Implicit Finite Difference Research Articles

Numerical Modeling of Two-Phase Fluid Filtration for Carbonate Reservoir in Two-Dimensional Formulation

This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation.

Алгоритм для численного решения диффузионно-реакционно-дрейфового уравнения с дробной производной по времени и координате

<p>The paper is devoted to the construction and program implementation of the computational algorithm for modeling a process of diffusion-drift nature based on the fractional diffusion approach. The mathematical model is formulated as an initial-boundary value problem for the time-space fractional diffusion-drift equation in a limited domain. Time and space fractional derivatives are considered in the sense of Caputo and Riemann – Liouville, respectively. A modified implicit finite-difference scheme is constructed. The concept of the considered mathematical problem provides an example of a deterministic model of the charging process of dielectric materials. An application program has been developed that implements the constructed numerical algorithm. The results were verified using the example of solving a test problem.</p>

Transient temperature and pressure coupling model of cementing injection stage in ultra deep wells considering segmental rheological model, casing eccentricity and U-tube effect

The geological environments of ultra deep wells such as high temperature and high pressure (HTHP), narrow safety density window pose significant challenges to cementing operation. Considering segmental rheological model and density model, casing eccentricity, U-tube effect and viscous dissipation, this paper established a transient temperature and pressure coupling model during cementing injection stage of ultra deep wells. The segmental rheological model can describe the rheological properties of cementing fluids under HTHP much better. Considering casing eccentricity and viscous dissipation can accurately model the variation of temperature and pressure profiles. Taking U-tube effect into the model, the formation process of vacuum section at top of the casing can be simulated. The fully implicit finite-difference approach was used to solve the coupling model, and the calculation results were compared with field measurement data and simulation data to verify its accuracy. The calculation errors of the developed model are respectively 10% and 5% compared to field measurement data and Wellplan’s simulation data. Numerical simulations were conducted to reveal the cementing fluid distribution, transient temperature distribution and pressure distribution during the cementing injection stage. The simulation results show that the variety of cementing fluids with different density, rheological properties and thermophysical parameters complicates the change mechanism of temperature and pressure. The temperature and pressure distributions both present unsteady characteristics during cementing injection stage, which puts forward higher requirements of pressure control for narrow safety density window formation. The influence of temperature and pressure on rheology of cementing fluids cannot be ignored, and segmental rheological model should be considered for accurate pressure prediction. Also the casing eccentricity cannot be ignored for cementing pressure calculation in deviated or horizontal wells. Managed pressure cementing (MPC) has a greater advantage in dealing with narrow density window formation. The synergistic control of cementing fluid density and back pressure can keep the equivalent circulating density (ECD) within the safe density window. Through the application of the transient temperature and pressure coupling model, we can simulate the variations of key dynamic parameters during cementing injection stage and optimize the cementing parameters.

Non-similar approach for enhanced heat and mass transfer in nanofluid using Keller box algorithm

The nanofluids provide various benefits over pure fluids in heat and mass transport applications; hence, their research is crucial. For instance, they can increase heat transfer rate by enhancing the fluid’s thermal conductivity and may enhance mass transfer rate by changing the surface characteristics. Furthermore, nanofluids are being demonstrated to effectively diminish pressure drops in exchangers for heat, which can lower energy consumption and operating expenses. In the existing literature, the majority of the theoretical studies considered self-similar flows. However, there are certain actual flow situations that do not allow for a self-similar solution. The current study considers such of those situations where the non-similarity of the transport phenomena is unavoidable. The non-similarity of the present problem is caused by the consideration of thermophoretic diffusion or the contribution of viscous dissipation when the wall temperature follows a power-law form. For a pure fluid, the same problem admits a self-similar solution in the absence of viscous dissipation effects. In this problem, the non-similarity is caused by the nature of the thermal transport process and not because of the momentum transport. Therefore, the consideration of viscous dissipation in the boundary layer of nanofluid is an interesting aspect to explore the behavior of thermal and mass transport phenomena. Moreover, the current analysis intends to investigate the transport enhancement in a non-similar flow of a nanofluid by utilizing the Buongiorno model. In the current nonsimilar modeling, possibilities for the existence of a self-similar solution are also highlighted. An implicit finite-difference numerical scheme, the Keller-Box method, is utilized. The problem involves several physical parameters of interest, such as the Eckert number, Lewis number, Brownian motion parameter, and thermophoresis parameter, whose potential impact on the non-similar nature of the problem and on thermal enhancement is analyzed and quantified.

