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Choosing each \delta v ^ { N } to be nonzero in turn expresses equilibrium as a balance of internal and external forces:
Equation 2.8.2-2
I ^ {N} - P ^ {N} = 0.
These discretized equilibrium equations, together with the continuity equation discussed in ``Continuity statement for the wetting liquid phase in a porous medium, '' Section 2.8.4, define the state of the porous medium. The equilibrium equations are written at the end of a time increment when implicit integration is used and, for all but the simplest cases, they are nonlinear. Newton's method is often used for their solution. Also, small, linear perturbations of the system are sometimes of interest (an example is the small vibration problem). These considerations imply a need for the Jacobian matrix of the system, which defines the variation of each term in the equations with respect to the basic variables of the discretized problem, which--for this case--are the nodal positions, x ^ { N } (or, equivalently, the displacements x ^ { N } - X ^ { N } ) , and the nodal wetting liquid pressure values, u _ { w } ^ { N } . Symbolically we write such a variation of a term, f say, as d f , meaning
d f \stackrel {\mathrm{def}} {=} \frac {\partial f}{\partial x ^ {N}} d x ^ {N} + \frac {\partial f}{\partial u _ {w} ^ {P}} d u ^ {P}.
From the variation of discretized equilibrium, Equation 2.8.2-2, the term d P ^ { N } gives rise to the mass matrix (for the d'Alembert forces) and the "load stiffness matrix" in the Jacobian. The load stiffness matrix is discussed in Chapter 3, "Elements," and Chapter 6, "Loading and Constraints," for particular load types. The load stiffness term associated with the weight of the wetting liquid is
- \int_ {V} \frac {1}{J} d \bigl [ J (s n + n _ {t}) \rho_ {w} \bigr ] \mathbf {N} ^ {N} \cdot \mathbf {g} d V,
where
J \stackrel {\mathrm{def}} {=} \left| \frac {d V}{d V ^ {0}} \right|
is the ratio of volume in the current configuration to volume in the reference configuration.
The term d I ^ { N } is
\begin{array}{l} d I ^ {N} = d \int_ {V} \boldsymbol {\beta} ^ {N}: \boldsymbol {\sigma} d V \\ = \int_ {V} \left[ \frac {1}{J} \pmb {\beta} ^ {N}: d (J \pmb {\sigma}) + \pmb {\sigma}: d \pmb {\beta} ^ {N} \right] d V. \\ \end{array}
The first term includes d ( J \pmb { \sigma } ) , which is the variation of stress caused by variations in nodal positions
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and pore liquid pressure values. In a continuum sense (that is, before the spatial discretization of the solution variables) this term is defined by the effective stress principle and by the constitutive assumptions used for the material and is discussed in more detail below. Introducing the spatial discretization into the second term provides a contribution to the initial stress matrix.
Since the effective stress, \overline { { \pmb { \sigma } } } , is generally stored as components associated with spatial directions, the rotation of the material during an increment must be included in the formulation. This issue is discussed in detail in ``Rate of deformation and strain increment,'' Section 1.4.3; ``Stress rates,'' Section 1.5.4; ``State storage,'' Section 1.5.5; and ``Solid element formulation,'' Section 3.2.2. For the purpose of the present development we assume that the variation of stress is
d (J \pmb {\sigma}) = d ^ {\nabla} (J (1 - n _ {t}) \overline {{\pmb {\sigma}}}) - d (J n _ {t} \overline {{p}} _ {t} \mathbf {I}) + J (d \pmb {\Omega} \cdot \pmb {\sigma} + \pmb {\sigma} \cdot d \pmb {\Omega} ^ {T}) - d (J \chi u _ {w}) \mathbf {I},
where d ^ { \nabla } ( J { \overline { { \pmb { \sigma } } } } ) is the variation in effective stress associated with constitutive response in the material (that is, caused by variations in the strain or other state variables) and d \Omega \stackrel { \mathrm { d e f } } { = } \mathrm { a s y m } ( \partial d \mathbf { x } / \partial \mathbf { x } ) is the spin of the material. Using this assumption, the Jacobian contribution from stress in the porous medium is
Equation 2.8.2-3
\begin{array}{l} d I ^ {N} = \int_ {V} \bigl [ \frac {1}{J} \pmb {\beta} ^ {N}: \{d ^ {\nabla} (J (1 - n _ {t}) \overline {{{\pmb {\sigma}}}}) - d (J n _ {t} \overline {{{p}}} _ {t} \mathbf {I}) \} + \pmb {\sigma}: (d \pmb {\beta} ^ {N} + 2 \pmb {\beta} ^ {N} \cdot d \pmb {\Omega}) \\ - \pmb {\beta} ^ {N}: \mathbf {I} (\chi + u _ {w} \frac {d \chi}{d u _ {w}}) d u _ {w} - \pmb {\beta} ^ {N}: \mathbf {I} \chi u _ {w} \mathbf {I}: d \pmb {\varepsilon} ] d V, \\ \end{array}
where d" \stackrel { \mathrm { d e f } } { = } \operatorname { s y m } ( \partial d \mathbf { x } / \partial \mathbf { x } ) is the strain rate (the "rate of deformation") so that
\frac {d J}{J} = \mathrm{trace} (d \pmb {\varepsilon}) = \mathbf {I}: d \pmb {\varepsilon}.
