Mathematical modeling of two-phase compressible fluid filtration based on modified adaptive method of minimum amendments

The work objective is to build and investigate the modified adaptive method of minimum amendments (MAMMA) which is destined for the numerical simulation of the two-phase compressible fluid filtration in porous media. This approach allows overcoming the known use limitations of other methods of the finite-difference equations solution, such as: crucial differential pressures acting on the oil-and-water bearing formation; and the compressibility of the medium at the considerable gas content in the oil phase. An approximation method - an explicit one for defining the function of water saturation, and an implicit one for the pressure function computation - is selected as the research basis. When setting the initial boundary value problem and its sampling, the process of the two-phase compressible fluid filtration in the space-dimensional domain with the lateral area bounded below by the subface of stratum, and above - by the bed top, is considered. A two-layer iterative method of the variational type - a modified method of minimal amendments adapted for solving finite-difference equations of the two-phase compressible fluid with a non-selfadjoint operator under the most general assumptions on the properties of the grid-problem operator is built. It is shown that a MAMMA has the asymptotic convergence rate characteristic of the “classical” alternate triangular method that does not use the Chebyshev acceleration technique and can be applied to the problems with a self-adjoint operator. Numerical experiments have confirmed the high efficiency of MAMMA. It is established that to achieve the specified accuracy, the number of iterations at the MAMMA reduces to 3-20 times as compared to the method of Seidel and the overrelaxation method

Sufficient conditions for convergence of positive solutions to linearized two-dimensional sediment transport problem

Introduction. The sediment transportation is one of the major processes that define the magnitude and back surface changing rate for water bodies. The most used prognostic studies in this field are based on the mathematical models that allow reducing, and in some cases, eliminating the expensive and often environmentally burdensome experiments. Spatially one-dimensional models are mainly used to predict changes in bottom topography. For actual coastal systems with irregular coastal configuration, the flow vector is generally not orthogonal to the tangent line for the coastline at each of its points. It also may not coincide with the wind stress vector. Therefore, to solve lots of practically important problems associated with the prediction of the dynamics of the back surface of water basins, it is necessary to use spatially two-dimensional models of sediment transportation and effective numerical methods of their implementation. Materials and Methods. The authors (A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko, and V.V. Sidoryakina) have earlier proposed a spatially two-dimensional model of sediment transport that satisfies the basic conservation laws (of material balance and momentum) which is a quasilinear equation of parabolic type. The linear difference schemes are constructed and studied; model and some practically important tasks are solved. However, the theoretical study on the proximity of solutions for the original nonlinear initial-boundary value problem and the linearized continuous task - on which basis a discrete model (difference scheme) was built - remained in the shadow. The study of the linearized problem correctness and the determination of sufficient conditions for positivity of solutions are of special interest because only positive solutions to this sediment transport problem have physical value within the framework of the considered models. Research Results. The investigated nonlinear two-dimensional model of sediment transport in the coastal zone of shallow reservoirs takes account of the following physically significant factors and parameters: soil porosity; critical value of the tangent stress at which load transport starts; turbulent interaction; dynamically varying of the bottom geometry; wind currents; and bottom friction. Linearization is carried out on the time grid - nonlinear coefficients of the parabolic equation are taken for the previous time grid step. Next, a chain of tasks connected by the initial data - final solutions of the linearized mixed Cauchy problems chain on a uniform time grid is constructed, and thus, the linearization for the initial 2D nonlinear model is carried out in large. Earlier, the authors have proved the existence and uniqueness of the solution to a linear tasks chain. Prior estimate of the proximity of the linearized problem chain solution to the initial non-linear task solution has been also obtained. The conditions of its solution positivity and their convergence to the nonlinear sediment transport problem are defined in the norm of the Hilbert space L1 with the rate O(τ) where τ is a time step. Discussion and Conclusions. The obtained research results of the spatially two-dimensional nonlinear sediment transport model can be used for predicting the nonlinear hydrodynamic processes, improving their accuracy and reliability due to the availability of new accounting functionality of physically important factors, including the specification of the boundary conditions

Additive two-dimensional splitting schemes for solving 3D suspension transport problems on optimal boundary-adaptive grids with uniform spacing’s in the vertical direction

The article considers3D matter transport model in dissolved and suspended forms (impurities) in coastal marine systems. The initial boundary value problem numerical solution is carried out on the basis of local2D splitting schemes. In this case, special attention is paid to the hig-hquality computational grid construction. As a rule, in the proposed methods, Cartesian grids are used, a variety of which are grids with boundary adaptation. The technology of constructing 3D vertically uniform boundary-adaptive grids on the basis of surface 2D grids, which is created using the procedure of minimizing the generalized Dirichlet functional, is presented. Previously, this approach has shown its effectiveness in constructing 2D non-degenerate regular grids containing the minimum number of cells (convex quadrangles) for test Z-shape regions, such as the “Maltese cross” and others, as well as in the 2Dhydrophysicsproblems numerical solution of coastal systems

Predictive modeling of wave hydrodynamics and relief formation in the presence of multi-scale turbulent exchange