Solution of the plane thermoelasticity problems based on a new set of equations

The interaction of thermal and mechanical fields in modern technologies for the synthesis and processing of materials attracts more and more attention. This is due to the need to predict the behavior and properties of new materials and products depending on the conditions of their creation. In the present work, attention is focused on the comparison of the solutions to the thermoelasticity plane problems with and without accounting for the temperature dependence of the material properties. It is assumed that an external heat source traveling over the surface with a specific velocity and following a given trajectory is responsible for a significant temperature increase. It may correspond to a laser source, for example. The new way of problem formulation in stresses is used. The temperature dependence of the characteristics leads to nonlinear thermoelasticity equations when the equilibrium equations, the requirement of strain compatibility, and the Duhamel-Neuman relation for the plane stress state case are combined. To determine the temperature field, a nonlinear thermal conductivity equation is used. To solve the problem numerically, the use of an implicit finite-difference scheme and coordinate splitting for the thermal conductivity equation and the finite-difference method of successive over-relaxation with Chebyshev acceleration for the equations for stresses was made. As a result, the difference in the solution of linear and nonlinear problems is demonstrated for the considered cases. Using the example of a mixture of titanium and aluminum powders, numerical modeling demonstrated that some stress components can reach values up to 3.5 times higher than those obtained in the case of temperature-independent material properties when there is a 500-degree local temperature increase.

Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, thenumerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.

Numerical modeling of reaction–diffusion e-epidemic dynamics

The objective of this paper is to understand the dynamics of virus spread in a computer network by e-epidemic reaction–diffusion model and applying an implicit finite difference (FD) scheme for a numerical solution. The SIR models are used in studies of epidemiology to predict the behavior of the propagation of biological viruses within the population. We divide the population of computer nodes into three parts i.e. susceptible (may catch the virus), infected 1 (infected but not completely), and infected 2 (completely infected). By using Taylor’s series expansion the consistency of the implicit scheme is proved. The unconditional stability of the implicit FD model is proved by using the Von Numan stability analysis. The qualitative analysis of the underlying model is also analyzed such as positivity and boundedness of the model. The numerical stability and bifurcation of is also analyzed. Likewise, identical modeling techniques are adopted to analyze the spread of the virus in digital networks. Because the computer virus behaves in the identical way as the biological virus behaves, this paper emphasizes the significance of diffusion in decreasing the gap between reality and theory, offering a more precise depiction of the spread of the virus within the digital network.

Computation of hydromagnetic tangent hyperbolic non-Newtonian flow from a rotating non-isothermal cone to a non-Darcy porous medium with thermal radiative flux

A theoretical and numerical study is conducted on nonlinear, steady-state thermal convection boundary layer flow of a magnetized incompressible Tangent Hyperbolic non-Newtonian fluid from a rotating cone to a non-Darcy porous medium. Power-law variation in temperature on the cone surface is considered and thermal radiation heat transfer is also present. The Brinkman-Darcy-Forchheimer model is deployed for the porous medium. The study is motivated by rotational (spin) coating with new emerging magnetic rheological polymers, a process which often utilizes filtration media and high temperatures. The transformed non-dimensional conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. Extensive visualization of axial, tangential velocity components and temperature distributions with variation in key parameters including Rosseland radiative number, Darcy number, Forchheimer number (non-Darcy inertial parameter), magnetic interaction parameter, tangent-hyperbolic non-Newtonian power-law index and Weissenberg (non-Newtonian) number is included. Additionally, axial and tangential (circumferential) skin friction and Nusselt number values are tabulated for variation in key control parameters. With increasing Weissenberg number, axial velocity is depleted near the cone surface, tangential velocity is suppressed throughout the boundary layer regime and temperature is strongly enhanced. Axial flow is strongly decelerated further from the cone surface with increasing tangent-hyperbolic power-law index and there is also a significant depletion in tangential (swirl) velocity. Temperature is however boosted throughout the boundary layer transverse to the cone surface with a rise in tangent-hyperbolic power-law index. Both tangential and axial velocity are suppressed with increment in magnetic interaction parameter whereas temperature and thermal boundary layer thickness are enhanced. With larger Darcy number (i.e. greater permeability), axial velocity is strongly increased near the cone surface with no tangible modification further from the cone; However tangential velocity is consistently elevated throughout the boundary layer with greater Darcy number whereas temperature is depleted. An increase in Forchheimer number substantially damps both axial and tangential velocity whereas it elevates temperature. Increasing radiative flux strongly energizes the magnetic polymer and elevates temperature but suppresses the axial and tangential velocities. With elevation in non-isothermal wall exponent, axial skin friction is suppressed whereas Nusselt number are elevated at the cone vertex. Further along the cone surface a similar response is observed but there is also a reduction in magnitudes of tangential skin friction.

