Filtration Equation Research Articles

Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.

Modeling of Filtration and Heat and Mass Transfer Processes in Groundwater

Purpose. The method of mathematical modeling is an important tool for solving complex problems involving the analysis of groundwater dynamics and heat and mass transfer processes in them when studying their contamination from various anthropogenic sources in the event of accidental spills of chemically hazardous substances, etc. The main purpose of the article is to develop a set of mathematical models for calculating the process of filtration of non-pressure groundwater, mass transfer of impurities and the process of heat transfer in groundwater. Methodology. The two-dimensional Boussinesq equation of filtration was used to predict the dynamics of groundwater. The two-dimensional equation of convective-diffusive transport of impurities was used to model the processes of mass transfer in groundwater. The process of freezing of individual sections of the groundwater flow is modeled using the Laplace equation for the velocity potential (calculation of the flow velocity field in a time-varying geometry) and the two-dimensional equation of heat transfer in groundwater. Finite difference schemes were used to solve the modeling equations of groundwater dynamics and heat and mass transfer. Findings. A set of mathematical models has been developed to calculate the process of filtration of non-pressure groundwater and its chemical contamination. The experiment has confirmed the adequacy of the constructed numerical model of filtration of a non-pressure groundwater flow. An effective mathematical model was developed that allows determining the temperature fields in groundwater during the operation of a well used to freeze certain sections of the flow. The results of computer modeling indicate the effectiveness of the developed mathematical models. Originality. Effective mathematical models for predicting the level of chemical contamination of groundwater, its dynamics and thermal regime are proposed. The constructed mathematical models make it possible to determine the dynamics of changes in the temperature regime of groundwater during the operation of wells through which refrigerant is supplied to freeze individual areas. A computer program has been developed that allows for a comprehensive assessment of groundwater conditions. Practical value. A set of computer programs has been developed to conduct a computational experiment to study the processes of filtration, chemical contamination of groundwater and heat transfer processes in them. This set of programs can be used for the scientific substantiation of engineering solutions aimed at protecting groundwater.

Immunosuppression and transplantation-related characteristics affect the difference between eGFR equations based on creatinine compared to cystatin C in kidney transplant recipients.

Previous studies show heterogeneity when applying estimated glomerular filtration (eGFR) equations to kidney transplant recipients (KTRs). However, research on the impact of transplantation-related characteristics on eGFR equations using creatinine (eGFRcr) compared to cystatin C (eGFRcys) is scarce. We conducted a comprehensive analysis with three eGFRcr equations (Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) 2009, European Kidney Function Consortium (EKFC) 2021, kidney recipient specific-glomerular filtration rate KRS-GFR) 2023), comparing them to two eGFRcys (CKD-EPI 2012 and EKFC 2023) in 596 KTRs. Bland-Altman plots demonstrated relative differences according to different eGFR-stages. Multivariable logistic regression identified transplantation-related characteristics independently associated with smaller or greater differences between eGFRcr and eGFRcys equations. 94.3% of the cohort were White individuals. Median eGFR differed as much as 9ml/min/1.73 m2 between equations. The median relative differences (Q2) were greater (more negative) when comparing the eGFRcr equations to eGFRcys CKD-EPI 2012, than when comparing them to eGFRcys EKFC 2023 (P<.001). Better average eGFR was associated with smaller mean relative differences in all comparisons but eGFRcr CKD-EPI 2009 with eGFR EKFC 2023 and eGFRcr EKFC 2021 with eGFRcys EKFC 2023. Living kidney donation and belatacept use were independent factors associated with a smaller difference (≥Q3) between eGFRcr and eGFRcys equations, while prednisone use or higher HbA1c were independently associated with a greater difference (≤Q1) between equations. Different eGFR-stages, donor, or recipient characteristics, along with immunosuppression such as belatacept or prednisone, contribute to differences between eGFRcr and eGFRcys. These effects need to be considered in the clinical management of KTRs.

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

Blow-Up Phenomena for a Non-Newton Filtration Equation with Local Linear Boundary Dissipation

In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of the blow-up solution with arbitrary high initial energy. Meanwhile, the upper and lower estimates of the blow-up time are discussed.

