A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture

The Pseudo Steady-State (PSS) constant bDpss is defined as the difference between the dimensionless wellbore pressure and dimensionless average pressure of a reservoir with a PSS flow regime. As an important parameter, bDpss has been widely used for decline curve analysis with Type Curves. For a well with a finite-conductivity fracture, bDpss is independent of time and is a function of the penetration ratio of facture and fracture conductivity. In this study, we develop a new semi-analytical solution for bDpss calculations using the PSS function of a circular reservoir. Based on the semi-analytical solution, a new conductivity-influence function (CIF) representing the additional pressure drop caused by the effect of fracture conductivity is presented. A normalized conductivity-influence function (NCIF) is also developed to calculate the CIF. Finally, a new approximate solution is proposed to obtain the bDpss value. This approximate solution is a fast, accurate, and time-saving calculation.

Pseudo steady-state (PSS) is a boundary-dominant flow regime created when pressure waves spread to the boundary in a closed drainage area.In this flow regime, the relationship of dimensionless wellbore pressure and dimensionless average pressure can be expressed as [20][21][22][23][24]: where b Dpss is the Pseudo Steady-State (PSS) constant [24].This PSS constant b Dpss has been widely used to define the appropriate dimensionless decline rate in many currently used production-decline rate analysis models [20,[25][26][27][28].
When the pressure disturbance reaches to the boundary, the PPS flow regime occurs and the PSS constant b Dpss can be obtained by running a long-term numerical simulation [20,21,23].Pratikno et al. [20] presented a numerical solution for the PSS constant for a well with a finite-conductivity fracture in a circular closed reservoir (Figure 1).For convenience, bDpss is usually approximated as an analytical expression [20,21,23].Pratikno et al. [20] proposed the following approximate expression: ( ) ( ) where: ( ) ( ) 2 ln 0.049298 0.43464 0.936268 1.00489 ψ 0.319733ψ 0.0423532 ψ 0.00221799 ψ 1 0.385539 ψ 0.0698865 ψ 0.0484653 ψ 0.00813558 and: fD C ln ψ = (5) Equation ( 3) is the pressure drop of a well with an infinite-conductivity fracture in the PSS flow regime [11].The second term in Equation (2), f(CfD), is the additional pressure drop caused by the effect of fracture conductivity.We define f(CfD) as the conductivity-influence function (CIF).Thus, the PSS constant bDpss can be expressed as the sum of the pressure drop of a well with an infiniteconductivity fracture and the conductivity-influence function (CIF).Note that the CIF in Equation ( 4) is only a function of the fracture conductivity.
Wang et al. [23] also introduced an approximate expression for f(CfD) using regression for a circular reservoir.
( ) 0.95 0.56 ψ 0.16 ψ 0.028 ψ 0.0028 ψ 0.00011ψ 1 0.094 ψ 0.093 ψ 0.0084 ψ 0.001ψ 0.00036 ψ For the low permeability and ultra-low permeability reservoirs, the elliptical boundary has been used to approximately represent circular a reservoir for calculation of bDpss.The corresponding conductivity-influence function for an elliptical reservoir was presented by Amini et al. [21]: In addition, assuming an elliptical boundary, two analytical solutions for a well with an infiniteconductivity fracture and a finite-conductivity fracture were developed by Prats et al. [29] and Lu et al. [24], respectively.
As analyzed from a previous statement, three problems occur with the bDpss calculation.For convenience, b Dpss is usually approximated as an analytical expression [20,21,23].Pratikno et al. [20] proposed the following approximate expression: where: and: Equation ( 3) is the pressure drop of a well with an infinite-conductivity fracture in the PSS flow regime [11].The second term in Equation (2), f (C fD ), is the additional pressure drop caused by the effect of fracture conductivity.We define f (C fD ) as the conductivity-influence function (CIF).Thus, the PSS constant b Dpss can be expressed as the sum of the pressure drop of a well with an infinite-conductivity fracture and the conductivity-influence function (CIF).Note that the CIF in Equation ( 4) is only a function of the fracture conductivity.
Wang et al. [23] also introduced an approximate expression for f (C fD ) using regression for a circular reservoir.
For the low permeability and ultra-low permeability reservoirs, the elliptical boundary has been used to approximately represent circular a reservoir for calculation of b Dpss .The corresponding conductivity-influence function for an elliptical reservoir was presented by Amini et al. [21]: In addition, assuming an elliptical boundary, two analytical solutions for a well with an infinite-conductivity fracture and a finite-conductivity fracture were developed by Prats et al. [29] and Lu et al. [24], respectively.
As analyzed from a previous statement, three problems occur with the b Dpss calculation.
(1) The assumption of elliptical flow is an approximate model of the circular boundary [29].For a fractured well, the real pressure front should be a circle instead of an ellipse during the late-time flow regime.(2) When C fD trends to infinity, i.e., the infinite-conductivity fracture, Equations ( 4), (6), and (7) cannot meet the following condition, meaning the limit of conductivity-influence function f (C fD ) should be zero.lim (3) For the existing approximate models [20,21,23], the conductivity-influence function f (C fD ) is only relative to the fracture conductivity.For different penetration fracture ratios I x , the distributions in the flow field around the fracture are different, which affects the value of conductivity-influence function (Figure 2).Thus, CIF is not only a function of conductivity, but also related to penetration ratio.
(1) The assumption of elliptical flow is an approximate model of the circular boundary [29].For a fractured well, the real pressure front should be a circle instead of an ellipse during the late-time flow regime.(2) When CfD trends to infinity, i.e., the infinite-conductivity fracture, Equations ( 4), (6), and (7) cannot meet the following condition, meaning the limit of conductivity-influence function f(CfD) should be zero.
( ) (3) For the existing approximate models [20,21,23], the conductivity-influence function f(CfD) is only relative to the fracture conductivity.For different penetration fracture ratios Ix, the distributions in the flow field around the fracture are different, which affects the value of conductivityinfluence function (Figure 2).Thus, CIF is not only a function of conductivity, but also related to penetration ratio.
(a) (b) In this paper, based on the assumption of a circular closed reservoir, we extended and corrected the work of Pratikno et al. [20].The contributions of our work include: (1) a semi-analytical method is developed to calculate the bDpss by use of the PSS function instead of the transient-pressure function [20]; (2) based on the results from the semi-analytical method, a new conductivity-influence function (CIF) is introduced considering the effect of penetration ratio and fracture conductivity has been established; and (3) a new normalized conductivity-influence function (NCIF) is introduced to calculate the value of bDpss.

