Transport in Porous Media

, Volume 109, Issue 1, pp 109–130

Multi-continuum Approach to Modelling Shale Gas Extraction

  • Anna Russian
  • Philippe Gouze
  • Marco Dentz
  • Alain Gringarten

DOI: 10.1007/s11242-015-0504-y

Cite this article as:
Russian, A., Gouze, P., Dentz, M. et al. Transp Porous Med (2015) 109: 109. doi:10.1007/s11242-015-0504-y


Production rates in horizontal shale gas wells display declines that are controlled by the low permeability and the intrinsic heterogeneity of the shale matrix. We present an original multi-continuum approach that yields a physical model able to reproduce the complexity of the decreasing gas rates. The model describes the dynamics of gas rate as function of the physical reservoir parameters and geometry, while the shale matrix heterogeneity is accounted for by a stochastic description of transmissivity field. From the 3D (Dimensional) problem setting, including the heterogeneous shale matrix, the fractures generated by the hydrofracking operations, as well as the production well characteristics, we establish an effective upscaled 1D model for the gas pressures in fracture and matrix as well as the volumetric flux. We analyse the decline curves behaviour, and we identify the time scales that characterize the dynamics of the gas rate decline using explicit analytical Laplace space solutions of the upscaled process model. Asymptotically, the flux curves decrease exponentially, while in an intermediate regime we find a power-law behaviour, in which the flux scales with a power law in time as \(t^{-\beta }\), where \(\beta \) reflects the medium heterogeneity. We use this solution to fit a set of real data displaying distinctly different decline trends and study the sensitivity of the model to the reservoir parameters in order to identify their respective controls at the different stages of the decline curve dynamics. Results indicate that the initial value of the gas rate is determined by the transmissivity of the fractures and the initial pressure of the gas in the shale matrix. The latter causes mainly a shift in the entire decline curve. The early time of decline curve shape is primarily controlled by the fracture properties (compressibility and transmissivity). During the main part of the economically valuable production times, i.e. before the production rate drops exponentially, the decline curve is strongly controlled by the properties of the shale rocks including their heterogeneity, which is modelled by two parameters describing the non-Fickian pressure diffusion effects in a stochastic framework.


Shale gas Modelling Multi-continuum model Shale reservoirs 

List of symbols

\(a, b_i\)

Rock and fluid compressibility (\(\hbox {Pa}^{-1}\))

\( k\)

Permeability (\(\hbox {L}^{2}\))

\(N_\mathrm{f} \)

Number of fracture stages (\(-\))

\( p_\mathrm{w}\)

Pressure in the horizontal well (\(\hbox {M L}^{-1}\hbox { T}^{-2}\))


spatially averaged pressure (\(\hbox {M L}^{-1}\hbox { T}^{-2}\))

\(\langle \overline{p} \rangle \)

Mean pressure averaged spatially and stochastically (\(\hbox {M L}^{-1}\hbox { T}^{-2}\))


Gas rate production (\(\hbox {L}^{3}\hbox { T}^{-1}\))


Radial distance from the edge of horizontal well (L)


Radius of horizontal well (L)


Radial extension of the fractured zone (L)


Effective compressibility (\(\hbox {Pa}^{-1}\))


Time (T)


Mean diffusion time in the fractured zone (T)


Mean diffusion time over the thickness of the matrix (T)


Local spatial distance from the fractured zone (L)

\(\alpha \)

Outflow constant [ \(\hbox {L}^{-3}\hbox { T}^{-1}\hbox { M}\))

\(\beta \)

Exponent of truncated power-law distribution (\(-\))

\( \mu \)

Viscosity (\(\hbox {M T}^{-2}\))

\( \lambda \)

Laplace variable (\(\hbox {T}^{-1}\))

\(\rho _i\)

Density (\(\hbox {M L}^{-3}\))

\(\tau _1\)

Lower limit truncated power-law distribution (T)

\(\tau _2\)

Cut-off truncated power-law distribution (T)

\(\tau _{a}\)

Activation time of the drainage matrix (T)

\(\varphi \)

Memory function of the multi-continuum model (\(\hbox {T}^{-1}\))

Derived quantities

\(K_i = k_i/\mu _i\)

Permeability divided by viscosity (\(\hbox {L}^{3}\hbox { M}^{-1}\hbox { T}\))

\(S_i=z_i s_i\)

Total effective compressibility (\(\hbox {m Pa}^{-1}\))

\(T_i=z_i K_i\)

Effective transmissivity (\(\hbox {L}^{4}\hbox { M}^{-1}\hbox { T}\))

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Anna Russian
    • 1
  • Philippe Gouze
    • 1
  • Marco Dentz
    • 2
  • Alain Gringarten
    • 3
  1. 1.Géosciences, Université de Montpellier 2, CNRSMontpellierFrance
  2. 2.Institute of Environmental Assessment and Water Research (IDAEA)Spanish National Research Council (CSIC)BarcelonaSpain
  3. 3.Centre for Petroleum StudiesImperial College London (ICL)LondonUK