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Article

Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs

1
School of Energy Resources, China University of Geosciences (Beijing), Beijing 100083, China
2
Beijing Key Laboratory of Unconventional Natural Gas Geological Evaluation and Development Engineering, Beijing 100083, China
3
Key Laboratory of Strategy Evaluation for Shale Gas, Ministry of Land and Resources, Beijing 100083, China
*
Author to whom correspondence should be addressed.
Energies 2017, 10(10), 1602; https://doi.org/10.3390/en10101602
Submission received: 14 September 2017 / Revised: 6 October 2017 / Accepted: 6 October 2017 / Published: 13 October 2017
(This article belongs to the Special Issue Flow and Transport Properties of Unconventional Reservoirs)

Abstract

:
Based on the porous flow theory, an extension of the pseudo-functions approach for the solution of non-linear partial differential equations considering adsorption-desorption effects was used to investigate the transient flow behavior of fractured wells in shale gas reservoirs. The pseudo-time factor was employed to effectively linearize the partial differential equations of the unsteady flow response. The production performance of vertically fractured wells in shale gas reservoirs under either constant flow rate or constant bottom-hole pressure conditions was analyzed using the composite flow model. The calculation results indicate that the non-linearities that develop in the gas diffusivity equation have significant effects on the unsteady response, leading to a larger pressure depletion and rate decline in the late-time period. In addition, gas desorption from the shale acts as a recharge source, which relieves the gas production rate of decline. Greater values for the Langmuir volumes or Langmuir pressures provide additional pressure support, leading to a lower rate decline while the flowing well bottom-hole pressure is maintained. The reservoir size mainly affects the duration of the pressure depletion and rate decline. In the case of ignoring the non-linearity and adsorption-desorption effect in the differential equation, a greater rate decline under constant bottom-hole pressure production can be obtained during the boundary-dominated depletion. This work provides a better understanding of gas desorption in shale gas reservoirs and new insight into investigating the production performances of fractured gas well.

1. Introduction

In recent years, shale gas reservoirs have gradually become the major sources of natural gas production around the world. In nature, shale can serve as both source and reservoir rock [1,2,3], and natural gases are stored in both the free gas and absorbed gas forms. Martin et al., stated that the amount of shale gas in place is controlled by the total organic contents (TOC), clays, and adsorption ability of methane on the internal surface of a solid [4]. In shale reservoirs, gas desorption can produce a considerable amount of gas. Tinni et al., presented a novel approach that can be used to evaluate the influence of adsorption on the gas production in shale gas reservoirs [5].The production performances can be altered by the influence of gas adsorption in unconventional reservoirs [6]. Mengal and Wattenbarger concluded that it is generally not possible to investigate the accurate production forecasts if the effects of desorption is neglected [7].Thompson et al. [8] proposed that gas desorption can have a great influence on the analysis of conventional Arps decline curves [9]. Recently, much of the research has focused on the adsorption-desorption effect in unconventional reservoirs [10,11,12,13,14,15]. Since a portion of the gas in shale reservoirs is stored in the adsorbed form, a detailed investigation on the contribution of gas adsorption can provide critical insights into the analysis of the transient flow behavior in gas reservoirs.
During the last few decades, increasing attention has been paid to the economical development of shale gas reservoirs using hydraulic fracturing [16,17,18,19,20,21,22,23,24]. In some cases, the adoption of the composite flow model can replace the application of a two-dimensional or three-dimensional flow model when analyzing the transient performance of a fractured well. In terms of the analysis on a composite flow model, Wattenbarger et al., claimed that the flow near production wells in tight gas reservoirs is dominated by a one-dimensional flow after hydraulic fracturing treatment, and they reported a rate decline analysis of gas wells using a one-dimensional flow model [25]. Cinco et al., performed an appropriate analysis of fractured wells according to the bilinear flow theory for the early-time pressure behavior [26]. Later, Cinco and Satnaniego proposed a new approach to analyze the pressure transient response of a vertical fractured well [27]. Brown et al., established an analytical trilinear flow model to investigate the production performance of a fractured well in an unconventional reservoir [28]. In recent years, the composite flow models have been applied to analyze the production performance of fractured wells [29,30,31,32,33]. Stalgorova et al., established an analytical model, as an extension of the trilinear flow solution, for unconventional reservoirs with multiply-fractured horizontal wells [34,35]. Yao et al., established a semi-analytical composite model for heterogeneous reservoirs [36]. Guo et al., presented an analytical model for the production decline analysis of a multi-stage fractured shale reservoir [37]. However, the significant influences of fluid properties changes on the fracture performance were not fully investigated in these studies.
Since the recent boom in gas production caused by the development of hydraulic fracturing technologies, many articles analyzing the transient performance of gas flow in unconventional reservoirs have been published [38,39,40,41]. A historical challenge in gas reservoir analysis is how to solve the highly non-linear gas partial differential equation, which fully considers the significant changes in gas properties during depletion. Overall, many researchers [42,43,44,45,46,47,48,49] have focused on the application of pseudo-functions to achieve the linearization and subsequent analytical treatment of the gas flow equations, replacing the pressure and time variables with pseudo-pressure and pseudo-time functions. With this method, the change in gas properties during production is also taken into consideration. On this basis, the main objective of this article is to explore the applicability of the pseudo-functions approach, which investigates the variable gas properties and significant desorption effect in shale gas reservoirs. Firstly, the extended pseudo-function is applied into a composite flow model to obtain the analytical solution. Then, type curves are constructed to analyze the effects of the fluid properties, gas desorption and reservoir size on the transient behaviors. The pseudo-time factor is employed to effectively linearize the partial differential equations of unsteady gas flow in shale gas reservoirs.

