Porous Solids

Overview

Simulation of fluid flow in porous media and of poromechanics, requires to define, along with fluid properties, the hydrodynamical properties of the solid matrix. Thus, for porous media flow and and poromecanical simulation in GEOS, two types of composite constitutive models can be defined to specify the characteristics of a porous material: (1) a CompressibleSolid model, used for flow-only simulations and which assumes that all poromechanical effects can be represented by the pressure dependency of the porosity; (2) a PorousSolid model which, instead, allows to couple any solid model with a BiotPorosity model and to include permeability’s dependence on the mechanical response.

Both these composite models require the names of the solid, porosity and permeability models that, combined, define the porous material. The following sections outline how these models can be defined in the Constitutive block of the xml input files and which type of submodels they allow for.

CompressibleSolid

This composite constitutive model requires to define a NullModel as solid model (since no mechanical properties are used), a PressurePorosity model and any type of Permeability model.

To define this composite model the keyword CompressibleSolid has to be appended to the name of the permeability model of choice, as shown in the following example for the ConstantPermeability model.

<Constitutive>
  <CompressibleSolidConstantPermeability name="porousRock"
                                         solidModelName="nullSolid"
                                         porosityModelName="rockPorosity"
                                         permeabilityModelName="rockPermeability"/>

 <NullModel name="nullSolid"/>

 <PressurePorosity name="rockPorosity"
                   referencePressure="1.0e27"
                   defaultReferencePorosity="0.3"
                   compressibility="1.0e-9"/>

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

</Constitutive>

PorousSolid

To run poromechanical problems, the total stress is decomposed into an “effective stress” (driven by mechanical deformations) and a pore fluid pressure component, following the Biot theory of poroelasticity. For single-phase flow, or multiphase problems with no capillarity, this decomposition reads

$\sigma_{ij} = \sigma\prime_{ij} - b p \delta_{ij}$

where $\sigma_{ij}$ is the $ij$ component of the total stress tensor, $\sigma\prime_{ij}$ is the $ij$ component of the effective (Cauchy) stress tensor, $b$ is Biot’s coefficient, $p$ is fluid pressure, and $\delta$ is the Kronecker delta.

The PorousSolid models simply append the keyword Porous in front of the solid model they contain, e.g., PorousElasticIsotropic, PorousDruckerPrager, and so on. Additionally, they require to define a BiotPorosity model and a ConstantPermeability model. For example, a Poroelastic material with a certain permeability can be defined as

<Constitutive>
  <PorousElasticIsotropic name="porousRock"
                          porosityModelName="rockPorosity"
                          solidModelName="rockSkeleton"
                          permeabilityModelName="rockPermeability"/>

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

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

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

Note that any of the previously described solid models is used by the PorousSolid model to compute the effective stress, leading to either poro-elastic, poro-plastic, or poro-damage behavior depending on the specific model chosen.