Basic Assumptions
As shown in Figure 1, a vertical fractured well is located in the center of the circular closed isotropic formation with radius re.The flow in the reservoir and fracture is assumed to be single phase and isothermal with a slightly compressible Newtonian fluid.The penetrate ratio Ix is defined as:

Semi-Analytical Model for bDpss Calculation
Different from the transient-pressure-function method presented by Pratikno et al. [20], we derive the semi-analytical model with the PSS function.The advantages of our method include: (1) we can obtain the PSS pressure directly instead of using the long-term approximation of transient pressure, and (2) without the numeric inversion, our method is more accurate and is less time consuming.

Basic Assumptions
As shown in Figure 1, a vertical fractured well is located in the center of the circular closed isotropic formation with radius r e .The flow in the reservoir and fracture is assumed to be single phase and isothermal with a slightly compressible Newtonian fluid.The penetrate ratio I x is defined as:

Semi-Analytical Model for b Dpss Calculation
Different from the transient-pressure-function method presented by Pratikno et al. [20], we derive the semi-analytical model with the PSS function.The advantages of our method include: (1) we can obtain the PSS pressure directly instead of using the long-term approximation of transient pressure, and (2) without the numeric inversion, our method is more accurate and is less time consuming.

Flow Model of the Reservoir
The fracture is equally divided into 2N segments.Each segment can be considered as a uniform-flux fracture [30].Therefore, the uniform-flux solution of a fracture located in a closed circular reservoir can be used to calculate the pressure [11].To consider the fracture symmetry, we focused on half of the fracture.According to the superposition principle, the dimensionless pressure of the ith segment in the reservoir can be written as: where F is the function presented by Ozkan [11] for a circular closed reservoir in the PSS flow regime.Equation ( 10) can be written in matrix form as:

Flow Model of the Fracture
Luo and Tang [30] derived a wing solution in the discretized form and the pressure of the ith segment, which can be expressed as: with Equation ( 13) can be written in matrix form as:

Semi-Analytical Solution for b Dpss
According to the continuity condition, which states that the pressure and flux must be continuous along the fracture surface, the following conditions must hold along the wing plane: Substituting Equation (15) into Equations ( 11) and ( 14) yields: Substituting Equation (1) into Equation ( 16), we obtain: In addition, the total flow rate satisfies the following: b Dpss can be obtained by solving Equations ( 17) and ( 18) with the Gauss elimination method.