2. Pseudo-Functions Approach

2.1. Derivation of the Pseudo-Functions

The presence of absorbed phases significantly affects the production performance and reserve evaluation of a shale gas reservoir. Consequently, the pressure depletion rapidly increases with the process of gas production in the late-time period, and it is necessary to take the adsorption-desorption effect into consideration. The equilibrium between absorbed phase and the solid phase at a given pressure is characterized by an adsorption isotherm. Sing et al., presented the detailed description of the six models of physical sorption isotherms [50]. There are also other types of adsorption isotherm models that have been applied to analyze the sorption data in the experimental process, such as the Freundilich-type isotherm [51] and Dubinin’s family of isotherms [52].However, these isotherm models have not been clearly accepted in analyzing the transient responses of unconventional gas reservoirs. By far, the adsorption isotherm that has been widespread applied to model the adsorption-desorption effect is the Langmuir isotherm [53] as given in Equation (1):
V g ( p ) = V L p p L + p
where Vg(p) is the gas volume of the adsorption at pressure p; VL is the Langmuir volume, referred to as the maximum gas volume of adsorption at an infinite pressure; and pL is the Langmuir pressure, which is the pressure corresponding to one-half of the Langmuir volume. Based on the equation describing the mass balance of gas flow in shale gas reservoirs proposed by Patzek et al., and Yu et al. [54,55], the one-dimensional continuity equation with the adsorption-desorption effect is given below:
( ρ g v g ) x = 1 α t [ ρ g S g ϕ + ( 1 ϕ ) ρ a ] t
where ρg is the free gas density; vg is the Darcy velocity of gas; Sg is the initial gas saturation; φ is the reservoir porosity, ρa is the adsorbed gas density; and αt = 3.6 × 24 × 10−3 is the conversion factor. When neglecting the elasticity of the porous media under isothermal conditions, a nonlinear governing equation for a one-dimensional transient flow with the gas desorption effect in a shale gas reservoir can be presented as:
x [ k g μ g ( p ) p Z ( p ) p x ] = 1 α t [ ϕ S g + ( 1 ϕ ) ρ a ρ g ] t ( p Z ( p ) )
where kg is the reservoir permeability, μg is the gas viscosity, and Z is the gas compressibility factor.
As given in Equation (3), the viscosity μg(p) and compressibility factor Z(p) are pressure-dependent parameters of natural gas, and Equation(3) is apparently nonlinear. In order to solve this nonlinear equation, the pseudo-pressure function [56] is defined as follows:
p P ( p ) = μ g i Z i p i p p i p μ g ( p ) Z ( p ) d p
where pP is the pseudopressure, μgi is the initial gas viscosity, Zi is the initial gas compressibility factor, and pi is the initial pressure. Substituting the pseudo-pressure function and Langmuir adsorption model into the diffusivity equation, the flow of a real gas through a shale formation can be expressed as follows:
2 p p ( p ) x 2 = ϕ μ g i c g i S g α t k g [ μ g ( p ) c g ( p ) μ g i c g i + μ g ( p ) ρ b μ g i c g i ϕ S g p s c Z ( p ) T p Z ( p s c ) T s c V L p L ( p L + p ) 2 ] p p ( p ) t
where cgi is the initial gas compressibility, cg is the gas compressibility, ρb is the bulk density of shale, and Zsc(psc) is the gas compressibility factor under the standard condition. Due to the residual presence of the μg(p)cg(p) pressure-dependent term on the right hand side of this diffusivity formulation, it is necessary to implement further handling of the nonlinearity in Equation (5). The traditional method is to approximate it as a constant, which will produce a large error in an analysis of the production performance.
In the initial stage, viscosity-compressibility changes do not dominate the unsteady state responses of the system, and μg(p)cg(p) is shown to represent a weak non-linearity. This is the same with as the phenomenon in liquid systems. However, the significant changes in μg(p)cg(p) during the boundary-dominated depletion cannot be ignored in gas reservoirs. In order to investigate the effect of pressure-dependent fluid properties on transient responses, pseudo-variables are needed to be implemented in unsteady state analysis. Recently, Ye and Ayala [57] proposed a density-based approach to analyze the unsteady state responses for natural gas reservoirs. This approach emphasized the significance of viscosity-compressibility changes from the pressure depletion based on the following depletion-driven dimensionless variables:
β * ( t ) = 1 t 0 t μ g i c g i μ g ( p avg ) c g ( p avg ) d t
where pavg is the average pressure in the reservoir. To effectively linearize the partial differential Equation (5) for the cases under study, pseudo-functions should be applied to re-express the pseudo-variables on the right hand side of the differential equation in a friendlier way. On this basis, according to the results of Fraim [58], the pseudo-time factor considering the adsorption-desorption effect in this work is defined as follows:
β ( t ) = 1 t 0 t 1 [ μ g ( p avg ) c g ( p avg ) μ g i c g i + μ g ( p avg ) ρ b μ g i c g i ϕ S g p s c Z ( p avg ) T p avg Z ( p s c ) T s c V L p L ( p L + p avg ) 2 ] d t
The integrand function λ(t) is defined as follows:
λ ( t ) = 1 [ μ g ( p avg ) c g ( p avg ) μ g i c g i + μ g ( p avg ) ρ b μ g i c g i ϕ S g p s c Z ( p avg ) T p avg Z ( p s c ) T s c V L p L ( p L + p avg ) 2 ]
Apparently, λ(t) and β(t) are dimensionless and the relationship between them can be presented as follows:
β ( t ) = 1 t 0 t λ ( t ) d t
Substituting the pseudo-function variable into the diffusivity Equation (5), the simplified version of the governing equation is shown below:
2 p P ( p ) x 2 = ϕ ( 1 S w i ) μ g i c g i α t K g p P ( p ) ( β t )
where β(t) is a depletion-driven time rescaling factor capturing the behavior of the gas desorption and viscosity-compressibility changes during pressure depletion. It should be noted that Equation (10) is an approximate version of Equation (3). The proposed approximation demonstrates that the introduction of the pseudo-time factor can successfully linearize the partial differential equation of the gas flow in porous media, making the analysis methods for the “liquid flow model” applicable to the gas flow in a shale reservoir.

