Computational Geomechanics. Manuel Pastor

Computational Geomechanics

evident for problems in which high‐frequency oscillations are important. As we shall show in the next section, these are of little importance for earthquake analyses.

In eliminating the variable w_i(w), we have neglected several terms but have achieved an elimination of two or three variable sets depending on whether the two‐ or three‐dimensional problem is considered. However, another possibility exists for obtaining a reduced equation set without neglecting any terms provided that the fluid (i.e. water in this case) is compressible.

With such compressibility assumed, Equation (2.16) can be integrated in time, provided that we introduce the water displacement upper U Subscript i Superscript normal upper R Baseline equals left-parenthesis bold upper U Superscript normal upper R Baseline right-parenthesis in place of the velocity w_i(w). We define

(2.22a) ModifyingAbove upper U With ampersand c period dotab semicolon Subscript i Superscript upper R Baseline equals w Subscript i Baseline slash n

(2.22b) ModifyingAbove bold upper U With ampersand c period dotab semicolon Superscript upper R Baseline equals bold w slash n

where the division by the porosity n is introduced to approximate the true rather than the averaged fluid displacement. We now can rewrite (2.16) after integration with respect to time as

(2.23a) p equals minus upper Q left-parenthesis italic alpha epsilon Subscript italic i i Baseline plus italic n upper U Subscript i comma i Superscript upper R Baseline right-parenthesis

(2.23b) p equals minus upper Q left-parenthesis alpha bold m Superscript normal upper T Baseline bold epsilon plus n nabla Superscript normal upper T Baseline bold upper U Superscript upper R Baseline right-parenthesis

and thus we can eliminate p from (2.11) and (2.13).

The resulting system which is fully discussed in Zienkiewicz and Shiomi (1984) is not written down here as we shall derive this alternative form in Chapter 3 using the total displacement of water U = U^R + u as the variable. It presents a very convenient basis for using a fully explicit temporal scheme of integration (see Chan et al. 1991) but it is not applicable for long‐term studies leading to steady‐state conditions, as the water displacement U then increases indefinitely.

It is fortunate that the inaccuracies of the u–p version are pronounced only in high‐frequency, short‐duration, phenomena, since, for such problems, we can conveniently use explicit temporal integration. Here a very small time increment can be used for the short time period considered (see Chapter 3).

Table 2.1 summarizes various forms of governing equations used.

2.2.3 Limits of Validity of the Various Approximations

It is, of course, important to know the degree of approximation involved in various differential equation systems. Thus, it is of interest to know under what circumstances undrained conditions can be assumed, to define the behavior of the material and when the simplified equation system discussed in the previous section is applicable, without introducing serious error. An attempt to answer these problems was made by Zienkiewicz et al. (1980). The basis was the consideration of a one‐dimensional set of linearized equations of the full systems (2.11), (2.13), and (2.16) and of the approximations (2.20) and (2.21). The limiting case in which w_i,i = 0 (representing undrained conditions) was also considered.

Table 2.1 Comparative sets of coupled equations governing deformation and flow.

u − w − p equations (exact) [(2.11), (2.13), and (2.16)]

StartLayout 1st Row bold upper S Superscript upper T Baseline sigma minus rho ModifyingAbove bold u With two-dots minus ModifyingBelow rho Subscript f Baseline left-bracket ModifyingAbove bold w With ampersand c period dotab semicolon plus bold w nabla Superscript normal upper T Baseline bold w right-bracket With bar plus rho bold b equals 0 2nd Row minus nabla p minus bold upper R minus rho Subscript f Baseline ModifyingAbove bold u With two-dots minus ModifyingBelow rho Subscript f Baseline left-bracket ModifyingAbove bold w With ampersand c period dotab semicolon plus bold w nabla Superscript upper T Baseline bold w right-bracket slash n With bar plus rho Subscript f Baseline bold b equals 0 3rd Row nabla Superscript upper T Baseline bold w plus alpha bold m Superscript upper T Baseline ModifyingAbove bold epsilon With ampersand c period dotab semicolon plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper Q EndFraction plus n StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript f Baseline Over rho Subscript f Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0 EndLayout

u − p approximation for dynamics of lower frequencies. Exact for consolidation [(2.20), (2.21)]

StartLayout 1st Row bold upper S Superscript upper T Baseline bold sigma minus rho ModifyingAbove bold u With two-dots plus rho bold b equals 0 2nd Row nabla Superscript upper T Baseline bold k left-parenthesis minus nabla p minus rho Subscript f Baseline ModifyingAbove bold u With two-dots plus rho Subscript f Baseline bold b right-parenthesis plus alpha bold m Superscript upper T Baseline ModifyingAbove bold epsilon With ampersand c period dotab semicolon plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper Q EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0 EndLayout

u − U, only convective terms neglected (3.72)

StartLayout 1st Row bold upper S Superscript upper T Baseline bold sigma plus italic alpha upper Q left-parenthesis alpha minus n right-parenthesis nabla left-parenthesis nabla bold u right-parenthesis bold plus italic alpha upper Q n nabla left-parenthesis nabla Superscript normal upper T Baseline bold upper U right-parenthesis minus left-parenthesis 1 minus n right-parenthesis rho ModifyingAbove bold u With two-dots minus rho Subscript f Baseline n ModifyingAbove bold upper U With two-dots plus rho bold b equals 0 2nd Row left-parenthesis alpha minus n right-parenthesis upper Q nabla left-parenthesis nabla Superscript upper T Baseline bold u right-parenthesis plus italic n upper Q nabla left-parenthesis nabla Superscript upper T Baseline bold upper U right-parenthesis minus bold k Superscript negative 1 Baseline left-parenthesis n bold upper U minus n bold u right-parenthesis minus rho Subscript f Baseline bold upper U plus rho Subscript f Baseline bold b equals 0 EndLayout

In all the above

σ^″ = σ + αm p and dσ^″ = D dε = DS du

For

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