Numerical study of thermo-viscous fingering in porous media

Thermo-viscous fingering (TVF) instability is encountered in a variety of applications involving thermal enhanced oil recovery and injection molding as well as in natural event of fissure eruption. Although TVF may impact the ultimate recovery of any thermal FOR process quite adversely, the factors controlling such instability have not so far been examined systematically. In the current study, numerical and mathematical tools were used to analyze TVF in both rectilinear and quarter five-spot geometries. The basic equations and the parameters governing the problem are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters βT and βC, respectively. Other involved parameters are the Lewis number, Le and a thermal lag coefficient, λ. With appropriate models developed, linear stability analysis of TVF is carried out only for rectilinear geometry, whereas full nonlinear simulations of TVF are conducted for both rectilinear and quarter five-spot geometries.;For nonlinear simulations in both geometries, vorticity-streamfunction formulation is applied to the appropriate model equations. Hartley transform based Pseudo-spectral method and semi-implicit time-stepping algorithm are adopted. Time evolutions of nonlinear fingers have been examined qualitatively by plotting concentration and temperature iso-surfaces. The effect of increase in βT as well as decrease in λ is examined first for a hypothetical value of Le=1. For fixed βC, an enhancement in instability is observed with the increase in βT for all values of λ in both geometries. At large value of Le, although the instability is seen to be strictly dominated by βC slight variation in the trend is observed in different types of geometry. In rectilinear geometry of unity λ, for the tested values of the other parameters, less complex finger structures are observed than in a reference isothermal case with the same βC but β T=0. In five-spot geometry of unity λ, however, the converging nature of the potential velocity field in the presence of strong thermal diffusion advances the fluid front little farther than the one in the reference isothermal case, thus destabilizing the front slightly. As λ is decreased, in both geometries, a highly diffuse thermal front is seen to lag farther behind the fluid front and the stabilizing/destabilizing effect of the strong thermal diffusion gets alleviated. Consequently, the fluid front becomes as unstable as the reference isothermal case. The qualitative observations are further substantiated quantitatively using various standard measures as well as through identifying the relative contributions of solutal and thermal vorticity components in the development of the instability. The conclusions drawn from the study involving rectilinear geometry are in complete qualitative agreement with the ones obtained from the linear stability analyses, particularly using IVC approach. Such agreement infers the superiority of IVC approach over QSSA approach in linear stability analysis of the problem under investigation.;For linear stability analysis, the governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady state approximation (QSSA) or initial-value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio fir enhances the instability for fixed values of βC, Le and λ. For fixed βC and βT, a decrease in the thermal lag coefficient and/or an increase in the Lewis number always attenuate the instability Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects for positive β T. Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity A flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same βC but βT=0. At practically small value of λ, however, the instability ultimately approaches that due to βC only.