2.2. Behaviors of the Pseudo-Time Factor

On the basis of the proposed approach, this section demonstrates the effects of the pseudo-time factor during reservoir depletion. Apparently, the behaviors of the pseudo-time factor over time depend on the correlated fluid properties and depletion patterns in the system. A full discussion of the production performances for these cases under study is presented using the production decline model of one-dimensional flow [25].Consider a vertical well intercepted by a uniform flux vertical fracture in the center of a homogeneous rectangular reservoir, as shown in Figure 1.The height, length, and width are h, xe, and ye, respectively. The half-length of the fracture is yf, and the length of the fracture is equal to the width of the reservoir.
If the fracture produces at a pressure of pwf, this leads to isothermal transient flows in the reservoir. The dimensionless quantities are defined as follows:
p D = p P ( p ) p P ( p w f ) , q D = q g ( t ) μ g i B g i α p k g h p p ( p w f ) , t D f = α t k g t ϕ ( 1 S w i ) μ g i c g i y f 2 , x D = x y f
where qg is the gas flow rate, Bgi is the gas formation volume factor under the standard condition, h is the reservoir thickness, pwf is the wellbore pressure, yf is the fracture half-length, and αp = 2π × 3.6 × 24 × 10−7 is the conversion factor. In these equations, pD is the dimensionless pseudo pressure, qD is the dimensionless flow rate, tDf is the dimensionless time, and xD is the dimensionless coordinate in the x direction. The production behavior under a constant bottom-hole pressure is given below:
q D ( β t D f ) = 4 π x e D n = 0 exp [ π 2 4 ( 2 n + 1 ) 2 ( β t D f ) x e D 2 ]
where xeD is the dimensionless reservoir length. In this formulation, the calculation of depletion-driven factor β(t) should be explicitly stated.
It should be noted that high accuracy can be obtained from Equation (7) by employing the material balance equation with the adsorption-desorption effect. A generalized material balance equation that investigates the equilibrium between the free and adsorbed gas phases was developed by King [59], who applied graphical and iterative algorithms for the solution of the generalized results. Since then, based on the volume conservation principle, Moghadam et al., presented a new format for the material balance equation accounting for the shale gas storage mechanisms [60]. In this paper, the material balance equation with the adsorption-desorption in a shale gas reservoir has been derived by integrating the continuity equation with definite conditions.
The definite conditions for Equation (2) are presented as follows:
( ρ g v g ) x = 0 = q g s c ( t ) ρ s c α p h w , ( ρ g v g ) x = x e = 0 , ( q g s c ) t = 0 = 0
where qgsc is the standard gas flow rate, and ρsc is the gas density under the standard condition. Then, the one-dimensional continuity equation with the adsorption-desorption effect in integral form is given by the following:
1 x e 0 x e ( ρ g v g ) x d x = 1 α t 1 x e 0 x e [ ρ g S g ϕ + ( 1 ϕ ) ρ a ] t d x
Substituting the Langmuir isotherm model into the continuity Equation (12), the material balance equation considering the gas desorption in a shale gas reservoir is given below:
G p ( t ) G s c p i Z i = ( p i Z i p avg Z avg ) + p s c T i Z s c T s c ρ b V L ϕ S g i [ p i p L + p i p avg p L + p avg ]
where Gp is the cumulative production, and Gsc is the geological reserves.
The time-dependence of β(t) is correlated with the associated average reservoir pressure pavg predicted by the material balance equation at every depletion step for every value of Gp(t). For a reservoir with a constant flow rate, the cumulative production is Gp = qsc × t. If the well has variable rate production, the trapezoidal numerical integral can be incorporated to obtain the accumulative production for a given time.
At every step in the isothermal depletion process, reservoir fluid properties such as the gas compressibility, viscosity, and gas volume of adsorption can be readily tracked as functions of the pressure and time. According to the definition of the pseudo-time factor in this work, which decouples the viscosity-compressibility changes and gas desorption from the pressure depletion in a shale gas reservoir, the transient response of a shale gas reservoir can be further analyzed. Based on the above derivation, the behaviors of pseudo-time factors β(t) and β*(t) can be calculated according to the isothermal depletion of a stated reservoir, as shown in Figure 2. On this basis, the production performances of liquid and gas solutions can be investigated by calculating Equation (11) with the use of Stehfest numerical inversion algorithm [61] (Figure 3).
Figure 2 depicts the curves of β(t) and β*(t) versus time with different reservoir sizes under a constant bottom-hole pressure. As shown in this figure, in the initial stage, the extent of reservoir depletion is not significant, that is β*(t) ≈ 1.0. The viscosity-compressibility changes have a weak effect on the unsteady state responses of the system. At a later production stage, the average reservoir pressure pavg decreases sharply, and the seepage behavior in the gas reservoirs would gradually deviate from that of its corresponding liquid system (β*(t) < 1.0).As the production time increases, the desorption effect on the reservoir pressure depletion is significant, which indicates that a recharge source has been built in a shale gas reservoir. The deviation between the seepage behavior in a shale gas reservoir and that of its corresponding liquid system, at a later production period, becomes more significant.
The impact of the gas desorption on production rate under a constant bottom-hole pressure is presented in Figure 3. As shown in this figure, at an early stage, the production rates simulated using the liquid model, and the gas model with and without desorption model, are very similar. This is because the reservoir depletion is small and cannot significantly affect the viscosity-compressibility values of natural gas. However, during the later production period, the gas responses gradually deviate from their corresponding liquid analytical model results. Thus, the production rate of a shale gas reservoir is higher than that of a slightly-compressible liquid reservoir. It should be noted that the flow rate decreases as the production time increases, while the bottom-hole pressure is maintained and production behaviors are significantly affected by the depletion-driven fluid properties and gas desorption in a shale gas reservoir. For the flow in the liquid analytical model, the adsorption-desorption effect and significant changes in the gas properties during depletion are neglected, and the rate declines faster than under the other two conditions. This is explained by the significant changes in the fluid properties during the reservoir depletion. In addition, the desorption effect of shale gas is equivalent to an energy supply in the reservoir. Consequently, if the gas desorption is not considered, the conventional gas model would underestimate the later stage production rate under a bottom-hole pressure condition.

