Aleksey S. Telyakovskiy

An Eulerian-Lagrangian Solution Technique for Single-Phase Compositional Flow in Three-Dimensional Porous Media

We develop an Eulerian-Lagrangian iterative solution technique for accurate and efficient numerical simulation of single-phase compositional flow through three-dimensional compressible porous media in the presence of multiple injection and production wells. In the numerical simulator, an Eulerian-Lagrangian method is used to solve the convection-diffusion transport equations and a mixed finite-element method is used to solve the pressure equation. Numerical experiments are presented to investigate the performance of the method. Moreover, because of the Lagrangian nature of the algorithm, the simulator is capable of using large time steps and coarse spatial grids to generate accurate and stable results.

A Component-Based Eulerian--Lagrangian Formulation for Multicomponent Multiphase Compositional Flow and Transport in Porous Media

We derive a Eulerian--Lagrangian formulation for multicomponent multiphase compositional flow and transport in porous media, by utilizing momentum balance to define a barycentric component velocity...

Regarding polynomial approximation for ordinary differential equations

In this article, we consider an application of the approximate iterative method of Dzyadyk [V.K. Dzyadyk, Approximation methods for solutions of differential and integral equations, VSP, Utrecht, The Netherlands, 1995] to the construction of approximate polynomial solutions of ordinary differential equations. We illustrate that this method allows construction of polynomials of low degree with sufficiently high accuracy by examples, and as a result such polynomials can be used in practical applications. Moreover, Dzyadyks method produces an a priori estimate for the polynomial approximation of the solution of Cauchy problems. For the application of this method a Cauchy problem should be rewritten as the corresponding integral equation, followed by the replacement of the integrand by its Lagrange interpolation polynomial and Picard iterations.

A characteristic method for porous medium flow

We present an efficient and accurate characteristic method for the solution of the unsteady-state advection-diffusion transport equations, which arise in the mathematical models for flow and transport processes in porous media. The advantages of this method include symmetrising the governing transport equation, treating boundary conditions naturally in the formulation, and conserving mass. Moreover, because of the Lagrangian nature of the algorithm used, the method generates accurate numerical solutions even when large time steps and coarse spatial grids are used in the simulation. Numerical experiments are presented to illustrate the performance of the method.

An Eulerian-Lagrangian localized adjoint method for compositional multiphase flow in the subsurface

We present an Eulerian-Lagrangian formulation for multiphase and multicomponent compositional flows through porous media. An Eulerian-Lagrangian localized adjoint method (ELLAM) is used to solve the molar mass balance PDEs for individual components. The molar mass balance equation of the fluid mixture is reformulated in terms of a total velocity and global pressure and a mixed finite element method is used to solve the corresponding equations. Numerical experiments are carried out to show the strong potential of the Eulerian-Lagrangian formulation developed.

Polynomial approximation of nonlinear differential systems with prefixed accuracy

Abstract This paper presents a method for constructing polynomial approximations of the solutions of nonlinear initial value systems of differential equations. Given an a priori chosen accuracy, the degree of the vector polynomial can be adapted so that the approximate solution has the required precision. The method is based on the AI-method of Dzyadyk developed for the scalar case, and the computational cost is shown to be competitive with other methods.

Closed-form approximate solutions to the porous-medium equation

We consider approximate similarity solutions to the porous-medium equation. Porous-medium equation models laminar filtration of gas through porous media, and it represents filtration of moisture through soils when diffusivity has a power-law form. We consider initial-boundary conditions that allow the introduction of dimensionless variables and the reduction of an original partial differential equation to a boundary value problem for a nonlinear ordinary differential equation. We consider a couple of classes of approximate solutions to this nonlinear ODE, and we compare the approximate results with the exact results calculated by Shampines method [9]. In addition we describe potential future directions of research.