2.8.3 Constitutive behavior in a porous medium
A porous medium in ABAQUS/Standard is considered to consist of a mixture of solid matter, voids that contain liquid and gas, and entrapped liquid attached to the solid matter. The mechanical behavior of the porous medium consists of the responses of the liquid and solid matter to local pressure and of the response of the overall material to effective stress. The assumptions made about these responses are discussed in this section.
Liquid response
For the liquid in the system (the free liquid in the voids and the entrapped liquid) we assume that
Equation 2.8.3-1
\frac {\rho_ {w}}{\rho_ {w} ^ {0}} \approx 1 + \frac {u _ {w}}{K _ {w}} - \varepsilon_ {w} ^ {t h},
where \rho _ { w } is the density of the liquid, \rho _ { w } ^ { 0 } is its density in the reference configuration, K _ { w } ( \theta ) is the
liquid's bulk modulus, and
\varepsilon_ {w} ^ {t h} = 3 \alpha_ {w} (\theta - \theta_ {w} ^ {0}) - 3 \alpha_ {w} | _ {\theta^ {I}} (\theta^ {I} - \theta_ {w} ^ {0})
is the volumetric expansion of the liquid caused by temperature change. Here \alpha _ { w } ( \theta ) is the liquid's thermal expansion coefficient, µ is the current temperature, \theta ^ { I } is the initial temperature at this point in the medium, and \theta _ { w } ^ { 0 } is the reference temperature for thermal expansion. Both u / K _ { w } and \varepsilon _ { w } ^ { t h } are assumed to be small.
Grains response
The solid matter in the porous medium is assumed to have the local mechanical response under pressure
Equation 2.8.3-2
\frac {\rho_ {g}}{\rho_ {g} ^ {0}} \approx 1 + \frac {1}{K _ {g}} \left(s u _ {w} + \frac {\overline {{p}}}{1 - n - n _ {t}}\right) - \varepsilon_ {g} ^ {t h},
where K _ { g } ( \theta ) is the bulk modulus of this solid matter, s is the saturation in the wetting fluid, and
\varepsilon_ {g} ^ {t h} = 3 \alpha_ {g} (\theta - \theta_ {g} ^ {0}) - 3 \alpha_ {g} | _ {\theta^ {I}} (\theta^ {I} - \theta_ {g} ^ {0})
is its volumetric thermal strain. Here \alpha _ { g } ( \theta ) is the thermal expansion coefficient for the solid matter and \theta _ { g } ^ { 0 } is the reference temperature for this expansion. \left| 1 - \rho _ { g } / \rho _ { g } ^ { 0 } \right| is assumed to be small.
It is important to distinguish K _ { g } and \alpha _ { g } as properties of the solid grains material. The porous medium as a whole will exhibit a much softer (and generally nonrecoverable) bulk behavior than is indicated by K _ { g } and will also show a different thermal expansion. These effects are partially structural, caused by the medium being made up of irregular grains in partial contact. They may also be caused by the system being only partially saturated, with the voids containing a mixture of relatively compressible gas and relatively incompressible liquid.
