arXiv subjects

Akif Ibragimov

Publications and source records attributed to Akif Ibragimov.

At least 19 recordsLinked to original sources

arXiv2026

Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport

We develop a nonlinear continuum transport model describing the formation of localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation in which displacement is generated by diffusion and by the gradient of a background-dependent transport mechanism. The resulting system couples diffusion, nonlinear gradient-driven transport, and depletion of the background phase. The proposed framework is applicable to general chemotactic transport and, in particular, to problems related to the formation of oil and gas deposits. We analyze traveling-wave solutions and establish the existence of coherent traveling bands in the transport-dominated regime. The mobile phase is shown to form a unique one-hump profile for any given reference time, while the background component undergoes a positive, bounded monotone logistic-type transition between asymptotic states. An explicit representation of the traveling profile is obtained, and uniqueness is proved up to translation. We further derive the linearized perturbation operator around the traveling band and establish finite-time perturbation bounds through a maximum-principle argument. Finally, the traveling-wave system is reduced to a nonlinear third-order ordinary differential equation for the background profile, providing an alternative characterization of the coherent structure.

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arXiv2026

Linear non-divergence parabolic equations in non-cylindrical space-time sets

We study linear parabolic equations of the second order in non-divergence form in a general set which is non-cylindrical with respect to the spatial and time variables. The restriction of the set on any bounded time interval is bounded, but the spatial diameter of each fixed-time cross section can be unbounded as time tends to infinity. For homogeneous problems, we obtain exponential, power and other intermediate decaying rates for the solutions in different scenarios. For inhomogeneous problems, we obtain all-time and asymptotic, as time tend to infinity, estimates for the solutions in terms of the data on the parabolic boundary and forcing functions. The analysis requires subtle properties of general space-time sets and an iteration scheme to bootstrap the Growth Lemma. The time steps for such an iteration need not be constant and are adapted to the growth of the diameter.

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arXiv2026

Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain

We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function.

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arXiv2026

Asymptotic estimates for solutions of inhomogeneous non-divergence diffusion equations with drifts

We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition. Starting with the reduced linear problem, we obtain the asymptotic estimates for the solutions, as time $t\to\infty$, depending on the asymptotic behavior of the forcing term and boundary data. These are established in both cases when the drifts are uniformly bounded, and unbounded as $t\to\infty$. For the nonlinear problem, we prove the convergence of the solutions under suitable conditions that balance the growth of the nonlinear term with the decay of the data. To take advantage of the diffusion in the non-divergence form, we prove an inhomogeneous version of the Landis-typed Growth Lemma and apply it to successive time-intervals. At each time step, the center for the barrier function is selected carefully to optimize the contracting factor. Our rigorous results show the robustness of the model.

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arXiv2025

A class of anisotropic diffusion-transport equations in non-divergence form

We generalize Einstein's probabilistic method for the Brownian motion to study compressible fluids in porous media. The multi-dimensional case is considered with general probability distribution functions. By relating the expected displacement per unit time with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form that contains a transport term. Under the Darcy law assumption, a corresponding nonlinear partial differential equations for the density function is obtained. The classical solutions of this equation are studied, and the maximum and strong maximum principles are established. We also obtain exponential decay estimates for the solutions for all time, and particularly, their exponential convergence as time tends to infinity. Our analysis uses some transformations of the Bernstein-Cole--Hopf type which are explicitly constructed even for very general equation of state. Moreover, the Lemma of Growth in time is proved and utilized in order to achieve the above decaying estimates.

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arXiv2023

Models of Chemotactic System by Einstein's Brownian Motion Method and its Analysis

We study the movement of the living organism in a band form towards the presence of chemical substrates based on a system of partial differential evolution equations. We incorporate Einstein's method of Brownian motion to deduce the chemotactic model exhibiting a traveling band. It is the first time that Einstein's method has been used to motivate equations describing the mutual interaction of the chemotactic system. We have shown that in the presence of limited and unlimited substrate, traveling bands are achievable and it has been explained accordingly. We also study the stability of the constant steady states for the system. The linearized system about a constant steady state is obtained under the mixed Dirichlet and Neumann boundary conditions. We are able to find explicit conditions for linear instability. The linear stability is established with respect to the L-2 norm, H1-norm, and L-infinity norm under certain conditions.

