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For partially saturated flow you must define the porous medium’s absorption/exsorption behavior (see “Sorption,” Section 26.6.4).
Gel swelling (“Swelling gel,” Section 26.6.5) and volumetric moisture swelling of the solid skeleton (“Moisture swelling,” Section 26.6.6) can be included in partially saturated cases. These effects are usually associated with modeling of moisture migration in polymeric systems rather than with geotechnical systems.
Thermal properties if heat transfer is modeled
In problems that model heat transfer, the thermal conductivity for either the solid material or the permeating fluid, or more commonly for both phases, must be defined. Only isotropic conductivity can be specified for the pore fluid. The specific heat and density of the phases must also be defined for transient heat transfer problems. Latent heat for the phases can be defined if changes in internal energy due to phase changes are important. See “Thermal properties: overview,” Section 26.2.1, for details on defining thermal properties in Abaqus. Examples of problems that model fully coupled heat transfer along with pore fluid diffusion and mechanical deformation can be found in “Consolidation around a cylindrical heat source,” Section 1.15.7 of the Abaqus Benchmarks Guide, and “Permafrost thawing–pipeline interaction,” Section 10.1.6 of the Abaqus Example Problems Guide.
The thermal properties can be defined separately for the solid material and the permeating fluid.
Input File Usage: To define the conductivity, specific heat, density, and latent heat of the permeating fluid, use the following options:
*CONDUCTIVITY, TYPE=ISO, PORE FLUID
*SPECIFIC HEAT, PORE FLUID
*LATENT HEAT, PORE FLUID
*DENSITY, PORE FLUID
To define the conductivity, specific heat, density, and latent heat of the solid material, use the following options:
*EXPANSION, TYPE=ISO or ORTHO or ANISO
*SPECIFIC HEAT
*DENSITY
*LATENT HEAT
Abaqus/CAE Usage: Defining the thermal properties and the density of the permeating fluid is not supported in Abaqus/CAE.
To define the conductivity, specific heat, density, and latent heat of the solid material, use the following options:
Property module: material editor:
Thermal→Conductivity: Type: Isotropic
Thermal→Specific Heat
General→Density
Thermal→Latent Heat
Thermal expansion
Thermal expansion can be defined separately for the solid material and for the permeating fluid. In such a case you should repeat the expansion material property, with the necessary parameters, to define the different thermal expansion effects (see “Thermal expansion,” Section 26.1.2). Thermal expansion will be active only in a consolidation (transient) analysis.
Input File Usage: To define the thermal expansion of the permeating fluid:
*EXPANSION, TYPE=ISO, PORE FLUID
To define the thermal expansion of the solid material:
*EXPANSION, TYPE=ISO or ORTHO or ANISO
Abaqus/CAE Usage: To define the thermal expansion of the permeating fluid:
Property module: material editor: Other→Pore Fluid→Pore
Fluid Expansion
To define the thermal expansion of the solid material:
Property module: material editor: Mechanical→Expansion
Elements
The analysis of flow through porous media in Abaqus/Standard is available for plane strain, axisymmetric, and three-dimensional problems. The modeling of coupled heat transfer effects is available only for axisymmetric and three-dimensional problems. Continuum pore pressure elements are provided for modeling fluid flow through a deforming porous medium in a coupled pore fluid diffusion/stress analysis. These elements have pore pressure degree of freedom 8 in addition to displacement degrees of freedom 1–3. Heat transfer through the porous medium can also be modeled using continuum coupled temperature–pore pressure elements. These elements have temperature degree of freedom 11 in addition to pore pressure degree of freedom 8 and displacement degrees of freedom 1–3. Stress/displacement elements can be used in parts of the model without pore fluid flow. 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, .
SAT Saturation, s.
GELVR Gel volume ratio, .
FLUVR Total fluid volume ratio, n _ { f }
FLVEL Magnitude and components of the pore fluid effective velocity vector, .
FLVELM Magnitude, , 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:
CFF Concentrated fluid flow at a node.
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 that fluid is entering the model.
