Geofluids

Volume 2017, Article ID 1806052, 18 pages

https://doi.org/10.1155/2017/1806052

## Numerical Validation of Analytical Estimates for Propagation of Thermal Waves Generated by Gas-Solid Combustion

^{1}National Laboratory for Scientific Computing (LNCC), Av. Getúlio Vargas, 333, 25651-070 Petrópolis, RJ, Brazil^{2}Department of Mathematics, Federal University of Juiz de Fora (UFJF), Rua José Lourenço Kelmer, s/n, Campus Universitário, 36036-900 Juiz de Fora, MG, Brazil

Correspondence should be addressed to Grigori Chapiro; rb.fjfu.eci@irogirg

Received 2 July 2016; Revised 16 September 2016; Accepted 27 October 2016; Published 12 January 2017

Academic Editor: Mark Tingay

Copyright © 2017 Weslley da Silva Pereira and Grigori Chapiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Gas-solid combustion appears in many applications such as in situ combustion, which is a potential technique for oil recovery. Previous work has analyzed traveling wave solutions and obtained analytical formulas describing combustion wave temperature, velocity, and gas velocity for one-dimensional gas-solid combustion model using geometrical singular perturbation theory. In the present work these formulas are generalized. Using numerical simulation we show that they can be adapted and then applied to describe more general two-dimensional models for in situ combustion in a nonhomogeneous porous medium.

#### 1. Introduction

There is a renewed interest in using combustion to recover medium- or high-viscosity oil. In situ combustion (ISC) is the oldest thermal recovery technique and it has been considered a potential method for the recovery of heavy oil [1, 2]. In this method, air is injected in the porous medium. Heavy and immobile components of the crude oil are used as fuel, producing in-place heat and consequently reducing the viscosity of the oil. Most of the oil is driven toward the producers. An operational advantage of this technique is the abundance of injection gas regardless of location.

Modeling the combustion in porous media presents a number of research challenges, mainly because of the physical complexity of the process. The crude oil is composed of many substances which directly affect the combustion process. Furthermore, the combustion reaction and other thermal reactions, like pyrolysis, change the chemical structure of the components along the process. Due to these reasons, a good model for this technique generally consists of many equations, which hinders its analytical and numerical study. That is why a number of works simplify ISC process to one of its main components, gas-solid combustion; see [3–7] and references therein.

Analytical solutions and estimates are simple and cheap tools which can be used for validation of existing numerical solutions and to obtain information about some aspects of oil recovery before investing in computational codes. They also help to better understand the physical process. Usually the combustion front is regarded as a traveling wave. For one-dimensional models there are two main methods used to study such wave: (1) the first method explores the strong nonlinearity of the Arrhenius factor in the reaction rate, which allows neglecting the reaction rate as soon as the temperature decreases [8]. Other works use this method to obtain estimates on the combustion wave speed, temperature, and oxygen concentration; see [3–5, 9] and references therein. (2) The second method considers that the reaction is active for all temperatures but heat losses are negligible; see [7] and references therein. The last method also allows obtaining physically consistent process parameters such as combustion wave speed and combustion front temperature. The previously mentioned works consider one-dimensional models that do not take the pressure equation into account. None of these is directly applicable to two-dimensional models, nor to models containing the pressure equation. The main goal of this work is to verify whether the analytical estimates proposed in [7] can be applied to more general two-dimensional nonhomogeneous models.

Other works address two-dimensional combustion. For example in [10], two-dimensional simulations were performed and it was observed that the combustion front assumes a curved profile when the domain has strongly spatially correlated characteristics, whereas it is almost linear if heterogeneity is random. In [11] the compositional flow in porous media was considered to model forward ISC. The viscous fingering effect was observed for heavy oil oxidation. There is a number of other works that study the filtration combustion in forward (e.g., [12]) or reverse (e.g., [13, 14]) regimes. Although many papers deal with multidimensional combustion models, there is a lack of analytical studies on this topic. Similar to [10], fingering instabilities observed in the present work appear only for correlated nonhomogeneous medium, when fingers increase in length over time.

In this work we follow [3, 7] and consider the combustion of immobile fuel. In Section 2 we present a two-dimensional gas-solid model containing an equation for pressure, balance laws for energy, oxygen content, fuel molar mass, and total gas mass, where the gas density is described by the ideal gas law and the average gas speed by Darcy’s law. In Section 3 we show that the model proposed in this work is indeed a generalization of the one-dimensional model proposed in [3]. In this section we also review and generalize some analytical results obtained in [7]. In order to simulate the gas-solid process we use the finite element method described in Section 4. Section 5 presents numerical results, which are compared with analytical estimates. Further numerical examples are performed in order to validate the numerical model. Finally, Section 6 summarizes final conclusions.

#### 2. Model

Air is injected into a porous medium initially filled with a nonreactive gas and immobile fuel, which can be solid or liquid at low saturations. We consider a combustion reaction that consumes oxygen and fuel, generating heat. We assume that only a small part of the void space is occupied by the fuel, so its consumption does not affect the porosity of the matrix. We consider the local thermal equilibrium resulting in equal temperatures for all phases (fuel, gas, and rock). Heat losses are neglected, which is a reasonable hypothesis for ISC in field conditions.

Some other assumptions are considered: heat transfer due to radiation and energy source due to pressure variation are neglected; the work related to surface and body forces is also neglected; the ideal gas law is the equation of state for the gas phase; gas heat capacity is negligible when compared to the heat capacity of the rock. To simplify the model, we neglect changes in the effective molecular weight of the gas phase due to the reaction by approximating this quantity by a constant. We do not consider the effect of gravity in the flow and the dispersion terms in the diffusion-dispersion tensor. Thermodynamic and transport properties such as thermal conductivity of the medium, heat capacity of the rock, and specific heat capacity of the gas are considered constants. Generally, the porosity is related with permeability through polynomial relations, it also changes within the combustion reaction. For simplicity, we consider isotropic porous medium with constant porosity and simplified combustion reaction: .

Let and be the time and space coordinates. The primary dependent variables are the temperature, (Kelvin), the oxygen concentration in terms of molar fraction, (moles of oxygen per moles of gas), the molar concentration of fuel, (moles of fuel per cubic meter), and molar density of gas, (moles of gas per cubic meter).

The system of equations extends the one-dimensional model from [6] by considering variable pressure, two-dimensional domain, and Darcy’s law, based on [15]. Parameter names and typical values are given in Table 1. We present the following equations under the assumptions described above. The energy balance law iswhere is the volumetric heat capacity of the rock, is the specific heat capacity of the gas, is the average gas speed, is the thermal conductivity, is the heat released due to combustion of each mole of fuel, and is the combustion rate. The molar mass balance of oxygen in the gas phase iswhere is the porosity of the medium and is the molecular diffusion coefficient. The molar mass balance of fuel isThe molar mass balance of total gas isThe reaction rate is described by the Arrhenius law combined with the linear Law of Mass Action [16].where is the reference pressure, is the activation energy, is the preexponential factor, and is the ideal gas constant. The gas phase equation of state is the ideal gas law:The average gas speed is given by Darcy’s law:where is the dynamic gas viscosity and is the absolute permeability of the porous medium.