Modeling of Non-Stationary Mass Transfer Process in Groundwater

Purpose. Infiltration of contaminated water and accidental spills of chemically hazardous substances into groundwater lead to the formation of large zones of man-made pollution in aquifers. Therefore, it is important to develop protection systems against groundwater pollution. To analyze the effectiveness of such protection systems at the design stage, it is necessary to have scientifically based information on the dynamics of changes in groundwater contamination zones. Such information can be obtained using the method of mathematical modeling. The study aims to create a numerical model for calculating the non-stationary process of geomigration when using chemical protection of groundwater from pollution. Methodology. To describe the dynamics of groundwater flows, two filtration equations are considered, which allow mathematical modeling of the filtration process both for solving planned problems and for solving problems of specialized filtration. A two-dimensional geomigration equation was used to analyze changes in groundwater quality. This equation takes into account the convective transfer of impurities in the filtration flow, dispersion, and the intensity of impurity infiltration into the groundwater flow. This equation is also used to calculate the movement of the neutralizer in groundwater. The numerical integration of the filtration equation was performed using finite difference methods. An implicit splitting scheme was used to numerically integrate the geomigration equation. Findings. A fast-applicable numerical model for calculating groundwater dynamics has been built. The model is also a platform for solving another important task – the calculation of geomigration processes. A numerical model for calculating the unsteady-state geomigration process is proposed, which makes it possible to assess not only the process of formation of contamination zones in the groundwater flow, but also to determine the effectiveness of the method of neutralizing the impurities in the groundwater flow. Originality. Effective numerical models for rapid assessment of changes in groundwater dynamics and quality under the influence of anthropogenic sources have been developed. These models take into account a set of important physical factors that affect the process of geomigration and the process of neutralizing the impurity in the groundwater flow. Practical value. A computer program has been developed that allows determining the effectiveness of the process of neutralizing an aggressive impurity in groundwater by a computational experiment to protect it from anthropogenic pollution.

MODELING OF GROUND WATER DYNAMICS AND ITS POLLUTION

Problem statement. Large accumulators of liquid waste (e.g., mine water ponds, tailing ponds, etc.) are long-term sources that change the hydrological regime. A negative consequence of this process is flooding of the territory. In addition, the infiltration of contaminated water from such hazardous sources changes the quality of groundwater. Therefore, it is important to analyze the impact of such anthropogenic sources on the process of flooding and deterioration of groundwater quality. To solve this problem, it is very important to use the method of mathematical modeling as an effective mean of researching problems of this class, since the use of physical modeling is practically impossible within the scope of problems of this class. The purpose of the article. Development of numerical models for predicting changes in the hydrological regime (flooding of the territory) and groundwater quality under the influence of anthropogenic pollution sources. Methodology. To assess the dynamics of changes in the hydrological regime, a two-dimensional equation of filtration of a non-pressure groundwater flow is used. A two-dimensional geomigration equation (planned model) is used to analyze changes in groundwater quality during infiltration of contaminated water from the settling pond. This equation takes into account the convective transfer of contaminants in the filtration flow, dispersion, and the intensity of contaminant infiltration into the groundwater flow. The method of total approximation is used for numerical integration of the filtration equation. For the numerical integration of the geomigration equation, an implicit splitting scheme is used. Scientific novelty. Effective numerical models for rapid assessment of changes in groundwater dynamics and quality under the influence of anthropogenic sources that change the hydrological regime are proposed. The constructed numerical models take into account a set of important physical factors that affect the process of geomigration and flooding of the territory, namely: filtration coefficient, variable depth of free-flowing groundwater, dispersion, intensity of the source of impurity emission into the groundwater flow. This makes it possible to obtain a comprehensive assessment of the process of flooding and groundwater pollution.. Practical significance. A computer code has been created that allows practical usage of the developed numerical models. This code is an effective tool for theoretical study of non-stationary processes of territory flooding and anthropogenic groundwater pollution. Conclusions. A numerical model for calculating groundwater dynamics has been developed. The model allows to predict the level of groundwater rise under the influence of a man-made source of wastewater infiltration from a settling pond. A numerical model for calculating the process of geomigration from an anthropogenic source of emissions has been developed. The model makes it possible to predict the dynamics of contamination zone formation in a non-pressure groundwater flow. The developed numerical models take into account the most important parameters that affect the formation of flooding zones and groundwater contamination.