RVT Reaction total fluid volume (computed only in a transient analysis). This value is the time integrated value of RVF.
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
*HEADING
...
**************************
**
** Material definition
**
**************************
*MATERIAL, NAME=soil
Data lines to define mechanical properties of the solid material
...
*EXPANSION
Data lines to define the thermal expansion coefficient of the solid grains
*EXPANSION, TYPE=ISO, PORE FLUID
Data lines to define the thermal expansion coefficient of the permeating fluid
*PERMEABILITY, SPECIFIC=γw
Data lines to define permeability, k, as a function of the void ratio, e
*PERMEABILITY, TYPE=SATURATION
Data lines to define the dependence of permeability on saturation, k_s(s)
*PERMEABILITY, TYPE=VELOCITY
Data lines to define the velocity coefficient, β(e)
*POROUS BULK MODULI
Data line to define the bulk moduli of the solid grains and the permeating fluid
*SORPTION, TYPE=ABSORPTION
Data lines to define absorption behavior
*SORPTION, TYPE=EXSORPTION
Data lines to define exsorption behavior
*SORPTION, TYPE=SCANNING
Data lines to define scanning behavior (between absorption and exsorption)
*GEL
Data line to define gel behavior in partially saturated flow
*MOISTURE SWELLING
Data lines to define moisture swelling strain as a function of saturation in partially saturated flow
*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
*SPECIFIC HEAT, PORE FLUID
Data lines to define specific heat of the permeating fluid if transient heat transfer is modeled
*DENSITY
Data lines to define density of the solid grains if transient heat transfer is modeled
*DENSITY, PORE FLUID
Data lines to define density of the permeating fluid if transient heat transfer is modeled
*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
...
**************************
**
** Boundary conditions and initial conditions
**
**************************
*BOUNDARY
Data lines to specify zero-valued boundary conditions
*INITIAL CONDITIONS, TYPE=STRESS, GEOSTATIC
Data lines to specify initial stresses
*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 saturation
*AMPLITUDE, NAME=name
Data lines to define amplitude variations
**************************
**
** Step 1: Optional step to ensure an equilibrium
** geostatic stress field
**
**************************
*STEP
*GEOSTATIC
*CLOAD and/or *DLOAD and/or *TEMPERATURE and/or *FIELD
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
**************************
**
** Step 2: Coupled pore diffusion/stress analysis step
**
**************************
*STEP (,NLGEOM)
** Use NLGEOM to include geometric nonlinearities
*SOILS
Data line to define incrementation
*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
*FILM
Data lines referring to film property table if heat transfer is modeled
*BOUNDARY
Data lines to specify displacements, pore pressures, or temperatures
*END STEP
6.8.2 GEOSTATIC STRESS STATE
Products: Abaqus/Standard Abaqus/CAE
References
• “Defining an analysis,” Section 6.1.2
• “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1
• *GEOSTATIC
• “Configuring a geostatic stress field procedure” in “Configuring general analysis procedures,” Section 14.11.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
A geostatic stress field procedure:
• is used to verify that the initial geostatic stress field is in equilibrium with applied loads and boundary conditions and to iterate, if necessary, to obtain equilibrium;
• accounts for pore pressure degrees of freedom when pore pressure elements are used, and accounts for temperature degrees of freedom when coupled temperature–pore pressure elements are used;
• is usually the first step of a geotechnical analysis, followed by a coupled pore fluid diffusion/stress (with or without heat transfer) or static analysis procedure; and
• can be linear or nonlinear.
Establishing geostatic equilibrium
The geostatic procedure is normally used as the first step of a geotechnical analysis; in such cases gravity loads are applied during this step. Ideally, the loads and initial stresses should exactly equilibrate and produce zero deformations. However, in complex problems it may be difficult to specify initial stresses and loads that equilibrate exactly.
Abaqus/Standard provides two procedures for establishing the initial equilibrium. The first procedure is applicable to problems for which the initial stress state is known at least approximately. The second, enhanced, procedure is also applicable for cases in which the initial stresses are not known; it is supported for only a limited number of elements and materials.
