Filtration Equation Research Articles

Effect of Pressure, Concentration and Sludge Volume on Two-phase Filtrate Volumes Using Bentonite Clay Sludge

The effect of pressure, solids concentration and volume of slurry on total and two-phase filtrate volumes was investigated to establish optimum conditions for routine laboratory filtration. Full factorial design with three levels was utilized to obtain 27 unique experiments. The slurry used was prepared by mixing crushed and sieved bentonite clay of 75 microns with distilled water at different concentrations as obtained from the design of experiment. A filter press was utilized and the results were used to calibrate a two-phase exponential equation for sludge filtration to extract the first and second stage filtrate volumes. The total filtrate volume improved with increasing applied pressure and decreasing solids concentration. A slurry volume of 0.22 litres was found to be ideal. Similar effects were noticed on the first and second stage filtrate volume except that an optimum was discovered at 0.18 litres of slurry. The second stage filtration produced an upward curve with a point of inflection at a range of 0.18 to 0.22 litres of slurry. The second stage filtrate volume is also discovered to be directly proportional to the total filtrate volume. This connection may be considered for use to assess the filterability of other slurries.

A New Blow-Up Criterion to a Singular Non-Newton Polytropic Filtration Equation

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy E(u0) was subcritical (E(u0)<d), critical (E(u0)=d), and supercritical (E(u0)>d), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if E(u0) is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.

Comparison Results for Filtration Equations on Manifolds via Schwarz Rearrangements

Comparison Results for Filtration Equations on Manifolds via Schwarz Rearrangements

The modeling of the process of stationary gas flow to the horizontal well

The development process of a gas deposit in the statinary mode with a horizontal well was modeled within combined equations of gas filtration in the formation, as well as the gas flow in the borehole and balance equation of gas intake from formation into the horizontal shaft and the effect of the geological-technological aspects on the selection of an optimum structure providing the efficiency of development was estimated. During the displacement of the horizontal shaft from the formation bottom towards its central part, the dynamics of the increase of well flow rate, and from the cenral part to the attic zone the dynamics of decrease of flow rate have been set. It is shown that in the same data of geological conditions, and formation properties, the maximum flow rate of a horizontal well is achieved in the isotropic layers, and after a definite length of wellbore, its extension does not affect the well flow rate.

Modification of Pulse Decay Method for Determination of Permeability of Crystalline Rocks

An improvement of the pulse decay method of rock permeability measurement is presented. The technique is based on fitting experimental data to analytical and numerical solutions of the filtration equations derived with regard to the variation of flowing gas properties with temperature and pressure. A special apparatus and software for the implementation of this method were developed. A single experiment in which gas is used as a flowing medium enables determining both the permeability of a sample to water and the Klinkenberg constant. The permeability measurements on the samples of different types of rock with various reservoir properties were carried out and demonstrated satisfactory accuracy and efficiency of the method. An effective method for anisotropic permeability measurement is proposed as a development of this technique.

Возникновение концентрационной конвекции в длинной прямоугольной области пористой среды

We solve a problem of the convective flow stability in an elongated rectangular region of a porous medium. The transport of a mixture is described using the MIM (mobile-immobile media) approach. This approach consists in dividing the concentration of the solute into mobile and immobile compo-nents. The effect of density inhomogeneity on the flow of a solution in a porous medium is taken in-to account by the filtration equation within the Darcy-Boussinesq approximation. The problem is solved numerically with the use of the finite difference method. The fields of pressure and concen-tration are obtained. The influence of the parameters of the problem on the disturbance profiles is analyzed. A modified problem of stability is solved. The modification is necessary for finding the oscillatory convective mode. To obtain the solution, the spectrum of Lyapunov exponents is calcu-lated with Gram-Schmidt orthogonalization. Neutral curves are found that make it possible to detect the threshold for the onset of convection. The ranges of parameters in which oscillatory perturba-tions are realized are obtained. The effect of sorption parameters on the occurrence of oscillations is analyzed.

Rheology of a Viscous-Plastic Liquid in a Porous Medium

The hydrodynamics of the capillary flow of a viscous-plastic liquid in cylindrical rectilinear pores is considered, as a result of which the structural velocity distribution over the pore cross section is obtained. Analytical solutions are proposed for the equations of hydraulic diffusion and nonlinear filtration for a non-Newtonian fluid in a cylindrical porous medium. It is noted that when a non-Newtonian fluid flows in a porous medium, the filtration equations take a nonlinear form due to the effective viscosity, shear, and yield stresses taken into account in its structure. The proposed solutions make it possible to evaluate the state of the porous medium and its main parameters (permeability, hydraulic diffusion, and effective viscosity coefficients). The obtained solutions are compared with existing experimental data for non-Newtonian oils.

