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A Three-Dimensional Hydro-mechanical Model for Simulation of Dilatancy Controlled Gas Flow in Anisotropic Claystone

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Abstract

Dilatancy controlled gas flow is characterized by a series of gas pressure-induced dilatant pathways in which the pathway aperture is a function of the effective stress within the solid matrix. In this paper, a three-dimensional hydro-mechanical model is presented to simulate the gas migration in initially saturated claystone with considerable anisotropy. The governing equations including mass conservation, momentum balance and energy conservation are presented for the unsaturated rock containing three phases, i.e., gas, water and solid grain. The constitutive model is proposed in which two conceptualized fracture sets with nonlinear mechanical behavior and cubic law controlled permeability are inserted, which have a direct effect on the hydro-mechanical behavior of the equivalent continuum. Finally, the developed model is validated against three gas injection tests on initially saturated Callovo–Oxfordian claystone. In general, the model is capable of capturing the main features of dilatancy controlled flow, i.e., anisotropic radial deformation, major gas breakthrough, and mechanical volume dilation of the sample. The proposed model offers additional insight into the relation between gas flow, solid matrix deformation and fracture opening/closure, which helps us get in-depth understanding of this gas transport mechanism.

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Fig. 1

(modified from Martinez et al. (2013))

Fig. 2

(modified from Yang et al. (2018))

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Abbreviations

a :

Fracture spacing

\(a_{s}\) :

Spacing of fracture set \(s\)

\(A_{k}\),\(n_{k}\) :

Fitting parameters related to pore size distribution

\(b_{hs}\) :

Hydraulic aperture of fracture set \(s\)

\(C_{s}\) :

Specific storage coefficient

\({\mathbb{C}}\) :

Equivalent stiffness tensor

\(D\) :

Diameter of the sample

\(e\) :

Internal energy of the mixture

\(e_{s}\) :

Internal energy of the skeleton

\(e_{\alpha }\) :

Internal energy of fluid \(\alpha\)

\(E\) :

Young’s modulus

\(f_{s}\) :

Roughness influence factor of fracture set \(s\)

\({\mathbf{g}}\) :

Gravitational acceleration

\(G\) :

Shear modulus

\(h_{\alpha }\) :

Specific enthalpy of fluid \(\alpha\)

\({\mathbf{I}}\) :

Second-order identity tensor

\(K_{\text{n}}\) :

Normal stiffness of fracture

\(K_{\text{ni}}\) :

Initial normal stiffness of fracture

\(K_{\text{s}}\) :

Bulk modulus of solid grain

\(K_{\text{fs}}\) :

Shear stiffness of fracture

\(K_{\phi }\) :

Unjacketed pore bulk modulus

\({\mathbf{k}}_{\text{in}}\) :

Intrinsic permeability tensor

\({\mathbf{k}}_{\text{m}}\) :

Intrinsic permeability tensor of matrix

\({\mathbf{k}}_{\text{f}}\) :

Intrinsic permeability tensor of fracture

\(k_{r\alpha }\) :

Relative permeability of fluid \(\alpha\)

\(k_{\text{fs}}\) :

Permeability through fracture set \(s\) oriented parallel to the flow direction

\(k_{\text{ref}}\) :

Reference intrinsic permeability

\(k_{0}\) :

Initial intrinsic permeability of sample

\(L\) :

Sample length

\(m\) :

Shape parameter of van Genuchten model

\(M\) :

Molar mass of gas

\(n\) :

Eulerian porosity

\(n_{\alpha }\) :

Volume fraction of fluid \(\alpha\)

\(\vec{n}\) :

Unit vector normal to fracture plane

\(\vec{n}_{s}\) :

Unit vector normal to plane of fracture set \(s\)

\(N\) :

Biot’s skeleton modulus

\(p_{1}\) :

Axial pressure

\(p_{3}\) :

Confining pressure

\(p_{\alpha }\) :

Pressure of fluid \(\alpha\)

\(p_{c}\) :

Capillary pressure

\(\bar{p}_{f}\) :

Averaged pore pressure

\(p_{\text{gev}}\) :

Gas entry value

\(p_{0}\) :

Initial air entry value

\(p_{\text{ref}}\) :

Reference gas entry value

\(R\) :

Universal gas constant

\(S_{\alpha }\) :

Saturation degree of fluid \(\alpha\)

\(S_{\text{e}}\) :

Effective water saturation degree

\({\mathbb{S}}\) :

Equivalent compliance tensor

\({\mathbb{S}}_{\text{f}}\) :

Compliance tensor of fracture

\({\mathbb{S}}_{m}\) :

Compliance tensor of matrix

\(T\) :

Absolute temperature

\(T_{s}\) :

Surface tension on air–water interface

\({\mathbb{T}}\) :

Transformation matrix

\({\mathbf{u}}\) :

Displacement tensor

\(u_{\text{n}}\) :

Mechanical aperture of fracture

\(u_{\text{ns}}\) :

Mechanical aperture of fracture set \(s\)

\(U\) :

Interfacial energy

\({\mathbf{v}}_{\text{s}}\) :

Velocity vector of solid

\({\mathbf{v}}_{\alpha }\) :

Velocity vector of fluid \(\alpha\)

\({\mathbf{v}}_{{\varvec{\upalpha}}}^{{\mathbf{D}}}\) :

Darcy’s velocity of fluid \(\alpha\)

\(V_{\text{m}}\) :

Maximum fracture closure

\({\varvec{\upalpha}}\) :

Biot’s coefficients tensor

\(\beta\) :

Rotation angle between local and global axis

\(\beta_{m}\) :

Rotation angle of bedding plane

\(\beta_{\text{fs}}\) :

Rotation angle of fracture set \(s\)

\({\varvec{\upvarepsilon}}\) :

Total strain tensor

\(\varepsilon_{v}\) :

Volumetric strain

\({\varvec{\upvarepsilon}}^{*}\) :

Local strain tensor

\(\mu_{\alpha }\) :

Dynamic viscosity of fluid \(\alpha\)

\(\nu\) :

Poisson’s ratio

\(\pi\) :

Equivalent pore pressure

\(\rho\) :

Density of the mixture

\(\rho_{\text{s}}\) :

Density of solid skeleton

\(\rho_{\alpha }\) :

Density of fluid \(\alpha\)

\({\varvec{\upsigma}}\) :

Total stress tensor

\({\varvec{\upsigma}}{\prime }\) :

Effective stress tensor

\(\sigma_{n} {\prime }\) :

Stress traction normal to fracture set

\({\varvec{\upsigma}}^{*}\) :

Local stress tensor

\(\phi\) :

Lagrangian porosity

\(\phi_{\text{ref}}\) :

Reference porosity

\(\chi_{\text{w}}\) :

Water compressibility

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Acknowledgements

The authors gratefully acknowledge funding from a joint program supported by the China Scholarship Council and University of Ottawa. Moreover, the authors thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for financially supporting this research.

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Authors and Affiliations

  1. Department of Civil Engineering, University of Ottawa, 161 Colonel by, Ottawa, ON, K1N 6N5, Canada

    Jianxiong Yang, Mamadou Fall & Guanlong Guo

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  1. Jianxiong Yang
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  2. Mamadou Fall
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  3. Guanlong Guo
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Correspondence to Mamadou Fall.

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Yang, J., Fall, M. & Guo, G. A Three-Dimensional Hydro-mechanical Model for Simulation of Dilatancy Controlled Gas Flow in Anisotropic Claystone. Rock Mech Rock Eng 53, 4091–4116 (2020). https://doi.org/10.1007/s00603-020-02152-w

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