Establishing equilibrium when the initial stress state is approximately known
The geostatic procedure requires that the initial stresses are close to the equilibrium state; otherwise, the displacements corresponding to the equilibrium state might be large. Abaqus/Standard checks for equilibrium during the geostatic procedure and iterates, if needed, to obtain a stress state that equilibrates the prescribed boundary conditions and loads. This stress state, which is a modification of the stress field defined by the initial conditions (“Initial conditions in Abaqus/Standard and Abaqus/Explicit,”
Section 34.2.1), is then used as the initial stress field in a subsequent static or coupled pore fluid diffusion/stress (with or without heat transfer) analysis.
If the stresses given as initial conditions are far from equilibrium under the geostatic loading and there is some nonlinearity in the problem definition, this iteration process may fail. Therefore, you should ensure that the initial stresses are reasonably close to equilibrium.
If the deformations produced during the geostatic step are significant compared to the deformations caused by subsequent loading, the definition of the initial state should be reexamined.
If heat transfer is modeled during the geostatic step through the use of coupled temperature–pore pressure elements, the initial temperature field and thermal loads, if specified, must be such that the system is relatively close to a state of thermal equilibrium. Steady-state heat transfer is assumed during a geostatic step.
Input File Usage: *GEOSTATIC
Abaqus/CAE Usage: Step module: Create Step: General: Geostatic
Establishing equilibrium when the initial stress state is unknown
To obtain equilibrium in cases when the initial stress state is unknown or is known only approximately, you can invoke an enhanced procedure. Abaqus automatically computes the equilibrium corresponding to the initial loads and the initial configuration, allowing only small displacements within user-specified tolerances. (The default tolerance is .) The procedure is available with a limited number of elements and materials and is intended to be used in analyses in which the material response is primarily elastic; that is, inelastic deformations are small.
The procedure is supported for both geometrically linear and geometrically nonlinear analyses. However, in general, the performance in the geometrically linear case will be better. Therefore, it might be advantageous to obtain the initial equilibrium in a geometrically linear step, even though a geometrically nonlinear analysis is performed in subsequent steps.
Input File Usage: Use the following option to invoke the enhanced procedure:
*GEOSTATIC, UTOL=displacement tolerance
Abaqus/CAE Usage: Step module: Create Step: General: Geostatic: Incrementation tabbed page: Automatic: Max. displacement change
Limitations
The following limitations apply to the enhanced procedure:
• It is supported only for a limited number of elements (see “Elements” below) and materials (see “Material options” below). When the procedure is used with nonsupported elements or material models, Abaqus issues a warning message. In this case it is the user’s responsibility to ensure that the displacement tolerances are larger than the displacements in the analysis; otherwise, convergence problems may occur.
• It can be used in a restart analysis only if it had been used in the previous analysis.
Optional modeling of coupled heat transfer
When coupled temperature–pore pressure elements are used, heat transfer is modeled in these elements by default. However, you may optionally choose to switch off heat transfer within these elements during a geostatic step. This feature may be helpful in reducing computation time if temperature and associated heat flow effects are not important.
Input File Usage: Use the following option to suppress heat transfer modeling:
* \text { GEOSTATIC, } \text { HEAT } = \text { NO }
Abaqus/CAE Usage: Switching off the heat transfer part of the physics is not supported in Abaqus/CAE.
Vertical equilibrium in a porous medium
Most geotechnical problems begin from a geostatic state, which is a steady-state equilibrium configuration of the undisturbed soil or rock body under geostatic loading. The equilibrium state usually includes both horizontal and vertical stress components. It is important to establish these initial conditions correctly so that the problem begins from an equilibrium state. Since such problems often involve fully or partially saturated flow, the initial void ratio of the porous medium, e ^ { 0 } , the initial pore pressure, u _ { w } , and the initial effective stress must all be defined.
If the magnitude and direction of the gravitational loading are defined by using the gravity distributed load type, a total, rather than excess, pore pressure solution is used (see “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1). This discussion is based on the total pore pressure formulation.
