Computational Geomechanics. Manuel Pastor

Computational Geomechanics

has been included in the force term of the computer code SWANDYNE‐II (Chan 1995) although it is neglected in the left‐hand side of the final algebraic equation when the symmetric solution procedure is used.

The above set defines the complete equation system for solution of the problem defined providing the necessary boundary conditions have been specified as in (2.18) and (2.19), i.e.

(3.18) StartLayout 1st Row bold t equals bold sigma period bold upper G equals bold t overTilde o n normal upper Gamma equals normal upper Gamma Subscript t Baseline 2nd Row bold u equals bold u overTilde o n normal upper Gamma equals normal upper Gamma Subscript u EndLayout

and

StartLayout 1st Row p Subscript w Baseline equals p Subscript w Baseline overbar o n normal upper Gamma equals normal upper Gamma Subscript p Baseline 2nd Row bold n Superscript normal upper T Baseline w equals bold n Superscript normal upper T Baseline bold k left-parenthesis minus italic nabla p Subscript w Baseline plus upper S Subscript w Baseline rho Subscript f Baseline bold b right-parenthesis equals w overTilde Subscript n Baseline o n normal upper Gamma equals normal upper Gamma Subscript w EndLayout

Assuming isotropic permeability, the above equation becomes

bold k left-parenthesis minus StartFraction partial-differential p Subscript w Baseline Over partial-differential bold n EndFraction plus upper S Subscript w Baseline rho Subscript f Baseline bold n Superscript normal upper T Baseline bold b right-parenthesis equals w overTilde Subscript n Baseline equals minus q overbar o n normal upper Gamma equals normal upper Gamma Subscript w

where q overbar Subscript n is the influx, i.e. having an opposite sign to the outflow w overbar Subscript n .

The total boundary Γ is the union of its components, i.e.

normal upper Gamma equals normal upper Gamma Subscript t Baseline union normal upper Gamma Subscript u Baseline equals normal upper Gamma Subscript p Baseline union normal upper Gamma Subscript w

3.2.2 Discretization of the Governing Equation in Space

The spatial discretization involving the variables u and p_w is achieved by suitable shape (or basis) functions, writing

(3.19)

Note that the nodal values of the pore pressures are indicated with a superscript.

We assume here that the expansion is such that the strong boundary conditions (3.18) are satisfied on Γ_u and Γ_p automatically by a suitable prescription of the (nodal) parameters. As in most other finite element formulations, the natural boundary condition will be obtained by integrating the weighted equation by parts.

To obtain the first equation discretized in space, we premultiply (3.8) by (N ^u)^T and integrate the first term by parts (see for details Zienkiewicz et al (2013) or other texts) giving:

(3.20)

where the matrix B is given as

(3.21) bold upper B identical-to bold upper S bold upper N Superscript u

and the “load vector” f ⁽¹⁾, equal in number of components to that of vector bold u overbar contains all the effects of body forces, and prescribed boundary tractions, i.e.

(3.22)

At this stage, it is convenient to introduce the effective stress see (3.12) now defined to allow for effects of incomplete saturation as

(3.23) bold sigma equals bold sigma double-prime minus italic alpha chi Subscript normal w Baseline bold m p Subscript w

The discrete, ordinary differential equation now becomes

(3.24) StartEnclose box bold upper M ModifyingAbove Above bold u overbar With two-dots plus integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold sigma Superscript double-prime Baseline d upper Omega minus bold upper Q bold p overbar Superscript w Baseline minus bold f Superscript left-parenthesis 1 right-parenthesis Baseline equals 0 EndEnclose

where

(3.25) bold upper M equals integral Underscript normal upper Omega Endscripts left-parenthesis bold upper N Superscript u Baseline right-parenthesis Superscript normal upper T Baseline rho bold upper N Superscript u Baseline d upper Omega

is the MASS MATRIX of the system and

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