Introduction. Reliable prediction of indicators of turbulent flows is a very difficult task, which is explained by the exceptional physical complexity of turbulence, in particular its probabilistic nature, a wide space-time spectrum and a fundamentally three-dimensional non-stationary nature. Despite conducting a wide range of studies focused on the problem under consideration, they did not fully reflect the totality of various factors and processes affecting the structure and parameters of vertical turbulent mixing. Materials and methods. The article is devoted to the study of spatial- three-dimensional wave processes in shallow water bodies, taking into account the features of turbulent exchange depending on the source and localization in the column of liquid, as well as the study of the influence of regular wave processes on turbulent exchange and vertically using a mathematical model of wave processes based on the system of Navier-Stokes equations, including three equations of motion in the with dynamically changing geometry of the computational domain. Results. Based on the developed software package, a scenario of changes in hydrodynamic wave processes of the coastal zone is constructed. Discussions and conclusions. The separation of the wave flow into a near-surface macroturbulent layer caused by wave motion and a lower layer with background hydrodynamic turbulence is proved, the strength and intensity of turbulence changed synchronously with wave oscillations, demonstrating a pronounced asymmetry of turbulence generation throughout the water column

Predictive modeling of wave hydrodynamics and relief formation in the presence of multi-scale turbulent exchange

Introduction. Reliable prediction of indicators of turbulent flows is a very difficult task, which is explained by the exceptional physical complexity of turbulence, in particular its probabilistic nature, a wide space-time spectrum and a fundamentally three-dimensional non-stationary nature. Despite conducting a wide range of studies focused on the problem under consideration, they did not fully reflect the totality of various factors and processes affecting the structure and parameters of vertical turbulent mixing. Materials and methods. The article is devoted to the study of spatial- three-dimensional wave processes in shallow water bodies, taking into account the features of turbulent exchange depending on the source and localization in the column of liquid, as well as the study of the influence of regular wave processes on turbulent exchange and vertically using a mathematical model of wave processes based on the system of Navier-Stokes equations, including three equations of motion in the with dynamically changing geometry of the computational domain. Results. Based on the developed software package, a scenario of changes in hydrodynamic wave processes of the coastal zone is constructed. Discussions and conclusions. The separation of the wave flow into a near-surface macroturbulent layer caused by wave motion and a lower layer with background hydrodynamic turbulence is proved, the strength and intensity of turbulence changed synchronously with wave oscillations, demonstrating a pronounced asymmetry of turbulence generation throughout the water column

Development and correctness analysis of the mathematical model of transport and suspension sedimentation depending on bottom relief variation

Introduction. The paper is devoted to the study on the three-dimensional model of transport and suspension sedimentation in the coastal area due to changes in the bottom relief. The model considers the following processes: advective transfer caused by the aquatic medium motion, micro-turbulent diffusion, and gravity sedimentation of suspended particles, as well as the bottom geometry variation caused by the particle settling or bottom sediment rising. The work objective was to conduct an analytical study of the correctness of the initial-boundary value problem corresponding to the constructed model.Materials and Methods. The change in the bottom relief aids in solution to the initial-boundary value problem for a parabolic equation with the lowest derivatives in a domain whose geometry depends on the desired function of the solution, which in general leads to a nonlinear formulation of the problem. The model is linearized on the time grid due to the “freezing” of the bottom relief within a single step in time and the subsequent recalculation of the bottom surface function on the basis of the changed function of the suspension concentration, as well as a possible change in the velocity vector of the aquatic medium.Research Results. For the linearized problem, a quadratic functional is constructed, and the uniqueness of the solution to the corresponding initial boundary value problem is proved within the limits of an unspecified time step. On the basis of the quadratic functional transformation, we obtain a prior estimate of the solution norm in the functional space L2 as a function of the integral time estimates of the right side, and the initial condition. Thus, the stability of the solution to the initial problem from the change of the initial and boundary conditions, the right-hand side function, is established.Discussion and Conclusions. The model can be of value for predicting the spread of contaminants and changes in the bottom topography, both under an anthropogenic impact and due to the natural processes in the coastal area

Study of 3D discrete hydrodynamics models using cell filling

Modern methods and tools for coastal hydrodynamics modeling indicate the necessity of constructing discrete analogs of models for ones the properties: balance and conservation laws (for mass, flows, impulse), stability, convergence and etc. have been fulfilled. The paper considers a continuous three-dimensional mathematical model of the hydrodynamics of water basins and its discretization. The pressure correction method at variable water medium density was used to solve the problem of hydrodynamics. The considered discrete mathematical models of hydrodynamics take into account the filling of control cells on rectangular grids. This increased the accuracy of the solution in the case of complex geometry by improving the boundary approximation. From the obtained estimates of the components of the velocity vector, it follows that there are no two or more stationary regimes in which all forces are balanced, and the solution to the discrete problem exists and is unique and tends to the solution of the continuous problem upon reaching the stationary regime. Also the balance of the flows for the discrete model has been proved as well as absence of non-conservative dissipative terms