can be performed to simplify the analysis and speed the calculation. The examples below illustrate the implementation of FLAC3D modeling approaches corresponding to different levels of fluid/mechanical coupling. Three main factors can help in the selection of a particular approach: 1. the ratio between simulation time scale and characteristic time of the diffusion
process; 2. the nature of the imposed perturbation (fluid or mechanical) to the coupled
process; and 3. the ratio of the fluid to solid stiffness. The expressions for characteristic time t f c in Eq. (1.89), diffusivity, c, in Eq. (1.96) and the stiffness ratio Rk in Eq. (1.100) can be used to quantify these factors. These factors are considered in detail below, and a recommended procedure to select a modeling approach based on these factors is given in Section 1.7.2.4. 1.7.2.1 Time Scale
We first consider the time scale factor by measuring time from the initiation of a perturbation. We define ts as the required time scale of the analysis, and tc as the characteristic time of the coupled diffusion process (defined using Eqs. (1.89) and (1.96)). Short-term behavior
If ts is very short compared to the characteristic time, tc, of the coupled diffusion process, the influence of fluid flow on the simulation results will probably be negligible, and an undrained simulation can be performed with FLAC3D (using either a “dry” or “wet” simulation—see Section 1.7.5. The “wet” simulation is carried out using CONFIG fluid and SET fluid off). No real time will be involved in the numerical simulation (i.e., ts <<< tc), but the pore pressure will change due to volumetric straining if the fluid modulus (M orKf ) is given a realistic value. The footing load simulation in Example 1.1 is an example of this approach. Long-term behavior If ts >>> tc, and drained behavior prevails at t = ts , then the pore-pressure field can be uncoupled from the mechanical field. The steady-state porepressure field can be determined using a fluid-only simulation (SET fluid on, SET mech off) (the diffusivity will not be representative), and the mechanical field can be determined next by cycling the model to equilibrium in mechanical mode with Kf = 0 (SET mech on, SET fluid off). (Strictly speaking, this engineering approach is only valid for an elastic material because a plastic material is path-dependent.)
Another way to describe the time scale is in terms of undrained or drained response. “Undrained” strictly means that there is no exchange of fluid with the outside world (where “outside world” means “outside the sample” in a lab test, and “other elements” in a numerical simulation or a field situation). “Drained” means that there is a full exchange of fluid with the outside world, which implies that the fluid pressure is able to equalize everywhere. The words are typically associated with “short-term” and “long-term,” respectively, because an undrained test usually can be done quickly, while a drained test requires a long time for excess fluid pressures to dissipate. In the field, “short-term” means that there is insignificant migration of fluid, and “long-term” means that almost all of the induced excess pressure-differences have become zero (which needs a long time). Note that in simulations conducted outside of the fluid configuration or within the fluid configuration but with fluid turned off, the total stress adjustments due to an imposed pore-pressure change (such as those resulting from the lowering of the water table) will not be done internally by the code.
The pore-pressure increments may, however, be monitored (using a FISH function, for instance) and used to decrement the total normal stresses before cycling to mechanical equilibrium. The saturated and unsaturated mass densities will also need to be adjusted if the water table has been moved within the grid, and the simulation is conducted outside of the fluid configuration.
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1.7.2.2 Nature of Imposed Perturbation to the Coupled Process
The imposed perturbation to a fully coupled hydromechanical system can be due to changes in either the fluid flow boundary condition or the mechanical boundary condition. For example, transient fluid flow to a well located within a confined aquifer is driven by the change in pore pressures at the well. The consolidation of a saturated foundation as a result of the construction of a highway embankment is controlled by the mechanical load applied by the weight of the embankment. If the perturbation is due to change in pore pressures, it is likely that the fluid flow process can be uncoupled from the mechanical process. This is described further below, and illustrated by Section 1.8.6. If the perturbation is mechanically driven, the level of uncoupling depends on the fluid versus solid stiffness ratio, as described below. 1.7.2.3 Stiffness Ratio
The relative stiffness Rk (see Eq. (1.100)) has an important influence on the modeling approach used to solve a hydromechanical problem: Relatively stiff matrix (Rk <<< 1)
If the matrix is very stiff (or the fluid highly compressible) and Rk is very small, the diffusion equation for the pore pressure can be uncoupled, since the diffusivity is controlled by the fluid (Detournay and Cheng 1993). The modeling technique will depend on the driving mechanism (fluid or mechanical perturbation): 1. In mechanically driven simulations, the pore pressure may be assumed to remain constant. In an elastic simulation, the solid behaves as if there were no fluid, whereas in a plastic analysis the presence of the pore pressure may affect failure. This modeling approach is adopted in slope stability analyses (see Section 1.7.3).