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arXiv2022

Einstein model of the movement of small particles in a stationary liquid revisited: Finite Propagation Speed

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality and the Ladyzhenskaya-Uraltseva iteration lemma.

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arXiv2022

Fundamentals in Peaceman Model for Well-Block radius For Non-Linear Flows Near Well

We consider sewing machinery between finite difference and analytical solutions defined at different scale: far away and near source of the perturbation of the flow. One of the essences of the approach is that coarse problem and boundary value problem in the proxy of the source model two different flows. We are proposing method to glue solution via total fluxes, which is predefined on coarse grid. It is important to mention that the coarse solution "does not see" boundary. From industrial point of view our report provide mathematical tool for analytical interpretation of simulated data for fluid flow around a well in a porous medium. It can be considered as a mathematical "shirt" on famous Peaceman well-block radius formula for linear (Darcy) radial flow but can be applied in much more general scenario. As an important case, we consider nonlinear Forchheimer flow. In the article we rigorously obtained well-block radius, explicitly depending on $β-$Forchheimer factor and total rate of the flow on the well, and provide generalization of the Dake Formula and evaluation of the $D-$factor.

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arXiv2022

Einstein's Brownian motion model for chemotactic system and traveling band

We study the movement of the living organism in a band form towards the presence of chemical substrate based on a system of partial differential evolution equations. We incorporate the Einstein's method of Brownian motion to deduce the chemotactic model exhibiting travelling band. It is the first time that Einstein method has been used to motivate equations describing mutual interaction of chemotactic system. In addition to considering chemotactic response and the random motion of organism, we also consider the formation of crowd by organism via interactions within or between the community. This crowd effect can also be seen as any organism travel or migrate in a herd or group in search of food. We have shown that in the presence of limited and unlimited substrate traveling bands are achievable and it has been explained accordingly.

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arXiv2019

Geometric model of the fracture as a manifold immersed in porous media

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness, on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in $\mathbb{R}^3$. Using operators of Laplace Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional BVP. We then couple the model with the porous media. Theoretical and numerical analysis have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close, and therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

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arXiv2017

Fracture Model Reduction and Optimization for Forchheimer Flows in Reservoir

In this study, we analyze the flow filtration process of slightly compressible fluids in fractured porous media. We model the coupled fractured porous media system, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. Flow in the fracture is modeled as a reduced low dimensional BVP which is coupled with an equation in the reservoir. We prove that the solution of the reduced model can serve very accurately to approximate the solution of the actual high-dimensional flow in reservoir fracture system, because the thickness of the fracture is small. In the analysis we consider two types of Forchhemer flows in the fracture: isotropic and anisotropic, which are different in their nature. Using method of reduction, we developed a formulation for an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and beta-Forchheimer coefficient.

math.AP↗

arXiv2016

Fluid Flows of Mixed Regimes in Porous Media

In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.

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arXiv2015

Productivity Index for Darcy and pre-/post-Darcy Flow (Analytical Approach)

We investigate the impact of nonlinearity of high and low velocity flows on the well productivity index (PI). Experimental data shows the departure from the linear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells and fractures is described by Forchheimer equations and is relatively well-studied. While low velocity flow receives much less attention, there is multiple evidence suggesting the existence of pre-Darcy effects for slow flows far away from the well. This flow is modeled via pre-Darcy equation. We combine all three flow regimes, pre-Darcy, Darcy and post-Darcy, under one mathematical formulation dependent on the critical transitional velocities. This allows to use our previously developed framework to obtain the analytical formulas for the PI for the cylindrical reservoir. We study the impact of pre-Darcy effect on the PI of steady-state flow depending on the well-flux and the parameters of the equations.