3. Mathematical Model

In this article, an analytical solution is presented to characterize the production performance of a fractured well in a shale gas reservoir. The composite flow model is simple, but flexible enough to embody the basic properties of an unconventional reservoir. For a gas reservoir, especially one with a relatively narrow drainage area, a composite flow model is an appropriate method to avoid the need to solve integral equations and analyze the transient flow in a finite-conductivity fracture coupled with the reservoir flow. One of the best advantages of the composite model is that it is convenient to derive the approximate solutions. Based on the above results, the production performance of a vertically fractured well in a shale gas reservoir under either constant flow rate or constant bottom-hole pressure conditions can be obtained by using the trilinear flow model.

3.1. Model Assumption

Assuming that a finite-conductivity fractured well with a bi-wing shape is completed in a homogeneous rectangular gas reservoir; its length, width, and height are xe, ye, and h respectively. The case of a slab transverse vertical fracture in the center of the reservoir is examined, where the height of the fracture is equal to the thickness of the reservoir. The shale gas flows into the wellbore from the reservoir through the fracture. It is assumed that the well produces at a constant flow rate, and an isothermal seeping process appears in the reservoir. Take the lower left corner of the gas reservoir as the origin of the coordinates (0, 0), as shown in Figure 4.
The dimensionless quantities are defined as follows:
p I D = p D ( p I ) p P ( p w f )    p I I D = p P ( p I I ) p P ( p w f )    p f D = p P ( p f ) p P ( p w f ) ,
y D = y y f    y e D = y e y f    y f D = y f y f    w f D = w f y f    c f D = k f w f k y f
where yD is the dimensionless coordinate in the y direction, yeD is the dimensionless reservoir width, ye is the reservoir width, wfD is the dimensionless fracture width, wf is the fracture width, cfD is the dimensionless fracture conductivity, kf is the fracture permeability, and wf is the fracture width.

