Computational study of adsorption effects on the miscible displacement in porous media

Abstract

Transport in porous media has remarkably influenced the propagation of fluids in a variety of chemical as well as physical systems. This thesis is primarily concerned with the mathematical modelling of the flow of miscible fluids through a porous medium. There exist numerous situations in miscible displacement when the solute concentration trigger the hydrodynamic instability related to unfavourable viscosity gradients within a fluid system. This instability typically known as Viscous Fingering (VF) affects the oil recovery, chemical processing, hydrology as well as CO2 sequestration in porous medium. The situation becomes even more complicated when the solute present in the fluid interacts with the porous medium via adsorption. This thesis focuses on the theoretical investigation of the underlying hydrodynamic instability in the presence of adsorption which is a physico-chemical mechanism of fluid flow phenomena having applications in Chromatographic separation and aquifers. The fluid system is considered to be miscible, incompressible and neutrally buoyant with a homogeneous medium. Based on the adsorption isotherms this thesis is divided into two parts. In first part of the thesis, the problem involving linear adsorption of the solute on the porous matrix is investigated. In this scenario the rate of adsorption is further assumed to be dependent on the composition of the fluid mixture of displacing fluid and sample solvent: it is called the solvent strength effect. The solute dynamics are first examined in pure dispersive profile by solving the governing equations semi-analytically, with the sample solvent having an analytical error function solution. The adsorbed solute concentration equation is solved numerically by Fourier-pseudo spectral method. The solution profile of the solute shows a bimodal distribution due to the solvent strength effect which results in different advection rate of the solute. Next, the coupling effects of viscous fingering with the solvent modulated adsorption of the solute on its propagation dynamics is examined. The model discussed is a generalised one, as all the studies carried previously for the solute retention become a particular case of the proposed model. The sample solvent is assumed to be more viscous than the displacing fluid, which results in unstable rear interface. The numerical simulations are performed by highly efficient Fourier pseudo-spectral method using Fast Fourier Transform (FFT) along with operator splitting and predictor-corrector techniques to advance in time. The findings show that the two perturbations, viscous fingering at the rear interface and solvent strength, acting simultaneously reduces the effect of each other. This anti-synergetic scenario motivated us to analyse the influence of less viscous sample solvent under the same conditions as discussed above. The numerical simulation results show that in the presence of less viscous sample solvent the two perturbations are enhancing the effect of each other. A statistical analysis is performed in order to quantify the solute transport with the influence of various governing flow parameters. The second part of the thesis discusses the transportation dynamics of a non-linearly adsorbed solute following Langmuir isotherm, but with different initial conditions on the saturation of the solute concentration. The non-linear flow problems are formulated mathematically and are solved analytically for uni-dimension and numerically for full non-linear system of equations. In this flow system, we first analyse the influence of Langmuir adsorption on step-down initial concentration. The viscosity of the fluid is assumed to be derived by the solute concentration in mobile phase. The analytical solutions obtained for unidimensional model, confirms the occurrence of shock wave (in absence of dispersion) and shock layer (in presence of dispersion). The numerical solution also captures the shock layer dynamics efficiently. It is observed that with the given initial condition, the Langmuir adsorption speed up the instability process. Moreover, the instability vanishes the shock layer if get formed before the onset of viscous fingering. Further, the analysis has been carried out for the step-up initial solute concentration. In this scenario the analytical solution reveals the formation of expanding wave also known as rarefaction wave. The simulation results shows a highly dispersive solute distribution, which spreads non-monotonically with the non-linear adsorption parameter. The viscous fingering dynamics are also investigated, which is found to be delayed by the rarefaction wave formation. However, for very large non-linear adsorption parameter or for fluids having high viscosity contrast, the onset of viscous fingering is shown to earlier than the linear adsorption case. In addition to the above studies, to understand the interaction dynamics of the non-linear waves (shock layer and rarefaction wave) the mathematical problem with a finite slice of solute undergoing Langmuir adsorption is formulated. The problem considered has application in chromatography as well as in localised contaminant zones in groundwater. The numerical simulation results, without considering any viscosity contrast between the interplaying fluids, show that the interaction of the rarefaction wave front with the shock layer results in formation of triangular peaks with peak height decreasing with time. The problem considering viscosity contrast between the interplaying fluids is discussed for viscous fingers originating from the rear interface and interaction with the shock layer as well as for fingers originating from the frontal interface and interacting with the rarefaction wave. In the first case, it is observed that two counter effects occur at the frontal interface: the self-sharpening effects damp the distortions, whereas the convective forces favours them. As a consequence, the fingers which intrude through the shock layer are observed to be pushed back. In the second case, the fingers originate from the shock layer are unable to intrude through the rarefaction wave front because of the low concentration gradient along the rarefaction wave. The mixing dynamics are discussed to analyse the after effects of interaction of fingers with the non-linear wave front. The study presented in this thesis may be helpful particularly to physico–chemists and biochemists using chromatographic techniques where viscous fingering and adsorption effects interplay. People working in environmental science can also be benefited from this study, as they explain to what extent spreading of chemicals or pollutants in underground soils can be affected by these processes. Eventually, our study will also be of interest to branches of mathematical physics studying pattern formation and fingering phenomena in porous media.