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Analytical Solution for Shale Gas Productivity of a Multiple-Fractured Horizontal Well Based on a Diffusion Model

  • Research Article - Petroleum Engineering
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Abstract

An analytical solution is developed for the shale gas productivity of a multiple-fractured horizontal well based on a diffusion model and a trilinear flow pattern. The shale gas reservoir is divided into three flow regions: hydraulic-fracture region, micro-fracture network or dual-porosity region, and pure-matrix region. For the pure-matrix region, a transient diffusion equation is solved based on our previous diffusivity model developed for the shale matrix. For the micro-fracture network region, a modified dual-porosity model is proposed wherein both the free and adsorbed gases in the shale matrix flow into the micro-fracture network through a pseudo-steady diffusion process. These gases then form conflux at the hydraulic fractures and continue to the wellbore. A dimensionless solution is obtained for the bottom-hole pressure in the Laplace domain considering the skin effect. An analytical solution is obtained for the gas production rate in a real-time domain through a partial Taylor series simplification and Laplace inverse transform. This analytical solution is compared with the field data of the shale gas produced from a fractured horizontal well located in southwestern China, and a good agreement is observed. Finally, a parametric study is conducted to quantify the effects of key parameters on the gas production rate. The parameters include the bottom-hole pressure, half-length of the hydraulic fracture, permeability of the hydraulic fracture, block size of the shale matrix, and pore size within the shale matrix. These results show that the analytical solution can be used to estimate the enhancement of the shale gas recovery through hydraulic fracturing.

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Abbreviations

a :

Shape factor, \(\hbox {m}^{-2}\), \(a={3\pi ^{2}}/{L^{2}}\)

\(d_\mathrm{m}\) :

Molecular collision diameter, m

\(d_\mathrm{p}\) :

Size of pores with shale matrix, m

\(D_\mathrm{m}\) :

Gas apparent diffusivity in shale matrix, \(\hbox {m}^{2}/\hbox {s}\)

\(D_\mathrm{f}\) :

Pseudo-gas diffusivity of micro-fractures network, \(\hbox {m}^{2}/\hbox {s}\)

\(D_\mathrm{F}\) :

Pseudo-gas diffusivity of hydraulic fractures, \(\hbox {m}^{2}/\hbox {s}\)

\(D_\mathrm{K}\) :

Knudsen diffusion coefficient, \(\hbox {m}^{2}/\hbox {s}\)

\(D_\mathrm{S}\) :

Surface diffusion coefficient, \(\hbox {m}^{2}/\hbox {s}\)

\(D_{\mathrm{f}'}\) :

Fractal dimension of the pores surface, dimensionless

h :

Reservoir thickness, m

\(k_\mathrm{f}\) :

Permeability of micro-fractures network, m\(^{2}\)

\(k_\mathrm{F}\) :

Permeability of hydraulic fractures, m\(^{2}\)

Kn :

Knudsen number, \(Kn=\frac{\kappa _{B}T}{\sqrt{2}\pi d_\mathrm{m}^{2} d_\mathrm{p} p_\mathrm{m}}\)

L :

The size of shale matrix block, m

\(M_\mathrm{g}\) :

Apparent molecular weight of shale gas, kg/mol

N :

Number of hydraulic fractures

\(p_\mathrm{i}\) :

Initial reservoir pressure, \(P_\mathrm{a}\)

\(P_\mathrm{L}\) :

Langmuir pressure constant, \(P_\mathrm{a}\)

\(p_{{\mathrm{sc}}}\) :

Gas pressure at standard condition, \(P_\mathrm{a}\)

\(p_\mathrm{w}\) :

Bottom-hole pressure, \(P_\mathrm{a}\)

\(q_{{\mathrm{Fsc}}}\) :

Hydraulic-fracture production rate at standard condition, \(\hbox {m}^{3}/\hbox {s}\)

\(r_\mathrm{w}\) :

Wellbore radius, m

s :

Variable of Laplace transformation, dimensionless

\(s_\mathrm{c}\) :

Horizontal well flow choking skin factor, dimensionless

t :

Production time, s

T :

Reservoir temperature, K

V :

Volumetric gas concentration, sm\(^{3}\)/m\(^{3}\)

\(V_\mathrm{i}\) :

Volumetric gas concentration at initial condition, sm\(^{3}\)/m\(^{3}\)

\(V_\mathrm{E}\) :

Equilibrium volumetric gas concentration, \({\hbox {m}^{3}}/{\hbox {m}^{3}}\)

\(V_\mathrm{L}\) :

Langmuir volume (at standard condition), m\(^{3}/\hbox {m}^{3}\)

\(w_\mathrm{F}\) :

Hydraulic-fracture width, m

\(x_\mathrm{e}\) :

Fractured reservoir volume half-width, m

\(x_\mathrm{F}\) :

Hydraulic-fracture half-length, m

\(y_\mathrm{e}\) :

Half-distance between hydraulic fractures, m

Z :

Z factor of shale gas, dimensionless

\(\phi \) :

Porosity for shale matrix 1, 2, micro-fractures system and hydraulic fracture, fraction

\(\sigma \) :

Adsorption index, dimensionless

\(\varepsilon \) :

Free index, dimensionless

\({\delta }'\) :

The ratio of normalized molecular size to local average pore diameter \({\delta }'=d_\mathrm{m}/d_\mathrm{p}\)

\(\kappa _\mathrm{B}\) :

Boltzmann constant, \(1.381\times 10^{-23}\hbox {J/K}\)

\(\omega _{K}\) :

The probability function of collisions with wall, \(\omega _K =\frac{Kn}{Kn+1}\)

\(\tau \) :

Gas diffusion time in shale matrix, s

\(\tau _h\) :

Tortuosity of pores of shale matrix, dimensionless

\(\rho _\mathrm{g}\) :

Shale gas density, kg/m\(^{3}\)

\(\rho _{{\mathrm{gsc}}}\) :

Shale gas density at standard condition, kg/m\(^{3}\)

\(\mu \) :

Gas viscosity, \(\hbox {Pa}\cdot \hbox {s}\)

\(\psi \) :

Pseudo-pressure, \(\hbox {P}_\mathrm{a}/\hbox {s}\)

\(\psi _\mathrm{L}\) :

Pseudo-Langmuir pressure constant, \(\hbox {P}_\mathrm{a}/\hbox {s}\)

D :

Dimensionless

i :

Initial condition

w :

Wellbore

m :

Matrix

f :

Micro-fractures

F :

Hydraulic fractures

sc :

Standard condition

L:

Langmuir’s constant

−:

Laplace transform

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Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    Jia Liu, J. G. Wang, Feng Gao & Yang Ju

  2. School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, 221116, China

    J. G. Wang & Feng Gao

  3. State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology at Beijing, Beijing, 100083, China

    Yang Ju

  4. School of Engineering, University of Tasmania, Hobart, TAS, 7001, Australia

    Xiaolin Wang

Authors
  1. Jia Liu
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  2. J. G. Wang
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  4. Yang Ju
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  5. Xiaolin Wang
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Correspondence to J. G. Wang.

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Liu, J., Wang, J.G., Gao, F. et al. Analytical Solution for Shale Gas Productivity of a Multiple-Fractured Horizontal Well Based on a Diffusion Model. Arab J Sci Eng 43, 2563–2579 (2018). https://doi.org/10.1007/s13369-017-2824-4

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