scholarly journals Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

2020 ◽  
Vol 25 (4) ◽  
pp. 74
Author(s):  
Fernando Alcántara-López ◽  
Carlos Fuentes ◽  
Fernando Brambila-Paz ◽  
Jesús López-Estrada
Keyword(s):  
Porous Media ◽  
Fractional Derivatives ◽  
Single Phase ◽  
Time Derivative ◽  
Transient Behavior ◽  
Matrix Permeability ◽  
The Matrix ◽  
Fractional Time Derivative ◽  
Porosity Model ◽  
Infinite Reservoir

The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered in a full-penetrating wellbore. Furthermore, the model relaxes classic hypotheses considering that matrix permeability has a significant impact on the pressure deficit from the wellbore, reaching the triple permeability and triple porosity model wich allows the wellbore to be fed by all the porous media and not exclusively by the fractured system; where it is considered a pseudostable interporous flow. In addition, it is considered the anomalous flow phenomenon from the pressure of each independent porous medium and as a whole, through the temporal fractional derivative of Caputo type; the resulting phenomenon is studied for orders in the fractional derivatives in (0, 2), known as superdiffusive and subdiffusive phenomena. Synthetic results highlight the effect of anomalous flows throughout the entire transient behavior considering a significant permeability in the matrix and it is contrasted with the effect of an almost negligible matrix permeability. The model is solved analytically in the Laplace space, incorporating the Tartaglia–Cardano equations.

Application of Fractional Derivatives for Simulating Diffusion Into Porous Matrix in Mathematical Modeling of the Contaminant Transport in a Confined Fractured Porous Aquifer

2006 ◽  
Author(s):  
Sergei Fomin ◽  
Vladimir Chugunov ◽  
Toshiyuki Hashida
Keyword(s):  
Solute Transport ◽  
Contaminant Transport ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Porous Matrix ◽  
Confined Aquifer ◽  
Closed Form Solutions ◽  
Advection Dispersion ◽  
Porous Aquifer ◽  
Fractional Time Derivative

Solute transport in the fractured porous confined aquifer is modeled by the advection-dispersion equation with fractional time derivative of order γ, which may vary from 0 to 1. Accounting for diffusion in the surrounding rock mass leads to the introduction of an additional fractional time derivative of order 1/2 in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties are modeled and analyzed.

Application of Scaling and Quasistatic Methods to Study Nonlinear Subdiffusion-Reaction Equations With Fractional Time Derivative

2009 ◽  
Author(s):  
Tadeusz Kosztolowicz ◽  
Katarzyna D. Lewandowska
Keyword(s):  
Reaction Front ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Time Derivative ◽  
Diffusion Reaction ◽  
Natural Order ◽  
Diffusion Region ◽  
Scaling Method ◽  
Fractional Time Derivative ◽  
Reaction Equations

We consider a subdiffusive system where transported particles of spieces A and B chemically react according to the formula A + B → 0̸. This process is described by the nonlinear subdiffusion-reaction equations with fractional time derivatives. We show that the scaling method, which is commonly used to study diffusion-reaction equations of natural order, is not applicable to the subdiffusion case due to the specific properties of fractional derivatives, unless very special assumptions are taken into account. Contrary to the scaling method, the quasistatic one provides the explicite solutions in the diffusion region and the time evolution of reaction front xf, which reads xf = Ktα/2, where α is the subdiffusion parameter and K is uniquely determined. We also present the numerical solutions of subdiffusion-reaction equations and show that the numerical results coincide with the analytical ones.

scholarly journals Non-local porous media equations with fractional time derivative

2021 ◽  
Vol 211 ◽  
pp. 112486
Author(s):  
Esther Daus ◽  
Maria Pia Gualdani ◽  
Jingjing Xu ◽  
Nicola Zamponi ◽  
Xinyu Zhang
Keyword(s):  
Porous Media ◽  
Time Derivative ◽  
Porous Media Equations ◽  
Non Local ◽  
Fractional Time Derivative

scholarly journals Critical parameters for reaction–diffusion equations involving space–time fractional derivatives

2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Sunday A. Asogwa ◽  
Mohammud Foondun ◽  
Jebessa B. Mijena ◽  
Erkan Nane
Keyword(s):  
Fractional Derivatives ◽  
Time Derivative ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Critical Parameters ◽  
Diffusion Type ◽  
Making Sense ◽  
Whole Space ◽  
Fractional Time Derivative