3.2. Solution for the Model

Referring to the definition of pseudo-time factor β(t) that has been presented in this paper, the dimensionless governing equation for the gas flow in the formation is as follows. Detailed derivation of the mathematical model is presented in Appendix A:
2 p D x D 2 + 2 p D y D 2 = p D ( β t D )
(1) In Region I, the linear flow is parallel to the surface of the fracture (y-direction), and Equation (14) is simplified as follows:
2 p I D y D 2 = p I D ( β t D f )
(2) The flow in the reservoir is mainly the linear flow vertical to the surface of the fracture in Region II (dominated by that in the x-direction).
2 p I I D x D 2 + 1 y f D p I D ( x D , 1 2 y e D + y f D , β t D f ) y D = p I I D ( β t D f )
(3) The governing equation can be calculated by the integral average along the x direction (the pressure function is still denoted as pfD), and the corresponding dimensionless governing equation is as follows:
d 2 p f D d y D 2 + 2 c f D p I I D ( 1 2 x e D + 1 2 w f D , y D , β t D f ) x D = 0
The pressure distribution function of a vertical fracture under a constant flow rate in the Laplace domain is obtained as follows:
s p ˜ f D ( y D , s ) = π c f D 1 D ( s ) cosh ( y D 1 2 y e D y f D ) D ( s ) sinh y f D D ( s )
where s is the time variable in Laplace domain. The relationship between the pressure solution at a constant rate and the flow rate solution under a constant bottom-hole pressure can be derived according to the superposition principle [62]:
p ˜ D ( s ) q ˜ D ( s ) = 1 s 2
where p ˜ D is dimensionless pseudo pressure pD of finite-conductivity fracture in Laplace domain, q ˜ D is dimensionless flow rate qD of finite-conductivity fracture in Laplace domain. Then, they can be inverted to the real domain numerous times by the use of an algorithm (such as that of the Stehfest numerical inversion) during the integration over the time and spatial domains.

3.3. Model Validation

As shown in Figure 5, the solution proposed in this paper was validated using HIS Fekete Harmony [63], which can provide solutions to support customers in various gas well production analysis and simulation services [64,65]. The composite method was applied to model the gas flow in a shale gas reservoir. The reservoir was assumed to be homogeneous. The reservoir had a finite length of 1200 m and width of 800 m. The value of the bottom-hole pressure was held at 10 MPa for the simulation. The fracture height was supposed to be equal to the formation thickness (47.2 m). The fracture half-length was fixed at 70 m. The adsorption effect was characterized by the Langmuir isotherm. The comparison suggested that there was a good agreement between the solutions derived in this article and the results from commercial software. Thus, the results validated the accuracy of our model.

4. Parametric Study on Type Curves

The dynamic characteristics under constant flow rate or constant bottom-hole pressure condition can be derived by illustrating the influence of the depletion-driven fluid properties and gas desorption in a shale gas reservoir. The comparison of the liquid and gas analytical solutions can be derived correspondingly. The fractured well, fluid, and formation properties associated with the generation of the type curves are listed in Table 1.

4.1. Effects of Depletion-Driven Fluid Propertiesand Gas Desorption

The effects of the depletion-driven fluid properties and gas desorption on the pressure depletion for a vertically fractured well under a constant flow rate are presented in Figure 6. As shown in this figure, in the initial stage, the curves agree well with each other. At a later production period, the curves bend upward, and the effects of the depletion-driven fluid properties and gas desorption on the curves become more significant. When neglecting the adsorption-desorption effect and significant changes in the gas properties during depletion, a greater pressure drop would be required to maintain the expected flow rate. As a result, the pressure depletion is closer to reality when considering the effects of the gas property changes and gas desorption during reservoir depletion, and provides extra information on shale gas production. The impact of the reservoir size on the pressure response under a constant flow rate is also shown in Figure 6. As expected, a larger pressure difference is required to maintain a constant flow rate in a smaller size reservoir; it also illustrates that a closer boundary distance is associated with a quicker appearance of an upward trend.
Figure 7 illustrates the impact of the depletion-driven fluid properties, gas desorption and reservoir size on the production behaviors under a constant bottom-hole pressure condition. As presented in this figure, in the early stage, the curves agree with each other. The flow rate decreases as the production time increases, while the flowing well bottom-hole pressure is maintained, and production behaviors can be significantly affected by the fluid property changes and gas desorption. For the flow in a liquid analytical model, the significant changes of in the gas properties during depletion are neglected, leading to a larger rate decline and smaller cumulative production. These results are compared in this figure against the analytical trilinear flow solution of Brown and Ozkan [28], which did not incorporate fluid properties corrections. In another case, a larger rate decline and smaller cumulative production can be obtained when neglecting the desorption effect. These results are compared in the same figure against the unmodified density-based solution of Ye and Ayala [57], which did not incorporate desorption corrections, yielding a poor prediction. For example, for a reservoir with a length of 150 m and width of 110 m, the rigorous solution predicts a rate decline from 8700 to 1000 m3 in 6.46 years. If the non-linearity is neglected in the differential equation, this decline in the flow rate is predicted to occur in 5.68 years; if the solution is derived without considering gas desorption, this decline in flow rate is predicted to occur in 5.9 years. At 15 years of production, the cumulative production values for the above two cases are calculated with errors of 6.7% and 4.2%, respectively. It is worth noting that changes in the fluid properties and gas desorption have a significant influence on gas production in the late-time period. Figure 7 also presents the rate decline curves with different reservoir sizes under a constant bottom-hole pressure condition. It can be seen that the reservoir size has a dominant effect in the later production period, and a larger reservoir would lead to a later downward trend.

