Poromechanics Solver

Introduction

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

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, flowSolverName, 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]

flowSolverName

groupNameRef

required

Name of the flow solver used by the coupled solver

initialDt

real64

1e+99

Initial time-step value required by the solver to the event manager.

isThermal

integer

0

Flag indicating whether the problem is thermal or not. Set isThermal=”1” to enable the thermal coupling

logLevel

integer

0

Log level

name

groupName

required

A name is required for any non-unique nodes

solidSolverName

groupNameRef

required

Name of the solid solver used by the coupled solver

targetRegions

groupNameRef_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"
      defaultGrainBulkModulus="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"
      flowSolverName="SinglePhaseFlowSolver"
      logLevel="1"
      targetRegions="{ Domain }">
      <LinearSolverParameters
        solverType="gmres"
        preconditionerType="mgr"/>
    </SinglePhasePoromechanics>

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

    <SinglePhaseFVM
      name="SinglePhaseFlowSolver"
      logLevel="1"
      discretization="singlePhaseTPFA"
      targetRegions="{ Domain }"/>
  </Solvers>