formula for different reservoirs, some researchers have recently developed several new analytical solutions for different types of the well-reservoir configurations [23-27]. As our main novelties, we present the objectives of this study as follows:

I.          Deriving a general analytical solution that can be used to model the pressure transient behavior of a PPW completed in a reservoir with a diversity of structures.

II.          Examining the validity of the presented technique by analyzing the responses of a PPW in a special case: homogenous reservoir with infinite radial extent.

The organization of this paper is as follows. In section 2, some basic concepts, which are required for the subsequent sections are introduced. In section 3, the analytical solution of the responses of a PPW is derived. Then, the analytical solution for the pressure behavior of a special case is given in section 4. Afterwards, the proposed solutions obtained in the previous sections are assessed and discussed in section 5. Finally, section 6 provides some concluding remarks.

2. Basic concepts

Before introducing the main results, some important preliminary concepts should be explained. These concepts are given in the following three subsections.

2.1. Source function

On the basis of the results obtained by Gringarten [28], if  and  are the source domain and boundary of the reservoir, respectively and  and  are dummy points in the source and boundary, respectively, then the pressure drop at location  and time , in the reservoir with initial pressure distribution , and a prescribed pressure or flux at the boundary , can be written as [23]:

(1)

where the pressure drop is defined as follows:

in which  denotes reservoir domain. For a reservoir with an initial uniform and constant pressure, , Eq. (2) can be expressed as:

The term  or  denotes the instantaneous Green’s function. The expression  is the withdrawal (or injection) rate per unit volume (). Moreover,  represents the directional derivative of the function  or  at a point  in the outward direction of the boundary , where the vector  is normal to the boundary . For an infinite reservoir, the second term on the right-hand side of Eq. (1) approaches zero [23].

For simplicity, it can be assumed that the withdrawal flow rate is uniform over the source volume for a given reservoir, . Since the second term on the right-hand side of Eq. (1) is not a function of the flow rate, it can be concluded that the given reservoir has a similar behavior to infinite reservoirs. Thus, Eq. (1) can be expressed as

where  is the instantaneous source function that is defined as:

The integral term on the right-hand side of Eq. (4) is called the continuous source function. For a source with constant fluid withdrawal rate, , Eq. (4) can be simplified to:

It should be noted that Eq. (5) can be developed by means of the superposition principle.

2.2. Newman’s method

The application of Newman’s product method in the context of petroleum engineering can be expressed as follows: for a given reservoir, which can be considered as the intersection of one (and or two) dimensional reservoirs, the instantaneous Green’s function equals to the multiplication of the instantaneous Green’s functions for the one (and or two) dimensional reservoirs. In a similar approach, for a source that can be considered as the intersection of one (and or two) dimensional sources, the instantaneous source function is equal to the multiplication of the instantaneous source functions for the one (and or two) dimensional sources [29]. For example, the intersection of two perpendicular infinite plane sources constitutes an infinite line source. Thus, by making use of the Newman’s product method, the instantaneous source function of an infinite line source is equal to the product of the instantaneous source functions of the corresponding plane sources:

And for another example, it can be considered a point source as the intersection of the three perpendicular infinite plane sources. Then, the instantaneous point source function can be obtained by means of the Newman’s product method as:

If the point source is constituted by the intersection of an infinite plane source perpendicular to an infinite line source, then the corresponding instantaneous source function can be obtained as:

(9)

2.3. Gaver-Stehfest algorithm

The Gaver–Stehfest algorithm for the numerical inversion of Laplace transform was developed in the late 1960s [30]. This well-known method can be expressed as follows: the unknown pressure  for given  and  is obtained by using the Gaver-Stehfest algorithm to numerically invert the Laplace solutions. The procedure is described by the following equations:

and the coefficients  are given by:

where

3. Analytical solution of the pressure transient behavior

The analytical solution of the pressure transient responses for a PPW flowing at a constant rate is now derived based on the source functions and the Newman’s product method. Before presenting the solutions, some useful dimensionless variables are defined as follows:

Dimensionless pressure: ,

Dimensionless time: ,

Dimensionless vertical distance: ,

Dimensionless midpoint of the perforated interval: ,

Dimensionless radius: ,

Dimensionless anisotropy group:  and

Penetration ratio:

The basic instantaneous source function for an infinite slab source in an infinite slab reservoir can be written as [24]:

where the top and bottom of the formation are impermeable. The instantaneous source functions for different boundary conditions can be found in [24]. It is worth to say that the following approach will work for a variety of the boundary conditions by choosing an appropriate source function. The total withdrawal rate from the well is:

where  is the flow rate per unit length of the source.

