Filtration Equation Research Articles

A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative

The paper considers the problem of finite-difference simulation of filtration processes on the base of a space-fractional-differential equation with a generalised Caputo derivative with respect to a function. To make calculations faster, we propose an algorithm of fractional derivative approximation which reduces the computational complexity of matrix operations that arise while performing simulations on uniform and non-uniform grids. The algorithm is based on Taylor’s series expansions of fractional integral kernel and special representation of the matrices of linear systems that arise during finite-difference approximation. Theoretical estimates of the proposed algorithm’s complexity and accuracy are presented. We describe and experimentally compare computational schemes for matrix-vector multiplication used for numerical solution of linear systems by the BicgStab algorithm and analyse factors that influence its convergence. The results of computational experiments which demonstrate an efficiency of the proposed computational scheme for solving a filtration equation with a generalised Caputo derivative with respect to a space variable on a large-sized grids are given.

MODEL RADIAL APPROXIMATIONS OF FILTERING BASED ON THE GREEN'S FUNCTIONS

The article discusses the problem of approximating solutions of differential equations describing the process of a two-dimensional fluid flow in porous media. The approximation is presented as a combination of radial basis functions based of the Green's function used to solve the Poisson equation with variable coefficients in the case of steady state filtration and parabolic equations in the transient regime. To illustrate the effectiveness of the proposed approximation obtained by the field pressure distribution in the reservoir with a network of injection and production wells. Comparing of approximated pressure and design points to a satisfactory accuracy of the results. It is revealed that the obtained model of approximation of the stationary filtration equation using Green as the basis functions provides the root-mean-square deviation of the approximated pressure from the calculated one at the level of 0.95 - 1.05 %. This type of basis functions is used in constructing approximating models for non-stationary modes of multiphase filtration. The maximum deviations are observed precisely in these cells, where the most abrupt (peak) pressure changes occur. The Neumann's boundary conditions, as follows from the presented results, are satisfied. It should be noted that the Neumann boundary conditions, especially for all boundaries, are the most complicated computational case. To approximate the nonstationary solution of the filtration equation, the Green function is used, which is shown as a product of two parts: one part only depends exponentially on time; the second part depends only on the spatial coordinates. The use of the background intensity of the sources made it possible to build approximators for the possibilities of non-stationary transformation of the intensity of the sources even with a non-zero equilibrium of the intensities.

On homogenized equations of filtration in two domains with common boundary

We consider an initial-boundary value problem describing the process of filtration of a weakly viscous fluid in two distinct porous media with common boundary. We prove, at the microscopic level, the existence and uniqueness of a generalized solution of the problem on the joint motion of two incompressible elastic porous (poroelastic) bodies with distinct Lamé constants and different microstructures, and of a viscous incompressible porous fluid. Under various assumptions on the data of the problem, we derive homogenized models of filtration of an incompressible weakly viscous fluid in two distinct elastic or absolutely rigid porous media with common boundary.

Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtration$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$If the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853].

Peculiarities of Excitation of Self-Oscillations in Geological Systems

Theory of self-oscillations in the filtration of gas-liquid mixtures in a porous media is presented. Self-oscillatory mode of filtration of hydrocarbon mixtures has described. The possibility of occurrence of similar modes without phase equilibrium (the effect of “gas bearing”) is discussing. System of filtration equations, including energy equation, is also explored is able to reveal such effect in filtration of geothermal waters.

Comparison of finite-dimensional approximations of the permeability field on solving the inverse problem for the equation of stationary single-phase fluid filtration

The problem of identifying the permeability coefficient for equation of liquid stationary single-phase filtration by the known values of bottomhole pressure on the wells is considered. This problem is reduced to residual function minimization.

Multiscale model reduction of fluid flow based on the dual porosity model

Construction of multiscale method for double porosity model is considered. Mathematical modelling of such type of models widely used for describe the flow in fractured porous media. Idea of double porosity model based on the interaction mechanism between fractured and porous continua. Each continuum considered as a separate homogeneous medium which the expressed through filtration equation with additional terms corresponding to the exchange flux. The solution method is based on the use of a generalized multiscale finite element method. The main idea is to build local-domain solutions that can be used to effectively solve the coarse grid.

