Reply to Discussion of "Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-Forchheimer Flow in Porous Media"

Abstract

The authors of SPE 893251 would like to thank FracTech and Halliburton ERC for their discussion of the paper. The data presented in their discussion provide excellent examples of what drove Stim-Lab to investigate non-Darcy flow in more depth and to arrive at the conclusions presented in paper SPE 89325. Although the data in the FracTech and ERC examples appear to fall on a straight line on the Forchheimer graph, the intercept value of permeability does not represent consistently accepted values of permeability for those proppants. For years, Stim-Lab had observed apparently linear Forchheimer graphs with an intercept value that failed to agree with the directly measured Darcy (low-rate) permeability. In this discussion we use ki as the apparent Darcy permeability from the intercept of the Forchheimer equation and kd as the directly measured Darcy absolute permeability at low rate. For Forchheimer’s equation to be valid, both slope and intercept, ki and β, must be equally valid. That is ki must equal kd and the straight line of slope β must intercept at ki = kd. The FracTech data in Fig. D-1, representing test results for 20/40-mesh white sand at 1,000-psi stress, was digitized, and the apparent permeability at each point was computed from the inverse permeability. The data were plotted here in Fig. R-1 on the log-log inertial graph as the magenta diamonds. The coefficients from the FracTech linear regression of their Forchheimer graph were used to calculate kapp over a range of values of ρv/µ. The results are shown in Fig. R-1 as the solid blue line. Note that the FracTech data fall on a slope that is much shallower than −1.0, as indicated by the solid black line. The Forchheimer fit of their data yields an apparent value of Τ=34.86, kd=91.1 darcies, and does not approach a unit slope until ρv/µ approaches 1,000. The red line on the graph is based on data from the Stim-Lab database and Predict-K. The database contains data sets for six 20/40-mesh sand proppants with 28 measurements of permeability at approximately 1,000 psi net stress. The average directly measured Darcy permeability for these 28 tests is 288 darcies, with no test showing less than 202 darcies. This curve fits the FracTech data better than their own regression, honors the actual Darcy permeability for this proppant, and yields values of β and Τ consistent with the mean particle diameter of the sample. In comparison, the reported Darcy permeability value from the FracTech data fails to correlate with any published or estimated Darcy permeability values for 20/40-mesh proppant at this stress. The separation between the red and blue lines in the figure shows the error in permeability generated by the flawed Forchheimer analysis.