Transient Coupled Flow Model for Matrix-Fracture in High-Pressure Gas Reservoirs Opened by Gas Drilling

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Introduction
The development of high-pressure tight gas reservoirs is an unsolved problem in the petroleum industry due to the low productivity and high pressure of gas wells.Producing gas from tight reservoirs presents a unique challenge due to the contamination of water from the fracturing and drilling fluids [1][2][3].As an advanced drilling technology, using air as a drilling fluid began in the 1950s [4][5][6].Gas drilling technology has shown to be promising to solve the problem, which is characterized by the easier discovery of reservoirs, low reservoir damage, and increased rate of penetration [7,8].
However, there are many problems during practice, especially the safety of gas drilling, which is very uncontrollable in many areas due to the lack of understanding of the transient flow law and pressure characteristics of high-pressure gas, when the high-pressure reservoir is opened by gas drilling [9,10].Therefore, it is very important for gas drilling engineers to understand the flow characteristics of gas in fractured highpressure reservoirs.
A matrix-fracture medium model is commonly used for natural fracture gas reservoirs.Warren and Root have developed a dual-porosity medium model, which is based on the assumption that the matrix is the main storage space, the natural fractures are the main seepage channels, and the matrix provides a gas source to fracture [11][12][13].Kazemi et al. have developed a double-porosity dual-permeability model; the gas of the matrix system and small-scale fractures not only can provide a gas source to fracture but also can provide a gas source to the wellbore [14].Although there are abundant studies of the coupled flow behavior for matrix-fracture reservoirs, few models research the characteristics of unstable complex flow for high-pressure reservoirs during gas drilling.
For naturally fractured gas reservoirs, most researchers have focused on transient flow behaviors.Slimani and Tiab [15] established a theoretical model, to investigate the influence of transient flow behaviors on type curves for partially penetrating vertical wells in naturally fractured reservoirs.Wu et al. researched a triple-porosity medium model to analyze the fluid flow behavior in matrix-fracture-cave reservoirs [16].Du et al. conducted a multiregion composite model, to investigate the transient flow behavior for fractured-cave reservoirs [17].For composite fractured gas reservoirs, Nie et al. [18] studied a transient flow model with the high-velocity non-Darcy effect, to simulate the non-Darcy flow for the inner region as a dual-porosity medium, and the Darcy flow for the outer region as a homogeneous medium.
With the development of hydraulic fracturing technology, which is one of the effective ways to develop tight sandstone and shale reservoirs with low permeability characteristics, many researchers attach importance to the study of complex fluid behaviors in such reservoirs [19].For a hydraulic fractured sandstone tight reservoir, a series of new fluid flow models are presented to help engineers understand the transient pressure characteristics of gas flow.A new mathematical model has been proposed to study the transient pressure characteristics for hydraulic fractured wells with nonuniform fracture geometry and permeability [20].The semianalytical model has been established and solved to analyze the transient pressure behavior for a fractured vertical well with hydraulic/ natural fracture networks by considering stress-sensitive effects [21].The model for multistage fractured horizontal wells with discretely distributed natural fractures has been researched to investigate pressure performance [22].The new fractally fractional diffusion model of composite dualporosity has been developed to evaluate the performance of multiple fractured horizontal wells with stimulated reservoir volume in tight gas reservoirs [23].The discrete fracturematrix method based on the numerical well testing model has been proposed to study the pressure transient responses of discretely distributed natural fractures [24].Due to the characteristics of ultralow porosity and ultralow permeability in shale gas fields, which contain both adsorbed gas and free gas, hydraulic fracturing technology and horizontal well technology are commonly used in shale gas development.In recent years, discrete fracture models have been vigorously developed to study gas seepage in gas reservoirs in shale [25,26], and many scholars believe that the changes in matrix permeability and fracture permeability with increasing effective stress cannot be ignored [27,28].A series of multiple-porosity model transient flow models are proposed, including the dualporosity dual-permeability model and triple-porosity model, to study the transient pressure behaviors [29,30].Although there are abundant studies of transient flow behavior for naturally and hydraulically fractured reservoirs, few models mentioned above have touched on the convection term and considered the influence of the energy equation.
In fact, owing to the high pressure of the reservoirs and high conductivity of the fractures, the transient flow of gas cannot ignore the influence of the gas convection term and energy equation.Based on the law of conservation of mass, momentum, and energy, a transient coupled flow in matrix-fracture for the high-pressure reservoirs opened by gas drilling model is established.The flow in fracture is considered as a radial flow to simulate the convection and diffusion process of gas, and the flow in matrix reservoir is considered a parallel seepage into fracture to simulate the non-Darcy flow, as shown in Figure 1.By using Shaher Momani's algorithm and CyabJiBe format, the analytical solution of the coupled transient flow in matrix-fracture model can be obtained [31][32][33].Then, the type curves with the influence of energy conversion and physical characteristics of matrix-fracture are plotted.