Liquid entrapment
Entrapment of liquid is associated with specific materials that absorb liquid and swell into a "gel." A simple model of this behavior is based on the idealization of this gel as a volume of individual spherical particles of equal radius r _ { a } . Tanaka and Fillmore (1979) show that, when a single sphere of such material is fully exposed to liquid, its radius change can be modeled as
r _ {a} = r _ {a} ^ {f} - \sum_ {N} a _ {N} \exp \left(- \frac {t}{\tau_ {N}}\right),
where r _ { a } ^ { f } is the fully swollen radius approached as t \to \infty and N , a _ { N } and \tau _ { N } are material parameters. Tanaka and Fillmore also show the first term in the series dominates, so the model can be simplified to
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r _ {a} = r _ {a} ^ {f} - a _ {1} \exp \left(- \frac {t}{\tau_ {1}}\right).
This provides the rate form
\dot {r} _ {a} = \frac {r _ {a} ^ {f} - r _ {a}}{\tau_ {1}}.
When the gel particles are only partially exposed to liquid (in an unsaturated system), it seems reasonable to assume that the swelling rate will be lessened according to the level of saturation. Further, we assume that the gel will swell only when the saturation of the surrounding medium exceeds the effective saturation of the gel, 1 - [ ( r _ { a } ^ { f } ) ^ { 3 } - ( r _ { a } ) ^ { 3 } ] / [ ( r _ { a } ^ { f } ) ^ { 3 } - ( r _ { a } ^ { \mathrm { d r y } } ) ^ { 3 } ] , where \boldsymbol { r } _ { a } ^ { \mathrm { d r y } } is the radius of a gel particle that is completely dry. We combine these into a simple, linear effect:
\dot {r} _ {a} = \frac {r _ {a} ^ {f} - r _ {a}}{\tau_ {1}} \left\langle s - 1 + \left(\frac {(r _ {a} ^ {f}) ^ {3} - (r _ {a}) ^ {3}}{(r _ {a} ^ {f}) ^ {3} - (r _ {a} ^ {\mathrm{dry}}) ^ {3}}\right) \right\rangle ,
where hf i = f if f > 0, hf i = 0 otherwise.
The packing density and swelling may cause the gel particles to touch. In that case the surface available to absorb and entrap liquid is reduced until, if the gel particles occupy the entire volume except for solid material, liquid entrapment must cease altogether. With k _ { a } gel particles per unit reference volume, the maximum radius that the gel particles can achieve before they must touch (in a face center cubic arrangement) is
r _ {a} ^ {t} \stackrel {\mathrm{def}} {=} \left(\frac {n ^ {0} J}{4 \sqrt {2} k _ {a}}\right) ^ {\frac {1}{3}},
and the volume is entirely occupied with gel and solid matter when the effective gel radius is
r _ {a} ^ {s} = \left(\frac {3}{4 \pi} \frac {n ^ {0} J}{k _ {a}}\right) ^ {\frac {1}{3}}.
The gel swelling behavior is, therefore, further modified to be
\dot {r} _ {a} = \frac {r _ {a} ^ {f} - r _ {a}}{\tau_ {1}} \left\langle s - 1 + \left(\frac {(r _ {a} ^ {f}) ^ {3} - (r _ {a}) ^ {3}}{(r _ {a} ^ {f}) ^ {3} - (r _ {a} ^ {\mathrm{dry}}) ^ {3}}\right) \right\rangle \left(1 - \left\langle \frac {r _ {a} - r _ {a} ^ {t}}{r _ {a} ^ {s} - r _ {a} ^ {t}} \right\rangle^ {2}\right).
Thus, in an unstressed medium the entrapped liquid volume is assumed to be
d V _ {t} = h _ {t} d V ^ {0},
where
Equation 2.8.3-3
h _ {t} \stackrel {\mathrm{def}} {=} \frac {4}{3} \pi \left(r _ {a} ^ {3} - (r _ {a} ^ {\mathrm{dry}}) ^ {3}\right) k _ {a},
where r _ { a } ( J , s ) is defined by the integration of Equation 2.8.3-3. This entrapped liquid can be compressed by pressure so that, when the porous medium is under stress, we assume
d V _ {t} = \left(1 - \frac {u _ {w}}{K _ {w}} + \varepsilon_ {w} ^ {t h}\right) h _ {t} d V ^ {0},
and thus
Equation 2.8.3-4
n _ {t} = \frac {d V _ {t}}{d V} = \frac {h _ {t}}{J} \left(1 - \frac {u _ {w}}{K _ {w}} + \varepsilon_ {w} ^ {t h}\right).