2. In pore-pressure driven elastic simulations (e.g., settlement caused by fluid extraction), volumetric strains will not significantly affect the pore-pressure field, and the flow calculation can be performed independently (SET fluid on, SET mech off). (In this case, the diffusivity will be accurate because for Rk <<<1, the generalized consolidation coefficient in Eq. (1.96) is comparable to the fluid diffusivity in Eq. (1.94).) In general, the pore-pressure changes will affect the strains, and this effect can be studied by subsequently cycling the model to equilibrium in mechanical mode (SET mech on, SET fluid off). Relatively soft matrix (Rk >>> 1)
If the matrix is very soft (or the fluid incompressible) and Rk is very large, then the system is coupled, with a diffusivity governed by the matrix. The modeling approach will also depend on the driving mechanism.
1. In mechanically driven simulations, calculations can be time-consuming. As discussed in note 5 of Section 1.7.1, it may be possible to reduce the value for M (or Kf ), such that Rk = 20, and not significantly affect the response.
2. In most practical cases of pore-pressure driven systems, experience shows that the coupling between pore pressure and mechanical fields is weak. If the medium is elastic, the numerical simulation can be performed with the flow calculation in flow-only mode (SET fluid on, SET mech off) and then in mechanical-only mode (SET mech on, SET fluid off) to bring the model to equilibrium. It is important to note that in order to preserve the true diffusivity (and hence the characteristic time scale) of the system.
1.7.2.4 Recommended Procedure to Select a Modeling Approach
It is recommended that the selection of a modeling approach for a fully coupled analysis follow the procedure outlined in Table 1.2. First, determine the characteristic time of the diffusion process
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for the specific problem conditions and properties (see Section 1.7.1), and compare this time to the actual time scale of interest. Second, consider whether the perturbation to the system is primarily pore-pressure driven or mechanically driven. Third, determine the ratio of the stiffness of the fluid to the stiffness of the solid matrix. Table 1.2 indicates the appropriate modeling approach based on the evaluation of these three factors. The table also indicates the required adjustment to the fluid modulus, Ma (or Ka f ), for each case. Finally, the table lists several examples from the manual that illustrate each modeling approach. Notes to Table 1.2:
1. The effective stress approach with no fluid flow is discussed in Section 1.7.3. In order to establish the initial conditions for this effective stress analysis, use the WATER table or INITIAL pp command, or a FISH function to establish steady-state pore pressures. Specify the correct wet density to zones below the water table, and dry density to zones above.
2. The effective stress approach with groundwater flow is discussed in Section 1.7.4. In order to establish the initial conditions for this effective stress analysis, use the INITIAL command or a FISH function to establish steady-state pore pressures, or specify SET fluid on mech off and step to steady state, if the location of the phreatic surface is not known. SetMa (or Kaf ) to a small value to speed convergence for a partially saturated system. Note that Ma (or Kaf ) should be greater than 0.3nLzρwg (or 0.3Lzρwg) to satisfy numerical stability (see Eq. (1.99)).
3. The pore-pressure generation approach is discussed in Section 1.7.5. In order to establish the initial conditions for the pore-pressure generation analysis, use the INITIAL command or a FISH function to establish steady-state flow, or specify SET fluid on mech off and step to steady state, if the location of the phreatic surface is not known. Set Ma (or Kaf ) to a small value to speed convergence for a partially saturated system. Note that Ma (or Kaf ) should be greater than 0.3nLzρwg (or 0.3Lzρwg) to satisfy numerical stability (see Eq. (1.99)) .
4. The uncoupled fluid-mechanical approach is described in Section 1.7.2. This approach is recommended for pore-pressure driven systems and should be used carefully if Rk >>> 1. Note that the adjusted value for Ma (or Kaf ) during the flow-only step should satisfy Eq. (1.103) so that the coupled diffusivity will be correct.
5. The fully coupled approach is discussed in Section 1.7.6. Note that for Rk >>> 1, if Ma (or Kaf ) is adjusted to reduce Rk = 20, the time response will be close (typically within 5%) to that for infinite M (or Kf ).
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