Conductivity-Influence Function (CIF)
The procedure to calculate the conductivity-influence function are as follows: (1) Calculate the PSS constant b Dpss for different I x and C fD using the semi-analytical model in Section 2.2.(2) Calculate the pressure drop of the infinite-conductivity fracture f (r eD ) for different I x with Equations ( 3) and ( 9).
(3) Calculate the difference of b Dpss and f (r eD ) with Equation ( 2) and the value of conductivity-influence function (CIF) can be obtained.
Figure 3 presents the CIF for different penetration ratios and fracture conductivities.Different than the solutions obtained by Pratikno et al. [20], the CIF is not only dependent on fracture conductivity but also has a strong relationship with penetration ratio, especially when C fD is less than 10.Additionally, fracture conductivity function tends to be zero when C fD is greater than 300.Table 1 presents the value of the conductivity-influence function for I x = 0.001-1 and C fD = 0.1-1000.
( ) In addition, the total flow rate satisfies the following: bDpss can be obtained by solving Equations ( 17) and (18) with the Gauss elimination method.

Conductivity-Influence Function (CIF)
The procedure to calculate the conductivity-influence function are as follows: (1) Calculate the PSS constant bDpss for different Ix and CfD using the semi-analytical model in Section 2.2.(2) Calculate the pressure drop of the infinite-conductivity fracture f(reD) for different Ix with Equations ( 3) and ( 9).
(3) Calculate the difference of bDpss and f(reD) with Equation ( 2) and the value of conductivity-influence function (CIF) can be obtained.

New Approximate Solution of Pseudo Steady-State Constant b Dpss
Based on the above work, we redefined the PSS constant as the sum of two parts: PSS constant for infinite fracture conductivity [11] and the conductivity-influence function.The PSS constant for infinite fracture conductivity b Dpss,FC is the function of the penetration ratio, and the conductivity-influence function is the function of the penetration ratio and conductivity.
We firstly calculate a specific case (I x = 0).As shown in Equation ( 9): If the circular closed reservoir is replaced by an infinite-acting reservoir, b Dpss can be approximately replaced by the dimensionless-pressure difference between the finite-conductivity fracture and infinite-conductivity fracture at the radial flow regime for the infinite reservoir [15].
Figure 4 illustrates the CIF of different conductivity at I x = 0 (blue) and I x = 1 (red).Two regression equations for CIF are obtained as follows.
For the case of I x = 0: where: and: For the case of I x = 1, the hydraulic fracture fully penetrates the reservoir. where: We notice that in our model, Equations ( 22) and ( 25) meet the following condition: lim Moreover, we define a normalized fracture conductivity function (NCIF) as follows: We calculate the CIF for I x = 0.001-1 and C fD = 0.1-1000 with the method in Sections 2.2 and 2.3.Ten values were calculated for each logarithmic period.Thus, 1200 values of CIF were obtained.The values of NCIF can be calculated using Equation (28). Figure 5 shows the relationship between NCIF and Ix.Notably, all data fall in the same straight line in logarithmic coordinates.This means that the NCIF is solely dependent on penetration ratio Ix.
A regression equation can be obtained for NCIF.
Note that if Ix = 0, the NCIF is equal to 0 in Equation (29).
Recasting Equation (29) yields: The limit of Equation ( 30) is equal to zero.
Substituting Equations ( 20) and (30) into Equation ( 19), we obtain: Based on our work, a new bDpss is presented in Equation (32). Figure 5 shows the relationship between NCIF and I x .Notably, all data fall in the same straight line in logarithmic coordinates.This means that the NCIF is solely dependent on penetration ratio I x .
A regression equation can be obtained for NCIF.
Note that if I x = 0, the NCIF is equal to 0 in Equation (29).
Recasting Equation (29) yields: The limit of Equation ( 30) is equal to zero.lim Substituting Equations ( 20) and (30) into Equation (19), we obtain: Based on our work, a new b Dpss is presented in Equation (32).

