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Fluid-driven transition from damage to fracture in anisotropic porous media: a multi-scale XFEM approach

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Abstract

In this paper, a numerical method is proposed to simulate multi-scale fracture propagation driven by fluid injection in transversely isotropic porous media. Intrinsic anisotropy is accounted for at the continuum scale, by using a damage model in which two equivalent strains are defined to distinguish mechanical behavior in the direction parallel and perpendicular to the layer. Nonlocal equivalent strains are calculated by integration and are directly introduced in the damage evolution law. When the weighted damage exceeds a certain threshold, the transition from continuum damage to cohesive fracture is performed by dynamically inserting cohesive segments. Diffusion equations are used to model fluid flow inside the porous matrix and within the macro-fracture, in which conductivity is obtained by Darcy’s law and the cubic law, respectively. In the fractured elements, the displacement and pore pressure fields are discretized by using the XFEM technique. Interpolation on fracture elements is enriched with jump functions for displacements and with level set-based distance functions for fluid pressure, which ensures that displacements are discontinuous across the fracture, but that the pressure field remains continuous. After spatial and temporal discretization, the model is implemented in a Matlab code. Simulations are carried out in plane strain. The results validate the formulation and implementation of the proposed model and further demonstrate that it can account for material and stress anisotropy.

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Abbreviations

\(\alpha (\beta )\) :

Mode I (mode II) cohesive law shape factor

\(\alpha (\varvec{x})\) :

Nonlocal weight function

\(\alpha _{ii}^t\) :

Material parameters controlling damage growth rate

\(\bar{p}\) :

Prescribed pore pressure

\(\bar{\varvec{t}}\) :

Prescribed traction on exterior boundary

\(\bar{\varvec{t}}_d\) :

Cohesive traction vector on fracture surfaces in global coordinate

\(\bar{\varvec{u}}\) :

Prescribed displacement

\(\bar{\epsilon }_i^{\rm eq}\) :

Nonlocal equivalent strain components

\(\bar{q}\) :

Prescribed flow rate

\(\bar{q}_d\) :

Flow rate difference across fracture

\(\varvec{\alpha }\) :

Biot coefficient tensor

\(\varvec{\epsilon }\) :

Strain tensor

\(\varvec{\omega }\) :

Damage variables

\(\varvec{\sigma }\) :

Stress tensor

\(\varvec{a}\) :

Enriched degree of freedom on displacement

\(\varvec{b}\) :

Enriched degree of freedom on pore pressure

\(\varvec{g}\) :

Gravity vector

\(\varvec{m}_{\varGamma _d}\) :

Unit tangent vector along fracture

\(\varvec{n}_{\varGamma _d}\) :

Unit normal vector along fracture

\(\varvec{q}\) :

Fluid flow rate

\(\varvec{s}\) :

Fracture natural coordinate

\(\varvec{v}\) :

Fluid velocity

\(\varvec{\kappa }_{\rm m}\) :

Matrix permeability tensor

\(\varDelta _n(\varDelta _t)\) :

Separations in the normal (shear) direction at current time

\(\delta _n(\delta _t)\) :

Separations in the normal (shear) direction at failure

\(\epsilon _i^{\rm eq}\) :

Local equivalent strain components

\(\epsilon _{12}^{s0}\) :

Initial out-of-bedding-plane shear strain threshold

\(\epsilon _{ii}^{t0}\) :

Initial tensile strain thresholds

\(\varGamma _{n}(\varGamma _{t})\) :

Mode I (mode II) energy constants

\(\kappa _i\) :

Internal state variables controlling damage evolution

\(\lambda _{n}(\lambda _{t})\) :

Mode I (mode II) initial slope indicator

\(\mathbb {C}\) :

Stiffness tensor

\(\mathbf M\) :

Damage operator in Voigt notation

\(\mathbf {S/C}\) :

Compliance/stiffness matrix in Voigt notation

\(\mu\) :

Fluid viscosity

\(\phi\) :

Matrix porosity

\(\phi (\varvec{x})\) :

Level set function

\(\phi _n(\phi _t)\) :

Mode I (mode II) cohesive energy release rate

\(\rho _{\text {f}}\) :

Fluid density

\(\sigma _{max}(\tau _{max})\) :

Mode I (mode II) cohesive strength

c :

Fracture hydraulic conductivity

\(H_s\) :

Helmholtz free energy

\(H_{\varGamma _d}\) :

Heaviside jump function

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

Fluid bulk modulus

\(l_c\) :

Nonlocal internal length parameter

M :

Biot modulus

m(n):

Mode I (mode II) non-dimensional exponents

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

Fluid mass

N :

Biot skeleton modulus

\(N_{ui}(N_{pi})\) :

Shape functions for displacement (pore pressure)

p :

Pore pressure

\(Q_{\rm in}\) :

Fluid injection rate

\(T_n(T_t)\) :

Normal (shear) cohesive traction in local coordinate

w :

Fracture aperture

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Acknowledgements

Financial support for this research was provided by the U.S. National Science Foundation, under Grant 1552368: “CAREER: Multiphysics Damage and Healing of Rocks for Performance Enhancement of Geo-Storage Systems—A Bottom–Up Research and Education Approach”

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  1. Energy and Environment Science & Technology Directorate, Idaho National Laboratory, Idaho Falls, ID, USA

    Wencheng Jin

  2. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Chloé Arson

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Jin, W., Arson, C. Fluid-driven transition from damage to fracture in anisotropic porous media: a multi-scale XFEM approach. Acta Geotech. 15, 113–144 (2020). https://doi.org/10.1007/s11440-019-00813-x

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