Splitting in systems of PDEs for two-phase multicomponent flow in porous media

Borazjani, Sara; Roberts, A. J.; Bedrikovetsky, Pavel

doi:10.1016/j.aml.2015.09.014

Cited by 39 publications

(13 citation statements)

References 17 publications

(56 reference statements)

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“…The developed averaging approach can be applied for two‐phase suspension‐colloidal‐nanotransport in porous media (Borazjani & Bedrikovetsky, ). For particles suspended in aqueous phase and attached to the surface, the splitting method separating two‐phase multicomponent system into one‐phase solute system and one equation for phase saturation yields one‐phase solute system (, ) (Borazjani et al, ). So the averaging procedure given by equations – is valid for two‐phase suspension‐colloidal systems.…”

Section: Discussionmentioning

confidence: 99%

Exact Upscaling for Transport of Size‐Distributed Colloids

Bedrikovetsky

Osipov

Кузьмина

et al. 2019

Water Resources Research

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The article investigates one-dimensional (1-D) suspension-colloidal transport of size-distributed particles with particle attachment. A population balance approach is presented for computing the particle transport and capture by porous media. The occupied area of each attached particle is particle size-dependent. The main model assumption is the retention rate dependency of the overall vacancy concentration for all particle sizes. For the first time, we derive an exact averaging (upscaling) procedure resulting in a closed system of large-scale equations for average concentrations of suspended and retained particles and of occupied rock surface area. The resulting large-scale 3 × 3 system significantly differs from the traditional 2 × 2 deep bed filtration model. However, the traditional model becomes a particular case that corresponds to an equal occupied area for all particles. The averaging yields the appearance of two empirical suspension and site occupation functions, which govern the kinetics of particle retention and site occupation, respectively. One-dimensional flow problems for the averaged equations are essentially nonlinear. However, they allow for exact solutions. We derive novel exact solutions for three 1-D problems: continuous injection of particulate colloidal suspension, injection of colloidal suspension bank with particle-free chase drive, and fines migration induced by high-rate flows. The analytical model for continuous injection closely matches three series of laboratory tests on nanofluid transport.

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Section: Discussionmentioning

confidence: 99%

Exact Upscaling for Transport of Size‐Distributed Colloids

Bedrikovetsky

Osipov

Кузьмина

et al. 2019

Water Resources Research

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“…Remark 1. Exact solutions of nonlinear diffusion and wave type PDEs can be found, for example, in [4,5,9,10,[13][14][15][16][17][18][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44].…”

Section: Concept Of 'Exact Solution' For Nonlinear Pdesmentioning

confidence: 99%

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Aksenov

Polyanin

2021

Mathematics

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This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to `ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

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“…The splitting technique can also be applied to EOR problems considering advective transport, parabolic terms and relaxation nonequilibrium equations. In cases where the auxiliary system allows the development of an analytical solution, the complete exact solution can be constructed …”

Section: Introductionmentioning

confidence: 99%

“…In cases where the auxiliary system allows the development of an analytical solution, the complete exact solution can be constructed. 13 Borazjani et al 14 presented an analytical solution for a nonselfsimilar, two-phase, one dimensional problem of displacement of oil by a polymer slug with changing salinity, showing that the low salinity front moves faster than the polymer due to adsorption. It was also presented the solution to the injection of a polymer slug in a low salinity system displaced by a low salinity or high salinity water drive.…”

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confidence: 99%

Water slug injection containingnpolymers in porous media

2019

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Injection of water containing dissolved chemicals is an efficient oil recovery technique. One of the problems of this method is the loss of the chemical components due to interactions between rock and fluid. In polymer injection, adsorption may happen and lead to low process efficiency. The interaction between rock and fluid is governed by the adsorption isotherm, which relates the polymer concentration in water with the adsorbed amount on the rock. In this paper the problem of oil displacement by a water slug containing n chemical components that may be adsorbed is analyzed. The system of conservation laws is solved and the structure of the solution for the case of Henry´s adsorption isotherm is completely described. The concentration profile of each component and the chromatographic cycle is calculated through simple expressions. The complete and detailed solution for the case of slug injection containing three chemical components is presented. The general solution developed can be used to model several Enhanced Oil Recovery techniques, in which the chemical components adsorb in porous media following Henry's adsorption isotherm. K E Y W O R D S chemical enhanced oil recovery, conservation laws, enhanced oil recovery, hyperbolic systems of partial differential equations, polymer flooding 1 | INTRODUCTION Different techniques can be employed to improve recovery in oil fields. Water injection is the most used and the one-dimensional mathematical problem was solved analytically. 1 It was considered immiscible and incompressible phases (oil and water). Adding polymer to the injection water reduces water mobility and modifies the fractional flow curve, hence increasing the sweep efficiency. Chemical enhanced oil recovery (EOR) has been applied in onshore and offshore petroleum fields. The most used chemical components dissolved in the injected water are polymer, surfactants, and alkalis. This technique recovers part of the remaining oil mainly due to a favorable mobility ratio change. One of the first offshore chemical EOR projects took place at West Bay and Quarantine Bay Field, in Louisiana shallow waters, in 1981. Later, other chemical EOR projects were applied in Cuadras

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Splitting in systems of PDEs for two-phase multicomponent flow in porous media

Cited by 39 publications

References 17 publications

Exact Upscaling for Transport of Size‐Distributed Colloids

Exact Upscaling for Transport of Size‐Distributed Colloids

Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions

Water slug injection containingnpolymers in porous media

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