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arXiv2015

Well Productivity Index for Compressible Fluids and Gases

In this paper we discuss the notion of the diffusive capacity for the generalized Forchheimer flow of fluid through porous media. The diffusive capacity is an integral characteristic of the flow motivated by the engineering notion of the productivity index (PI), Dake 1983, Raghavan 1993, Christopher et al. 2014. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. We study its time dynamics for two types of fluids: slightly compressible and strongly compressible fluid (ideal gas). In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, Aulisa et al. 2009, 2011, 2012. Here we generalize our results from Aulisa et al. 2012 on long term dynamics of the PI in case of arbitrary order of the nonlinearity of the flow. In this paper we study the mathematical model of the PI for compressible gas flow for the first time. In contrast to slightly compressible fluid this functional mathematically speaking is not time-invariant. At the same time it stays "almost" constant for a long period of time, but then it rapidly blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The "greater" are the initial reserves, the larger is this critical value. We present numerical and analytical results for the time asymptotic of the PI and its stability with respect to the initial data. Using comparison theorems for porous media equation from Vázquez 2007 we obtain estimates between the PI's for the original gas flow and auxiliary flow with a distributed source. The latter one generates the time independent PI, and can be calculated using formula similar to one in case of slightly compressible fluid.

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arXiv2014

The expanded mixed finite element method for generalized Forchheimer flows in porous media

We study the expanded mixed finite element method applied to degenerate parabolic equations with the Dirichlet boundary condition. The equation is considered a prototype of the nonlinear Forchheimer equation, a inverted to the nonlinear Darcy equation with permeability coefficient depending on pressure gradient, for slightly compressible fluid flow in porous media. The bounds for the solutions are established. In both continuous and discrete time procedures, utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions we prove the optimal error estimates in $L^2$-norm for solution. The error bounds are established for the solution and divergence of the vector variable in Lebesgue norms and Sobolev norms under some additional regularity assumptions. A numerical example using the lowest order Raviart-Thomas ($RT_0$) mixed element are provided agreement with our theoretical analysis.

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arXiv2013

A family of steady two-phase generalized Forchheimer flows and their linear stability analysis

We model multi-dimensional two-phase flows of incompressible fluids in porous media using generalized Forchheimer equations and the capillary pressure. Firstly, we find a family of steady state solutions whose saturation and pressure are radially symmetric and velocities are rotation-invariant. Their properties are investigated based on relations between the capillary pressure, each phase's relative permeability and Forchheimer polynomial. Secondly, we analyze the linear stability of those steady states. The linearized system is derived and reduced to a parabolic equation for the saturation. This equation has a special structure depending on the steady states which we exploit to prove two new forms of the lemma of growth of Landis-type in both bounded and unbounded domains. Using these lemmas, qualitative properties of the solution of the linearized equation are studied in details. In bounded domains, we show that the solution decays exponentially in time. In unbounded domains, in addition to their stability, the solution decays to zero as the spatial variables tend to infinity. The Bernstein

math.AP↗

arXiv2013

Upscaling of Nonlinear Forchheimer Flows

In this work we propose upscaling method for nonlinear Forchheimer flow in highly heterogeneous porous media. The generalized Forchheimer law is considered for incompressible and slightly-compressible single-phase flows. We use recently developed analytical results Aulisa et al. (2009) and write the resulting system in terms of a degenerate nonlinear flow equation for the pressure with the nonlinearity that depends on the pressure gradient. The coarse scale parameters for the steady state problem are determined so that the volumetric average of velocity of the flow in the domain on fine scale and on coarse scale are close enough. A flow-based coarsening approach is used, where the equivalent permeability tensor is first evaluated following the streamline of the existing linear cases, and successively modified in order to take into account the nonlinear effects. Compared to previous works Durlofsky and Karimi-Fard (2009) and Peszynska et al. (2009), our approach relies on recent analytical results of Aulisa et al. (2009) and combines it with rigorous mathematical upscaling theory for monotone operators. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities in the domain of computation. Direct numerical computations for average velocity and productivity index justify the usage of the coarse scale parameters obtained for the special steady state case in the fully transient problem. Analytical upscaling formulas in stratified domain are obtained for the nonlinear case. They correlate with high accuracy with numerical results.

math.NA↗

arXiv2013

Geometric framework for modeling nonlinear flows in porous media, and its applications in engineering

This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an important tool for evaluating technological parameters in reservoir engineering.

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