Estimating Maximum Gas Rates For Darcy-Type Flow

Introduction Quite often gas well test interpretations are complicated because laminar or Darcy-type flow conditions may not exist. In many cases the analysis procedure can be simplified if the testing conditions could be controlled so as to promote a situation conducive to a Darcy inflow behavior. The purpose of this paper is to present a correlation to estimate the maximum gas flow rate at or below which Darcy flow conditions are likely. Knowledge of this flow rate a priori may be useful in the design of a particular well test. Also, the computation of this flow rate a poste-priori allows for the modification of the conventional plot used to estimate the mechanical skin factor of the well. As indicated in this study, the conventional plot (s′=s+Dq) May sometimes under- or over estimate theactual near-welbore condition. Theory Consider a homogeneous, isotropic porous medium completely saturated with a single-phase gas. The medium is assumed to be comprised of rock grains of spherical shape. The Kozeny-Carman equation, as reported by Pirson1 and Scheidegger2, can be used to relate the average grain diameter d and porosity φ to the permeability k: Equation 1 Available in full paper (Note: All units are given in the Nomenclature.) This equation is derived in the Appendix; solving for d, we have Equation 2 Available in full paper Using the definition of Reynold number (i.e., N R = dv ρ/ μ), The following form for N R can be developed (see the Appendix): Equation 3 Available in full paper Equation (2) is next introduced into equation (3) yielding Equation 4 Available in full paper It is generally accepted3 that Darcy-type flows are attained at Reynolds number less than unity (based on the units of equation (4). Denoting by qcc the flow rate obtained at this critical Reynolds number, equation (4) becomes Equation 5 Available in full paper Note that the formation thickness h, and not the perforated interval hp is used in equation (5), as suggested in recent studies.8 A correlation similar to equation (5) has been presented by Whitson.9 Actually, his expression is a special case of equation (5), obtained by approximating the group Mathematical Expression Available in full paper as 1107; he also used the value 0.5mm for the grain diameter d. No derivations or justifications were given. Equation (5) also allows for the following interesting observation. It has become standard practice 4 to estimate the mechanical skin factor s by constructing a plot of s′ versus the gas flow rate q (where s′=s+Dq), and taking s as the y-intercept of the line (i.e., at q=0). Implicit in this procedure, however, is the assumption that the inflow behavior is of the non-Darcy type for all q<0. Perhaps a more plausible, and certainly intuitive approach would be the assertion that Darcy-type inflows are attained up to some critical flowrate qc (where D=0 and hence s′=s=constant); the standard plot4 would then be valid for all q larger than qc This rate qc may be estimated from equation (5).