Akif Ibragimov

Analysis of generalized Forchheimer flows of compressible fluids in porous media

This work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by Darcy’s law. We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws. The generalized Forchheimer equation is inverted to a nonlinear Darcy equation with implicit permeability tensor depending on the pressure gradient. This results in a degenerate parabolic equation for the pressure. Two classes of boundary conditions are considered, given pressure and given total flux. In both cases they are allowed to be unbounded in time. The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed. The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters. Some numerical simulations are also included.

Structural stability of generalized Forchheimer equations for compressible fluids in porous media

We study the generalized Forchheimer equations for slightly compressible fluids in porous media. The structural stability is established with respect to either the boundary data or the coefficients of the Forchheimer polynomials. A weighted Poincare–Sobolev inequality related to the nonlinearity of the equation is used to study the asymptotic behaviour of the solutions. Moreover, we prove a perturbed monotonicity property of the vector field associated with the resulting non-Darcy equation, where the correction is explicit and Lipschitz continuous in the coefficients of the Forchheimer polynomials.

On a mathematical model of the productivity index of a well from reservoir engineering

Motivated by the reservoir engineering concept of the productivity index of a produc- ing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann, or third boundary condition) is investigated using a known variational principle and confirmed numer- ically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and unique- ness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.

Generation of a chemical gradient across an array of 256 cell cultures in a single chip

A microfluidic diffusion diluter to create stable chemical gradients across an array of cell cultures was demonstrated. The device enabled concentration based studies to be conducted at 256 different concentrations across individual, low shear cell cultures. A gradient of staurosporine on cells stained with Mitotracker Deep Red (MTDR) showed a concentration-based effect on cell apoptosis across the cell culture array.

A New Method for Evaluating the Productivity Index of Nonlinear Flows

Summary This paper addresses the effects of nonlinearity of flows on the value of the productivity index (PI) of the well. Experimental data show that, during the dynamic process of hydrocarbon recovery, the PI stabilizes to some constant value, which, in general, is a nonlinear function of both the pressure drawdown and the production rate. Linear Darcy flow is well understood, and excellent approximate formulas are available for the PI in various well/reservoir geometries. To handle the more realistic nonlinear situation, the current practice is to solve the nonlinear problem multiple times for different values of production rate and then add ad-hoc corrective parameters in the linear formulas to reproduce the nonlinear nature of the flow. In this paper, we propose a rigorous framework to measure the PI of a well for nonlinear Forchheimer flows. Our approach, based on recent progress in the modeling of transient Forchheimer flows, uses both analytical and numerical techniques. It provides, for a wide class of reservoir geometries, an accurate relation between the PI for nonlinear Forchheimer flows and the PI for linear Darcy flows. The proposed method of building look-up tables and analytical formulas serves as an effective tool for fast PI evaluation in nonlinear cases.

Long-term dynamics for well productivity index for nonlinear flows in porous media

Motivated by the reservoir engineering concept of the well productivity index (PI) we study a time dependent functional for general nonlinear Forchheimer equation. The PI of the well characterizes the well capacity with respect to drainage area of the well. Unlike the linear case for which this concept is well developed, there are only a few recent publications dedicated to the PI for nonlinear case. In this paper the PI is comprehensively studied both theoretically and numerically. The impact of the nonlinearity of the flow filtration on the value of the PI is analyzed. Exact formula for the so called “skin factor” in radial case is derived depending on the rate of the flow, the order of nonlinearity and the geometric parameters. Dynamics of the PI for the class of boundary conditions is studied and its convergence to the specific value of steady state PI was justified. Developed framework is applied to obtain nonlinear analog of Peaceman formula for the well-block pressure in unstructured grid. Numerica...

Stability analysis of a model of atherogenesis: an energy estimate approach II

Atherosclerosis is a disease of the vasculature that is characterized by chronic inflammation and the accumulation of lipids and apoptotic cells in the walls of large arteries. This disease results in plaque growth in an infected artery typically leading to occlusion of the artery. Atherosclerosis is the leading cause of human mortality in the US, much of Europe, and parts of Asia. In a previous work, we introduced a mathematical model of the biochemical aspects of the disease, in particular the inflammatory response of macrophages in the presence of chemoattractants and modified low density lipoproteins. Herein, we consider the onset of a lesion as resulting from an instability in an equilibrium configuration of cells and chemical species. We derive an appropriate norm by taking an energy estimate approach and present stability criteria. A bio-physical analysis of the mathematical results is presented.

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. First, 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. Second, 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...

STABILITY ANALYSIS OF A REACTION-DIFFUSION SYSTEM MODELING ATHEROGENESIS ∗

This paper presents a linear, asymptotic stability analysis for a reaction-diffusion-convection system modeling atherogenesis, the initiation of atherosclerosis, as an inflammatory instability. Motivated by the disease paradigm articulated by Ross, atherogenesis is viewed as an inflammatory spiral with a positive feedback loop involving key cellular and chemical species interacting and reacting within the intimal layer of muscular arteries. The inflammatory spiral is initiated as an instability from a healthy state which is defined to be an equilibrium state devoid of certain key inflammatory markers. Disease initiation is studied through a linear, asymptotic stability analysis of a healthy equilibrium state. Various theorems are proved, giving conditions on system parameters guaranteeing stability of the health state, and a general framework is developed for constructing perturbations from a healthy state that exhibit blow-up, which are interpreted as corresponding to disease initiation. The analysis rev...

Pre-Darcy flow revisited under experimental investigation

BackgroundSufficient literature has been published about Pre-Darcy flow in non-petroleum disciplines. Investigators dissent about the significance of deviation of Darcy’s law at very low fluid velocities. Most of their investigations are based on coarse, unconsolidated porous media with an aqueous fluid. However little has been published regarding the same for consolidated oil and gas reservoirs. If a significant departure from Darcy’s law is observed, then this could have multiple implications on: reservoir limit tests, under prediction of reserves, unrecognized prospecting opportunities etc.MethodsThis study performs a comprehensive review of the literature. Experiments were conducted to confirm the presence and the significance of Pre-Darcy flow effect in petroleum rocks.ResultsThe review of literature and experiments indicate the presence of Pre-Darcy effect. Contributing factors to Pre-Darcy effect are discussed and some reasons causing this effect are postulated. The experiments also show that this effect is significant.ConclusionPre-Darcy effect is significant because it is the dominant flow regime in typical petroleum reservoirs.