Combining this with Equation 2.8.3-1 and neglecting small terms compared to unity then provides
Equation 2.8.3-5
J \frac {\rho_ {w}}{\rho_ {w} ^ {0}} n _ {t} \approx h _ {t}.
We assume that, in the initial state, the effective saturation of the gel is the same as the saturation of the surrounding medium:
r _ {a} ^ {0} = \left[ (r _ {a} ^ {f}) ^ {3} - \left((r _ {a} ^ {f}) ^ {3} - (r _ {a} ^ {\mathrm{dry}}) ^ {3}\right) (1 - s ^ {0}) \right] ^ {\frac {1}{3}}.
The constitutive behavior of the gel containing entrapped fluid is given by the elastic bulk relationship
\overline {{p}} _ {t} = - K _ {w} (\overline {{\varepsilon}} _ {v o l} - \varepsilon_ {w} ^ {t h}),
where \overline { { p } } _ { t } is the average pressure stress in the gel fluid and \overline { { \mathcal { E } } } _ { v o l } is its volumetric effective strain.
Effective strain
From Equation 2.8.3-2 we see that the volumetric strain - u _ { w } / K _ { g } + \varepsilon _ { g } ^ { t h } represents that part of the total volumetric strain caused by pore pressure acting on the solid matter in the porous medium and by thermal expansion of that solid matter. In addition, entrapment of liquid in the medium may cause an additional volume change ratio:
1 + \frac {d V _ {t} - d V _ {t} ^ {0}}{d V ^ {0}} = 1 + J n _ {t} - n _ {t} ^ {0}.
Finally, \pmb { \varepsilon } ^ { m s } \left( s \right) is a saturation driven moisture swelling strain that represents the volumetric swelling of the solid skeleton in partially saturated flow conditions. This moisture swelling can be isotropic or
anisotropic. The remaining part of the strain in the medium,
Equation 2.8.3-6
\overline {{\pmb {\varepsilon}}} \stackrel {\mathrm{def}} {=} \pmb {\varepsilon} + \left(\frac {1}{3} \big (\frac {s u _ {w}}{K _ {g}} - \varepsilon_ {g} ^ {t h} \big) - \frac {1}{3} \ln (1 + J n _ {t} - n _ {t} ^ {0})\right) \mathbf {I} - \pmb {\varepsilon} ^ {m s} (s)
is the strain that is assumed to modify the effective stress in the medium. That is, we assume
¾ = ¾(history of ", µ, state variables, etc) :
Specific constitutive models of this type are discussed in Chapter 4, "Mechanical Constitutive Theories." From this assumption, and using Equation 2.8.3-5, we can write the Jaumann rate of change of the effective stress in terms of the rate of change of the kinematic and pore liquid pressure variables as
Equation 2.8.3-7
\begin{array}{l} d ^ {\nabla} (J (1 - n _ {t}) \overline {{{\pmb {\sigma}}}}) = J (1 - n _ {t}) \overline {{{\pmb {\mathbf {D}}}}}: \left(d \pmb {\varepsilon} + \left(\frac {s}{3 K _ {g}} + \frac {u _ {w}}{3 K _ {g}} \frac {d s}{d u _ {w}} + \frac {h _ {t}}{3 K _ {w} (1 + J n _ {t} - n _ {t} ^ {0})}\right) \mathbf {I} d u _ {w}, \right. \\ \left. - \frac {d \pmb {\varepsilon} ^ {m s}}{d s} \frac {d s}{d u _ {w}} d u _ {w}\right), \\ \end{array}
where D is defined for each particular model in Chapter 4, "Mechanical Constitutive Theories."
Also, for the effective pressure stress of the fluid entrapped in the gel,
Equation 2.8.3-8
d (J n _ {t} \overline {{p}} _ {t} \mathbf {I}) = J n _ {t} K _ {w} \mathbf {I I}: \left(d \pmb {\varepsilon} + \left(\frac {s}{3 K _ {g}} + \frac {u _ {w}}{3 K _ {g}} \frac {d s}{d u _ {w}} + \frac {h _ {t}}{3 K _ {w} (1 + J n _ {t} - n _ {t} ^ {0})}\right) \mathbf {I} d u _ {w} \right.
