Leis de potência no regime estacionário do escoamento bifásico em meios porosos

Abstract

The flow of two immiscible fluids in porous media occurs spontaneously in nature but can also be induced by human actions. The most obvious example of this latter case is the injection of a fluid (usually water or gas) into an oil reservoir. In fact, due to its technological application, this theme has been the object of research for decades. However, most of those studies focus on the displacement of a phase due to another, where the interface between them moves in a transient regime until it reaches a given region (usually the producing well). This perspective is important in a macroscopic approach where the interest is delaying the breakthrough time, that is, the time that the injected fluid takes until reaching the producing well. On the other hand, in a mesoscopic approach, in the pore size scale, the interface is not abrupt and the porous channels are fed by both fluids. This produces a disorganized competition in the flow, but which for a short time can lead to a steady flow where the velocity of each fluid, and hence the relative permeability of each phase, does not change with time. This type of steady flow and its influence on the macroscopic spread of the invasion front is still not well known. In this paper, we are interested in investigating the displacement of two fluids in the steady state biphasic flow regime in porous media. To obtain the steady state we consider a two dimensional porous medium with periodic boundary conditions in both directions. Therefore, our system can be understood as two phases moving on the surface of a porous torus. The steady state is then defined when the velocity (which is related to the relative permeability) of both phases oscillates around an average value, which does not change with time. We then show that in this regime the oscillations in the time series of velocity have characteristic power laws whose exponent depends on the quantity (saturation) and properties of the fluids. For example, the avalanche distribution of velocities occurring due to the formation of fluid bubbles of different sizes that become trapped in the porous channels for a time and are suddenly released into the stream thereby altering the velocity and the relative permeability of the phase, depends on the saturation of the fluid phases involved and also on the value of the interfacial tension.