Numerical analysis of a pressure distribution field and fluid flow velocity vectors near cumulative perforation holes

Relevance. The need to determine a reliable pressure distribution field and fluid filtration velocity vectors inside the perforation hole and the reservoir rock surrounding it. Aim. On the basis of numerical finite element modeling of the fluid flow inside the perforation holes and its filtration in the surrounding reservoir rock, to reveal the patterns of pressure distribution and fluid filtration vectors in the perforation hole, on its walls and in the near-wellbore zone. Objects. Near-wellbore zone of a limestone reservoir of one of the oil fields in the south of the Perm Region, including perforations. Methods. Numerical finite element method for calculating the flow and filtration of liquid in the near-wellbore zone, taking into account the geometry of perforation holes. Results. The paper considers the main relationships used in numerical simulation of fluid flow and filtration in the ANSYS finite element modeling software package. The authors have developed the finite element scheme of the near-wellbore zone, including cumulative perforation holes and taking into account their geometric parameters, as well as the fact that inside the holes the fluid flow is modeled in open space using the Navier–Stokes equations, and in the surrounding reservoir rock based on the filtration equations and Darcy's law. Numerical calculations were carried out, on the basis of which the distribution of pressure, flow velocities and fluid filtration inside the holes and in the near-wellbore zone as a whole was obtained. Calculations were made with varying pressure in the well (or pressure drawdown), as well as for different values of reservoir permeability. The calculation results showed that for the actual drawdown values of 10 MPa and the reservoir permeability of 50 mD, the value of pressure change inside the perforation holes will not exceed 0.01 MPa, i.e. it can be assumed that it practically does not change inside the hole. It is noted that the maximum value of the filtration rate corresponds to the top of the perforation holes and then its value decreases as it approaches the borehole wall. It is concluded that in further modeling of the stress-strain state of the near-wellbore zone, taking into account the holes of cumulative perforation on the surface of the holes, it is permissible to set a constant pressure value equal to the pressure in the well, and not logarithmic or any other distribution of it.

Scale up validation tests for Buckingham Pi theorem applied to leaf test experiments for iron ore slurry

Bench tests were conducted on ore slurry filtration using the leaf test method, varying scales, pressures, and water/solids ratios. These results were compared with analytical outcomes derived from applying the dimensionless Buckingham pi method to the general equation of slurry filtration with incompressible cakes under constant pressure. The aim was to validate the tests by corroborating experimental findings with calculated parameters for scale-up purposes. Vacuum pressure was employed to facilitate slurry dewatering, forming a cake comprising solids from the slurry. This cake’s thickness evolved over time, introducing specific resistivity that impeded filtering efficiency. Parameters such as cake resistance and filter media resistance were computed. Filtrate volume was measured over time to determine the filtration rate. These values were then adjusted using factors obtained through dimensional analysis. These factors facilitated comparisons to validate the feasibility of employing the Buckingham pi method for predicting slurry filtration results at larger scales. The results indicated lower errors when analyzing the filtrate rate over time with a solids/water ratio of 20/80%. When evaluating cake resistance, the method yielded lower errors at a ratio of 70/30%, while results for medium resistance were inconclusive. It’s important to consider the ratio and associated errors when predicting the behavior of an industrial filter based on bench test results, aiming to achieve theoretical scaled filtration rates and cake resistances. However, it’s not advisable to rely on the Buckingham pi method for scale-up purposes when evaluating medium resistance.

Comparison of the mathematical modelling results of the relationship between cerebral ventricular size and capillary pressure based on experimental and clinical data

