The initial-boundary value problem for the system of one-dimensional motion of viscous liquid in a deformable viscous porous medium is considered. The introduction presents the relevance of a theoretical study of this problem, scientific novelty, theoretical and practical significance, methodology and research methods, a review of publications on this topic. The first paragraph shows the conclusion of the model and the statement of the problem. In paragraph 2, we consider the case of motion of a viscous compressible fluid in a poroelastic medium and prove the local theorem on the existence and uniqueness of the problem. In the case of an incompressible fluid, the global solvability theorem is proved in the Holder classes in paragraph 3. In paragraph 4, an algorithm for the numerical solution of the problem is given. Mathematical models of fluid filtration in a porous medium apply to a broad range of practical problems. The examples include but are not limited to filtration near river dams, irrigation, and drainage of agricultural fields, oil and gas production, in particular, the dynamics of hydraulic fractures, problems of degassing coal and shale deposits in order to extract methane; magma movement in the earth's crust, geotectonics in the study of subsidence of the earth's crust, processes occurring in sedimentary basins, etc. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastic properties.