Estimating Maximum Gas Rates For Darcy-Type Flow

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.

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 q_cc 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 h_p 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 q_c (where D=0 and hence s′=s=constant); the standard plot⁴ would then be valid for all q larger than q_c This rate q_c may be estimated from equation (5).