On homogenized equations of filtration in two domains with common boundary

Рассматривается начально-краевая задача, описывающая фильтрацию слабо вязкой жидкости в двух различных пористых средах с общей границей. Доказывается на микроскопическом уровне теорема существования и единственности обобщенного решения задачи о совместном движении двух несжимаемых упругих пористых (пороупругих) тел с различными постоянными Ламе, с различной микроструктурой и вязкой несжимаемой поровой жидкости. При различных предположениях на данные задачи выводятся усредненные модели фильтрации несжимаемой слабовязкой жидкости в двух различных пористых упругих или абсолютно твердых средах, имеющих общую границу. Библиография: 21 наименование.

A Numerical Algorithm for Solving Inverse Filtration Problems with the Pointwise Overdetermination

The inverse problems of recovering the right-hand side in a pseudoparabolic equations of filtration with the use of the pointwise overdetermination are studied. We expose some existence and uniqueness theorems which are the base of an numerical algorithm of recovering the right-hand side (the source function) and a solution. The problem is well-posed and the stability estimates hold. It can be reduced to a Volterra-type integral equation, where the operator has a small norm for small time segments. The finite element method is used to reduce the problem to a system of ordinary differential equations which is solved by the finite difference method. The idea of the predictor-corrector method is employed in the algorithm. The results of numerical experiments are presented. They show a good convergence of an approximate solutions to a solution.

On a Deep Bed Filtration Problem with Finite Blocking Time

We consider an initial–boundary value problem for a simple semilinear filtration equation with nonunique characteristics and prove that uniqueness nevertheless holds for the solution of this problem. The solution is then constructed by quadratures.

NATURAL FILTRATION EQUATIONS. FIASCO “OF DARCY'S LAW”

The theory of natural filtration equations is given. The naturalness of the new filtration equations is that they are the exact consequences of the fundamental laws of physics, directly take into account the density and porosity of the soil, the viscosity and density of the filtration fluid, drainage, the influence of gravity, etc.the falsity of the traditional continuity equation in the filtration theory is Established. New filtration equations are derived from the equation of continuum dynamics in stresses, including the density and viscosity of the liquid and the porosity of the soil.Inadequacy of the modeling filter equations with the friction law of Newton. The efficiency of simulation of filtration by Jakupova equations based on the power laws of friction with odd exponents is numerically confirmed, with the use of which the calculations of filtration in the well, drainage under the influence of gravity, displacement of oil by water from the underground area through two symmetrically located pits are carried out.

Well-posedness of quasilinear singular diffusion equations with boundary degeneracy

This paper concerns the well-posedness problems of a class ofquasilinear singular diffusion equations with boundary degeneracy.In addition to the possibledegeneracy and singularity likethe non-Newtonian filtration equations, the equations aredegenerate on a portion of the boundary due to the diffusion coefficients.The diffusion coefficients are anisotropic.For the well-posedness problems of such equations,the boundary condition may not be prescribed on the entire lateral boundary.The prescription of the boundary condition is determined bythe degeneracy of the diffusion coefficients along the outernormal direction of the spatial domain.The well-posedness problems of the equations are formulatedand the existence and uniqueness of weak solutions to the problems are proved.

A new blow-up criterion for non-Newton filtration equations with special medium void

This paper deals with the finite time blow-up of solutions to the initial boundary value problem of a non-Newton filtration equation with special medium void. A new criterion for the solutions to blow up in finite time is established by using the Hardy inequality. Moreover, the upper and lower bounds for the blow-up time are also estimated. The results solve an open problem proposed by Liu in 2016.

Problems of heat and mass transfer in the snow-ice cover

The snow-ice cover is considered as a three-phase continuous medium consisting of water, air, and ice. The mathematical model is based on the equations of mass conservation for each phase, the two-phase filtration equation for water and air in a movable porous ice skeleton, the rheological equation for porosity, the balance of forces and the equation of energy conservation for the ice-water-air system. The paper presents the results on analytic and numerical solutions of the problems under consideration.

The Well-Posedness of the Solutions Based on the L1 Initial Value Condition

The non-Newtonian polytropic filtration equation ut=div(a(x)|∇um|p-2∇um) is considered. Only if u0(x)∈L1(Ω), the well-posedness of solutions is studied. If the diffusion coefficient is degenerate on the boundary, then stability of the weak solutions is proved only depending upon the initial value conditions.

