scholarly journals Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

2021 ◽  
Vol 6 (1) ◽  
pp. 21
Author(s):  
Valentina Anna Lia Salomoni ◽  
Nico De Marchi
Keyword(s):  
Heavy Tails ◽  
Field Experiments ◽  
Heterogeneous Media ◽  
Numerical Solutions ◽  
Solute Concentration ◽  
Finite Difference Approximation ◽  
Diffusion Reaction ◽  
Difference Approximation ◽  
Solute Flux ◽  
Advection Dispersion

Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

General Deformations of Neo-Hookean Membranes

1973 ◽  
Vol 40 (1) ◽  
pp. 7-12 ◽  
Author(s):  
W. H. Yang ◽  
C. H. Lu
Keyword(s):  
Differential Equations ◽  
Strain Energy ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Special Cases ◽  
Convergent Algorithm

A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

2017 ◽  
Vol 27 (8) ◽  
pp. 1814-1850 ◽  
Author(s):  
Sapna Pandit ◽  
Manoj Kumar ◽  
R.N. Mohapatra ◽  
Ali Saleh Alshomrani
Keyword(s):  
Nonlinear Term ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Finite Difference Approximation ◽  
Haar Wavelets ◽  
Difference Approximation ◽  
Content Type ◽  
Gauss Elimination ◽  
Gauss Elimination Method ◽  
Better Than

Purpose This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave. Design/methodology/approach First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method. Findings Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions. Originality/value The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

Analytical-Numerical Method in Waterflooding Predictions

1965 ◽  
Vol 5 (03) ◽  
pp. 247-258 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Finite Difference ◽  
Exact Solutions ◽  
Numerical Method ◽  
Analytical Method ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Scale Factor ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation

Abstract Methods of predicting the influence of pattern geometry and mobility ratio on waterflooding recovery predictions are discussed. Two methods of calculation are used separately or concurrently. The analytical method yields exact solutions in a convenient form for a unit mobility ratio piston-like displacement. A few typical pressure distributions, sweep efficiencies and oil recoveries are presented for various patterns. For non-unit mobility ratio, one may resort to a numerical method, such as that of Sheldon and Dougherty. Because the domains of applicability of the analytical and numerical techniques overlap, the exact solutions provide estimates of the errors in the numerical procedures. The advantages of the analytical and numerical methods can be combined. To develop a numerical technique as independent of geometry as possible, the physical space is transformed into a standard rectangle. The entire effect of geometry is rendered through one term, the "scale-factor", derived from mapping relations. The scale factor can be calculated from the exact unit-mobility ratio solution for the particular pattern of interest. By this means recovery performances for arbitrary mobility ratio can be obtained for many patterns. A sample of results obtained in this manner is presented. Introduction Pattern geometry and mobility ratio are two major factors in making a waterflood recovery prediction. Because assisted recovery has become increasingly important to the oil industry, pattern configuration and mobility ratio also assume a greater significance in the assessment of the economic value of recovery projects. The influence of pattern geometry and mobility ratio in shaping a recovery curve and on the other quantities of interest to the reservoir engineer is the main subject of this paper. Much effort has already been spent on estimating quantitatively the influence of either pattern or mobility ratio or both on oil recovery. The literature reports many investigations of this nature. However, many results or methods of recovery prediction presented in the literature cannot be considered fully satisfactory. Even for unit mobility ratio and piston-like displacement, where analytical solutions are available, the literature shows discrepancies. For non-unit mobility ratio, the divergence in the results is extreme. For infinite mobility ratio in a repeated five-spot, depending on the investigator, the sweep efficiency ranges from 0 per cent to 60 per cent. With respect to the influence of pattern on recovery, only the repeated five-spot has received much attention. Other confined patterns and pilot configurations have received very little attention. Two calculation methods are presented in this paper, either separately or concurrently: the analytical method of potential theory and the numerical method of finite-difference approximation. The analytical method is more restricted in scope than the finite-difference method, but it has the definite advantage of providing exact solutions within its range of applicability. If a unit-mobility ratio piston-like displacement is assumed, the analytical approach is possible. A few typical results are reported in this paper; the detailed description of the general method and of a great variety of results will be the subject of other articles. For non-unity mobility ratio, we must resort to a numerical scheme. The numerical technique is that which was described by Sheldon and Dougherty. It is not limited to piston-like displacement. However, mainly single interface results will be presented here. Because the respective domains of applicability of the analytical and the numerical method overlap, useful comparisons of exact and numerical solutions can be made for a variety of patterns. The advantages of the analytical and numerical approaches can be combined. SPEJ P. 247ˆ

scholarly journals Time-space dependent fractional viscoelastic MHD fluid flow and heat transfer over accelerating plate with slip boundary

