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Multiphysics Lattice Discrete Particle Modeling (M-LDPM) for the Simulation of Shale Fracture Permeability

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Abstract

A three-dimensional multiphysics lattice discrete particle model (M-LDPM) framework is formulated to investigate the fracture permeability behavior of shale. The framework features a dual lattice system mimicking the mesostructure of the material and simulates coupled mechanical and flow behavior. The mechanical lattice model simulates the granular internal structure of shale, and describes heterogeneous deformation by means of discrete compatibility and equilibrium equations. The network of flow lattice elements constitutes a dual graph of the mechanical lattice system. A discrete formulation of mass balance for the flow elements is presented to model fluid flow along cracks and intact materials. The overall computational framework is implemented with a mixed explicit–implicit integration scheme and a staggered coupling method that makes use of the dual lattice topology enabling the seamless two-way coupling of the mechanical and flow behaviors. The proposed model is used for the computational analysis of shale fracture permeability behavior by simulating triaxial direct shear tests on Marcellus shale specimens under various confining pressures. The simulated mechanical response is calibrated against the experimental data, and the predicted permeability values are also compared with the experimental measurements. Furthermore, the paper presents the scaling analysis of both the mechanical response and permeability measurements based on simulations performed on geometrically similar specimens with increasing size. The simulated stress strain curves show a significant size effect in the post-peak due to the presence of localized fractures. The scaling analysis of permeability measurements enables prediction of permeability for large specimens by extrapolating the numerical results of small ones.

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Abbreviations

\(\epsilon _N, \epsilon _M, \epsilon _L\) :

Facet normal (N) and shear (M, L) strains

\(t_N, t_M, t_L\) :

Facet normal (N) and shear (M, L) stresses

\(\delta _N, \delta _M, \delta _L\) :

Mesoscale crack opening: normal (N) and shear (M, L)

\(\ell\) :

Tetrahedron edge associated with a facet

\(\mathbf {n}, \mathbf {m}, \mathbf {l}\) :

Unit vector normal and tangential to a facet

\(A_{\text {f}}\) :

Projected facet area

\(\mathbf {t}\) :

Total stress vector on each facet

\(\mathbf {t}_{{s}}\) :

Facet stress vector applied to solid phase

\(\mathbf {t}_{{w}}\) :

Fluid pressure vector on each facet

\(E_N, \alpha\) :

Facet normal modulus and shear-normal coupling parameter

\(\sigma _{{t}}, \ell _{{t}}\) :

Facet tensile strength and characteristic length

\(\sigma _{{s}}, \mu _0\) :

Facet shear strength and internal friction coefficient

b :

Biot coefficient

\(\ddot{\mathbf {U}}, \ddot{\mathbf {\Theta }}\) :

Translational and rotational acceleration of nodes

\(M_{{u}}, M_{\theta }\) :

Mass and moment of inertia of each cell

L :

Length of flow edge element

\(A, A_n\) :

Area and projected area of the face of two adjacent tetrahedra (or control volume)

\(V_i\) :

Control volume i (\(i =1, 2\)) associated with the flow edge element

\(g_i\) :

Faction coefficient

\(\rho _{{{w}}}, \rho _{{{w}}0}\) :

Fluid density in the current and reference state

\(p_i, p_0\) :

Fluid pressure in control volume \(V_i\) and its reference value

\(K_{{w}}\) :

Bulk modulus of fluid

\(M_{{wu}}^i\) :

Fluid mass in uncracked volume \(V_i\)

\(M_{{wc}}^i\) :

Fluid mass stored in cracked volume \(V_{{ci}}\)

\(Q_{{wu}}^{12}\) :

Fluid mass flux from uncracked volume \(V_1\) into \(V_2\)

\(Q_{{wc}}^{12}\) :

Fluid mass flux from \(V_1\) into \(V_2\) through cracks

\(M_{{b}}\) :

Biot modulus

\(\kappa _0\) :

Intrinsic permeability of intact materials

\(\mu _{{w}}\) :

Dynamic viscosity of fluid

\(\mathbf {C}, \mathbf {D}, \mathbf {S}\) :

Capacity, conductance, and source matrices for the flow model

\(\mathbf {p}\) :

Unknown pressure vector

H, D, R :

Height, diameter, and radius of the simulated cylindrical specimen

\(\sigma _{{c}}^\prime\) :

Effective confining pressure

\(p_{\text {in}}, p_{\text {out}}\) :

Inlet and outlet pressures on top and top of specimens

Q :

Absolute value of downstream flow rate

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

Macroscopic apparent permeability along specimen core

\(K_{\text {in}}, K_{{b}}\) :

Macroscopic permeability of the intact material and the fracture localization band

h :

Width of the permeable fracture localization band

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Acknowledgements

The work of WL and GC was supported by the Center for Sustainable Engineering of Geological and Infrastructure Materials (SEGIM) and the Quest high performance computing facility at Northwestern University. The work of XZ was supported with ES3 R&D resources. The work of WC and LF was supported by the Department of Energy’s Basic Energy Sciences program under Grant DE-AC52-06NA25396/FWP-LANL20171450.

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Authors and Affiliations

  1. Theoretical and Applied Mechanics, Northwestern University, Evanston, IL, 60208, USA

    Weixin Li & Gianluca Cusatis

  2. ES3, 550 West C St., San Diego, CA, 92101, USA

    Xinwei Zhou

  3. Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    J. William Carey & Luke P. Frash

  4. Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL, 60208, USA

    Weixin Li & Gianluca Cusatis

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Li, W., Zhou, X., Carey, J.W. et al. Multiphysics Lattice Discrete Particle Modeling (M-LDPM) for the Simulation of Shale Fracture Permeability. Rock Mech Rock Eng 51, 3963–3981 (2018). https://doi.org/10.1007/s00603-018-1625-8

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