Physical Model and Assumptions
Figure 1 is the schematic of the physical model, which represents the gas transient flow process in fractured reservoirs opened by gas drilling with no damage.The gas flow in reservoirs is divided into two regions, which have different flow characteristics and different reservoir characteristics.The first gas flow area is the gas transient radial flow at high velocity from fracture to wellbore, considering the convective and diffusion terms.The second gas flow area is the gas transient parallel flow perpendicular to the fracture from the matrix.The main assumptions of this model are the following: (1) The fracture has the same geometric characteristics.(2) In the second flow area, the matrix with homogeneous characteristics provides a gas source to fracture vertically.(3) The permeability of the matrix is independent of pressure.(4) The influence of the secondary gradient on the flow in the matrix is ignored.

Mathematical Model for the Gas Transient Parallel Flow
Perpendicular to Fracture from Matrix.Owing to the high pressure of the reservoirs, the influence of non-Darcy flow cannot be ignored for the gas flow in the matrix; the equation is as follows [18]: In Equation ( 1), K represents the selected reservoir matrix permeability.v and μ represent gas velocity and viscosity.ρ represents gas density.n represents the value related to the properties of the porous medium.β represents the non-Darcy flow factor.

Geofluids
For plane flow in homogeneous formation, Equation ( 1) can be rewritten as follows: In Equation ( 2), δ represents the inertial-turbulence correction coefficient factor and z m represents vertical distance.
The continuity equation of gas transient parallel flow perpendicular to fracture from the matrix is as follows: Binding Equation (2), Equation ( 3) can be written as follows (the detailed derivation of Equation ( 4) is presented in the appendix: The governing equations of the model are as follows (the detailed derivation of Equations ( 5)-( 7) is presented in the appendix) [34][35][36].
The continuity equation is as follows: The momentum conservation equation is as follows: The energy conservation equation is as follows: In the equation, energy and tangential stress τ rr = 2/3μð2∂rv r /r∂rÞ − 2μv r /r.According to Equations ( 5)-( 7), governing equations of the model are shown in Equations ( 8)- (10).
Continuity equation for transient coupled flow in the matrix-fracture system [34][35][36]: In Equation ( 8), _ q represents the gas source which is continuously provided from matrix to fracture.
Momentum conservation equation for transient coupled flow in the matrix-fracture system: The energy conservation equation for transient coupled flow in the matrix-fracture system: Parallel fow 3 Geofluids 3.4.Mathematical Model Solutions.In this section, Equations ( 7)-( 9) are rewritten as the vector form of the convection diffusion equation; the mathematical model is as follows:

Radial fow Matrix
The vector parameter u, a, b, c, f ðr, tÞ in Equation ( 11) are as follows: Shaher Momani's algorithm and CyabJiBe format for solving the convection-diffusion equation with a nonlinear source term is used to address the mathematical model.The basic definitions of the algorithm are as follows.
In the equation, J can be found in [37].
Differential operator D is described in detail in Reference [38].
ð15Þ Shaher Momani's algorithm for solving the convectiondiffusion equation consists in finding the solution in the form: The operator J and the inverse of the operator D are applied to solve Equation ( 11): The components u n recursively is defined as follows: The exact series solution Equation ( 16) is defined by Equations ( 18)- (20), which give an approximate solution to Equation (11).Finally, the standard characteristic curves of fluid velocity with time and pressure with space in fracture can be plotted by Mathcad as computing software, when the fractured reservoir is opened by gas drilling.The outer boundary pressure of the matrix reservoir is constant pressure p i = 60 MPa.The inner boundary pressure of the matrix p m l z=0 = p is the dynamic boundary pressure, which is the outer boundary pressure of the fracture.The inner boundary pressure of the fracture is the constant pressure p m l r=0 = 58 MPa.The matrix permeability is 0.01 mD.