4.2. Effect of Langmuir Volume

Figure 8 shows the effect of the Langmuir volume on the production behavior under a constant bottom-hole pressure condition. For the reservoir with a certain amount of gas content, under the same pressure condition, a larger Langmuir volume value leads to a greater adsorption capacity in a shale gas reservoir. Due to the presence of adsorbed gas, the gas reservoir can receive support from the additional gas source, which leads to larger gas production in shale gas reservoirs. As shown in this figure, the effect of desorption is minimal at early times. As the depletion progresses, greater Langmuir volumes lead to additional pressure support, and thus less rate decline and larger cumulative production while the flowing well bottom-hole pressure is maintained. When ignoring the adsorption-desorption effect of shale gas, a greater rate decline will appear under a constant bottom-hole pressure.
Figure 9 and Figure 10 show the effects of the Langmuir volume on pseudo-time variables β(t) and λ(t) under a constant bottom-hole pressure condition. Due to the presence of adsorbed gas, the gas reservoir can receive support from the additional gas source. As a result, the changes in the Langmuir volume can significantly affect the behaviors of β(t) and λ(t). Greater Langmuir volumes provide additional pressure support leading to lower β(t) and λ(t) values.

4.3. Effect of Langmuir Pressure

Figure 11 shows the effect of the Langmuir pressure on the production behavior under a constant bottom-hole pressure condition. In a shale gas reservoir, the Langmuir pressure is used to characterize the adsorption capacity of the reservoir, which is related to the nature and temperature of the reservoir and gas. As shown, the effect of desorption is minimal at early times. As the depletion progresses, greater Langmuir pressures will lead to a smaller rate decline and larger cumulative production while the flowing well bottom-hole pressure is maintained. If the adsorption-desorption effect is neglected, a greater rate decline will appear under the constant bottom-hole pressure condition.
Figure 12 and Figure 13 show the curves of β(t) and λ(t) versus time with Langmuir pressure values under a constant bottom-hole pressure, respectively. The desorption effect of shale gas can provide an additional source of support for the reservoir. As a result, changes in the Langmuir pressure can significantly affect the behaviors of β(t) and λ(t). Greater Langmuir pressures will lead to lower β(t) and λ(t) values. It should be noted that production behavior can be affected by gas desorption in shale gas reservoirs.

4.4. Example Calculation

In this paper, we attempt to apply the analytical solutions to the transient performance of a hydraulic fractured shale gas well in a Sichuan field. The daily rates from the early years have been used for plots. The available reservoir and fracture parameters are listed in Table 2. The gas flow rate is characterized by a decreasing trend for a long time. This indicates that the shale gas is produced with a constant bottom-hole pressure. Figure 14 depicts a log-log decline curve for the transient responses of this example well. The production data has been further applied in type-curve matching that can provide a quick estimation for reservoir and fracture properties, such as the formation permeability, fracture conductivity, and fracture half-length. The best match of the data with the type curve can be obtained by determining the key parameters from those match points in Figure 14. The formation permeability interpreted by our model is 0.0023 mD, and the value of fracture conductivity is 450.6 in the interpretation of matching results. Besides, the fracture half-length calculated by the solutions in this work (48.25 m) can have a good agreement with the designed half-length (45 m), which can further indicate the accuracy of our model. For this well, we require a further analysis and confirmation because the decline curve can be affected by the potential variation of the pressures and flow rates. In spite of this, the decline curve is considered to be a practical and convenient method to analyze our example well.

5. Conclusions

In this article, we established a mathematical model for a fractured well in a shale reservoir that accounted for the non-linearities and desorption effects in partial differential equations. The detailed conclusions based on our work are summarized as follows:
(1)
In this work, the application of the pseudo-functions approach has been extended to solve the nonlinear flow problems of shale gas. This is accomplished by the definition of the pseudo-time factor accounting for both the viscosity-compressibility changes and desorption effect during reservoir depletion. The best advantage of this approach is that some partial differential equations can be effectively linearized, which contributes to the comprehensive investigation of the production performance of a fractured well in a shale gas reservoir.
(2)
The material balance equation with gas desorption is derived by the integration of the continuity equation with definite conditions, which can be used to obtain the analytical results of material balance equation in the application of well testing.
(3)
The modified formulation is validated and verified with the commercial software, and the successful analytical match demonstrates that the proposed model can effectively capture the production performance of gas reservoirs with significant desorption effect.
(4)
At a later production period, the production behaviors are significantly affected by the depletion-driven fluid properties and gas desorption in a shale gas reservoir. The shale gas reservoir can receive support from desorption effect in this period. A larger Langmuir volume or larger Langmuir pressure leads to a greater energy supply and less rate decline under a constant bottom-hole pressure condition.

Acknowledgments

This article was supported by the Fundamental Research Funds for the Central Universities. The careful reviews and detailed comments by the anonymous reviewers and editors are greatly appreciated.

