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Production rate analysis of multiple-fractured horizontal wells in shale gas reservoirs by a trilinear flow model

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Abstract

Most multiple-fractured horizontal wells experience long-term linear flow due to the ultralow permeability of shale gas reservoirs. Considering the existence of natural fractures caused by compression and shear stresses during the process of tectonic movement or the expansion of high-pressure gas, a shale gas reservoir can be more appropriately described by dual-porosity medium. Based on the assumption of slab dual-porosity, this paper uses the trilinear flow model to simulate the transient production behavior of multiple-fractured horizontal wells in shale gas reservoirs, which takes the desorption of adsorbed gas, Knudsen diffusion and gas slippage flow in the shale matrix into consideration. Production decline curves are plotted with the Stehfest numerical inversion algorithm, and sensitivity analysis is done to identify the most influential reservoir and hydraulic fracture parameters. It was found that the density and permeability of the natural fracture network are the most important parameters affecting the production dynamics of multiple-fractured horizontal wells in shale gas reservoirs. The higher the density and permeability of the natural fractures are, the shorter the time is required to exploit the same amount of reserve, which means a faster investment payoff period. The analytical model presented in this paper can provide some insight into the reserve evaluation and production prediction for shale gas reservoirs.

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Abbreviations

A :

Cross-sectional area, πR 2m (m)

B g :

Volume factor of shale gas (m3/Sm3)

c t :

Total compressibility (P/a)

D k :

Knudsen diffusion coefficient (m2/s)

d F :

Distance between hydraulic fractures (m)

F :

Gas slippage factor, dimensionless

h :

Reservoir thickness, h =h mt + h ft (m)

h k :

Equilibrium thickness of kerogen (m)

h f :

Thickness of each natural fracture (m)

h m :

Thickness of each matrix slab (m)

h ft :

Total thickness of natural fracture (m)

h mt :

Total thickness of slab matrix, h mt = nh m (m)

J D :

Mass flux caused by Knudsen diffusion (kg/(m2 s))

J a :

Mass flux caused by slippage flow (kg/(m2 s))

k :

Permeability (m2)

k 1f :

Permeability in the fracture system of region 1 (m2)

k 2f :

Permeability in the fracture system of region 2 (m2)

K n :

Knudsen number, decimal

L e :

Length of horizontal wellbore (m)

m :

Pseudo-pressure

M :

Molecular weight of gas (kg/kmol)

N :

Number of natural fractures

n F :

Number of hydraulic fractures

p :

Pressure (Pa)

p L :

Langmuir pressure (Pa)

q F :

One-fourth flow rate of hydraulic fracture (m3/s)

R :

Gas constant, 8.314 × 103 (Pa m3/(kmol K))

R m :

Pore radius (m)

S :

Laplace transform parameter

SV:

Specific surface area (1/m)

t :

Time (s)

t a :

Pseudo-time (s)

T :

Temperature (K)

u :

Flow velocity caused by Darcy’s flow (m/s)

V L :

Langmuir volume (m3/kg)

w F :

Width of hydraulic fracture (m)

x F :

Half-length of hydraulic fracture (m)

x e :

Outer boundary of outer region (m)

y e :

Outer boundary of inner region (m)

Z :

Z-factor of real gas, dimensionless

Θ :

Apparent permeability coefficient, dimensionless

α :

The tangential momentum accommodation coefficient

ρ g :

Average gas density (kg/m3)

ρ bi :

Rock density (kg/m3)

ρ gsc :

Gas density under standard condition (kg/m3)

μ g :

Gas viscosity (Pa s)

ω :

Storativity ratio, defined in Eq. (4)

λ:

Interporosity flow coefficient, defined in Eq. (5)

∅:

Porosity, decimal

η :

Transmissibility factor (m2/s)

1:

Outer region

2:

Inner region

avg:

Average

D:

Dimensionless

f:

Natural fracture

F:

Hydraulic fracture

m:

Matrix

mi:

Matrix system at initial condition

fi:

Fracture system at initial condition

i:

Initial condition

sc:

Standard condition

t:

Total

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Acknowledgements

The authors are grateful for the support provided by the National Natural Science Foundation of China (Key Program) (Grant No. 51534006), and the National Science Fund for Distinguished Young Scholars of China (Grant No. 51125019).

