Leonid Pankratov - Academia.edu (original) (raw)
Papers by Leonid Pankratov
Kompʹûternye issledovaniâ i modelirovanie, Jun 1, 2023
Ключевые слова: двойная пористость, усреднение, двухфазное течение, капиллярная неравновесность, ... more Ключевые слова: двойная пористость, усреднение, двухфазное течение, капиллярная неравновесность, динамическое капиллярное давление, динамические относительные фазовые проницаемости Работа выполнена при поддержке Российского фонда фундаментальных исследований (грант № 20-01-00564).
Physics of Fluids, Feb 1, 2022
A homogenized model of non-isothermal compressible two-phase flow accompanied by dissociation of ... more A homogenized model of non-isothermal compressible two-phase flow accompanied by dissociation of a gas hydrate in a fractured porous medium is derived. The equations of the homogenized model contain non-local in time source terms corresponding to the contribution of the gas hydrate in the matrix blocks. The model of phase behavior of a gas hydrate is used that admits coexistence of the hydrate and products of its decomposition in a finite region of the phase diagram. This possible behavior is a consequence of non-linearity of the skeleton potential dependence on the depth of phase transition.
Journal of Differential Equations, Dec 1, 2021
The paper deals with homogenization of a model problem describing an immiscible compressible twop... more The paper deals with homogenization of a model problem describing an immiscible compressible twophase flow in random statistically homogeneous porous media. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
Zeitschrift für Angewandte Mathematik und Physik, Nov 29, 2006
The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion ... more The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.
Physics of Fluids, 2021
A homogenized model of incompressible two-phase flow accompanied by a gas-producing reaction in a... more A homogenized model of incompressible two-phase flow accompanied by a gas-producing reaction in a double porosity medium with a chemically active skeleton is derived. The equations of the homogenized model contain non-local in time source terms corresponding to the contribution of the gas-producing chemical reaction in the matrix blocks. The time non-locality, which manifests itself as the appearance of a time delay between the change in reactant concentrations and the reaction rate, is shown to stimulate the instability of the one-dimensional two-phase flow initiated by injection of the acid solution into the double porosity medium with chemically active matrix blocks. The instability results in the development of the self-oscillating mode of the reaction wave propagation.
arXiv (Cornell University), Nov 10, 2022
The paper is devoted to the derivation, by linearization, of simplified (fully homogenized) homog... more The paper is devoted to the derivation, by linearization, of simplified (fully homogenized) homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified dual porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified dual porosity model. We also present numerical simulations which show that the matrix-fracture exchange term based on this new linearization procedure gives a better approximation of the exact one than the corresponding exchange term obtained earlier by the authors.
HAL (Le Centre pour la Communication Scientifique Directe), 2010
Operator Theory and Its Applications, 2000
ABSTRACT
Applicable Analysis, 2020
In this paper, we consider a liquid-gas system with two components: water and hydrogen flow model... more In this paper, we consider a liquid-gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy-Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion-convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
Mathematics eJournal, 2001
We study by homogenization the various macroscopic models associated to the diffusion through a f... more We study by homogenization the various macroscopic models associated to the diffusion through a fissured media, i.e., a set of low conductivity blocks crossed by a net of highly conducting fissures. According to the fissure thickness, δe, range we obtain different models. For this, first we homogenize by seeking the limit when e, the small parameter associated to the blocks size, goes to zero and then we study the limit when δ goes to zero. In each situation we prove the convergence to the corresponding macroscopic model.
Discrete & Continuous Dynamical Systems - B, 2018
This paper presents a study of immiscible incompressible twophase flow through fractured porous m... more This paper presents a study of immiscible incompressible twophase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ε θ , where ε is the size of a typical porous block and θ > 0 is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.
The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-... more The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-phase immiscible flow in fractured-porous reservoirs when non-equilibrium phenomena occur in the matrix blocks, only. It is shown that the homogenized model can be represented as usual equations of two-phase incompressible immiscible flow, except for the addition of two source terms calculated by a solution to a local problem which is a boundary value problem for a non-equilibrium imbibition equation given in terms of the real saturation and a non-equilibrium parameter.