FILTRATION EQUATIONS FOR PORE AND CAPILLARY SYSTEM TAKING INTO ACCOUNT PRESSURE DIFFUSION

Biological media are characterized by a complex system of pores and capillaries, the interaction between which leads to features in the transport of biological fluids. At low flow velocities, the mutual influence of convection and diffusion becomes important, as well as the influence of pressure gradient on diffusion. To take into account the complex structure of porous space, models of fractured porous media, models of media with double and triple porosity, are known in the literature. In this paper, we propose a model of a medium with double porosity in which, in addition to diffusion and filtration, barodiffusion is taken into account. Definitive relations follow from thermodynamics of irreversible processes. We consider particular versions of the equations: for a medium with predominant convection in both pore systems; for media with predominantly diffusion transport mechanism, typical for nanoporous materials; and for a medium in which convection prevails in one pore system and diffusion in the second.

Radon measure-valued solutions of unsteady filtration equations

<p style='text-indent:20px;'>We study Radon measure-valued solutions of a Cauchy-Dirichlet problem in one space dimension that describes nonlinear convection-diffusion phenomena in heterogeneous media. Well-posedness results are proven, which extend those previously known for the space homogeneous case.</p>

Use of jet pump in vertical drainage systems

Usually, vertical drains are widely used to improve land reclamation and construction of urban planning, transport and road construction, and agriculture. Vertical drainage systems are drainage systems that use pumped wells, open dug wells, or tube wells. There are calculation methods based on pumping curves and calculation methods based on the filtration equation of hydraulic calculation of vertical drainage. In the proposed system, jet apparatus is used in vertical drainage systems. Jet pumps are designed in such a way that they transfer the impulse through the high-speed primary to the secondary flow. Jet devices are widely used in agriculture, mining, oil extraction, and various industries. In addition, they can be used as an independent units or together with pump units.

Interior second derivatives estimates for nonlinear diffusions

<p style='text-indent:20px;'>By obtaining some new estimates for classical solutions of filtration equations and singular parabolic equations we derive consequences for fully nonlinear parabolic equations of dual filtration type, with nonlinearities that can be both singular and degenerate elliptic and also non-homogeneous. Such equations appear in the theory of option pricing with market impact.</p>

Travelling wave solutions to two-velocity deep bed filtration equations

Строятся решения с бегущими волнами для системы уравнений объемной фильтрации суспензии в пористой среде в рамках модели с различными скоростями несущей жидкости и взвешенных подвижных частиц. В случае, когда скорости частиц и несущей жидкости различаются на множитель, произвольным образом зависящий от концентрации, решение получено в квадратурах. В ряде частных случаев в пространстве определяющих параметров найдены области, соответствующие физически реализуемым решениям. Приведены примеры решений, описывающих структуру волны засорения.

АНАЛИЗ СТОХАСТИЧЕСКОЙ СИСТЕМЫ ВЕНТЦЕЛЯ УРАВНЕНИЙ ФИЛЬТРАЦИИ ЖИДКОСТИ В КРУГЕ И НА ЕГО ГРАНИЦЕ

Wentzell boundary condition problems for linear elliptic equations of second order have been studied by various methods. Over time, the condition has come to be understood as a description of a process occurring on the boundary of a domain and affected by processes inside the domain. Since Wentzell boundary conditions in the mathematical literature have been considered from two points of view (in the classical and neoclassical cases), the aim of this paper is to analyse the stochastic Wentzell system of filtration equations in a circle and on its boundary in the space of differentiable K-“noise”. In particular, we prove the existence and uniqueness of the solution that determines quantitative predictions of changes in the geochemical regime of groundwater in the case of non-pressure filtration at the boundary of two media (in the region and on its boundary).

Approximation of BSDE with hidden forward equation and unknown volatility

The focus is on the approximation of the solution of BSDE in the case where the solution of forward equation is observed in the presence of small Gaussian noise. The volatility of the forward equation is considered to depend on some unknown parameter. This approximation is made in several steps. First a preliminary estimator of the unknown volatility is obtained, then using Kalman-Bucy filtration equations and Fisher-score device one-step MLE-process of this parameter is constructed. The solution of BSDE is approximated by means of the solution of PDE and the One-step MLE-process. The error of approximation is described in different metrics.

