# Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection

Mohamed El-Amin (Department of Mathematics and Computer Science, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia and Effat University, Jeddah, Saudi Arabia and Mathematics Department, Faculty of Science, Aswan University, Aswan, Egypt)
Jisheng Kou (School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, China)
Shuyu Sun (Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

ISSN: 0961-5539

Article publication date: 2 October 2017

116

## Abstract

### Purpose

This paper aims to introduce modeling, numerical simulation and convergence analysis of the problem of nanoparticles’ transport carried by a two-phase flow in a porous medium. The model consists of equations of pressure, saturation, nanoparticles’ concentration, deposited nanoparticles’ concentration on the pore-walls and entrapped nanoparticles concentration in pore-throats.

### Design/methodology/approach

A nonlinear iterative IMPES-IMC (IMplicit Pressure Explicit Saturation–IMplicit Concentration) scheme is used to solve the problem under consideration. The governing equations are discretized using the cell-centered finite difference (CCFD) method. The pressure and saturation equations are coupled to calculate the pressure, and then the saturation is updated explicitly. Therefore, the equations of nanoparticles concentration, the deposited nanoparticles concentration on the pore walls and the entrapped nanoparticles concentration in pore throats are computed implicitly. Then, the porosity and the permeability variations are updated.

### Findings

Three lemmas and one theorem for the convergence of the iterative method under the natural conditions and some continuity and boundedness assumptions were stated and proved. The theorem is proved by induction states that after a number of iterations, the sequences of the dependent variables such as saturation and concentrations approach solutions on the next time step. Moreover, two numerical examples are introduced with convergence test in terms of Courant–Friedrichs–Lewy (CFL) condition and a relaxation factor. Dependent variables such as pressure, saturation, concentration, deposited concentrations, porosity and permeability are plotted as contours in graphs, whereas the error estimations are presented in a table for different values of the number of time steps, number of iterations and mesh size.

### Research limitations/implications

The domain of the computations is relatively small; however, it is straightforward to extend this method to the oil reservoir (large) domain by keeping similar definitions of CFL number and other physical parameters.

### Originality/value

The model of the problem under consideration has not been studied before. Also, both solution technique and convergence analysis have not been used before with this model.

## Citation

El-Amin, M., Kou, J. and Sun, S. (2017), "Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 No. 10, pp. 2289-2317. https://doi.org/10.1108/HFF-05-2016-0210

## Publisher

:

Emerald Publishing Limited