In this paper we give a new proof of the homogenization result for an immiscible incompressible t... more In this paper we give a new proof of the homogenization result for an immiscible incompressible two-phase flow in double porosity media obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikeli\'c (1996) and in the paper of L. M. Yeh (2006) under some restrictive assumptions. The microscopic model consists of the usual equations derived from the mass conservation laws for both fluids along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium is discontinuous. The important difference with respect to the results of the cited papers is that the global pressure function as well as the saturation are also discontinuous. This makes the init...
The paper deals with homogenization of reactive immiscible incompressible two-phase flow in doubl... more The paper deals with homogenization of reactive immiscible incompressible two-phase flow in double porosity media. The mathematical model is given by a coupled system of two-phase flow equations under isothermal condition. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat and the capillary pressure laws along with the additional source terms corresponding to the chemical reactions in the reservoir. The medium is made of two superimposed continua, a connected fracture system and an ε-periodic system of disjoint matrix blocks. We assume that the permeability of the fissures is of order one, while the permeability of the blocks is of order ε2. We derive the global behavior of the model by passing to the limit as ε→ 0 and obtain the global model of the reactive flow. It is shown that the homogenized model can be represented as the usual equations of a reactive immiscible incompressible two-phase flow except for the ad...
Nonlinear Analysis: Real World Applications, 2018
Discrete & Continuous Dynamical Systems - B, 2018
We consider complex Ginzburg-Landau (GL) type equations of the form: ∂tu = (1 + αi)∆u + R u + (1 ... more We consider complex Ginzburg-Landau (GL) type equations of the form: ∂tu = (1 + αi)∆u + R u + (1 + βi) |u| 2 u + g, where R, β, and g are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems. Bibliography: 52 titles.
Kompʹûternye issledovaniâ i modelirovanie, Jun 1, 2023
Ключевые слова: двойная пористость, усреднение, двухфазное течение, капиллярная неравновесность, ... more Ключевые слова: двойная пористость, усреднение, двухфазное течение, капиллярная неравновесность, динамическое капиллярное давление, динамические относительные фазовые проницаемости Работа выполнена при поддержке Российского фонда фундаментальных исследований (грант № 20-01-00564).
Physics of Fluids, Feb 1, 2022
A homogenized model of non-isothermal compressible two-phase flow accompanied by dissociation of ... more A homogenized model of non-isothermal compressible two-phase flow accompanied by dissociation of a gas hydrate in a fractured porous medium is derived. The equations of the homogenized model contain non-local in time source terms corresponding to the contribution of the gas hydrate in the matrix blocks. The model of phase behavior of a gas hydrate is used that admits coexistence of the hydrate and products of its decomposition in a finite region of the phase diagram. This possible behavior is a consequence of non-linearity of the skeleton potential dependence on the depth of phase transition.
Journal of Differential Equations, Dec 1, 2021
The paper deals with homogenization of a model problem describing an immiscible compressible twop... more The paper deals with homogenization of a model problem describing an immiscible compressible twophase flow in random statistically homogeneous porous media. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
Zeitschrift für Angewandte Mathematik und Physik, Nov 29, 2006
The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion ... more The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.
Physics of Fluids, 2021
A homogenized model of incompressible two-phase flow accompanied by a gas-producing reaction in a... more A homogenized model of incompressible two-phase flow accompanied by a gas-producing reaction in a double porosity medium with a chemically active skeleton is derived. The equations of the homogenized model contain non-local in time source terms corresponding to the contribution of the gas-producing chemical reaction in the matrix blocks. The time non-locality, which manifests itself as the appearance of a time delay between the change in reactant concentrations and the reaction rate, is shown to stimulate the instability of the one-dimensional two-phase flow initiated by injection of the acid solution into the double porosity medium with chemically active matrix blocks. The instability results in the development of the self-oscillating mode of the reaction wave propagation.
arXiv (Cornell University), Nov 10, 2022
The paper is devoted to the derivation, by linearization, of simplified (fully homogenized) homog... more The paper is devoted to the derivation, by linearization, of simplified (fully homogenized) homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified dual porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified dual porosity model. We also present numerical simulations which show that the matrix-fracture exchange term based on this new linearization procedure gives a better approximation of the exact one than the corresponding exchange term obtained earlier by the authors.
