The time dimension appears in the form of a first order time derivative in the constitutive mechanical model (Eqs. 11.3, 11.4) and in the diffusion problems though the storage term (Eq. 11.6). We will here discuss the time integration procedure and the accuracy and stability problems that are involved.
11.2.6.1 Time Integration — Diffusion Problems
The period to be considered is divided into time steps. Linear development of the basic variable with respect to the time is generally considered within a time step:
t — tA tB — t
p = — PB + — B PA (11.26)
tB — tA tB — tA
where the subscripts A, B denote, respectively, the beginning and the end of a time step. Then the pressure rate is:
dp _ Pb — Pa _ A p dt tB — tA At
This time discretisation is equivalent to a finite difference scheme. It allows the evaluation of any variable at any time within a time step.
The balance equation should ideally be satisfied at any time during any time step. Of course this is not possible for a discretised problem. Only a mean assessment of the balance equation can be obtained. Weighted residual formulations have been proposed in a similar way as for finite elements (Zienkiewicz et al., 1988). However, the implementation complexity is too high with respect to the accuracy. Then the easiest solution is to assess only the balance equation at a given time, denoted tT, inside the time step tA to tB, such that a time variable, т, is defined:
tT tA
T =
tB — tA
All variables have then to be evaluated at the reference time, tT. Different classical schemes have been discussed for some decades:
• Fully explicit scheme — t = 0: all variables and the balance are expressed at the time step beginning, where everything is known (from the solution of the preceding time step). The solution is, therefore, very easily obtained.
• Crank-Nicholson scheme or mid-point scheme — t = 1/2
• Galerkin’s scheme — t = 2/3
• Fully implicit scheme — t = 1
The last three schemes are functions of the pore pressure/temperature/concentration at the end of the time step, and may need to be solved iteratively if non-linear problems are considered.
For some problems, phase changes, or similar large variations of properties, may occur abruptly. For example, icing or vaporising of water is associated with latent heat consumption and abrupt change of specific heat and thermal conductivity. Such rapid change is not easy to model. The change in specific heat may be smoothed using an enthalpy formulation, because enthalpy, H, is an integral of the specific heat, c. Then the finite difference of the enthalpy evaluated over the whole time step gives a mean value, c, and so allows an accurate balance equation:
|
= cdT |
(11.29) |
|
T |
|
|
Hb — Ha |
(11.30) |
|
tB — tA |