Aim of the study was to compare the results of mathematical modelling of the dependence between brain ventricle size and capillary pressure for humans and animals based on the equations of multicomponent poroelastic filtration for brain parenchyma. Material and methods. The study included two groups of animals - 4 male mice of each inbred line C57Bl/6 and BALB/C at the age of 12 weeks – and 4 healthy volunteers. The brain and cerebrospinal fluid system images of mice were obtained using an 11.7 T horizontal MR scanner, group of humans were examined using the Ingenia 3.0 T MRI scanner. An axial section at the level of –0.5 mm from bregma in the mouse groups and a frontal slice at the level of the middle of the bodies of the lateral and third ventricles, posterior to the foramen of Monroe in the human group were chosen as the geometry for mathematical modelling. Mathematical modelling is based on the stationary mathematical model of multicomponent poroelastic filtration. Multiple linear regression of mean ventricular wall displacement on fluid media interaction parameters was constructed to compare results obtained. Regression coefficients were compared using nonparametric analysis of variance based on the Kraskell–Wallis criterion and post-hoc Dunn’s criterion with Hill’s correction Results. A qualitative coincidence in the behavior of capillary pressure and mean ventricular wall displacement was demonstrated for the human and mouse groups. No significant differences were found between the two animal lines. For the animals characterized by small ventricular size (BALB/c), greater similarity to humans is observed than for the genetic line with hypertrophied ventricles (C57Bl/6). A significant difference between humans and mice is observed only for capillary-venous interaction. Conclusions. The low variance within groups and insignificant discrepancy between groups indicate the possibility of further accumulation of empirical data to establish correction coefficients of the animal model, which will bring it more in line with the model for humans. Thus, the analyzed models are sufficiently comparable with each other.

ОСОБЛИВОСТІ РОБОТИ НАПІРНИХ ПОХИЛЬНИХ ЗБІРНИХ ДРЕНАЖНИХ ТРУБОПРОВОДІВ ЗА НАЯВНОСТІ ПОХИЛУ РІВНЯ ҐРУНТОВИХ ВОД

The development of a reliable methodology for calculating drainage pipelines in reclamation systems is crucial for en-hancing agricultural production efficiency and optimizing the use of water and land resources.Many existing calculation methods do not consider the simultaneous influence of pipe slope and groundwater level slope on the characteristics and operating conditions of collecting drainage pipelines. Addressing this issue will contribute to the advancement of modern hydraulic engineering and hydromelioration.The purpose of the article is to develop a methodology for calculating the parameters of pressure collecting drainage pipelines in reclamation systems, which are laid with a slope and operate in the presence of a groundwater level slope.A system of differential equations describing the liquid motion in collecting drainage pipelines laid with a slope and operating with a certain groundwater level slope is presented in the article. This system includes a variable mass hydraulicsequation and a modified filtration equation. The original system is reduced to a dimensionless form by introducing new vari-ables. The solution to the equations system results in straightforward and easy-to-use analytical dependencies for calculating the main hydraulic and structural characteristics of precast drainage pipes.The analysis employs the concept of an infinitely long drainage pipeline operating with a groundwater level slope, equivalent to an inclined pipeline with an infinitely filtering capacity of the side surface walls. The impact of drainage pipe geometric slope and groundwater level on its design charac-teristics is evaluated.The obtained dependencies for calculating collecting drainage pipelines will enhance the efficiency and reliability of drainage systems, improve resistance to extreme weather conditions, reduce energy consumption and enhance adaptation to various operating conditions

Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content

Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content

STUDY ON THE INITIAL BOUNDARY VALUE PROBLEM FOR AFRACTIONAL DIFFERENTIAL EQUATION WITH A FRACTIONALDERIVATIVE OF VARIABLE ORDER

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O(τ2+hk+1) for the case α(x)ϵ(0,1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

One of the approaches to modeling the process of blood distribution in soft tissues of living organisms

An approach to building a mathematical model of pressure distribution in soft tissues of living organisms is proposed. Soft tissues are modeled by a porous medium that has the shape of a hollow cylinder of a certain length and radius. The movement of the fluid contained in the porous medium is described by the filtration equation undergiven boundary conditions. A numerical experiment was conducted, the results of which qualitatively coincide with real processes.

Effect of Race-Free Estimated Glomerular Filtration (eGFR) Equation on CKD Prevalence in the US Military Health System (MHS)

Effect of Race-Free Estimated Glomerular Filtration (eGFR) Equation on CKD Prevalence in the US Military Health System (MHS)

Study on filtration equations based on filtrate viscosity

ABSTRACT Classical filtration equations incorporate microscopic parameters, which pose challenges when applied in industrial settings. Previous extensive research has established the impact of filter cake thickness and variations in filtration pressure on the filtration rate. Moreover, the thickness of the filter cake is influenced by various factors, such as the filtration chamber’s design dimensions and the coal slurry’s properties. Thus, by expressing the properties of the coal slurry at a macroscopic level, it becomes possible to achieve a macroscopic representation of the filtration rate. This study presents the derivation of a macroscopic filtration equation that relies on the viscosity of the filtrate. The derivation process is mainly based on Darcy’s law, analyzing the relationship between the permeability of each layer and the permeability of cake under constant pressure and explaining and analyzing the method of determining the equation parameters.