Author Contributions

Xiaodong Wang proposed this topic and supervised the work. Xiaodong Wang and Xiaoyang Zhang were in charge of model establishment and analytical solution. Xiaochun Hou and Wenli Xu analyzed the data. Xiaoyang Zhang wrote the paper.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

Dimensionless Variables

vtDfdimensionless time
pDdimensionless pseudo pressure
qDdimensionless flow rate
cfDdimensionless fracture conductivity
xDdimensionless coordinate in the x direction
yDdimensionless coordinate in the y direction
xeDdimensionless reservoir length
yeDdimensionless reservoir width
wfDdimensionless fracture width
stime variable in Laplace domain, dimensionless
p ˜ D dimensionless pseudo pressure pD of finite-conductivity fracture in Laplace domain
q ˜ D dimensionless flow rate qD of finite-conductivity fracture in Laplace domain

Field Variables

x, yplane coordinates
wffracture width, m
yffracture half-length, m
xelateral boundary of reservoir, m
yevertical boundary of reservoir, m
ppressure, MPa
piinitial pressure, MPa
pwfbottom-hole producing pressure, MPa
pffracture pressure, MPa
pLLangmuir pressure, MPa
pPpseudo pressure, MPa
pavgaverage pressure in reservoir, MPa
Titemperature in reservoir, K
qggas flow rate, 104 m3/d
qgscstandard gas flow rate, 104 m3/d
kggas reservoir permeability, 10−3 μm2
kffracture permeability, 10−3 μm2
cfDfracture conductivity, dimensionless
hreservoir thickness, m
μggas viscosity, mPa·s
BgFormation volume factor, m3/m3
φreservoir porosity, fraction
tduration, day
cgisothermal gas compressibility factor, 1/MPa
Swiirreducible water saturation, %
γgspecific gravity, fraction
ρgfree gas density, kg/m3
ρaadsorbed gas density, kg/m3
ρbbulk density of shale, kg/m3
vgDarcy velocity of gas, m/s
Vggas volume of adsorption, m3/kg
VLLangmuir volume, m3/kg
Zgas compressibility factor, fraction
GPcumulative gas production, 104 m3
Gscoriginal gas in place, 104 m3
βpseudo-time factor, dimensionless
αtcoefficient, 3.6 × 24 × 10−3
αpcoefficient, 2π × 3.6 × 24 × 10−7

Special Subscripts:

Ddimensionless
ggas property
iinitial condition
ffracture property
scstandard condition

Appendix A. Derivation of the Model

The analytical solution to the gas flow in a shale gas reservoir can be derived according to the governing equation in porous media:
x [ k g μ g ( p ) p Z ( p ) p x ] + y [ k g μ g ( p ) p Z ( p ) p y ] = 1 α t [ ϕ S g + ( 1 ϕ ) ρ a ρ g ] t ( p Z ( p ) )
Substituting the pseudo-pressure function into the Equation (A1), the equation that governs the flow in a shale formation is:
2 p p ( p ) x 2 = ϕ μ g i c g i S g α t k g [ μ g ( p ) c g ( p ) μ g i c g i + μ g ( p ) ρ b μ g i c g i ϕ S g p s c Z ( p ) T p Z ( p s c ) T s c V L p L ( p L + p ) 2 ] p p ( p ) t
Substituting the definition of pseudo-time factor β(t) into the Equation (A2), the dimensionless governing equation can be simplified as follows:
2 p D x D 2 + 2 p D y D 2 = p D ( β t D )
Definite conditions are:
p D ( x D , y D , 0 ) = 0
p D ( x e D , y D , β t D ) x D = 0 , p D ( 0 , y D , β t D ) x D = 0
p D ( x D , y e D , β t D ) y D = 0 , p D ( x D , 0 , β t D ) y D = 0
As shown in Figure 4, we can obtain that pD = pID and pD = pIID in Region I and Region II respectively. Equation (A3) can be simplified as a group of one-dimensional equations.
(1) In Region I, the linear flow is parallel to the surface of fracture (y-direction), Equation (A3) is simplified as:
2 p I D y D 2 = p I D ( β t D f )
Initial condition:
p I D ( x D , y D , 0 ) = 0
Boundary condition:
p I D ( x D , y e D , β t D f ) y D = 0
Interface conditions:
p I D ( x D , y e D 2 + y f D , β t D f ) = p I I D ( x D , y e D 2 + y f D , β t D f )
p I D ( x D , y e D 2 + y f D , β t D f ) y D = p I I D ( x D , y e D 2 + y f D , β t D f ) y D
(2) The flow in the reservoir is mainly the linear flow vertical to the surface of fracture in Region II (dominated in x-direction). The flow in the reservoir can be simplified as follows:
2 p I I D x D 2 + 1 y f D p I D ( x D , y e D 2 + y f D , β t D f ) y D = p I I D ( β t D f )
Initial condition:
p I I D ( x D , y D , 0 ) = 0
Boundary condition:
p I I D ( x e D , y D , β t D f ) x D = 0
Interface conditions:
p I I D ( 1 2 x e D + 1 2 w f D , y D , β t D f ) = p f D ( y D , β t D f )
K μ p I I D ( 1 2 x e D + 1 2 w f D , y D , β t D f ) x D = K f μ p f D ( 1 2 x e D + 1 2 w f D , y D , β t D f ) x D
(3) It is believed that the steady flow of fluid in the fracture is symmetric (Cinco, 1978). Compared with the entire effective drainage area of the well, the width of the fracture is relatively small. The corresponding dimensionless governing equation is simplified as follows:
d 2 p f D d y D 2 + 2 c f D p I I D ( 1 2 x e D + 1 2 w f D , y D , β t D f ) x D = 0
Outer boundary condition:
d p f D ( 1 2 y e D + y f D ) d y D = 0
Inner boundary condition (constant flow rate or constant bottom-hole pressure):
d p f D ( 1 2 y e D ) d y D = π c f D
p f D ( 1 2 y e D ) = 1
The Laplace transform and superposition principle are used to deal with Equations (14)–(17) in Section 3.2 of this article. Then, the pressure distribution function and production behavior of an infinite-conductivity fractured well in the Laplace domain can be derived.
s p ˜ f D ( y D , s ) = π c f D 1 D ( s ) cos h ( y D 1 2 y e D y f D ) D ( s ) sinh y f D D ( s )
s q ˜ w D ( y D , s ) = c f D π D ( s ) sinh y f D D ( s ) cos h ( y D 1 2 y e D y f D ) D ( s )
where:
D ( s ) = 2 C ( s ) / c f D ; C ( s ) = B ( s ) tan h ( x e D 1 2 x e D 1 2 w f D ) B ( s ) ;
B ( s ) = s + 1 y f D A ( s ) ; A ( s ) = s tan h ( y e D 1 2 y e D y f D ) s .