From the definition of the dimensionless pressure drop, it can be concluded that:

where the subscript  stands for a fully penetrating well (FPW).  Since the dimensionless pressure drop is obtained by integrating the instantaneous source function with respect to time from  to , taking the time derivative of Eq. (15) yields:

(16)

where  represents the instantaneous source function for a PPW in -direction. On the other hand, the dimensionless pressure drop for a FPW can be written as

(17)

Differentiating both sides of Eq. (17), the instantaneous source function of a FPW can be obtained as follows:

Substituting Eq. (18) into Eq. (16) yields the instantaneous source function of a partially penetrating as:

By using the Newman’s product method, the instantaneous source function of the PPWs can be attained as:

According to Eq. (20), the instantaneous source function for a well in partial penetration can be captured by combining Eq. (19) with Eq. (13); i.e,

(21)

which can be simplified to the following from:

where

Thus, the pressure response for a PPW produced at a constant rate can be written as:

(24)

By setting Eq. (21) into Eq. (24), the dimensionless pressure can be obtained as:

(25)

The calculation of the pressure response of a PPW from Eq. (25) in an integration form is a computationally expensive process. This complexity arises from two reasons. First, the time derivative of  must be evaluated. Second, the integration term must also be computed. These critical problems can be solved by transforming Eq. (25) to Laplace space, since the Laplace transformation can eliminate the derivative expression  and the integral term. The Laplace transform of Eq. (25) yields

(26)

where

(27)

is the Laplace transform of Eq. (23), and  is the Laplace variable. The simple poles of Eq. (27) are located at . So, the final result of the pressure drop without wellbore storage and skin can be written as:

(28)

The dimensionless pressure at the wellbore radius can be obtained by setting  in Eq. (28):

(29)

or

(30)

Partial completion of a well causes an additional pressure drop other than the pressure drop resulting from the full penetration, . This additional pressure drop is called geometrical skin or pseudoskin. The first term on the right-hand side of Eq. (30) is the dimensionless pressure drop for a FPW. So, it can be concluded that the pseudo skin pressure drop due to partial penetration is given by:

(31)

or

(32)

It can be shown that the dimensionless wellbore pressure can be computed at , where  [31-33]. Therefore, Eqs. (30) and (32) can be written as:

(33)

and

(34)

with given penetration ratio, , dimensionless anisotropy group, , dimensionless midpoint of perforated interval, , and the calculated dimensionless vertical distance, .

Dimensionless wellbore pressure with storage and skin for PPWs can be obtained by means of the superposition theorem which was given by van Everdingen and Hurst [34] and Agarwal et al. [35]:

where  and  are the dimensionless wellbore pressure with wellbore storage and skin effects, and  dimensionless sandface pressure for the constant-rate case without the wellbore storage and skin effects, respectively. Moreover,

steady-state skin factor,          

,

, dimensionless wellbore storage,

, dimensionless sandface rate

The Laplace transform of Eq. (35) yields

(36)

Replacing the  term in Eq. (36) by the right-hand side of Eq. (33) yields the dimensionless pressure for PPWs with wellbore storage and skin. Dimensionless wellbore pressure values can be calculated from Eq. (36) by means of the Gaver-Stehfest numerical Laplace transform inversion algorithm [25].

4. Application of the proposed method

By applying the presented technique in section 3 to different types of reservoirs for analyzing the pressure behavior of PPWs, various mathematical models may be obtained. The simplest case is a homogenous and an infinite reservoir that is completed partially and is affected by wellbore storage and skin. Applying the proposed method to this case gives the following formulations.