Influence of Nanofluids on Boundary Layer Flow over an Inclined Stretching Sheet in a Porous Media along with Magnetic Field

The current study intends to analyse the magnetohydrodynamics boundary layer flow of a water-based nanofluid of Silver, Copper and Ferrous Ferric Oxide nanoparticles over a Permeable inclined stretched sheet in a porous media. A similarity transformation is utilized to convert the governing partial differential equations into non-dimensional, non-linear ordinary differential equations. The Keller box finite difference implicit approach is then applied to solve these nonlinear equations numerically. The impacts of several parameters, namely, inclination parameter, magnetic parameter, Soret Number, Eckertnumber and nano particles volume fractionon velocity, temperature and nanoparticle concentration are explored.

Simplified implicit finite-difference method of spatial derivative using explicit schemes with optimized constant coefficients based on L1 norm

The explicit finite-difference (EFD) method is widely used in numerical simulation of seismic wave propagation to approximate spatial derivatives. However, the traditional and optimized high-order EFD methods suffer from the saturation effect, which seriously restricts the improvement of numerical accuracy. In contrast, the implicit FD (IFD) method approximates the spatial derivatives in the form of rational functions and thus can obtain much higher numerical accuracy with relatively low orders; however, its computational cost is expensive due to the need to invert a multidiagonal matrix. We derive an explicit strategy for the IFD method to reduce the computational cost by constructing the IFD method with the discrete Fourier matrix; then, we transform the inversion of the multidiagonal matrix into an explicit matrix multiplication; next, we construct an objective function based on the [Formula: see text] norm to reduce approximation error of the IFD method. This explicit strategy of the IFD method can avoid inverting the multidiagonal matrix, thus improving the computational efficiency. This constant coefficient optimization method reduces the approximation error in the medium-wavenumber range at the cost of tolerable deviation (smaller than 0.0001) in the low-wavenumber range. For the 2D Marmousi model, the root-mean-square error of the numerical results obtained by this method is one-fifth that of the traditional IFD method with the same order (i.e., 5/3) and one-third that of the traditional EFD method with much higher orders (i.e., 72). The significant reduction of numerical error makes the developed method promising for numerical simulation in large-scale models, especially for long-time simulations.

Mixed Convection Boundary Layer Flow over a Horizontal Circular Cylinder in AL2O3-Ag/Water Hybrid Nanofluid with Viscous Dissipation

This paper investigated the mathematical modelling for mixed convection boundary layer flow over a horizontal circular cylinder in hybrid nanofluid with viscous dissipation. The transformed partial differential equations (PDEs) are numerically solved using an implicit finite-difference approach known as the Keller-box method. The numerical solutions for the reduced Nusselt number, , local skin friction coefficient, , temperature profile, and velocity profiles are found and graphically presented in detail. Effects of the Eckert number, Richardson number and nanoparticle volume fraction are all examined and explained. It is found that the increase of volume fraction of nano material in nanofluid has increased the value of skin friction coefficient. The low density of nano oxides such as alumina in hybrid nanofluids also contribute to reduce friction between fluid and body surface. Based on numerical analysis, the combination of nanoparticles in the form of hybrid nanofluid may reduce skin friction phenomena while sustaining heat transfer characteristics comparable to nanofluid. The results in this paper are original and will assist researchers working in the field of boundary layer flow. It can also be utilised as a reference in experimental studies to reduce operating costs.

Two-dimensional mathematical model of heat and moisture transfer processes during the natural storage and drying of heterogeneous agricultural products

Developed a two-dimensional mathematical model of simultaneous heat and moisture transfer processes during the natural storage and drying of heterogeneous agricultural products in open areas. The mathematical model includes time-space variability of temperature and humidity, internal heat and moisture release, heat and moisture exchange with the environment, and also solar radiation. Based on an implicit finitedifference scheme with the second order in time and space variables, we developed an effective numerical solution for solving the problem. For studying and predicting the processes of heat and moisture transfer, software was created and experiments were done on raw cotton, making it possible to identify temperature and humidity changes at arbitrary points on the highhumidity raw cotton. Keywords: mathematical model, moisture transfer, heat transfer, heterogeneous porous media, finite-difference scheme, cotton

Pinning-depinning transition of droplets on inclined substrates with a three-dimensional topographical defect.

