# Horizontal equilibrium in a porous medium In many geotechnical applications there is also horizontal stress, typically caused by tectonic action. If the pore fluid is under hydrostatic equilibrium and $\tau _ { x z } = \tau _ { y z } = 0$ , equilibrium in the horizontal directions requires that the horizontal components of effective stress do not vary with horizontal position: $\overline { { \sigma } } _ { h } ( z )$ only, where $\overline { { \sigma } } _ { h }$ is any horizontal component of effective stress. # Initial conditions The initial effective geostatic stress field, , is given by defining initial stress conditions. Unless the enhanced procedure is used, the initial state of stress must be close to being in equilibrium with the applied loads and boundary conditions. See “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1. You can specify that the initial stresses vary only with elevation, as described in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1. In this case the horizontal stress is typically assumed to be a fraction of the vertical stress: those fractions are defined in the x- and y-directions. In problems involving partially or fully saturated porous media, initial pore fluid pressures, $u _ { w } .$ , void ratios, $e ^ { 0 }$ , and saturation values, s, must be given (see “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1). In partially saturated cases the initial pore pressure and saturation values must lie on or between the absorption and exsorption curves (see “Sorption,” Section 26.6.4). A partially saturated problem is illustrated in “Wicking in a partially saturated porous medium,” Section 1.9.3 of the Abaqus Benchmarks Guide. You may also specify initial temperatures in the model if heat transfer is modeled during the geostatic procedure. # Boundary conditions Boundary conditions can be applied to displacement degrees of freedom 1–6 and to pore pressure degree of freedom 8 (“Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1). If coupled temperature–pore pressure elements are used, boundary conditions on temperature degree of freedom 11 can also be applied to nodes belonging to these elements. If the enhanced procedure is used and nonzero boundary conditions are applied, it is the user’s responsibility to ensure that the displacements corresponding to the tolerances specified are larger than the displacements in the analysis; otherwise, the displacements at the nonzero boundary nodes will be reset to zero with the tolerances specified. The boundary conditions should be in equilibrium with the initial stresses and applied loads. If the horizontal stress is nonzero, horizontal equilibrium must be maintained by fixing the boundary conditions on any nonhorizontal edges of the finite element model in the horizontal direction or by using infinite elements (“Infinite elements,” Section 28.3.1). If heat transfer is modeled, the temperature boundary conditions should be in equilibrium with the initial temperature field and applied thermal loads. # Loads The following loading types can be prescribed in a geostatic stress field procedure: • Concentrated nodal forces can be applied to the displacement degrees of freedom (1–6); see “Concentrated loads,” Section 34.4.2. • Distributed pressure forces or body forces can also be applied; see “Distributed loads,” Section 34.4.3. The distributed load types available with particular elements are described in Part VI, “Elements.” The magnitude and direction of gravitational loading are defined by using the gravity or body force distributed load types. • Pore fluid flow is controlled as described in “Pore fluid flow,” Section 34.4.7. If heat transfer is modeled, the following types of thermal loading can also be prescribed (“Thermal loads,” Section 34.4.4). These loads are not supported in Abaqus/CAE during a geostatic analysis. • Concentrated heat fluxes. • Body fluxes and distributed surface fluxes. • Convective film conditions and radiation conditions; film properties can be made a function of temperature. # Predefined fields The following predefined fields can be specified in a geostatic stress field procedure, as described in “Predefined fields,” Section 34.6.1: • For a geostatic analysis that does not model heat transfer and uses regular pore pressure elements, temperature is not a degree of freedom and nodal temperatures can be specified. • Predefined temperature fields are not allowed in a geostatic analysis that also models heat transfer. Boundary conditions should be used instead to specify temperatures, as described earlier. • The values of user-defined field variables can be specified; these values affect only field-variabledependent material properties, if any. # Material options Any of the mechanical constitutive models available in Abaqus/Standard can be used to model the porous solid material. However, the enhanced procedure can be used only with the elastic, porous elastic, extended Cam-clay plasticity, and Mohr-Coulomb plasticity models. Use of a nonsupported material model with this procedure may lead to poor convergence or no convergence if displacements are larger than the displacements corresponding to the tolerances specified. Abaqus will issue a warning message if the procedure is used with a nonsupported material model. If a porous medium will be analyzed subsequent to the geostatic procedure, pore fluid flow quantities such as permeability and sorption should be defined (see “Pore fluid flow properties,” Section 26.6.1). If heat transfer is modeled, thermal properties such as conductivity, specific heat, and density should be defined for both the solid and the pore fluid phases (see “Thermal properties if heat transfer is modeled” in “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1, for details on how to specify separate thermal properties for the two phases). # Elements Any of the stress/displacement elements in Abaqus/Standard can be used in a geostatic procedure. Continuum pore pressure elements can also be used for modeling fluid in a deforming porous medium. These elements have pore pressure degree of freedom 8 in addition to displacement degrees of freedom 1–3. However, the enhanced procedure can be used only with continuum and cohesive elements with pore pressure degrees of freedom and the corresponding stress/displacements elements. Use of nonsupported elements with this procedure may lead to poor convergence or no convergence if displacements are larger than the displacements corresponding to the tolerances specified. Abaqus will issue a warning message if the procedure is used with a nonsupported element. Continuum elements that couple temperature, pore pressure, and displacement can be used if heat transfer needs to be modeled. These elements have temperature degree of freedom 11 in addition to pore pressure degree of freedom 8 and displacement degrees of freedom 1–3. See “Choosing the appropriate element for an analysis type,” Section 27.1.3, for more information. # Output The element output available for a coupled pore fluid diffusion/stress analysis includes the usual mechanical quantities such as (effective) stress; strain; energies; and the values of state, field, and user-defined variables. In addition, the following quantities associated with pore fluid flow are available:

VOIDR	Void ratio, e.
