Poromechanics Solver¶

Introduction¶

This section describes the use of the poroelasticity models implemented in GEOSX.

Theory¶

Governing Equations¶

In our model, the geomechanics (elasticity) equation is expressed in terms of the total stress $\mathbf{\sigma}$ :

$\nabla \mathbf{\sigma} + \rho_b \mathbf{g} = 0$

where it relates to effective stress $\mathbf{\sigma\prime}$ and pore pressure $p$ through Biot’s coefficient $b$ :

$\mathbf{\sigma} = \mathbf{\sigma\prime} - b p\mathbf{I}$

The fluid mass conservation equation is expressed in terms of pore pressure and volumetric (mean) total stress:

$\left( \frac{1}{M} + \frac{b^2}{K_{dr}} \right) \frac{\partial p}{\partial t} + \frac{b}{K_{dr}} \frac{\partial \sigma_v}{\partial t} + \nabla \cdot \mathbf{v}_f = f$

where $M$ is the Biot’s modulus and $K_{dr}$ is the drained bulk modulus.

Unlike the conventional reservoir model that uses Lagranges porosity, in the coupled geomechanics and flow model, Eulers porosity $\phi$ is adopted so the porosity variation is derived as:

$\partial \phi = \left( \frac{b-\phi}{K_s}\right) \partial p + \left( b-\phi \right) \partial \epsilon_v$

where $K_{s}$ is the bulk modulus of the solid grain and $\epsilon_v$ is the volumetric strain.

Parameters¶

The poroelasticity model is implemented as a main solver listed in <Solvers> block of the input XML file that calls both SolidMechanicsLagrangianSSLE and SinglePhaseFlow solvers. In the main solver, it requires the specification of solidSolverName, fluidSolverName, and couplingTypeOption.

The following attributes are supported:

Name	Type	Default	Description
cflFactor	real64	0.5	Factor to apply to the CFL condition when calculating the maximum allowable time step. Values should be in the interval (0,1]
discretization	string	required	Name of discretization object (defined in the Numerical Methods) to use for this solver. For instance, if this is a Finite Element Solver, the name of a Finite Element Discretization should be specified. If this is a Finite Volume Method, the name of a Finite Volume Discretization discretization should be specified.
fluidSolverName	string	required	Name of the fluid mechanics solver to use in the poromechanics solver
initialDt	real64	1e+99	Initial time-step value required by the solver to the event manager.
logLevel	integer	0	Log level
name	string	required	A name is required for any non-unique nodes
porousMaterialNames	string_array	required	The name of the material that should be used in the constitutive updates
solidSolverName	string	required	Name of the solid mechanics solver to use in the poromechanics solver
targetRegions	string_array	required	Allowable regions that the solver may be applied to. Note that this does not indicate that the solver will be applied to these regions, only that allocation will occur such that the solver may be applied to these regions. The decision about what regions this solver will beapplied to rests in the EventManager.
LinearSolverParameters	node	unique	Element: LinearSolverParameters
NonlinearSolverParameters	node	unique	Element: NonlinearSolverParameters

couplingTypeOption: defines the coupling scheme.

The solid constitutive model used here is PoroLinearElasticIsotropic, which derives from ElasticIsotropic and includes an additional parameter: Biot’s coefficient. The fluid constitutive model is the same as SinglePhaseFlow solver. For the parameter setup of each individual solver, please refer to the guideline of the specific solver.

An example of a valid XML block for the constitutive model is given here:

  <Constitutive>
    <PorousElasticIsotropic
      name="porousRock"
      solidModelName="skeleton"
      porosityModelName="skeletonPorosity"
      permeabilityModelName="skeletonPerm"/>

    <ElasticIsotropic
      name="skeleton"
      defaultDensity="0"
      defaultYoungModulus="1.0e4"
      defaultPoissonRatio="0.2"/>

    <CompressibleSinglePhaseFluid
      name="fluid"
      defaultDensity="1"
      defaultViscosity="1.0"
      referencePressure="0.0"
      referenceDensity="1"
      compressibility="0.0e0"
      referenceViscosity="1"
      viscosibility="0.0"/>

    <BiotPorosity
      name="skeletonPorosity"
      grainBulkModulus="1.0e27"
      defaultReferencePorosity="0.3"/>

    <ConstantPermeability
      name="skeletonPerm"
      permeabilityComponents="{ 1.0e-4, 1.0e-4, 1.0e-4 }"/>
  </Constitutive>

Example¶

    <SinglePhasePoromechanics
      name="PoroelasticitySolver"
      solidSolverName="LinearElasticitySolver"
      fluidSolverName="SinglePhaseFlowSolver"
      porousMaterialNames="{ porousRock }"
      logLevel="1"
      discretization="FE1"
      targetRegions="{ Domain }">
      <LinearSolverParameters
	solverType="gmres"
        preconditionerType="mgr"/>
    </SinglePhasePoromechanics>

    <SolidMechanicsLagrangianSSLE
      name="LinearElasticitySolver"
      timeIntegrationOption="QuasiStatic"
      logLevel="1"
      discretization="FE1"
      targetRegions="{ Domain }"
      solidMaterialNames="{ skeleton }"/>

    <SinglePhaseFVM
      name="SinglePhaseFlowSolver"
      logLevel="1"
      discretization="singlePhaseTPFA"
      targetRegions="{ Domain }"
      fluidNames="{ fluid }"
      solidNames="{ porousRock }"
      permeabilityNames="{ skeletonPerm }"/>
  </Solvers>