Homogenization of the Equations of Filtration of a Viscous Fluid in Two Porous Media

A homogenized model of filtration of a viscous fluid in two domains with common boundary is deduced on the basis of the method of two-scale convergence. The domains represent an elastic medium with perforated pores. The fluid, filling the pores, is the same in both domains, and the properties of the solid skeleton are distinct.

MANEJO DE LA ALBUMINURIA. MOMENTO DE DERIVACIÓN AL NEFRÓLOGO

En los últimos años surgieron nuevas observaciones que permitieron entender que el curso clínico de la enfermedad renal por diabetes podría ser diferente a la descripción tradicional. Uno de los primeros conceptos actuales es la recomendación de reemplazar el término de micro y macroalbuminuria por el de albuminuria, la cual se califica según su severidad en estadios A1, A2 y A3. La detección de la albuminuria como único marcador de enfermedad renal parece no ser suficiente para el diagnóstico temprano; es necesario además el monitoreo de la función renal usando ecuaciones de estimación del filtrado glomerular (FG). El control estricto de la glucemia, la presión arterial (PA) y el uso de IECA o ARA II continúan siendo el tratamiento fundamental de la albuminuria. Para el control glucémico podría ser conveniente tener en cuenta los beneficios adicionales de los medicamentos antidiabéticos. La derivación al especialista en Nefrología debería plantearse de acuerdo al estadio de albuminuria y deterioro del FG.

Filtration Behaviour of Cement-Based Grout in Porous Media

Filtration behaviour of cement particles, especially under the high grouting pressure with a rapid grout flow velocity, has a significant effect on the grout injection. However, there have been few studies on this field where the governing equation of this behaviour remains unclear. In the present study, a novel experimental procedure for grout injection was adopted to acquire the spatial and temporal variations in porosity and viscosity of high-speed grout flow in coarse sand. Experimental observations showed that there were dramatic variations in viscosity and porosity during the grout penetration within the first 50 s, suggesting that the high velocity had a significant influence on the distribution of the filtration coefficient. A model based on the Stokes–Brinkman (S–B) equation and advection–filtration equations was established to describe the filtration of grout flow in porous media. Meanwhile, numerical solutions from both the proposed model and traditional Darcy’s law were compared with experimental results. The comparative results showed that the proposed approach can match the laboratory tests well; the analysis indicated that Darcy’s law was unable to properly describe high-speed grout flow in porous media due to the lack of a shear force and the inertial term. Nonuniform filtration behaviour of cement grout flowing in porous media was revealed. Due to the nonuniform distribution of the pore velocity isoline caused by Poiseuille flow, it led to a heterogenous distribution of porosity as well. Parametric studies on the applicability of Darcy’s law and S–B equation for grout flow were discussed, in which an error of less than 10% was calculated when the Reynolds number was less than 2.5.

Inertial and Darcy’s Terms Ratio in Boundary Layer at Fluid–Porous Medium Interface

The study considers the forced boundary-layer flow overlying the Darcy–Brinkman porous medium and gives a quantitative analysis of the nonlinear inertial terms in the Brinkman filtration equation. The inertial terms are shown to be larger than the Darcy’s drag near the porous medium interface. The applicability range of boundary-layer approach is determined. It is suitable in high-permeable media with moderate velocities of an external flow. If it is slow enough, the inertial terms can be omitted in spite of interface effect. On the other hand, fast external flow produces the filtration with large pore-scale Reynolds number; therefore, the Forchheimer’s drag should be taken into account. It is shown the Brinkman term as well as inertial terms have a significant role in boundary-layer formation within the porous medium.

Evolution of interfaces for the nonlinear double degenerate parabolic equation of turbulent filtration with absorption

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction–diffusion equation of turbulent filtration with strong absorption ut=(|(um)x|p−1(um)x)x−buβ,mp>1,β>0.Full classification is pursued in terms of the nonlinearity parameters m,p,β and asymptotics of the initial function near its support. Numerical analysis using a weighted essentially nonoscillatory (WENO) scheme with interface capturing is implemented, and comparison of numerical and analytical results is presented.