2017 ◽  
Vol 21 (6 Part A) ◽  
pp. 2337-2345
Author(s):  
Shengting Chen ◽  
Liancun Zheng ◽  
Chunrui Li ◽  
Jize Sui
Keyword(s):  
Heat Transfer ◽  
Numerical Solutions ◽  
Fluid Model ◽  
Mhd Flow ◽  
Constitutive Relationship ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Slip Boundary ◽  
Flow And Heat Transfer ◽  
Time Space

The MHD flow and heat transfer of viscoelastic fluid over an accelerating plate with slip boundary are investigated. Different from most classical works, a modified time-space dependent fractional Maxwell fluid model is proposed in depicting the constitutive relationship of the fluid. Numerical solutions are obtained by explicit finite difference approximation and exact solutions are also presented for the limiting cases in integral and series forms. Furthermore, the effects of parameters on the flow and heat transfer behavior are analyzed and discussed in detail.

scholarly journals On the numerical solutions for a parabolic system with blow-up

2021 ◽  
Vol 6 (11) ◽  
pp. 11749-11777
Author(s):  
Chien-Hong Cho ◽  
◽  
Ying-Jung Lu ◽  
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Numerical Solution ◽  
Parabolic System ◽  
Numerical Solutions ◽  
Blow Up ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Blow Up Rate ◽  
Axisymmetric Solutions

<abstract><p>We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.</p></abstract>

Determination of Main-Term and Cross-Term Gas Diffusivities in Heavy Oil Systems Considering Local Oil Swelling Effect

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Hyun Woong Jang ◽  
Daoyong Yang
Keyword(s):  
Liquid Phase ◽  
Heavy Oil ◽  
Molecular Diffusion ◽  
Numerical Solutions ◽  
The Other ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Dissolved Gas ◽  
Cross Term ◽  
The Individual

Abstract To inject gas into a heavy oil reservoir, molecular diffusion of the dissolved gas into heavy oil is one of the crucial mechanisms to lower its viscosity while swelling the diluted oil. Various efforts have been made to predict the diffusivity of such gas dissolved in heavy oil with or without considering the oil swelling. Practically, the oil swelling is always considered in an excessively simplified manner so that such swelling is not able to exhibit its true effect on the estimated diffusivity. In most studies where the oil swelling is considered, the liquid-phase hydrocarbon is assumed to swell equally at every location because the height of liquid-phase in a diffusion vessel is simply extended proportionally to the oil swelling direction. Such a proportional swell is often realized during numerical solutions by uniformly extending the numerical cells, regardless of the amount of dissolved gas contained in each of them. In addition, no studies have been made to examine the contribution of one gas over the other for a gas mixture-liquid system. In this study, a pragmatic approach is proposed to determine the main- and cross-term diffusivities of gas–liquid systems considering local swelling effect. More specifically, diffusivities of CO2 and a CO2–C3H8 mixture in a Lloydminster heavy oil are respectively estimated by implementing the finite difference approximation (FDA) with the face-centered explicit scheme. For the CO2–C3H8 mixture, the individual diffusivity of each gas in the mixture is firstly computed independent of the other gas in the mixture. Then, the cross-term diffusivity is included to verify the effect of the other gas in heavy oil for the diffusion of one gas, while the local oil swelling is implemented during the estimation of the individual gas diffusivities. It is found that the obtained diffusivities of pure CO2 and each individual component of the CO2–C3H8 mixture in the Lloydminster heavy oil are reasonable and accurate to reproduce the measured oil swelling factors obtained from the dynamic volume analysis (DVA) tests.

Sign in / Sign up

Export Citation Format

Share Document