ВПЛИВ ВЕЛИЧИНИ ПОХИЛУ НА РОЗРАХУНКОВІ ПАРАМЕТРИ ЗБІРНИХ ДРЕНАЖНИХ ТРУБОПРОВОДІВ

System of two differential equations, which describe the liquid motion in a drainage pipeline with variable flow rate and the conditions of liquid entry from the surrounding soil through the walls of the pipe in filtration mode, is considered. It consists of a variable mass hydraulics equation and a modified filtration equation. The explored pipeline is laid with a direct slope "i". It is shown that in this case, the second term of modified filtration equation can be neglected without significant error. By introducing new variables, the original system is reduced to a dimensionless form. The solution of this equations system in dimensionless form is presented. In this case the solution of the original equations system depends on the value of three main factors: the resistance factor of the collecting drainage pipeline "ζl"; the generalized parameter "A", which comprehensively takes into account the structural and filtering characteristics of the stream; the geometric slope of pipeline laying "i". The analysis used the concept of an infinitely long drainage pipeline, which is laid with a slope, or, what is the same, an inclined pipeline with an infinite filtering capacity of the side surface walls. It is noted that such pipeline will have the maximum throughput compared to the same pipeline of limited length. Relatively simple and easy-to-use analytical dependencies were obtained on the basis of the conducted analysis. They allow to calculate the nature of the flow rate variations and pressure drop along the length of the drainage pipeline laid with a certain slope. A series of calculations of the explored pipelines main characteristics were carried out according to the proposed formulas at different values of the slope. Corresponding graphic dependencies were constructed for clarity. It is shown that the value of the geometric slope of the pressure drainage pipeline, along with the resistance factor and the generalized parameter, significantly affects the calculated parameters of such pipelines.

Бесполостная дрена составного профиля, усиленная дренажной трубой

Purpose: Calculated substantiation of cavityless drain structure that provides for effective draining of poorly permeable soils. Hydraulic calculation methodology of composite profile cavityless drain, laid with a slope but without inclusion thitherwards of a drainage pipe, was developed earlier. Drainage pipe reinforces drainage capability of cavityless drain with the preservation of all its other advantages. Methods: Known differential equation of liquid filtration in a soil is used. It’s of special importance that the calculation method may be used at all modes of water movement in a drainage: laminar, transient or turbulent. The dependencies substantiated with experiments, held by Northern Scientific-Research Institute of Hydrotechnique and Melioration, were applied for transient and turbulent modes. The complex form of drainage transection is taken into account with the help of special coefficient which leads it to equivalent by area rectangular section. It’s proved that such transfer doesn’t lead to significant inaccuracy of results obtained. Results: Dependencies have been obtained allowing to calculate consumptions, flowing along drain cavityless part and along drain pipe. If drain is long, it’s feasible to divert a water periodically from it in lateral direction by collectors. Dependencies are derived which according to, the distance between intermediary collectors, whereto a water comes from drains that’re laid with a slope, and the water depth in drain cavityless part in characteristic sections are defined. Practical significance: Cavityless drainage is applied in agriculture, in the systems of engineering protection from environmental sinking and pollution. Cavityless drains of composite profile have attracted railwaymen’s attention providing effective draining of subgrade, they simultaneously increase its bearing capacity. Its application is especially effective on poorly permeable soils when laying in the zone of seasonable freezing of soils.

Modeling the remediation of aquifers from the lenses of petroleum products

An improved algorithm for modeling the dynamics of petroleum product lenses based on the planned filtration equations is presented. The program code «Oilfem 2», which implements the algorithm, is supplemented with a generator of non-orthogonal grids, the ability to create groundwater models or import them, the dependence between the pumping rate of the lens of petroleum products and its thickness is taken into account. The improved program code was applied on a test task to eliminate oil pollution, which is a round lens of petroleum products. The results of this work showed a good correspondence between the two solutions according to two criteria: the change in the flow rate of pumping oil products and the change in the volume of free oil products over time.

The Simulation of the Nonlinear Consolidation of Porous Media

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.

Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions

This study concerns with the quenching features of solutions of the non-Newtonian filtration equation. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined are certain cases. Numerical experiments are provided to illustrate and provide additional validation of the theoretical predictions to the quenching rates and times.

Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells

A method is proposed to solve the identification problem of the permeability field of a three-dimensional reservoir under conditions of three-phase fluid filtration. The peculiarity of the method lies in the fact that the wells retain the proportionality of the values of the permeability coefficients of the layers obtained a priori from the results of geophysical studies. The approximation of the permeability field was carried out in layers using surface splines. The problem of identifying the permeability field belongs to the class of inverse coefficient problems for a system of nonlinear partial differential equations describing the process of fluid flow in a porous medium. The solution of the problem was reduced to minimizing the residual function constructed on the total liquid production rate at the wells. Minimization of the residual function was carried out by the Levenberg–Marquardt method. The black oil model was used to describe the fluid filtration process. The filtration equations were solved numerically by the method of the simultaneous solution, and the control volume method was used to approximate the equations by spatial variables. The proposed approach was used to solve a model problem. The influence of various types of errors on the results of identification was investigated.