Scale-up in poroelastic systems and applications to reservoirs

James G. Berryman

berryman@sep.stanford.edu

ABSTRACT

A fundamental problem of heterogeneous systems is that the macroscale behavior is not necessarily well-described by equations familiar to us at the meso- or microscale. In relatively simple cases like electrical conduction and elasticity, it is true that the equations describing macroscale behavior take the same form as those at the microscale. But in more complex systems, these simple results do not hold. Consider fluid flow in porous media where the microscale behavior is well-described by Navier-Stokes' equations for liquid in the pores while the macroscale behavior instead obeys Darcy's equation. Rigorous methods for establishing the form of such equations for macroscale behavior include multiscale homogenization methods and also the volume averaging method. In addition, it has been shown that Biot's equations of poroelasticity follow in a scale-up of the microscale equations of elasticity coupled to Navier-Stokes. Laboratory measurements have shown that Biot's equations indeed hold for simple systems but heterogeneous systems can have quite different behavior. So the question arises whether there is yet another level of scale-up needed to arrive at equations valid for the reservoir scale? And if so, do these equations take the form of Biot's equations or some other form? We will discuss these issues and show that the double-porosity equations play a special role in the scale-up to equations describing reservoir behavior, for fluid pumping, geomechanics, as well as seismic wave propagation.