Iterative methodsof solving the coupled filtration problem

Vestnik MGSU

Sheshenin Sergey Vladimirovich¹  Kakushev Eldar Ramazanovich²  Zakalyukina Irina Mikhailovna³ 

[1] Lomonosov MoscowState University (MSU);

[2] Lomonosov Moscow State University(MSU);

[3] Moscow State University of Civil Engineering (MGSU);

关键词: conjugate gradient method;    LBB condition;    Darcy's law;    coupled filtration problem;    Biot's filtration model;   

DOI  :  

来源: DOAJ

【 摘 要 】

This paper represents a summary of the iterative solution to the problem of linearized coupledfiltration. The formulation of the coupled filtration problem can be applied for the purposes of simulationof the land surface subsidence caused by the pumping of the fluid out of a well located near theland surface. The pumping process causes pressure redistribution and, consequently, undesirablesubsidence 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 beavailable. With this assumption in hand, coupled differential equations are derived on the basis ofthe Biot's filtration model and the Darcy's law.First, spatial discretization is based on the finite element method, while the finite-differencescheme is used to assure discretization within the course of time. Discretization of the linear coupledproblem leads to the generation of a linear saddle system of algebraic equations. It is well-knownthat the stability condition of such a system is usually formulated as the LBB condition (inf-supcondition). The condition is satisfied for a differential problem (to say more accurately, for a variationalproblem). The validity of the stability condition for an algebraic system depends on the finiteelements used for the purpose of the problem discretization. For example, the LBB condition is notalways satisfied for most simple Q1-Q1 elements. Therefore, first of all, stability of the finite elementsystem is studied in the paper. The filtration problem has a number of parameters; therefore, it isnot easy to identify analytically the domain in which the stability condition is satisfied. Therefore, thestability condition is under research that includes some numerical tests and examination of physicaldimensionality. The analysis completed by the authors has ended in the derivation of the formulathat determines the stability condition formulated on the basis of the problem parameters.Second, solution methods are explored numerically in respect of sample 3D problems. Dimensionsof domains under consideration are typically as far as 20 km in length and width and up to5 km in depth. Thus, the resulting linear system is rather large, as it is composed of hundreds ofthousands to millions of equations. Direct methods of resolving these saddle systems can hardly besuccessful and they are definitely inefficient. Therefore, the only choice is the iterative method. Thesimplest and the most robust method is the Uzawa method applied in combination with the conjugategradients iteration method used for the Schur complement system solution. The computer codethat implements iterative solution methods is written in FORTRAN language of programming. Theconjugate gradients method is compared to its alternatives, such as the Richardson iteration and theminimal 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 ofthe aforementioned iteration methods.

【 授权许可】

Unknown