This work is focused on the analysis of non-linear 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 non-linear Forchheimer equation is inverted to a non-linear 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.

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. An inhomogeneous Poincare-Sobolev inequality related to the non-linearity of the equation is used to study the asymptotic behavior 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 linear in the coefficients of the Forchheimer polynomials.

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We construct a mathematical model of the early formation of an atherosclerotic lesion based on a simplification of Russell Ross' paradigm of atherosclerosis as a chronic inflammatory response. Atherosclerosis is a disease characterized by the accumulation of lipid-laden cells in the arterial wall. This disease results in lesions within the artery that may grow into the lumen restricting blood flow and, in critical cases, can rupture causing complete, sudden occlusion of the artery resulting in heart attack, stroke and possibly death. It is now understood that when chemically modified low-density lipoproteins (LDL cholesterol) enter into the wall of the human artery, they can trigger an immune response mediated by biochemical signals sent and received by immune and other cells indigenous to the vasculature. The presence of modified LDL can also corrupt the normal immune function triggering further immune response and ultimately chronic inflammation. In the construction of our mathematical model, we focus on the inflammatory component of the pathogenesis of cardiovascular disease (CVD). Because this study centres on the interplay between chemical and cellular species in the human artery and bloodstream, we employ a model of chemotaxis first given by E. F. Keller and Lee Segel in 1970 and present our model as a coupled system of non-linear reaction diffusion equations describing the state of the various species involved in the disease process. We perform numerical simulations demonstrating that our model captures certain observed features of CVD such as the localization of immune cells, the build-up of lipids and debris and the isolation of a lesion by smooth muscle cells.

Motivated by the reservoir engineering concept of the well Productivity Index, we introduced and analyzed a functional, denoted as "diffusive capacity", for the solution of the initial-boundary value problem (IBVP) for a linear parabolic equation.21This IBVP described laminar (linear) Darcy flow in porous media; the considered boundary conditions corresponded to different regimes of the well production. The diffusive capacities were then computed as steady state invariants of the solutions to the corresponding time-dependent boundary value problem.Here similar features for fast or turbulent nonlinear flows subjected to the Forchheimer equations are analyzed. It is shown that under some hydrodynamic and thermodynamic constraints, there exists a so-called pseudo steady state regime for the Forchheimer flows in porous media. In other words, under some assumptions there exists a steady state invariant over a certain class of solutions to the transient IBVP modeling the Forchheimer flow for slightly compressible fluid. This invariant is the diffusive capacity, which serves as the mathematical representation of the so-called well Productivity Index. The obtained results enable computation of the well Productivity Index by resolving a single steady state boundary value problem for a second-order quasilinear elliptic equation. Analytical and numerical studies highlight some new relations for the well Productivity Index in linear and nonlinear cases. The obtained analytical formulas can be potentially used for the numerical well block model as an analog of Piecemann.

We study the long term asymptotics of the diffusive capacity, the integral characteristic of a domain with respect to a nonlinear Forchheimer flow in porous media. Conditions on the boundary are given in terms of the total flux and constraints on the pressure trace on the boundary. We prove that, if the total flux is stabilizing, then the difference between the pressure average inside the domain and the pressure average on the boundary is stabilizing as well. This result can be used for calculating the productivity index of the well, an important characteristic of the well performance. To obtain the main theorem, a refined comparison of the fully transient pressure with the pseudo-steady state pressure (the time derivative of pressure is constant) was performed. These results can be effectively used in reservoir engineering and can also be applied to other problems modeled by nonlinear diffusive equations. Bibliography: 26 titles.

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. Introduction The PI is one of the more basic characteristics of well performance not requiring assumptions about the equations of the flow motion and the state of the fluid. The concept of PI expresses the following: Once the well production is, in some sense, stabilized, then the ratio between the production rate and the pressure drawdown (difference between the reservoir average pressure and the well average pressure) is practically independent from the production history or even from the operating conditions (Muskat 1937; Dietz 1965; Dake 1978; Raghavan 1993; Larsen 2001). The higher the value of the PI is, the better the performance of the well in the reservoir is. We are particularly interested in the asymptotic (late-time) value of the PI. For an isolated reservoir with no-flow condition on the outer boundary and constant production rate, stabilization of the PI means that the pressure drawdown becomes time invariant; this flow regime is called pseudosteady state (PSS). In the case of constant wellbore pressure, both the production and reservoir pressure change in time, but the PI asymptotically stabilizes to a constant value, leading to the flow regime called boundary-dominated (BD) (Dietz 1965; Dake 1978; Raghavan 1993; Larsen 2001).

Motivated by the reservoir engineering concept of the well Productivity Index (PI) we study a time dependent functional for general non-linear 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 non-linear analog of Peaceman formula for the well-block pressure in unstructured grid. Numerical simulations sustain theoretical results.