An Axi-Symmetric Problem of Suspensions Filtering with the Formation of a Cake Layer

In this paper, we consider a vertically positioned cylindrical filtering element. Filtering occurs in the radial direction, therefore, the direction of the velocities of the liquid and suspended particles coincide with this radial direction. The flow can be considered to be one-dimensional and radially axisymmetric. To describe such a filtering process, the axisymmetric Stefan problem will be formulated. The radial mass balance formalism and Darcy’s law are utilized to obtain a basic equation for cake filtration. The boundary condition at the moving surface is derived and the cake filtration is formulated in a Stefan problem. Equations are derived that describe the dynamics of cake growth in the cake filtration, and they are numerically solved. The influence of different model parameters on the compression and fluid pressure across the cake and the growth of its thickness are studied.

Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation

ABSTRACT Let d ∈ {1, 2, 3, . . .} and Ω ⊂ ℝ d be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem P { u t | x | 4 + Δ ( | Δ u | m − 2 Δ u ) = k ( t ) | u | p − 1 u ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = ∂ x j u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , j ∈ { 1 , … , d } , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where T &gt; 0, m ∈ [2, ∞), p ∈ (1, ∞) and 0 ≠ u 0 ∈ W 0 2 , m ( Ω ) ∩ L p + 1 ( Ω ) $0 \ne u_0 \in W^{2,m}_0(\Omega) \cap L^{p+1}(\Omega)$ . We investigate the upper and lower bounds on the blow-up time of a weak solution to (P).

Fines migration during coal bed methane production: mathematical and laboratory modelling, field cases

Fines migration in coalbed methane (CBM) fields comprises a serious environmental and gas-production challenge. The literature widely reports two kinds of fines: potential coal fines, which are a part of the coal body and can be detached by breakage under a significant drag force exerted from the inflowing water, and detrital coal fines, which are attached to the coal body by electrostatic forces. The theory for detrital coal fines migration is well developed. A theory for potential coal fines, where the drag deforms the coal asperities and detaches fines by rock failure, is not available. The objectives of this study are (1) to derive the governing equations for fines generation by breakage using failure criteria, and (2) to predict well productivity during dewatering and gas production using laboratory-based modelling. The micro-model developed is based on beam theory and comprises static rock deformation by the flow-through water and calculating failure criteria by tensile and shear stresses. The failure condition determines the number of fines that detach after the application of each flow rate, allowing determining the maximum retention function of potential coal fines. The breakage micro-model is incorporated into filtration equations that account for fines mobilisation, migration, straining and consequent permeability decline. Eight series of lab flooding data with coal cores have been treated. The close match between the lab and model validates the model developed. The model allows predicting productivity decline due to permeability reduction by fines breakage and straining.

Calculation of distribution drainage pipelines laid with a slope

Quite simple and easy-to-use analytical dependencies for calculation of the main hydraulic and structural characteristics of distribution drainage pipelines, laid with a certain slope, are presented. They are proposed on the basis of the analysis of the differential equations system, which consists of the variable mass hydraulics equation and the modified filtration equation, and describes the fluid motion in such pipelines. The solutions of the original equations system are obtained under the assumption of neglecting the term that takes into account the effect of the flow rate variation along the path. The analysis uses the concept of an infinitely long distribution drainage pipeline, which is laid with a slope, or, what is the same, an inclined pipeline with an infinite filtration capacity of the side surface walls. Series of calculations of the main pipelines characteristics at different slope values were carried out according to the proposed formulas. Corresponding graphic dependencies were constructed for clarity. It is shown that the geometric slope of the pressure distribution drainage pipeline, the resistance factor and the generalized parameter A significantly affects the design parameters of such pipes.