Abstract We will look at reaction–diffusion type equations of the following type, $$\begin{aligned} \partial ^\beta _tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }]. \end{aligned}$$ ∂ t β V ( t , x ) = - ( - Δ ) α / 2 V ( t , x ) + I t 1 - β [ V ( t , x ) 1 + η ] . We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $$0<\eta \leqslant \eta _c$$ 0 < η ⩽ η c , there is no global solution other than the trivial one while for $$\eta >\eta _c$$ η > η c , non-trivial global solutions do exist. The critical parameter $$\eta _c$$ η c is shown to be $$\frac{1}{\eta ^*}$$ 1 η ∗ where $$\begin{aligned} \eta ^*:=\sup _{a>0}\left\{ \sup _{t\in (0,\,\infty ),x\in \mathbb {R}^d}t^a\int _{\mathbb {R}^d}G(t,\,x-y)V_0(y)\,\mathrm{d}y<\infty \right\} \end{aligned}$$ η ∗ : = sup a > 0 sup t ∈ ( 0 , ∞ ) , x ∈ R d t a ∫ R d G ( t , x - y ) V 0 ( y ) d y < ∞ and $$G(t,\,x)$$ G ( t , x ) is the heat kernel of the corresponding unforced operator. $$V_0$$ V 0 is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.

Improvements in Simulation of Naturally Fractured Reservoirs

1983 ◽  
Vol 23 (04) ◽  
pp. 695-707 ◽  
Author(s):  
James R. Gilman ◽  
Hossein Kazemi
Keyword(s):  
Fluid Flow ◽  
Single Phase ◽  
Reservoir Rock ◽  
Storage Volume ◽  
Naturally Fractured ◽  
Matrix Blocks ◽  
The Matrix ◽  
Porosity Model ◽  
Made In ◽  
The Continuum

Abstract Simulation of multiphase flow in heterogeneous two-porosity reservoirs such as naturally fractured systems is a difficult problem. In the last several years much progress has been made in this area. This paper focuses on progress has been made in this area. This paper focuses on the practical aspects of that technology. It describes a stable, flexible, fully implicit, finite-difference simulator in heterogeneous, two-porosity reservoirs. Flow rates and wellbore pressures are solved simultaneously along with fracture and matrix fluid saturations and pressures at all grid points. Hydrodynamic pressure gradient is maintained at formation perforations in the wellbore. The simulator is accurate enough to match analytical solutions to single-phase problems. The equations have been extended to include polymer flooding and tracer transport with nine-point connection for determining severe local channeling and directional tendencies. It is shown that the two-porosity model presented in this paper will produce essentially the same answers as the paper will produce essentially the same answers as the common single-porosity model of a highly heterogeneous system but with a substantial reduction of computing time. In addition, this paper describes in detail several two-porosity parameters not fully discussed in previous publications. previous publications. Introduction Naturally fractured reservoir simulators are developed to simulate fluid flow in systems in which fractures are interconnected and provide the main flow path to injection and production wells. The fractures have high permeability and low storage volume, the reservoir rock permeability and low storage volume, the reservoir rock (matrix blocks) has low permeability and high storage volume. The idealization of assuming one porosity as the continuum can apply to many heterogeneous systems where one porosity provides the main path for fluid flow and the other porosity acts as a source. Throughout this paper, the fracture should be thought of as the continuum paper, the fracture should be thought of as the continuum and the matrix perceived as the adjacent sources or sinks. For single-phase flow of a gas or liquid, fluid compression and viscous forces control fluid movement. Gravity and capillary forces are not pertinent. Several single-phase idealizations that produce essentially the same practical engineering answers are discussed in the literature. Fig. 1 shows a model with both vertical and horizontal fractures. Separate nodes are used for fracture and matrix. For this case, 77 nodes are used to model the system. Fig. 2 also shows a model that allows vertical and horizontal fractures. However, this model requires far fewer nodes because areas in which the matrix blocks behave similarly are grouped in a single node. Each gridblock may contain many matrix blocks. The matrix blocks act as sources that feed into the fractures in a gridblock. The fractures can be thought of as a system of connected pipes. This model was proposed by Warren and Root. The boundary conditions used can make a dramatic difference in simulation results. Generally, we assume that only the fractures produce into the wellbore and are the path of fluid flow from one gridblock to the next. For multiphase flow, three forces must be properly accounted for--viscous, gravity, and capillary. In this case, we might require that the matrix blocks be further divided into grid blocks to obtain better definition of saturation distribution (Fig. 3). However, this will lead to additional work and may not be required. Kazemi et al. extended the Warren-Root model to multiphase systems to account for capillary and gravity forces. SPEJ P. 695

scholarly journals Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Nor Hafizah Zainal ◽  
Adem Kılıçman
Keyword(s):  
Fourier Series ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fourier Series Method ◽  
Fractional Time Derivative

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

Compositional Modeling of Discrete-Fractured Media Without Transfer Functions by the Discontinuous Galerkin and Mixed Methods

SPE Journal ◽  
2006 ◽  
Vol 11 (03) ◽  
pp. 341-352 ◽  
Author(s):  
Hussein Hoteit ◽  
Abbas Firoozabadi
Keyword(s):  
Porous Media ◽  
Discontinuous Galerkin ◽  
Numerical Approach ◽  
Fractured Media ◽  
Fracture Model ◽  
Discrete Fracture Model ◽  
Compositional Modeling ◽  
Discrete Fracture ◽  
The Matrix ◽  
Porosity Model