The z-axis points vertically in this discussion, and atmospheric pressure is neglected. We assume that the pore fluid is in hydrostatic equilibrium during the geostatic procedure so that
\frac {d u _ {w}}{d z} = - \gamma_ {w},
where \gamma _ { w } is the user-defined specific weight of the pore fluid (see “Permeability,” Section 26.6.2). (The pore fluid is not in hydrostatic equilibrium if there is significant steady-state flow of pore fluid through the porous medium: in that case a steady-state coupled pore fluid diffusion/stress analysis must be performed to establish the initial conditions for any subsequent transient calculations—see “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1.) If we also take \gamma _ { w } to be independent of z (which is usually the case, since the fluid is almost incompressible), this equation can be integrated to define
u _ {w} = \gamma_ {w} (z _ {w} ^ {0} - z),
where z _ { w } ^ { 0 } is the height of the phreatic surface, at which u _ { w } = 0 and above which u _ { w } < 0 and the pore fluid is only partially saturated.
We usually assume that there are no significant shear stresses \tau _ { x z } , \tau _ { y z } . Then, equilibrium in the vertical direction is
\frac {d \sigma_ {z z}}{d z} = \rho g + s n ^ {0} \gamma_ {w},
where \rho is the dry density of the porous solid material (the dry mass per unit volume), g is the gravitational acceleration, n ^ { 0 } is the initial porosity of the material, and s is the saturation, 0 \leq s \leq 1 . 0 (see “Permeability,” Section 26.6.2). Since porosity is the ratio of pore volume to total volume and the void ratio is the ratio of pore volume to solids volume, n ^ { 0 } is defined from the initial void ratio by
n ^ {0} = \frac {e ^ {0}}{1 + e ^ {0}}.
Abaqus/Standard requires that the initial value of the effective stress, { \overline { { \sigma } } } , , be given as an initial condition (“Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). Effective stress is defined from the total stress, , by
\overline {{\boldsymbol {\sigma}}} = \boldsymbol {\sigma} + s u _ {w} \mathbf {I},
where is a unit matrix. Combining this definition with the equilibrium statement in the z-direction and hydrostatic equilibrium in the pore fluid gives
\frac {d \overline {{\sigma}} _ {z z}}{d z} = \rho g - \gamma_ {w} \left(s (1 - n ^ {0}) - \frac {d s}{d z} (z _ {w} ^ {0} - z)\right) \quad \mathrm{for} \quad z < z _ {1} ^ {0}, \quad \mathrm{and}
\frac {d \overline {{\sigma}} _ {z z}}{d z} = \rho g \quad \mathrm{for} \quad z _ {1} ^ {0} \leq z,
again using the assumption that \gamma _ { w } is independent of z. z _ { 1 } ^ { 0 } is the position of the surface that separates the dry soil from the partially saturated soil. The soil is assumed to be dry ( s = 0 ) for z _ { 1 } ^ { 0 } < z , and it is assumed to be partially saturated for z _ { w } ^ { 0 } < z < z _ { 1 } ^ { 0 } and fully saturated for z \le z _ { w } ^ { 0 } .
In many cases s is constant. For example, in fully saturated flow s = 1 . 0 everywhere below the phreatic surface. If we further assume that the initial porosity, n ^ { 0 } , and the dry density of the porous medium, \rho , are also constant, the above equation is readily integrated to give
\overline {{\sigma}} _ {z z} = \rho g (z - z ^ {0}) - \gamma_ {w} s (1 - n ^ {0}) (z - z _ {w} ^ {0}) \quad \mathrm{for} \quad z < z _ {1} ^ {0}, \quad \mathrm{and}
\overline {{\sigma}} _ {z z} = \rho g (z - z ^ {0}) \quad \mathrm{for} \quad z _ {1} ^ {0} \leq z,
where z ^ { 0 } is the position of the surface of the porous medium, z _ { w } ^ { 0 } < z ^ { 0 } .
In more complicated cases where { \pmb s } , n ^ { 0 } , and/or \rho vary with height, the equation must be integrated in the vertical direction to define the initial values of \overline { { \sigma } } _ { z z } \left( z \right) .