The pressure response of a homogeneous, isotropic and infinite radial system with a FPW without wellbore storage and skin is given by:

Substituting Eq. (37) into Eq. (28) gives the pressure response of a PPW:

Equation (38) reduces to the following equation at the wellbore by considering  and :

(39)

where the pseudoskin is

(40)

Substituting Eq. (39) into Eq. (36) yields the dimensionless wellbore pressure with wellbore storage and skin. Figure 1 shows the typical responses of a PPW in an infinite reservoir. For example, for a well that is perforated in the half-bottom interval where , Eqs. (39) and (40) can be simplified to:

(41)

and

(42)

In addition, Eqs. (41) and (42) can be used when the half-top is perforated if the -direction in the Cartesian coordinate is reversed.

5. Results and discussion

The pressure responses of a PPW in a reservoir with impermeable lower and upper boundaries have been presented in Figures 1, 2 and 3 with  versus  dimensionless variables. The well properties of the provided examples are summarized in Table 1. Figure 1 shows the pressure behavior of a PPW with different values of penetration ratio, given the values of dimensionless midpoint of perforated interval and dimensionless anisotropy group, where the value  implies the FPW. Figure 2 depicts the effect of dimensionless midpoint of perforation interval with constant penetration ratio and dimensionless anisotropy group. The influence of dimensionless anisotropy group is analyzed by considering three different values, where the perforation ratio and the dimensionless midpoint of the perforation interval are given both by  (Figure 3). The dimensionless wellbore storage coefficient and skin have been assumed as  and , respectively.

Table 1. The well and reservoir properties of the study cases

As can be seen from Figures 1, 2 and 3, two radial flow regimes and one spherical flow between them can be observed in the responses for a PPW. At early times, is the initial radial flow over the perforated interval , with  proportional to  and a first derivative stabilization. Then, the spherical flow can be observed with  proportional to   and a negative half unit slope straight line on the derivative log-log curve. The flow lines are established in both the horizontal and vertical directions, until the lower and upper boundaries are reached, which is the end of the spherical flow regime. Finally, the flow becomes radial in the entire formation thickness, with  proportional to  and a second derivative stabilization.

Figure 4 presents the pseudoskin due to the partial penetration for three different penetration ratios, which indicates that the smaller  causes the higher pseudoskin. Based on the results shown in Figures 1 and 4, as the penetration ratio decreases, the pseudoskin and pressure drops increase. The results of Figure 3 show that with low dimensionless anisotropy group, the contribution of vertical flow is limited and the spherical flow is started later ( and  are assumed constant). When the completed interval is not centered within the entire formation thickness, the spherical flow ends when the closest bottom or top of the formation is reached. A hemi-spherical flow regime is then observed instead of the spherical flow, until the second boundary is reached (Figure 2). Furthermore, Figure 2 reveals that the pressure and its derivative results for the PPW are overlapped for the cases  and . It implies that if the well is symmetrically penetrated either in the half-top or half-bottom interval, the fluid flow in the reservoir will behave in a similar way.

The presented three different flow regimes can be analyzed to estimate the well and reservoir parameters. The permeability-thickness product for the perforated interval , and the skin,  can be obtained by analyzing the initial radial flow regime. The analysis of the spherical flow regime yields the permeability anisotropy. In addition, the permeability-thickness product of the reservoir , and the total skin  can be determined from the second radial flow regime.

Figure 1. Comparison of the responses of fully and partially penetrating wells for different values of penetration ratio

Figure 2. Effect of dimensionless midpoint of the perforated interval on dimensionless pressure drop and its logarithmic derivative

Figure 3. Effect of dimensionless anisotropy group on dimensionless pressure drop and its logarithmic derivative

Figure 4. Pseudoskin for different values of penetration ratio when  and .

6. Conclusions

Based on the results presented and discussed earlier in this study, the following conclusions can be drawn:

1)    A new analytical constant flow rate solution is derived based on the Green’s function approach in Laplace space that describes the pressure behavior of partially penetrating (limited entry) wells at the wellbore when wellbore storage and skin effects are significant.

2)    For any reservoir type and boundary condition, solutions of the PPWs can be obtained from the FPWs responses. Generality and simplicity of this technique can be used to define the responses for a PPW completed at any position in any region.

3)    The pseudoskin pressure drop due to partial penetration increases as the spherical flow develops and then reaches its maximum value at large times. This is different from the mechanical skin, which is caused by formation damage in the drilling and completion operations.