POR	Pore pressure, $u_{w}$ .
SAT	Saturation, s.
GELVR	Gel volume ratio, $n_{t}$ .
FLUVR	Total fluid volume ratio, $n_{f}$ .
FLVEL	Magnitude and components of the pore fluid effective velocity vector, f.
FLVELM	Magnitude, f, of the pore fluid effective velocity vector.
FLVELn	Component n of the pore fluid effective velocity vector (n=1, 2, 3).

If heat transfer is modeled, the following element output variables associated with heat transfer are also available:

HFL	Magnitude and components of the heat flux vector.
HFLn	Component n of the heat flux vector (n=1, 2, 3).
HFLM	Magnitude of the heat flux vector.
TEMP	Integration point temperatures.

The nodal output available includes the usual mechanical quantities such as displacements, reaction forces, and coordinates. In addition, the following quantities associated with pore fluid flow are available: POR Pore pressure at a node. RVF Reaction fluid volume flux due to prescribed pressure. This flux is the rate at which fluid volume is entering or leaving the model through the node to maintain the prescribed pressure boundary condition. A positive value of RVF indicates fluid is entering the model. If heat transfer is modeled, the following nodal output variables associated with heat transfer are also available: NT Nodal point temperatures. RFL Reaction flux values due to prescribed temperature. RFLn Reaction flux value n at a node (n=11, 12, …). CFL Concentrated flux values. CFLn Concentrated flux value n at a node (n=11, 12, …). All of the output variable identifiers are outlined in “Abaqus/Standard output variable identifiers,” Section 4.2.1. Input file template ```txt *HEADING ... *MATERIAL, NAME=mat1 Data lines to define mechanical properties of the solid material ... *DENSITY Data lines to define the density of the dry material *PERMEABILITY, SPECIFIC=γw Data lines to define permeability, k, as a function of the void ratio, e *CONDUCTIVITY Data lines to define thermal conductivity of the solid grains if heat transfer is modeled *CONDUCTIVITY,TYPE=ISO, PORE FLUID Data lines to define thermal conductivity of the permeating fluid if heat transfer is modeled *SPECIFIC HEAT Data lines to define specific heat of the solid grains if transient heat transfer is modeled in a subsequent step *SPECIFIC HEAT,PORE FLUID Data lines to define specific heat of the permeating fluid if transient heat transfer is modeled in a subsequent step *DENSITY Data lines to define density of the solid grains if transient heat transfer is modeled in a subsequent step *DENSITY,PORE FLUID Data lines to define density of the permeating fluid if transient heat transfer is modeled in a subsequent step ``` \*LATENT HEAT Data lines to define latent heat of the solid grains if phase change due to temperature change is modeled \*LATENT HEAT,PORE FLUID Data lines to define latent heat of the permeating fluid if phase change due to temperature change is modeled \*INITIAL CONDITIONS, TYPE=STRESS, GEOSTATIC Data lines to define the initial stress state \*INITIAL CONDITIONS, TYPE=PORE PRESSURE Data lines to define initial values of pore fluid pressures \*INITIAL CONDITIONS, TYPE=RATIO Data lines to define initial values of the void ratio \*INITIAL CONDITIONS, TYPE=SATURATION Data lines to define initial saturation \*INITIAL CONDITIONS, TYPE=TEMPERATURE Data lines to define initial temperature \*BOUNDARY Data lines to define zero-valued boundary conditions \*\* \*STEP \*GEOSTATIC \*CLOAD and/or \*DLOAD and/or \*DSLOAD Data lines to specify mechanical loading \*FLOW and/or \*SFLOW and/or \*DFLOW and/or \*DSFLOW Data lines to specify pore fluid flow \*CFLUX and/or \*DFLUX Data lines to define concentrated and/or distributed heat fluxes if heat transfer is modeled \*BOUNDARY Data lines to specify displacements or pore pressures \*END STEP # 6.9 Mass diffusion analysis • “Mass diffusion analysis,” Section 6.9.1 # 6.9.1 MASS DIFFUSION ANALYSIS Products: Abaqus/Standard Abaqus/CAE # References • “Defining an analysis,” Section 6.1.2 • \*MASS DIFFUSION • “Configuring a mass diffusion procedure” in “Configuring general analysis procedures,” Section 14.11.