A New Approach to Instability Theory in Porous Media

Early work in the area of instability theory is limited in that it is based on first-order perturbation theory and the concept of a velocity potential. Thus, while it can deal with an incipient finger, it cannot deal with the subsequent growth of a finger. This paper develops a new approach to the instability theory that overcomes this limitation. The new approach, like earlier work, is based on the assumption that the immiscible displacement of one fluid by another can be treated as a moving-boundary problem. Therefore, two solutions arise, one for each side of the plane interface that initially separates the two fluids. plane interface that initially separates the two fluids. Because the new approach makes use of a force potential rather than a velocity potential, it is possible to impose several new conditions on these two solutions. As a consequence, further extensions to the stability theory have been obtained. In particular, it is now possible to predict the steady-state velocity at which a finger propagates and, consequently, the breakthrough recovery obtained not only when the displacement is stable, but also when it is pseudostable. pseudostable. Introduction It has been recognized for some time 1 that viscous fingering can play a significant role in the immiscible displacement of one fluid by another. This led a number of people to attempt a theoretical description of viscous fingering. Many of these attempts are based on first-order perturbation theory. When this approach is taken, it is usual to postulate the existence of a velocity potential. Then, if the divergence of the perturbation velocity is identically equal to zero, the postulated velocity potential will satisfy Laplace's equation. There are several problems with this approach. First, a velocity potential exists only for fields of flow involving a fluid of constant density and viscosity and a porous medium that is homogeneous and isotropic throughout. Thus it is debatable whether results based on the concept of a velocity potential have any application when the porous medium is not ideal. Second, the concept of a porous medium is not ideal. Second, the concept of a velocity potential has only kinematical significance and gives no insight whatsoever into the dynamic properties of the flow. As a consequence, when a velocity potential is used, one cannot infer how variations in rock and fluid properties might affect the perturbation velocity. properties might affect the perturbation velocity. Finally, the divergence of the perturbation velocity is zero only when a plane interface separates the displaced fluid from the displacing fluid. As soon as the interface is perturbed, two fluids will occupy a region where there was previously only one, and the divergence of the perturbation velocity will no longer be zero. Rather, the perturbation velocity will no longer be zero. Rather, the divergence of the sum of the perturbation velocities, one for each fluid, will be zero. Thus, while first-order perturbation theory can deal with an incipient finger, it perturbation theory can deal with an incipient finger, it cannot deal with the subsequent growth of the finger. The purpose of this paper is to develop a theory for viscous purpose of this paper is to develop a theory for viscous fingering that does not suffer from the aforementioned limitations that arise from the use of a velocity potential. The new theory, which is based on the concept of a force potential, not only explicitly accounts for the dynamic flow potential, not only explicitly accounts for the dynamic flow properties of both fluids, but also is capable of delineating properties of both fluids, but also is capable of delineating the entire life-span of the growing finger. Theory Fluid Displacement. To develop the theory presented in subsequent sections, it is necessary to have a working model of how one fluid displaces another. Such a model must be simple to keep the mathematics tractable. Consequently, it is usual to assume that the displaced and displacing fluids are separated initially by a plane interface and that the porous medium is ideal. If the porous medium is truly ideal (in the sense that the size and shape of all flow paths are identical), the displaced and displacing fluids will continue to be separated during the course of a displacement by a saturation discontinuity (or shock), provided the displacement is stable. 4 Thus stable provided the displacement is stable. 4 Thus stable displacements of this kind can be treated as a moving boundary problem. However, any real porous medium will contain a distribution of pore sizes. As a consequence, different parts of an initially plane interface will move at different parts of an initially plane interface will move at different velocities through the porous medium; i.e., a distribution of saturations will evolve to separate that part of the porous medium where only displacing fluid is flowing from that where only displaced fluid is flowing. The length and shape of this transition zone will depend not only on the pore-size distribution of the porous medium, but also on pore-size distribution of the porous medium, but also on the balance between viscous, gravity, and capillary forces that exists at each point along the saturation profile during the course of the displacement. Moreover, the shape of the saturation profile can be predicted by conventional displacement theory "if the displacement is stable. Even in an ideal porous medium, the displacement front separating the fluids may become unstable under certain conditions. Consequently, a distribution of saturations will evolve again to separate the two regions where only one fluid is flowing. However, the length and shape of this transition region will differ markedly from that which results from a distribution of pore sizes. SPEJ P. 765