- \left. \frac {d \varepsilon^ {m s}}{d s} \frac {d s}{d u _ {w}} d u _ {w}\right).
Then, from Equation 2.8.2-3,
Equation 2.8.3-9
d I ^ {N} = \int_ {V} \left[ \pmb {\beta} ^ {N}: \{(1 - n _ {t}) \overline {{\mathbf {D}}} + n _ {t} K _ {w} \mathbf {I I} \}: d \pmb {\varepsilon} \right.
- \pmb {\beta} ^ {N}: \bigg ((\chi + u _ {w} \frac {d \chi}{d u _ {w}}) \mathbf {I}
- \left(\frac {s}{3 K _ {g}} + \frac {u _ {w}}{3 K _ {g}} \frac {d s}{d u _ {w}} + \frac {h _ {t}}{3 K _ {w} (1 + J n _ {t} - n _ {t} ^ {0})}\right) \left\{(1 - n _ {t}) \overline {{\mathbf {D}}} + n _ {t} K _ {w} \mathbf {I I} \right\}: \mathbf {I}
\left. + \frac {d s}{d u _ {w}} \{(1 - n _ {t}) \overline {{{\mathbf {D}}}} + n _ {t} K _ {w} \mathbf {I I} \}: \frac {d \pmb {\varepsilon} ^ {m s}}{d s}\right) d u _ {w}
\left. + \pmb {\sigma}: (d \pmb {\beta} ^ {N} + 2 \pmb {\beta} ^ {N} \cdot d \pmb {\Omega}) - \pmb {\beta} ^ {N}: \mathbf {I} _ {\chi} u _ {w} \mathbf {I}: d \pmb {\varepsilon} \right] d V.
2.8.4 Continuity statement for the wetting liquid phase in a porous medium
ABAQUS/Standard provides capabilities for particular cases of fluid flow through a porous medium. These cases are associated with having a relatively incompressible wetting liquid present in the medium. The medium may be wholly or partially saturated, with this liquid. When the medium is only partially saturated the remainder of the voids is filled with another fluid. An example is a geotechnical problem, with soil containing water and air: continuity is written for the water phase.
The wetting liquid can attach to and, thus, be trapped by certain solid particles in the medium: this volume of trapped liquid attached to solid particles forms a "gel."
A porous medium is modeled approximately in ABAQUS by attaching the finite element mesh to the solid phase. Liquid can flow through this mesh. A continuity equation is, therefore, required for the liquid, equating the rate of increase in liquid mass stored at a point to the rate of mass of liquid flowing into the point within the time increment. This continuity statement is defined in this section. It is written in a variational form as a basis for finite element approximation. The liquid flow is described by introducing Darcy's law or, alternatively, Forchheimer's law. The continuity equation is satisfied approximately in the finite element model by using excess wetting liquid pressure as the nodal variable (degree of freedom 8), interpolated over the elements. The equation is integrated in time by using the backward Euler approximation. The total derivative of this integrated variational statement of continuity with respect to the nodal variables is required for the Newton iterations used to solve the nonlinear, coupled, equilibrium and continuity equations. This expression is also derived in this section.
Consider a volume containing a fixed amount of solid matter. In the current configuration this volume occupies space V with surface S. In the reference configuration it occupied space V ^ { 0 } . Wetting liquid can flow through this volume: at any time the volume of such "free" liquid (liquid that can flow if driven by pressure) is written V _ { w } . Wetting liquid can also become trapped in the volume, by absorption into the gel. The volume of such trapped liquid is written V _ { t } .
The total mass of wetting liquid in the control volume is
\int_ {V} \rho_ {w} \left[ d V _ {w} + d V _ {t} \right] = \int_ {V} \rho_ {w} (n _ {w} + n _ {t}) d V,
where \rho _ { w } is the mass density of the liquid.
The time rate of change of this mass of wetting liquid is
\frac {d}{d t} \left(\int_ {V} \rho_ {w} (n _ {w} + n _ {t}) d V\right) = \int_ {V} \frac {1}{J} \frac {d}{d t} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) d V.