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Figure 1. Diagram of one-dimensional fluid flow.
Figure 1. Diagram of one-dimensional fluid flow.
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Figure 2. Change relationship of β(t) and β*(t) over time.
Figure 2. Change relationship of β(t) and β*(t) over time.
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Figure 3. Influences of depletion-driven fluid properties and gas desorption on gas production with one-dimensional flow model.
Figure 3. Influences of depletion-driven fluid properties and gas desorption on gas production with one-dimensional flow model.
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Figure 4. Diagram of composite flow model for vertically fractured well.
Figure 4. Diagram of composite flow model for vertically fractured well.
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Figure 5. Comparison of flow rate results of this model and commercial software.
Figure 5. Comparison of flow rate results of this model and commercial software.
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Figure 6. Effects of depletion-driven fluid properties, gas desorption, and outer boundary on pressure behavior.
Figure 6. Effects of depletion-driven fluid properties, gas desorption, and outer boundary on pressure behavior.
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Figure 7. Effects of depletion-driven fluid properties, gas desorption, and outer boundary on production behavior.
Figure 7. Effects of depletion-driven fluid properties, gas desorption, and outer boundary on production behavior.
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Figure 8. Effects of Langmuir volume value on production behavior.
Figure 8. Effects of Langmuir volume value on production behavior.
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Figure 9. Effects of Langmuir volume on pseudo-time factor β(t).
Figure 9. Effects of Langmuir volume on pseudo-time factor β(t).
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Figure 10. Effect of Langmuir volume on pseudo-time variable λ(t).
Figure 10. Effect of Langmuir volume on pseudo-time variable λ(t).
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Figure 11. Effects of Langmuir pressure value on production behavior.
Figure 11. Effects of Langmuir pressure value on production behavior.
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Figure 12. Effects of Langmuir pressure on pseudo-time factor β(t).
Figure 12. Effects of Langmuir pressure on pseudo-time factor β(t).
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Figure 13. Effects of Langmuir pressure on pseudo-time variable λ(t).
Figure 13. Effects of Langmuir pressure on pseudo-time variable λ(t).
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Figure 14. Log-log decline curve for field example.
Figure 14. Log-log decline curve for field example.
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Table 1. Data used in discussion.
Table 1. Data used in discussion.
ParameterValueUnit
kg0.000810−3 µm2
φ14%
Swi10%
h25m
γg0.6Value
yf50m
pi34.5MPa
Ti327.6K
ρb2.63 × 103kg/m3
cfD1.5Value
Table 2. Reservoir and fracture data.
Table 2. Reservoir and fracture data.
ParameterValueUnit
Initial pressure pi16.3MPa
Initial temperature Ti338.15K
Formation thickness h39.7m
Porosity φ5%
Water saturation Swi34.75%
Bottom-hole pressure Pwf4.82MPa
Langmuir volume VL3m3/t
Langmuir pressure pL2.8MPa
Initial gas compressibility cgi0.0592MPa−1
Designed fracture half-length yf45m

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Zhang, X.; Wang, X.; Hou, X.; Xu, W. Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs. Energies 2017, 10, 1602. https://doi.org/10.3390/en10101602

AMA Style

Zhang X, Wang X, Hou X, Xu W. Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs. Energies. 2017; 10(10):1602. https://doi.org/10.3390/en10101602

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Zhang, Xiaoyang, Xiaodong Wang, Xiaochun Hou, and Wenli Xu. 2017. "Rate Decline Analysis of Vertically Fractured Wells in Shale Gas Reservoirs" Energies 10, no. 10: 1602. https://doi.org/10.3390/en10101602

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