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Correspondence to Shu-yong Hu.

Appendices

Appendix 1: Governing equation in shale matrix

Based on the mass conservation law, and considering Knudsen diffusion and the slippage flow, we can derive the following equation to describe gas flow in a shale matrix:

$$\frac{\partial }{\partial z}\left( {D_{\text{k}} \frac{{\partial \rho_{\text{g}} }}{\partial z} - uF\rho_{\text{g}} } \right) - \frac{{\partial q_{\text{d}} }}{\partial t}\frac{2}{{R_{\text{m}} }} = \frac{{\partial \left( {\rho_{\text{g}} \hat{\phi }_{\text{m}} } \right)}}{\partial t}$$
(33)

where q d is the mass of gas desorbed per unit rock, and can be expressed as

$$q_{\text{d}} = \frac{{\rho_{\text{bi}} \rho_{\text{gsc}} }}{{SV_{\text{k}} }}\frac{{V_{\text{L}} p}}{{p_{\text{L}} + p}}$$
(34)

For gas reservoirs, gas properties are always functions of reservoir pressure, thus Eq. (33) is a linear partial differential equation. Pseudo-pressure and pseudo-time are adopted to linearize the equation.

With the equation of state and the definitions of pseudo-pressure and pseudo-time, Eq. (33) becomes

$$k_{\text{am}} \frac{{\partial^{2} \Delta m_{\text{m}} }}{{\partial z^{2} }} = \left( {\hat{\phi }c_{\text{t}} \mu_{\text{g}} } \right)_{\text{mi}} \left( {1 + \frac{2}{{R_{\text{m}} }}\frac{{\rho_{\text{bi}} B_{\text{g}} }}{{\phi_{\text{m}} c_{\text{t}} {\text{SV}}_{\text{k}} }}\frac{{V_{\text{L}} p_{\text{L}} }}{{\left( {p_{\text{L}} + p_{\text{m}} } \right)^{2} }}} \right)\frac{{\partial \Delta m_{\text{m}} }}{{\partial t_{\text{a}} }}$$
(35)

where \(k_{\text{am}} = \tilde{k}_{\text{m}} \left( {F + \frac{{\mu_{\text{g}} c_{\text{g}} D_{\text{k}} }}{{\tilde{k}_{\text{m}} }}} \right) = \tilde{k}_{\text{m}} \theta_{\text{m}}\) and \(\theta_{\text{m}} = F + \frac{{\mu_{\text{g}} c_{\text{g}} D_{\text{k}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{\text{m}} }}\).

The definitions of pseudo-time and pseudo-pressure are

$$t_{\text{a}} = \left( {c_{\text{t}} \mu_{\text{g}} } \right)_{{\xi {\text{i}}}} \int\limits_{0}^{t} {\frac{1}{{\left( {c_{\text{t}} \mu_{\text{g}} } \right)_{\xi } }}} {\text{d}}t\quad \left( {\xi = 1{\text{m}},\,1{\text{f}},\,2{\text{m}},\,2{\text{f}},\,{\text{F}}} \right)$$
(36)

and

$$m_{\xi } (p_{\xi } ) = 2\int\limits_{0}^{{p_{{_{\xi } }} }} {\theta_{\xi } } \frac{{p^{'} }}{{\mu_{\text{g}} Z}}{\text{d}}p^{'}$$
(37)

If we define the following group of parameters as an adsorption index,

$$\sigma = \frac{{\rho_{\text{bi}} B_{\text{g}} }}{{\phi_{\text{m}} c_{\text{tm}} {\text{SV}}_{\text{k}} }}\frac{2}{{R_{\text{m}} }}\frac{{V_{\text{L}} p_{\text{L}} }}{{\left( {p_{\text{L}} + p_{\text{m}} } \right)^{2} }}$$
(38)

Eq. (35) can be simplified as

$$\frac{{\partial^{2} \Delta m_{\text{m}} }}{{\partial z^{2} }} = \frac{{\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\phi } c_{\text{t}} \mu_{\text{g}} } \right)_{\text{mi}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{\text{m}} \theta_{\text{m}} }}\left( {1 + \sigma } \right)\frac{{\partial \Delta m_{\text{m}} }}{{\partial t_{\text{a}} }}$$
(39)

With the definitions of dimensionless parameters given in “Definitions of parameters” section, one can obtain the dimensionless governing equation for a shale matrix.