Droplets on inclined substrates can depin and slide freely above a critical substrate inclination angle. Pinning can be caused by topographical defects on the substrate, and understanding the influence of defect geometry on the pinning-depinning transition is important for diverse applications such as fog harvesting, droplet-based microfluidic devices, self-cleaning surfaces, and inkjet printing. Here, we develop a lubrication-theory-based model to investigate the motion of droplets on inclined substrates with a single three-dimensional Gaussian-shaped defect that can be in the form of a bump or a dent. A precursor-film/disjoining-pressure approach is used to capture contact-line motion, and a nonlinear evolution equation is derived which describes droplet thickness as a function of the position along the substrate and time. The evolution equation is solved numerically using an alternating direction implicit finite-difference scheme to study how the defect geometry influences the critical inclination angle and the shape of a pinned droplet. It is found that the critical substrate inclination angle increases as the defect becomes taller/deeper or wider along the direction lateral to the droplet-sliding direction. However, the critical inclination angle decreases as the defect becomes wider along the sliding direction. Below the critical inclination angle, the advancing contact line of the droplet at the droplet centerline is pinned to the defect at the point having maximum negative slope. Simple scaling relations that reflect the influence of defect geometry on the droplet retention force arising from surface tension are able to account for many of the trends observed in the numerical simulations.

Impact of chemical reaction, viscous dissipation and thermal radiation on MHD flow past an oscillating plate

This study analyzes the consequences of first-order chemical reactions, radiation, and viscous dissipation on the unsteady magnetohydrodynamic natural convective flow of a viscous incompressible fluid over a vertically positioned semi-boundless oscillating plate with uniform mass diffusion and temperature. An implicit finite-difference technique is employed to solve a set of dimensionless, coupled, nonlinear partial differential equations. The numerical results for fluid velocity, concentration, and temperature at the plate under different dimensionless parameters are graphically displayed. Due to first-order homogeneous chemical reactions, it has been discovered that the velocity rises at the time of a generative reaction and drops during a destructive reaction. A decline in velocity is observed with an increase in the phase angle, radiation parameter, and chemical reaction parameter. Further, it has been revealed that plate oscillation, radiation, chemical reactions, and the magnetic field significantly influence the flow behavior.

О математической модели взаимодействия высокотемпературного потока плазмы с поверхностью древесины

In this study, the processes of thermal decomposition of wood during its treatment with a plasma flow are considered. To develop a mathematical model of these processes, a differential thermogravimetric analysis of the heating of larch samples in an argon atmosphere at a rate of 10–20 degrees/min is carried out. Based on the results obtained, a mathematical model of thermal decomposition is proposed, including four stages. The wood during heating is represented by a mixture of six components. At each stage, the kinetic parameters of the reactions are determined by processing measurements at a heating rate of 20 degrees/min. The equations of chemical kinetics describing changes in the mass of wood components are numerically solved using the finite-difference implicit Euler method and the obtained reaction parameters. The numerical solution to the equations of chemical kinetics with these parameters shows satisfactory agreement with the data from the corresponding experiment. The calculation performed at a heating rate of 10 degrees/min with the same kinetic parameters also shows satisfactory agreement with the measurements. Thus, the obtained reaction parameters do not depend on the heating rate in the considered range. The proposed model can be used in the mathematical description of changes in the structure and thermal state of wood samples exposed to high-temperature plasma flow treatment.