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide • “Creating and modifying prescribed conditions,” Section 16.4 of the Abaqus/CAE User’s Guide • “Defining a concentrated concentration flux,” Section 16.9.33 of the Abaqus/CAE User’s Guide, in the HTML version of this guide • “Defining a body concentration flux,” Section 16.9.35 of the Abaqus/CAE User’s Guide, in the HTML version of this guide • “Defining a surface concentration flux,” Section 16.9.34 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview A mass diffusion analysis: • models the transient or steady-state diffusion of one material through another, such as the diffusion of hydrogen through a metal; • requires the use of mass diffusion elements; and • can be used to model temperature and/or pressure-driven mass diffusion. # Governing equations The governing equations for mass diffusion are an extension of Fick’s equations: they allow for nonuniform solubility of the diffusing substance in the base material and for mass diffusion driven by gradients of temperature and pressure. The basic solution variable (used as the degree of freedom at the nodes of the mesh) is the “normalized concentration” (often also referred to as the “activity” of the diffusing material), $\phi \ { \stackrel { \mathrm { d e f } } { = } } \ c / s ,$ where c is the mass concentration of the diffusing material and s is its solubility in the base material. Therefore, when the mesh includes dissimilar materials that share nodes, the normalized concentration is continuous across the interface between the different materials. For example, a diatomic gas that dissociates during diffusion can be described using Sievert’s law: $c = s p ^ { \frac { 1 } { 2 } }$ , where p is the partial pressure of the diffusing gas. Combining Sievert’s law with the definition of normalized concentration given earlier, $\phi = c / s = p ^ { \frac { 1 } { 2 } }$ . Equilibrium requires the partial pressure to be continuous across an interface, so normalized concentration will be continuous as well. If an expression other than Sievert’s law defines the relationship between concentration and partial pressure for a diffusing material, solubility should be defined accordingly. The diffusion problem is defined from the requirement of mass conservation for the diffusing phase: $$ \int_ {V} \frac {d c}{d t} d V + \int_ {S} \mathbf {n} \cdot \mathbf {J} d S = 0, $$ where V is any volume whose surface is S, is the outward normal to S, is the flux of concentration of the diffusing phase, and is the concentration flux leaving S. Diffusion is assumed to be driven by the gradient of a general chemical potential, which gives the behavior $$ \mathbf {J} = - s \mathbf {D} \cdot \left[ \frac {\partial \phi}{\partial \mathbf {x}} + \kappa_ {s} \frac {\partial}{\partial \mathbf {x}} \bigg (\ln (\theta - \theta^ {Z}) \bigg) + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}} \right], $$ where $\mathbf { D } ( c , \theta , \mathbf { f } )$ is the diffusivity; $s ( \theta , \mathbf { f } )$ is the solubility; $\kappa _ { s } ( c , \theta , \mathbf { f } )$ is the “Soret effect” factor, providing diffusion because of temperature gradient; is the temperature; $\theta ^ { Z }$ is the value of absolute zero on the temperature scale being used; $\kappa _ { p } ( c , \theta , \mathbf { f } )$ is the pressure stress factor, providing diffusion driven by the gradient of the equivalent pressure stress, $p \ { \stackrel { \mathrm { d e f } } { = } } \ - \operatorname { t r a c e } ( { \pmb { \sigma } } ) / 3 ;$ is stress; and are any predefined field variables. Whenever D, $\kappa _ { s }$ , or $\kappa _ { p }$ depends on concentration, the problem becomes nonlinear and the system of equations becomes nonsymmetric. In practical cases the dependence on concentration is quite strong, so the nonsymmetric matrix storage and solution scheme is invoked automatically when a mass diffusion analysis is performed (see “Defining an analysis,” Section 6.1.2). # Fick’s law Mass diffusion behavior is often described by Fick’s law (Crank, 1956): $$ \mathbf {J} = - \mathbf {D} \cdot \frac {\partial c}{\partial \mathbf {x}}. $$ Fick’s law is offered in Abaqus/Standard as a special case of the general chemical potential relation. To establish the relationship between Fick’s law and the general chemical potential, we write Fick’s law as $$ \mathbf {J} = - \mathbf {D} \cdot \left(s \frac {\partial \phi}{\partial \mathbf {x}} + \phi \frac {\partial s}{\partial \mathbf {x}}\right). $$ In most practical cases $s = s ( \theta )$ , and we can write $$ \mathbf {J} = - s \mathbf {D} \cdot \frac {\partial \phi}{\partial \mathbf {x}} - \mathbf {D} \cdot \frac {c}{s} \frac {\partial s}{\partial \theta} \frac {\partial \theta}{\partial \mathbf {x}}. $$ The two terms in this equation describe the normalized concentration and temperature-driven diffusion, respectively. The normalized concentration-driven diffusion term is identical to that given in