Results and Discussion
Figure 2 shows the velocity behavior characteristics of coupled flow for matrix-fracture in a high-pressure reservoir for gas drilling.The three flow stages can be represented by curves.Figure 3 shows the amplification characteristic curve of the first flow stage.
The first stage describes gas velocity characteristic in the fractured reservoir, which is opened by gas drilling instantaneously.The magnification curve is shown in Figure 3.In this flow stage, the gas flow velocity in fracture is much higher than that of the second and third stages, it is extremely unstable, and it may have a huge impact on the safety of the bottom hole.The flowing gas mainly comes from the gas stored in fracture, and the matrix will not provide gas to the fracture.Because the total amount of gas stored in the fracture of reservoirs is much less than that in the matrix, the duration of the flow process is extremely short, which is less than 1 second.
In the second stage, the flowing gas in the fracture mainly includes the gas from the reservoir in the fracture and the gas supplied by the matrix to the fracture.The gas flowing velocity drops rapidly, and it is quasisteady flow.The third stage reflects the flow characteristics of the stable flow stage.The flowing gas in the fracture is supplied mainly from the matrix.In this process, the pressure and the flow velocity drop very slowly.
Figure 4 shows the characteristics of the pressure funnel in fracture from 0.05 seconds to 1 second after being opened by gas drilling.The pressure attenuation mainly exists in the fracture near the wellbore, that is, within 10 m from the wellbore.The closer to the wellbore, the more serious the pressure attenuation.As time goes on, the pressure in the fracture becomes lower and lower.The pressure drop is mainly consumed near the wellbore, during the fluid flowing from the reservoir edge to the bottom of the well.

Impact of the Parameters.
In this section, the characteristic parameters of the transient coupled flow model in the matrix-fracture reservoir are analyzed in detail, including the influence of the energy conversion, the matrix permeability k, the fracture aperture L f , and the fracture outlet pressure p.These parameters have different influences on the characteristics of gas flow velocity and pressure in the fracture, as shown in Figures 5-9 5, by comparing the flow velocity in fracture without considering the energy conversion and considering the energy conversion.In the first stage, it is found that if the energy conversion process is ignored, the flow velocity cannot peak rapidly when the fracture is opened by gas drilling but rises slowly, and the characteristics of the curve cannot reflect the real flow situation.In the second and third flow stages, the characteristics of flow velocity without considering energy conversion are higher than those considering energy conversion.In the condition of high velocity of gas flow in the fracture, the part of kinetic energy can be converted into internal energy.According to the law of energy conservation, the velocity of gas flow decreases due to the reduction of kinetic energy.Therefore, the effect of energy conversion cannot be ignored in the study of the characteristics of gas flow at high velocity for high-pressure reservoirs.6 Geofluids reason is that the flowing gas mainly comes from the gas source stored in the fracture, while the gas in the matrix does not flow.Figure 7 illustrates the impact of matrix permeability (k) on the curves of the pressure drop funnel in fracture for the transient coupled flow model.The values of matrix permeability (k) are shown in Figure 7.The characteristic curves of the pressure drop funnel also shows that matrix permeability (k) affects the attenuation of fracture pressure.The larger the matrix permeability (k), the slower the fracture pressure falls.If the matrix permeability (k) is increased, the supply of gas from the matrix to the fracture can increase, so the rate of pressure drop decreases.4.2.3.Impact of the Fracture Aperture L f .Figure 8 shows the influence of different fracture aperture (L f ) on the curves of flow velocity with time in fracture simulated by the transient coupled flow model.The values of the fracture aperture (L f ) are shown in Figure 8.The curves also illustrate that the fracture aperture (L f ) influences the overall three flow stages.The larger the fracture aperture (L f ), the higher the instantaneous peak for gas production in the first stage and the greater the gas velocity in the second and third fracture flow stages.Therefore, the larger the fracture aperture (L f ) encountered during gas drilling, the greater the impact on the wellbore safety.The reason is that the smaller the fracture aperture, the smaller the flow channel and the greater the flow resistance.Undoubtedly, the fracture outlet pressure determines the driving force of the flow system, including gas flow system from matrix to fracture and gas flow system from fracture to wellbore.The lower the fracture outlet pressure, the higher the driving force of the fluid system, the more gas supplied from matrix to fracture, and the greater the gas flow velocity in the fracture.