Optimizing solar energy for wood drying under various climates: A comparative study of flat plate and photovoltaic thermal solar collectors

Wood drying is an important operation in many sectors, and harnessing solar energy for this purpose has received a lot of interest due to its sustainability and cost-effectiveness. In this work, we compare how two solar collector configurations namely, Flat Plate Collector and Photovoltaic Thermal collector, perform in a solar wood drying system. The moisture content of wood was reduced from 73% to 12% in four different regions: Errachidia, Meknes, Tangier, and Rabat. The Gauss–Seidel method is used to solve the governing equations for heat transfer after discretizing them via implicit finite differences. The effectiveness of the solar collectors was simulated through the TRNSYS software, and a numerical program was generated using the Python programming language. According to our research, the FPC beats the PVT collector because of its higher output temperatures dramatically shorten the time it takes for wood to dry. The study emphasizes how temperature and humidity affect drying, showing that faster drying occurs in hotter, drier locations. Furthermore, drying time can be saved by using FPC and PVT collectors compared to conventional open sun drying.The primary conclusions of this research are: In the FPC solar dryer, the drying time needed in Errachidia, Meknes, Rabat and Tangier was 144 h, 162 h and 171 h, respectively. While in the PVT solar dryer, it required 162 h in Errachidia, 171 h in Meknes, 180 h in Rabat and 189 h in Tangier. Further, the FPC collectors showed a range of 293 K to 347 K for the outlet temperature. The PVT collectors, however, showed a variable range between 295.9 K and 320 K. The drying periods and temperature ranges examined revealed that Errachidia was the most advantageous location, regardless of whether FPC or PVT collectors were used. In fact, the drying time that can be saved by using FPC is 41% while PVT is 31% compared to traditional open sun drying.

Constructions of the soliton solutions to coupled nonlinear Schrödinger equation with advanced mathematical techniques

In our research paper, we explore the application of mathematical techniques, both analytical and numerical, to solve the coupled nonlinear Schrödinger equation. To obtain accurate solutions, we use the improved, modified, extended tanh-function method. By breaking down the Schrödinger equation into real and imaginary components, we derive four interconnected equations. We analyze these equations using the generalized tanh method to find precise solutions. This set of equations is of great importance in quantum mechanics and helps us understand the behavior of quantum systems. We provide an analytical and numerical solution using the implicit finite difference. Our method is second-order in both space and time, and we have verified its stability through von Neumann’s stability analysis.

Three-Dimensional Boundary-Layer Flow Across A Wedge: A Numerical Investigation

The present work investigates the flow of a continuous, laminar, 3-D boundary-layer over a fixed wedge, where the outside free-stream flows are approximately approximated by a power of distances. The controlling partial differential equations are transformed into coupled nonlinear ordinary differential equations with the necessary boundary conditions using surface similarity transformations. These equations involve two physical variables: pressure gradient and shear-to-strain rate. The resulting equations are numerically solved by the implicit finite-difference scheme known as the Keller-box approach. The obtained results are compared with those reported in the literature for a few special cases. Our numerical results indicate that the flow zone is divided into two areas: near-field (close to the wedge surface) and far-field field (mostly controlled by inviscid flow).

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions.

Forced convective flows of Casson hybrid nanofluid and entropy production over diverging channel with variable thermophysical properties

The heat transfer occurring on boundary layer flows during dilute suspension of Cu–Al2O3 nanoparticles-based non-Newtonian Casson hybrid nanofluid over diverging channel is to be characterized in this model. The governing equations comprise continuity, momentum, energy and concentration equations which incorporated variable viscosity and magnetic effects in the momentum equation, variable thermal conductivity, chemical reaction, thermal radiative heat flux, uniform magnetic field, Joule heating and viscous dissipative effects in the energy equation and Brownian motion and thermophoretic effects in the concentration equation. The modeled equations are nondimensionalized using nonsimilar variables and linearized using quasilinearization technique. The system of linearized partial differential equations is solved numerically using implicit finite difference and successively iterated with the help of Varga’s algorithm. The physical impacts of viscous dissipation parameter (Ec), Brownian motion ([Formula: see text]) on velocity, temperature, concentration and Schmidt number (Sc) influences on skin friction, Nusselt number, Sherwood number profiles, Bejan lines and total entropy production profiles are simulated graphically. The velocity profiles are enhanced and the temperature profile declines for augmented values of Eckert number. Moreover, for augmented values of Schmidt number the heat and mass transfer rates are enhanced and the Bejan lines dropped a decreasing trend whereas total entropy production is augmented near the wall region. The improved values of the Schmidt number physically increased Sc in the heat transfer rate. The ascending profile of the mass transfer rate with increased values of the viscous dissipation parameter Ec demonstrated that the graph is raised for larger values of the viscous dissipating parameter (Ec). The physical behavior of incremental mass transfer rate led to the conclusion that the relative diffusivity of nanoparticles is characterized by both nondimensional numbers Ec and Sc which are directly proportional to the viscous forces.