Appendix 2: Governing equation in a natural fracture system

Based on the mass conservation law, the following equation can be obtained for a natural fracture system in the outer region:

$$\frac{{\partial \left( {\rho \phi } \right)_{{ 1 {\text{f}}}} }}{\partial t} + \frac{{\partial \left( {\rho v_{x} } \right)_{{ 1 {\text{f}}}} }}{\partial x} - \rho q_{{ 1 {\text{m}}}} = 0$$
(40)

where q 1m is the source item, representing the volumetric gas flow rate from a unit of shale matrix to a natural fracture system, and can be expressed as

$$q_{{ 1 {\text{m}}}} = - \frac{2}{{h_{{ 1 {\text{m}}}} }}\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 1 {\text{m}}}} }}{{\mu_{\text{g}} }}\left. {\frac{{\partial p_{{ 1 {\text{m}}}} }}{\partial z}} \right|_{{z = {{h_{\text{m}} } \mathord{\left/ {\vphantom {{h_{\text{m}} } 2}} \right. \kern-0pt} 2}}}$$
(41)

Substituting Eq. (41) into Eq. (40), and with the equation of state and the definitions of pseudo-pressure and pseudo-time, we can obtain the final expression of a governing equation for a natural fracture system in the outer region:

$$\frac{{\partial^{2} \Delta m_{{ 1 {\text{f}}}} }}{{\partial x^{2} }} - \frac{2}{{h_{\text{m}} }}\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 1 {\text{m}}}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 1 {\text{f}}}} }}\left. {\frac{{\partial \Delta m_{1m} }}{\partial z}} \right|_{{z = {{h_{\text{m}} } \mathord{\left/ {\vphantom {{h_{\text{m}} } 2}} \right. \kern-0pt} 2}}} = \frac{{\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\phi } c_{\text{t}} \mu_{\text{g}} } \right)_{{ 1 {\text{f i}}}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 1 {\text{f}}}} }}\frac{{\partial \Delta m_{{1{\text{f}}}} }}{{\partial t_{\text{a}} }}$$
(42)

Similarly, the final expression of a governing equation for a natural fracture system in the inner region can be obtained as follows:

$$\frac{{\partial^{2} \Delta m_{{ 2 {\text{f}}}} }}{{\partial y^{2} }} - \frac{2}{{h_{\text{m}} }}\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{2m} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 2 {\text{f}}}} }}\left. {\frac{{\partial \Delta m_{2m} }}{\partial z}} \right|_{{z = {{h_{\text{m}} } \mathord{\left/ {\vphantom {{h_{\text{m}} } 2}} \right. \kern-0pt} 2}}} + \frac{1}{{x_{F} }}\frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{1{\text{f}}}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 2 {\text{f}}}} }}\left. {\frac{{\partial \Delta m_{{ 1 {\text{f}}}} }}{\partial x}} \right|_{{x = x_{\text{F}} }} = \frac{{\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\phi } c_{\text{t}} \mu_{\text{g}} } \right)_{{ 2 {\text{f i}}}} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k}_{{ 2 {\text{f}}}} }}\frac{{\partial \Delta m_{{ 2 {\text{f}}}} }}{{\partial t_{\text{a}} }}$$
(43)

With the definitions of dimensionless parameters given in “Definitions of parameters” section, one can obtain the dimensionless forms of Eqs. (42) and (43).

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Hu, Sy., Zhu, Q., Guo, Jj. et al. Production rate analysis of multiple-fractured horizontal wells in shale gas reservoirs by a trilinear flow model. Environ Earth Sci 76, 388 (2017). https://doi.org/10.1007/s12665-017-6728-0

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