HAL (Le Centre pour la Communication Scientifique Directe), 2010
Operator Theory and Its Applications, 2000
ABSTRACT
Applicable Analysis, 2020
In this paper, we consider a liquid-gas system with two components: water and hydrogen flow model... more In this paper, we consider a liquid-gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy-Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion-convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence.
HAL (Le Centre pour la Communication Scientifique Directe), 2009
Mathematics eJournal, 2001
We study by homogenization the various macroscopic models associated to the diffusion through a f... more We study by homogenization the various macroscopic models associated to the diffusion through a fissured media, i.e., a set of low conductivity blocks crossed by a net of highly conducting fissures. According to the fissure thickness, δe, range we obtain different models. For this, first we homogenize by seeking the limit when e, the small parameter associated to the blocks size, goes to zero and then we study the limit when δ goes to zero. In each situation we prove the convergence to the corresponding macroscopic model.
Discrete & Continuous Dynamical Systems - B, 2018
This paper presents a study of immiscible incompressible twophase flow through fractured porous m... more This paper presents a study of immiscible incompressible twophase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ε θ , where ε is the size of a typical porous block and θ > 0 is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.
The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-... more The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-phase immiscible flow in fractured-porous reservoirs when non-equilibrium phenomena occur in the matrix blocks, only. It is shown that the homogenized model can be represented as usual equations of two-phase incompressible immiscible flow, except for the addition of two source terms calculated by a solution to a local problem which is a boundary value problem for a non-equilibrium imbibition equation given in terms of the real saturation and a non-equilibrium parameter.
In this paper we give a new proof of the homogenization result for an immiscible incompressible t... more In this paper we give a new proof of the homogenization result for an immiscible incompressible two-phase flow in double porosity media obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikeli\'c (1996) and in the paper of L. M. Yeh (2006) under some restrictive assumptions. The microscopic model consists of the usual equations derived from the mass conservation laws for both fluids along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium is discontinuous. The important difference with respect to the results of the cited papers is that the global pressure function as well as the saturation are also discontinuous. This makes the init...
The paper deals with homogenization of reactive immiscible incompressible two-phase flow in doubl... more The paper deals with homogenization of reactive immiscible incompressible two-phase flow in double porosity media. The mathematical model is given by a coupled system of two-phase flow equations under isothermal condition. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat and the capillary pressure laws along with the additional source terms corresponding to the chemical reactions in the reservoir. The medium is made of two superimposed continua, a connected fracture system and an ε-periodic system of disjoint matrix blocks. We assume that the permeability of the fissures is of order one, while the permeability of the blocks is of order ε2. We derive the global behavior of the model by passing to the limit as ε→ 0 and obtain the global model of the reactive flow. It is shown that the homogenized model can be represented as the usual equations of a reactive immiscible incompressible two-phase flow except for the ad...
Nonlinear Analysis: Real World Applications, 2018
Discrete & Continuous Dynamical Systems - B, 2018
We consider complex Ginzburg-Landau (GL) type equations of the form: ∂tu = (1 + αi)∆u + R u + (1 ... more We consider complex Ginzburg-Landau (GL) type equations of the form: ∂tu = (1 + αi)∆u + R u + (1 + βi) |u| 2 u + g, where R, β, and g are random rapidly oscillating real functions. Assuming that the random functions are ergodic and statistically homogeneous in space variables, we prove that the trajectory attractors of these systems tend to the trajectory attractors of the homogenized equations whose terms are the average of the corresponding terms of the initial systems. Bibliography: 52 titles.