The mass of wetting liquid crossing the surface and entering the volume per unit time is
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- \int_ {S} \rho_ {w} n _ {w} \mathbf {n} \cdot \mathbf {v} _ {w} d S,
where { \bf v } _ { w } is the average velocity of the wetting liquid relative to the solid phase (the seepage velocity) and n is the outward normal to S.
Equating the addition of liquid mass across the surface S to the rate of change of liquid mass within the volume V gives the wetting liquid mass continuity equation
\int_ {V} \frac {1}{J} \frac {d}{d t} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) d V = - \int_ {S} \rho_ {w} n _ {w} \mathbf {n} \cdot \mathbf {v} _ {w} d S.
Using the divergence theorem and because the volume is arbitrary, this provides the pointwise equation
\frac {1}{J} \frac {d}{d t} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) + \frac {\partial}{\partial \mathbf {x}} \cdot (\rho_ {w} n _ {w} \mathbf {v} _ {w}) = 0.
The equivalent weak form is
\int_ {V} \delta u _ {w} \frac {1}{J} \frac {d}{d t} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) d V + \int_ {V} \delta u _ {w} \frac {\partial}{\partial \mathbf {x}} \cdot (\rho_ {w} n _ {w} \mathbf {v} _ {w}) d V = 0,
where \delta u _ { w } is an arbitrary, continuous, variational field. This statement can also be written on the reference volume:
\int_ {V ^ {0}} \delta u _ {w} \frac {d}{d t} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) d V ^ {0} + \int_ {V ^ {0}} \delta u _ {w} J \frac {\partial}{\partial \mathbf {x}} \cdot (\rho_ {w} n _ {w} \mathbf {v} _ {w}) d V ^ {0} = 0.
In ABAQUS/Standard this continuity statement is integrated approximately in time by the backward Euler formula, giving
\begin{array}{l} \int_ {V ^ {0}} \delta u _ {w} \bigg [ \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) _ {t + \Delta t} - \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) _ {t} \bigg ] d V ^ {0} \\ + \Delta t \int_ {V ^ {0}} \delta u _ {w} \left[ J \frac {\partial}{\partial \mathbf {x}} \cdot (\rho_ {w} n _ {w} \mathbf {v} _ {w}) \right] _ {t + \Delta t} d V ^ {0} = 0, \\ \end{array}
which, over the current volume, is
\begin{array}{l} \int_ {V} \delta u _ {w} \bigg [ \big (\rho_ {w} (n _ {w} + n _ {t}) \big) _ {t + \Delta t} - \frac {1}{J _ {t + \Delta t}} \big (J \rho_ {w} (n _ {w} + n _ {t}) \big) _ {t} \bigg ] d V \\ + \Delta t \int_ {V} \delta u _ {w} \left[ \frac {\partial}{\partial \mathbf {x}} \cdot (\rho_ {w} n _ {w} \mathbf {v} _ {w}) \right] _ {t + \Delta t} d V = 0. \\ \end{array}
We now drop the subscript t+¢t by adopting the convention that any quantity not explicitly
associated with a point in time is taken at t { + \Delta } t .
The divergence theorem allows the equation to be rewritten as
\begin{array}{l} \int_ {V} \left[ \delta u _ {w} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} (n _ {w} + n _ {t}) - \frac {1}{J} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} (J (n _ {w} + n _ {t})\right) _ {t}\right) - \Delta t \frac {\rho_ {w}}{\rho_ {w} ^ {0}} n _ {w} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {v} _ {w} \right] d V \\ + \Delta t \int_ {S} \delta u _ {w} \frac {\rho_ {w}}{\rho_ {w} ^ {0}} n _ {w} \mathbf {n} \cdot \mathbf {v} _ {w} d S = 0, \\ \end{array}
where--for convenience--we have normalized the equation by the density of the liquid in the reference configuration, \rho _ { w } ^ { 0 } .
