Analytical solutions for some chemical transport processes in porous media
Stamatiou, Alexandros Viktor Angelos
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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.