Iterative methods of solving the coupled filtration problem
This paper represents a summary of the iterative solution to the problem of linearized coupled filtration. The formulation of the coupled filtration problem can be applied for the purposes of simulation of the land surface subsidence caused by the pumping of the fluid out of a well located near the land surface. The pumping process causes pressure redistribution and, consequently, undesirable subsidence of the land surface. The filtration problem considered by the authors is a direct problem, therefore, domain dimensions, ground properties and pumping characteristics are supposed to be available. With this assumption in hand, coupled differential equations are derived on the basis of the Biot's filtration model and the Darcy's law. First, spatial discretization is based on the finite element method, while the finite-difference scheme is used to assure discretization within the course of time. Discretization of the linear coupled problem leads to the generation of a linear saddle system of algebraic equations. It is well-known that the stability condition of such a system is usually formulated as the LBB condition (inf-sup condition). The condition is satisfied for a differential problem (to say more accurately, for a variational problem). The validity of the stability condition for an algebraic system depends on the finite elements used for the purpose of the problem discretization. For example, the LBB condition is not always satisfied for most simple Q1-Q1 elements. Therefore, first of all, stability of the finite element system is studied in the paper. The filtration problem has a number of parameters; therefore, it is not easy to identify analytically the domain in which the stability condition is satisfied. Therefore, the stability condition is under research that includes some numerical tests and examination of physical dimensionality. The analysis completed by the authors has ended in the derivation of the formula that determines the stability condition formulated on the basis of the problem parameters. Second, solution methods are explored numerically in respect of sample 3D problems. Dimensions of domains under consideration are typically as far as 20 km in length and width and up to 5 km in depth. Thus, the resulting linear system is rather large, as it is composed of hundreds of thousands to millions of equations. Direct methods of resolving these saddle systems can hardly be successful and they are definitely inefficient. Therefore, the only choice is the iterative method. The simplest and the most robust method is the Uzawa method applied in combination with the conjugate gradients iteration method used for the Schur complement system solution. The computer code that implements iterative solution methods is written in FORTRAN language of programming. The conjugate gradients method is compared to its alternatives, such as the Richardson iteration and the minimal residue methods. All three methods were tested as methods of solving the model problems. The authors provide their numerical results and conclusions based on the comparative analysis of the aforementioned iteration methods.