Summary In a recent work, we introduced a numerical approach that combines the mixed-finite-element (MFE) and the discontinuous Galerkin (DG) methods for compositional modeling in homogeneous and heterogeneous porous media. In this work, we extend our numerical approach to 2D fractured media. We use the discrete-fracture model (crossflow equilibrium) to approximate the two-phase flow with mass transfer in fractured media. The discrete-fracture model is numerically superior to the single-porosity model and overcomes limitations of the dual-porosity model including the use of a shape factor. The MFE method is used to solve the pressure equation where the concept of total velocity is invoked. The DG method associated with a slope limiter is used to approximate the species-balance equations. The cell-based finite-volume schemes that are adapted to a discrete-fracture model have deficiency in computing the fracture/fracture fluxes across three and higher intersecting-fracture branches. In our work, the problem is solved definitively because of the MFE formulation. Several numerical examples in fractured media are presented to demonstrate the superiority of our approach to the classical finite-difference method. Introduction Compositional modeling in fractured media has broad applications in CO2, nitrogen, and hydrocarbon-gas injection, and recycling in gas condensate reservoirs. In addition to species transfer, the compressibility effects should be also considered for such applications. Heterogeneities and fractures add complexity to the fluid-flow modeling. Several conceptually different models have been proposed in the literature for the simulation of flow and transport in fractured porous media. The single-porosity approach uses an explicit computational representation for fractures (Ghorayeb and Firoozabadi 2000; Rivière et al. 2000). It allows the geological parameters to vary sharply between the matrix and the fractures. However, the high contrast and different length scales in the matrix and fractures make the approach unpractical because of the ill conditionality of the matrix appearing in the numerical computations (Ghorayeb and Firoozabadi 2000).The small control volumes in the fracture grids also add a severe restriction on the timestep size because of the Courant-Freidricks-Levy (CFL) condition if an explicit temporal scheme is used.

Influence of Heat Treatments on Microstructures in an Fe-Co-V Alloy

1974 ◽  
Vol 32 ◽  
pp. 512-513
Author(s):  
S. Mahajan ◽  
M. R. Pinnel ◽  
J. E. Bennett
Keyword(s):  
Single Phase ◽  
Alloy Composition ◽  
Supersaturated Solid Solution ◽  
Cell Formation ◽  
Heat Treatments ◽  
Air Cooling ◽  
Iced Brine ◽  
Transmission Electron ◽  
The Matrix ◽  
Bcc Structure

The microstructural changes in an Fe-Co-V alloy (composition by wt.%: 2.97 V, 48.70 Co, 47.34 Fe and balance impurities, such as C, P and Ni) resulting from different heat treatments have been evaluated by optical metallography and transmission electron microscopy. Results indicate that, on air cooling or quenching into iced-brine from the high temperature single phase ϒ (fcc) field, vanadium can be retained in a supersaturated solid solution (α2) which has bcc structure. For the range of cooling rates employed, a portion of the material appears to undergo the γ-α2 transformation massively and the remainder martensitically. Figure 1 shows dislocation topology in a region that may have transformed martensitically. Dislocations are homogeneously distributed throughout the matrix, and there is no evidence for cell formation. The majority of the dislocations project along the projections of <111> vectors onto the (111) plane, implying that they are predominantly of screw character.

Analysis of process of emulsions transport in hydrophilic/oleophilic granular porous media driven by capillary force

2018 ◽  
Vol 2 (21) ◽  
pp. 85-101
Author(s):  
Olga Shtyka ◽  
Łukasz Przybysz ◽  
Mariola Błaszczyk ◽  
Jerzy P. Sęk
Keyword(s):  
Porous Media ◽  
Experimental Research ◽  
Granular Media ◽  
Single Phase ◽  
Influential Factor ◽  
Dispersed Phase ◽  
Porous Structures ◽  
Phase Concentration ◽  
Granular Porous Media ◽  
Kinetics Of

The research focuses on the issues concerning a process of multiphase liquids transport in granular porous media driven by the capillary pressure. The current publication is meant to introduce the results of experimental research conducted to evaluate the kinetics of the imbibition and emulsions behavior inside the porous structures. Moreover, the influence of the dispersed phase concentration and granular media structure on the mentioned process was considered. The medium imbibition with emulsifier-stabilized emulsions composed of oil as the dispersed phase in concentrations of 10 vol%, 30 vol%, and 50 vol%, was investigated. The porous media consisted of oleophilic/hydrophilic beads with a fraction of 200–300 and 600–800 μm. The experimental results provided that the emulsions imbibition in such media depended stronger on its structure compare to single-phase liquids. The increase of the dispersed phase concentration caused an insignificant mass decreasing of the imbibed emulsions and height of its penetration in a sorptive medium. The concentrations of the imbibed dispersions exceeded their initial values, but reduced with permeants front raise in the granular structures that can be defined as the influential factor for wicking process kinetics.

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