Alexei Kushner

Petrov invariants of Hamiltonian systems with a control parameter

The problem is considered of the classification of Hamiltonian systems with a scalar control parameter relative to feedback transformations. Differential invariants of these systems, which are called Petrov invariants, are set up. The dimensions of algebras of these invariants are found. The conditions of global equivalence of regular Hamiltonian systems with the control parameter are found in terms of Petrov invariants.

Geometric theory of special modes in the distributed-parameter control systems

The present paper is the first in the projected series of publications illustrating the application of the geometric theory of singular solutions of the nonlinear partial differential equations to the description of special modes in the distributed-parameter control systems. Consideration was given to the problem of biphase unidimensional filtering of fluids (oil and water) in the porous media of the natural oil pools.The paper is an extended version of the report presented at the International Conference PACO’2012 prepared for publication on recommendation of the Program Committee [1].

Attractors in models of porous media flow

A constructive method is proposed for finding finite-dimensional submanifolds in the space of smooth functions that are invariant with respect to flows defined by evolutionary partial differential equations. Conditions for the stability of these submanifolds are obtained. Such submanifolds are constructed for generalized Rapoport–Leas equations that arise in the theory of porous media flows.

Nonlinear Models of Oil Frontal Displacement and Shock Waves

Abstract The goal of this paper is to use geometrical theory of nonlinear partial differential equations and singularities theory to obtain the general scheme for control of fronts of petroleum displacement by active reagent. We illustrate our geometric approach to evolutionary models of petroleum deposits. Original simulation methods for the frontal displacement of oil by water and solutions of active reagents in porous medium of oilfield reservoirs with and without capillary forces has been developed. For simplicity and clarity of presentation the only one-dimensional models of fluid filtration between parallel batteries of production and injection wells (Buckley-Leverett and Rapoport-Liss models) has been considered. The model for oil displacement by solutions of active reagents in hot water has been also analyzed. The proposed results are generalized for two- and three-dimensional filtration models in reservoir engineering using vertical and horizontal wells, which greatly facilitates the statement and development of simulation and optimal control methods for reservoir engineering in general (see Akhmetzyanov (2008a,b)).

Control of displacement front in a model of immiscible two-phase flow in porous media

For the Buckley–Leverett equation describing the flow of two immiscible fluids in porous media, an exact parametric representation of the solution is constructed with the help of the Bäcklund transformation. As a result, the advance of the displacement front can be controlled to a high degree of accuracy. The method is illustrated using an example of a typical oil well with actual parameters.

Finite dimensional attractors of differential evolutionary equations

A method of constructing finite dimensional invariant submanifolds (= dynamics) for second order evolutionary differential equations is described. This method allows one to construct attractors of such equations and investigate stability of their solutions.

Thermodynamics and Optimal Control of Porous Media Flows in Oil Field Development

The nonisothermal two-phase porous media flow of oil and hot water in a horizontal oil reservoir is considered. An asymptotic method for calculating the flow and solving related optimal control problems is proposed. Namely, the problem of choosing optimal control actions to maximize the oil production for a given level of financial costs and to minimize the costs for a given level of oil production is considered.

Optimal Management of Oil Field Development in the Buckley–Leverett Model

We consider the problem of two-phase filtering (oil and water) in a horizontal layer of an oil deposit. We propose an asymptotic method for calculating both the filtering process and related optimal control problems, namely the problems of choosing optimal control actions to achieve maximum oil production at a given level of water resource consumption or a minimum water flow rate that provides the level of oil production required by the plan.

Finite dimensional dynamics for nonlinear filtration equation

Abstract We construct new finite dimensional submanifolds in the solution space of nonlinear differential filtration equations and describe the corresponding evolutionary dynamics. This method is implemented in a computer program of symbolic computations Maple.

Buckley-Leverett's filtration model and time-optimal oil fields development

Time-optimal management of the development of oil fields discussed. We use one-dimensional Buckley-Leverett filtering model.