Numerical Method Of Positive And Inverse Problem For Fluid Flow In Porous Media In Confined Aquifer

The flow of fluid through a porous medium is called seepage.Seepage mechanics is a science that studies the laws of fluid flow in porous media.The fluid flow in a confined aquifer can be described by an unsteady diffusion equation.Although the seepage problem of the confined aquifer with simple regular geometry can be solved by analytical method,the geometry of porous media in engineering is usually complex,and in many cases,the analytical solution of pressure head is not available.In recent decades,with the development of computer,the numerical method for solving seepage in porous media has become one of the hot issues in this field.In practical problems,the permeability coefficient is sometimes a function of space and time.It is also a research subject with important engineering application value to calculate the permeability coefficient based on the measured pressure head value and information of source and sink.In this work,the numerical schemes in the classical finite volume method for unsteady seepage problems are compared,and their advantages and disadvantages are pointed out.According to the different weighting parameters in the discrete equation,these schemes can be divided into explicit scheme,Crank-Nicolson scheme(hereinafter referred to as C-N scheme)and full implicit scheme.The accuracy of the explicit and full implicit schemes is only the first-order in terms of truncation error,and that of C-N scheme is second-order.Therefore,the calculation accuracy of C-N scheme is the highest.The numerical examples in this paper show this point well.For the seepage problem with the first type of boundary conditions,the partial derivative at the boundary can be obtained by constructing the mirror point;for the seepage with the second type of boundary conditions,the partial derivative of the node point near the boundary is directly given by the boundary conditions,which reflects the advantages of the finite volume method.Next,a higher-order scheme proposed is applied to solve the partial differential equation of the unsteady seepage.In this scheme,similar to the finite volume method,the seepage equation is firstly integrated over the control volume of each node and the time variable.In order to discretize the time integration,the undetermined weighting parameter is introduced.Then,the truncation error of the scheme is obtained.By making the truncation error as small as possible,the ratio of the time step to the square of space step is determined,and the exact value of the undetermined weighting parameter is given.Numerical examples show that the higher-order scheme is better than the C-N scheme for the seepage problems with the first kind of boundary condition.Finally,the explicit and full implicit schemes in the finite volume method are improved,and the improved explicit and full implicit schemes are used to solve the permeability coefficient based on the measured head value and the known source and sink.The numerical examples of the inverse problem show that the accurate seepage coefficient can be obtained without considering the measurement error,no matter for the first kind of boundary condition or for the second and third kind of boundary conditions,when taking the appropriate number of grids.