# Mode: `GEOMECHANICS`

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In PFLOTRAN, linear elasticity model is assumed as the constitutive model for deformation of the rock. Biot’s model is used to incorporate the effect of flow on the geomechanics. In addition, the effect of temperature on geomechanics is considered via coefficient of thermal expansion. The following governing equations are used:

where \({\boldsymbol{u}}\) is the unknown displacement field,
\({\boldsymbol{\sigma}}\) is the Cauchy stress tensor,
\(\lambda\), \(\mu\) are Lamé parameters (Young’s modulus and
Poisson’s ratio can be related to these two parameters),
\({\boldsymbol{b}}\) is the specific body force (which is gravity in
most cases), \({\boldsymbol{n}}\) is the outward normal to the
boundary \(\Gamma^N\). Also, \({\boldsymbol{u}}^p\) is the
prescribed values of \({\boldsymbol{u}}\) on the Dirichlet part of
the boundary \(\Gamma^D\), and \({\boldsymbol{t}}^p\) is the
prescribed traction on \(\Gamma^N\). Additionally, \(\beta\) is
the Biot’s coefficient, \(\alpha\) is the coefficient of thermal
expansion, \(p\), \(T\) are the fluid pressure and temperature,
obtained by solving subsurface flow problem. Also, \(p_0\) and
\(T_0\) are set to initial pressure and temperature in the domain,
\({\boldsymbol{\varepsilon}}\) is the strain tensor and
\(\text{tr}\) is the trace of a second order tensor, \(\Omega\)
is the domain, and \({\boldsymbol{I}}\) is the identity tensor. Note
that stress is assumed *positive under tension*. The above equation
also assumes that the resulting stresses and strains are relative to
the undeformed configuration. The effect of
deformation on the pore structure is accounted for via

Note that the above equations are solved using the finite element method (Galerkin finite element) with the displacements solved for at the vertices. Since, the flow equations are solved via the finite volume method with unknowns such as pressure and temperature solved for at the cell centers, in order to transfer data from the subsurface to geomechanics grid without interpolation, the geomechanics grid is constructed such that the vertices of the geomechanics grid coincide with the cell centers of the subsurface mesh. That is, the dual mesh of the subsurface mesh is used for the geomechanics solve.

Also, the geomechanics grid must be read in as an unstructured grid. Even if one needs to work with a structured grid, the grid must be set up in the unstructured grid format.