Results
In this section, we will compare our approximate model with our semi-analytical model and Pratikno et al.'s approximate solutions [20] for different penetrate ratio and fracture conductivity.Table 2 presents the values of bDpss obtained by our semi-analytical method, Pratikno et al.'s method and our approximate model.As shown in Table 2, the maximum relative error is 0.4% at Ix = 0.1 and 1.93% at Ix = 1 between our approximate model and semi-analytical model.However, huge differences between Pratikno et al.'s method and our semi-analytical method can be observed at large penetration ratio, for example, relative error 16.57% at CfD = 0.631 and Ix = 1.
We further show the comparisons in Figure 6.The circles with crosses, red lines, and blue lines correspond to the semi-analytical solutions (accurate solutions), Pratikno et al.'s solutions, and approximate solutions, respectively.As shown in Figure 6, an excellent agreement among the three methods is visible when the dimensionless fracture conductivity CfD is greater than 10 for all penetration ratios.The lines cross the centers of circles at low penetration ratios, such as Ix values less than 0.3.For these cases, the differences among the three models can be ignored.Our approximate solutions (blue lines) match very well with semi-analytical solutions (circles with cross) for all penetration ratios and fracture conductivities.The red lines deviate from the circles and blue lines and huge differences between Pratikno et al.'s solutions and our solutions are noticeable at low conductivity (CfD < 10) with a high penetration ratio (Ix > 0.5).

Results
In this section, we will compare our approximate model with our semi-analytical model and Pratikno et al.'s approximate [20] for different penetrate ratio and fracture conductivity.Table 2 presents the values of b Dpss obtained by our semi-analytical method, Pratikno et al.'s method and our approximate model.As shown in Table 2, the maximum relative error is 0.4% at I x = 0.1 and 1.93% at I x = 1 between our approximate model and semi-analytical model.However, huge differences between Pratikno et al.'s method and our semi-analytical method can be observed at large penetration ratio, for example, relative error 16.57% at C fD = 0.631 and I x = 1.
We further show the comparisons in Figure 6.The circles with crosses, red lines, and blue lines correspond to the semi-analytical solutions (accurate solutions), Pratikno et al.'s solutions, and approximate solutions, respectively.As shown in Figure 6, an excellent agreement among the three methods is visible when the dimensionless fracture conductivity C fD is greater than 10 for all penetration ratios.The lines cross the centers of circles at low penetration ratios, such as I x values less than 0.3.For these cases, the differences among the three models can be ignored.Our approximate solutions (blue lines) match very well with semi-analytical solutions (circles with cross) for all penetration ratios and fracture conductivities.The red lines deviate from the circles and blue lines and huge differences between Pratikno et al.'s solutions and our solutions are noticeable at low conductivity (C fD < 10) with a high penetration ratio (I x > 0.5).

Conclusions
The following conclusions can be drawn from this study: (1) Pratikno et al. [20] stated that only the fracture conductivity affected the conductivity-influence function (CIF) described in Equation (4).Based on our work, we found that both the fracture conductivity and penetration ratio exerted significant influences on CIF.As Figure 3 shows, CIF decreases with increasing fracture conductivity, and CIF tends to be zero when CfD is greater than 300.Additionally, CIF has obviously differences, especially when CfD is less than 10; the CIF decreases with increasing penetration ratio.(2) Based on the PSS function [11], a new semi-analytical model was proposed to directly calculate the bDpss, which consumes less time.Besides, our method is more accurate due to simultaneously considering both fracture conductivity and penetration ratio.(3) A new conductivity-influence function (CIF), considering the effect of penetration ratio and fracture conductivity, was developed.A normalized conductivity-influence function (NCIF) was also developed to calculate the value of CIF.Our work provides a fast, accurate, and time-saving method to evaluate bDpss for a well with a finite-conductivity fracture in a circular closed reservoir.

Figure 1 .
Figure 1.Schematic of a vertical well with a fracture in a circular reservoir.

Figure 1 .
Figure 1.Schematic of a vertical well with a fracture in a circular reservoir.

Figure 2 .
Figure 2. Schematic of the flow field around a fracture in a circular reservoir at late-time regime: (a) a short fracture with a low penetration ratio and (b) a long fracture with high penetration ratio.

Figure 2 .
Figure 2. Schematic of the flow field around a fracture in a circular reservoir at late-time regime: (a) a short fracture with a low penetration ratio and (b) a long fracture with high penetration ratio.

Figure 4 .
Figure 4. Regression of conductivity-influence function when the penetration ratio (Ix) equals 0 and 1.

Figure 4 .
Figure 4. Regression of conductivity-influence function when the penetration ratio (I x ) equals 0 and 1.

Figure 6 .
Figure 6.Comparisons of bDpss with our solutions and Pratikno et al.'s solutions.

Table 1 .
Conductivity-influence function (CIF) of different penetration ratio (I x ) and fracture conductivity (C fD ).

Table 2 .
Comparisons of b Dpss for different calculation methods.