Analytical solutions for some chemical transport processes in porous media

Abstract

Flow and transport through porous media is applicable in many areas of industrial and geoscientific importance, e.g. flow through packed bed reactors in chemical engineering, reactive transport in porous membranes in biology, flow of chemicals through porous rocks in enhanced oil recovery (EOR) and in chemical treatments. Most mathematical models for the transport of chemicals in porous media are formulated as systems of several partial differential equations. Very often, these systems are so complex that they can only be solved numerically and, in general, analytical solutions to these equations cannot be found. In this thesis, we will examine two models for which analytical solutions are available. The first of these is the well-known polymer flood model in enhanced oil recovery, which describes the dis placement of oil by polymer-enriched water. An analytical solution for this problem exists in the literature in the form of a “solution algorithm”. In this work, we have applied this algorithm to reveal all possible solution profiles that can occur under certain assumptions, which the original authors of the algorithm did not present. However, the main emphasis of this PhD thesis lies in the construction of analytical solutions to several (simplifications of) models describing the transport of scale inhibitor in oil reservoirs. These chemicals are employed to slow down the formation and deposition of mineral scale. The most complex model considered here assumes both kinetic precipitation/dissolution and kinetic adsorption/desorption of scale inhibitor into an aqueous phase flowing at constant velocity. This general model consists of a system of three partial differential equations and must be solved numerically, but there are two important sub-cases for which analytical solutions can be found. The first of these only considers the kinetic precipitation mechanism. The second, much more complicated case assumes kinetic precipitation together with equilibrium adsorption. Both problems consist of two first-order partial differential equations relating the mobile phase concentration (C) and the amount of scale inhibitor precipitate (Π). The central idea for the construction of analytical solutions is the existence of an “invariant” relationship between C and Π. Together with the method of characteristics, this relationship enables us to build solution profiles consisting of several different regions. A key feature of these profiles is the motion of a boundary point, x = αΠ(t), which divides the domain into a region where there is precipitate (Π > 0) and a region where the precipitate has been completely used up by the dissolution process (Π = 0). The velocity of this boundary point in relation to the concentration flux velocity is of importance when determining the corresponding concentration level. Knowledge of C on the boundary is another essential building block in the development of the solution. In treating the various cases, a powerful solution method emerges which may be applicable to the analysis of other chemical transport models in which one of the unknown quantities can be completely depleted, thereby altering the underlying system of first-order partial differential equations. It appears that this work is the first in which this solution methodology has been brought to bear on this type of internal moving boundary problem; it has certainly never been applied in any problem in oilfield chemistry or enhanced oil recovery.