Appendix Governing Equation
In this section, a detailed derivation of the governing equation of the gas transient radial flow from fracture to the wellbore region is provided.
(1) The three-dimensional N-S equation of compressible fluid is as follows [34][35][36]: The vector parameter U, E, F, G, V x , V y , V z , τ xx , τ yy , τ zz , τ xy , τ xz , q x , q y , q z in Equation (A.1) are as follows: (2) In the cylindrical coordinates, Equation (A.1) is rewritten as follows: = S: ðA:3Þ  ðA:5Þ In the equation, u x represents velocity in the x direction, u y represents velocity in the y direction, u z represents velocity in the z direction, u θ represents velocity in θ angular tangential, u r represents radial velocity in the r direction, q x represents volume flow in the x direction, u y represents volume flow in the y direction, and u z represents volume flow in the z direction.

3. 2 .
Mathematical Model for the Gas Transient Radial Flow from Fracture to Wellbore.In the study of transient radial flow in fracture, the influence of the convection term and energy equation is mostly ignored; the transient flow characteristics cannot be truly reflected.Therefore, the convection term and energy equation are introduced to describe the transient flow characteristics of natural gas in fractures, during the instantaneous opening of fractured reservoirs by gas drilling.The transient mathematical model of gas transient radial flow in fractures is established.

3. 3 .
Mathematical Model for Transient Coupled Flow Model in Matrix-Fracture.The transient coupled flow model in matrix-fracture is applied to describe and analyze the coupling flow characteristics of gas in matrix and fracture, when the fractured reservoir is opened by gas drilling, including the radial flow characteristics of gas from fracture to wellbore and the flow characteristics of gas flowing vertically from matrix to fracture.The gas source is continuously supplied from matrix to fracture with the decrease of fracture pressure.And the convection term and energy equation are introduced to the transient coupled flow model.The physical model is shown in Figure 1.

Figure 1 :
Figure 1: Physical model of gas flow process in matrix-fracture reservoirs opened by gas drilling.

4. 1 .
Flow Stage Analysis.The conditions of the model are as follows. .
Figure 5 compares the influence of considering energy equation and without energy equation on curves of the transient coupled flow in matrix-fracture model.As a compressible fluid, the gas flow process has the conversion of kinetic energy and internal energy under the condition of high pressure difference and high flow velocity, as shown in Figure

Figure 2 :Figure 3 :
Figure 2: Three-stage characteristics of transient coupled flow in matrix-fracture.
Figure 6 shows the influence of different matrix permeability (k) on the curves of flow velocity with time in fracture simulated by the transient coupled flow model.The values of the matrix permeability (k) are shown in Figure 6.Due to the fracture supplied with a source of gas from the matrix, it can be observed that the matrix permeability mainly influences the second and third flow stages of the type curves for the transient coupled flow model.In these two flow stages, when the matrix permeability (k) is increased, the flow velocity curves rises.On the contrary, if the matrix permeability (k) is decreased, the flow velocity curves goes down.In the first flow stage, the matrix permeability cannot affect the flow characteristic of the transient coupled flow model; the main

4. 2 . 4 .
Influence of the Fracture Outlet Pressure.Figure 9 illustrates the influence of the fracture outlet pressure on the curves of flow velocity with time in fracture for the transient coupled flow model.The parameters of the fracture outlet pressure are shown in Figure 9.It can be seen in the curves of flow velocity with time; the fracture outlet pressure influences the overall three flow stages during gas drilling.

Figure 7 :
Figure 7: Influence of matrix permeability (k) on pressure drop funnel in fractures.
xx + u θ τ θy + u r τ rx − q x xθ + u θ τ θθ + u r τ rθ − q θ xr + u θ τ θr + u r τ rr − q r Ignoring the tangential velocity u θ and vertical velocity u x , Equation (A.3) is rewritten to the N-S equation of radial flow and expanded as follows: