A. V. Akhmetzyanov

Computational aspects in controlling filtration of fluids and gases in porous media

To control filtration of gases, liquid hydrocarbons, oil, and water in homogeneous and heterogeneous porous media, it is necessary to obtain a multivariate solution to systems of nonlinear or quasi-linear equations in partial differentials of the parabolic type that define hydrodynamic (mathematical) models of control objects. In this paper, we propose hierarchical multigrid variants of balance and variational methods together with methods of domain decomposition, splitting with respect to physical processes and spatial coordinates. The paper consists of two parts. In the first part we consider a model of single-phase filtration of gas in gas-field developing; in the second, a model of two-phase filtration of oil and water, i.e., oil fields. This statement proves the universal character of the proposed results. Due to the multilevel partition of the original initial boundary value problem, we can construct economical solution algorithms using multiprocessor computer systems of the cluster type of parallel action for equations of the model.

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.

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.

Large-Scale Nonlinear Multivariable Systems (Decomposition, Modeling and Control Problems)

Abstract A class of multivariable systems is considered where modeling and control problems related to real physical processes can be solved only using approximate computational approach. The simulation processes are defined as solving mesh (finite-difference and finite-element) approximations of initial-boundary problems corresponding to original equations of mathematical physics for proper physical processes. The high dimensionality issues arising in the frame of such approach are overcome by means of decomposition and partitioning combined with multigrid spatial versions of approximating operator equations in function spaces. Multilevel computational methods for modeling and solving optimal control problems are oriented to using multiprocessor computer systems with parallel computing in message passing interface environment. The proposed results are actual both in theoretical and applied aspects. For instance, using the proposed approach to resolving problems of natural hydrocarbon deposit development simulation and optimal control opens wide capabilities for choosing efficient strategic decisions.

Optimal Choice of Coordinates for Oil Well Drilling

Methods and algorithms for determining coordinates for drilling new wells on an admissible set are developed by extending approaches of [1]. The cases in which (1) time-changes in oil saturation can be neglected and (2) pressure and oil saturation distributions in time and space described by a common system of differential equations are studied separately. Special attention is paid to methods that involve the determination of optimal controls.

Operational Planning and Control of Oil Production in Oil-Field Development

For oil production planning, oil-well operation modes must be chosen such that the water content in pumped products is minimized under certain technological constraints. Planning time is such that the variation in oil saturation in beds can be neglected. Oil-well operation modes are maintained by an automatic control system.

The 2F3D-Model of Filtration in the Bottom Zones of Horizontal Oil Wells. I. Qualitative Studies

The work consists of two parts. The first part (Qualitative Studies) considered the three-dimensional (3D) mathematical model of the two-phase (2F, oil–water) filtration in porous medium of the bottom zone of a horizontal well. The model was intended for controlling oil extraction from the reservoirs of the oil fields. Some results presented in the paper were concerned with the qualitative theory of partial differential equations underlying the models under consideration. The second part (Numerical Methods) presented numerical methods of solution and identification of the parameters of the model equation system.

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.

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.