Since n _ { w } = s n , this is the same as
Equation 2.8.4-1
\begin{array}{l} \int_ {V} \left[ \delta u _ {w} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} (s n + n _ {t}) - \frac {1}{J} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} J (s n + n _ {t})\right) _ {t}\right) - \Delta t \frac {\rho_ {w}}{\rho_ {w} ^ {0}} s n \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {v} _ {w} \right] d V \\ + \Delta t \int_ {S} \delta u _ {w} \frac {\rho_ {w}}{\rho_ {w} ^ {0}} s n \mathbf {n} \cdot \mathbf {v} _ {w} d S = 0. \\ \end{array}
Constitutive behavior
The constitutive behavior for pore fluid flow is governed either by Darcy's law or by Forchheimer's law. Darcy's law is generally applicable to low fluid flow velocities, whereas Forchheimer's law is commonly used for situations involving higher flow velocities. Darcy's law can be thought of as a linearized version of Forchheimer's law. Darcy's law states that, under uniform conditions, the volumetric flow rate of the wetting liquid through a unit area of the medium, s n \mathbf { v } _ { w } , is proportional to the negative of the gradient of the piezometric head (Bear, 1972):
s n \mathbf {v} _ {w} = - \widehat {\mathbf {k}} \cdot \frac {\partial \phi}{\partial \mathbf {x}},
where \widehat { \mathbf k } is the permeability of the medium and \phi is the piezometric head, defined as
\phi \stackrel {\mathrm{def}} {=} z + \frac {u _ {w}}{g \rho_ {w}},
where z is the elevation above some datum and g is the magnitude of the gravitational acceleration, which acts in the direction opposite to z. On the other hand, Forchheimer's law states that the negative of the gradient of the piezometric head is related to a quadratic function of the volumetric flow rate of the wetting liquid through a unit area of the medium (Desai, 1975):
s n \mathbf {v} _ {w} (1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}}) = - \widehat {\mathbf {k}} \cdot \frac {\partial \phi}{\partial \mathbf {x}},
where \beta ( \mathbf { x } , e ) is a "velocity coefficient" (Tariq, 1987) which may be dependent on the void ratio of the
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material. This dependence is defined using the *PERMEABILITY option. We see that, as the fluid velocity tends to zero, Forchheimer's law approaches Darcy's law. Also, if \beta = 0 , the two flow laws are identical.
kb can be anisotropic and is a function of the saturation and void ratio of the material. \widehat { \mathbf k } has units of velocity (length/time). [Some authors refer to \widehat { \mathbf k } as the hydraulic conductivity (Bear, 1972) and define the permeability as
\widehat {\mathbf {K}} = \frac {\nu}{g} \frac {1}{(1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}})} \widehat {\mathbf {k}},
where º is the kinematic viscosity of the fluid (the ratio of the fluid's dynamic viscosity to its density).]
We assume that g is constant in magnitude and direction, so
\frac {\partial \phi}{\partial \mathbf {x}} = \frac {1}{g \rho_ {w}} \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right),
where \mathbf { g } \stackrel { \mathrm { d e f } } { = } - g \partial z / \partial \mathbf { x } is the gravitational acceleration (we assume that \rho _ { w } varies slowly with position).
The permeability of a particular fluid in a multiphase flow system depends on the saturation of the phase being considered and on the porosity of the medium. We assume these dependencies are separable, so
\widehat {\mathbf {k}} = k _ {s} \mathbf {k},
where k _ { s } ( s ) provides the dependency on saturation, with k _ { s } ( 1 ) = 1 . 0 and { \bf k } ( { \bf x } , e ) is the permeability of the fully saturated medium.
Nguyen and Durso (1983) observe that, in steady flow through a partially saturated medium, the permeability varies with s ^ { 3 } . We, therefore, take k _ { s } = s ^ { 3 } by default. Different behavior for k _ { s } is defined by using the *PERMEABILITY option.
Introducing the flow constitutive law allows the mass continuity equation ( Equation 2.8.4-1) to be written
Equation 2.8.4-2
\begin{array}{l} \int_ {V} \left[ \delta u _ {w} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} (s n + n _ {t}) - \frac {1}{J} \left(\frac {\rho_ {w}}{\rho_ {w} ^ {0}} J (s n + n _ {t})\right) _ {t}\right) \right. \\ + \Delta t \frac {k _ {s}}{\rho_ {w} ^ {0} g (1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}})} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \Biggr ] d V \\ + \Delta t \int_ {S} \delta u _ {w} \frac {\rho_ {w}}{\rho_ {w} ^ {0}} s n \mathbf {n} \cdot \mathbf {v} _ {w} d S = 0. \\ \end{array}