Reservoir Engineering–General - Further Discussion on the Development of Stability Theory for Miscible Liquid-Liquid Displacement

The author applies the theory of hydrodynamic stability to fluid flow in porous media and concludes that the displacement of miscible liquids is stable if the stability coefficient, as defined by Eq. 19.2 or Eq. 27.0, is positive. The conclusion is invalid, however, since it is based on a faulty argument. To demonstrate this, the reader is referred to Eqs. 4.1 through 4.5 of the paper which describe, in differential form, the first-order disturbances c, u, v, w and p. In these same equations, cx, cz, gl, g3 and are functions of the stable concentration c. This concentration c is known and depends on x, z and t (the author assumes cy = 0). It follows that cx, cz, gl, g3 and µ are also functions of x, z and t. With this in mind, we turn to Eqs. 8.1 through 8.5 which were derived from Eqs. 4.1 through 4.5 by substituting Eq. 6.0. The significance of this substitution is that the author thereby limits the solutions of Eqs. 4.1 through 4.5 to equations of the form of Eq. 6.0. Eqs. 8.1 through 8.5 represent an infinite linear system with five unknowns: c, u, v, w and p. The coefficients cx, cz, gl, g3 and µ in this system are known at each point and at each instant; the coefficients containing derivatives of f are unknown. If this infinite linear system in c, u, v, w and p is to have a nontrivial solution, then it is indeed necessary (as the author states) that the determinant of the coefficient matrix of Eqs. 8.1 through 8.5, i.e., ( A 1, be equal to zero. This condition, however, is not sufficient. A non-trivial solution exists if, and only if, the rank of the coefficient matrix of the entire infinite system is smaller than five. In other words. all minors of order five must vanish and only if all the coefficients in Eqs. 8.1 through 8.5 are constant is this equivalent to the condition |A| =0. Constant coefficients means that gl, g3 and µ, for instance, are constant; and, since they depend on c it follows that c must be constant. We conclude that the condition I A I = 0, on which the author bases his calculations, is necessary and sufficient for a nontrivial solution only if the concentration c is constant. In other words, the author does not investigate the stability of a miscible displacement but, rather, only the stability of the flow of a homogeneous liquid. Using the initial condition and the fact that the coefficients are constant, we find