This paper presents a comparison of the accuracy of the pressure-squared and pseudo pressure formulations of the Forchheimer equation for simultaneous determination of the permeability and non-Darcy flow coefficient from high-velocity flow tests using core plugs. We show that the pressure-squared formulation must satisfy two contradictory conditions. The core length should be sufficiently small so that the average viscosity and real gas deviation factor, which are dependent on the pressure drop, approach the actual values. The core length, however, should be long enough to be representative of the characteristic length of the porous media. Because these two conditions cannot be met simultaneously, the pressure-squared formulation is less accurate. We show that these effects are more pronounced for tight formations because of higher pressure drop across the core. The pseudo pressure formulation requires only that the core length should approach the representative core length, and, therefore, it provides more accurate interpretation of the high-velocity core flow tests and generates accurate values of the permeability and non-Darcy flow coefficient and the representative core length.
Previous studies, including Firoozabadi et al.,1 have facilitated the integral forms of the Forchheimer2 equation as a convenient means of determining the permeability, k, and non-Darcy flow coefficient, ß, from high-velocity flow data. However, as Civan3 and Civan and Evans4 state, the core length averaged k and ß are functions of length, because the viscosity, µ, and the real gas deviation factor, z, are averaged over the pressure drop across the core ends to obtain µ and z, respectively. In theory, the pressure-squared function that many used, including Firoozabadi et al.,1 has inherent limitations because it must satisfy two contradictory conditions to obtain accurate estimation of permeability and inertial flow coefficient from laboratory core tests. The first condition requires very short cores for average viscosity and real gas deviation factor to be close to actual values. The second conditions requires sufficiently long cores to approximate the representative core length correctly. Mathematically, Eqs. 1 and 2 can express these conditions, respectively:
Equations 1 and 2
where LR is a representative elemental core length. The pseudo pressure formulation by Civan and Evans4 involve only µ, which is almost constant for practical purposes because the effect of pressure on the gas viscosity is negligible. Therefore, the pseudo pressure formulation alleviates the need for satisfying both of these contrasting conditions. Thus, only the condition stated by Eq. 1 needs to be satisfied. Therefore, in the pseudo pressure formulation, the limit is taken with respect to the representative length only. Civan and Evans4,5 present the details of the formulations and the method that forms the basis for this paper elsewhere. We present the application and the verification of the method in this paper.
This paper presents a comparison of the pressure-squared and pseudo pressure formulations, a demonstration of the effect of the core length, and determination of the representative core length for simultaneous measurement of permeability and non-Darcy flow coefficient.
We checked the values of the inertial flow coefficients determined by Firoozabadi et al.1 against those predicted by the Liu et al.6 correlation given by
in which ß is in ft–1, k is in md, and t is the tortuosity. Because Firoozabadi et al.1 do not report any values, the tortuosity of the sandstone was approximated as 2 following Carman.7 As Table 1 shows, the Liu et al.6 correlation can predict the inertial flow coefficients with two significant digits. This is within the accuracy of the pressure-squared function (i.e., Eq. 2 of Firoozabadi et al.1).
We used existing in-house data (Evans and Civan8) to demonstrate the effect of the core length on permeability and non-Darcy flow coefficients. A series of different length berea cores have been used to generate the pressure differential vs. flow rate experimental data at steady-state conditions. We then plotted these data for each different core length, and Table 2 shows the determined permeability and the non-Darcy flow coefficient values, which we then plotted against core length to determine the sensitivity owing to the core length. The results that Fig. 1 gives indicate that the core-length average permeability and the non-Darcy flow coefficient are dependent on the core length. We obtained the representative values of permeability and the non-Darcy flow coefficient by extrapolation of the core-length average values to the representative elemental core length for which these values reach the limiting values given, respectively, as
Equations 4 and 5
As can be seen, the representative elemental core length necessary for accurate measurements is simultaneously estimated to about 10 cm. One should be cautioned, however, that the extrapolated values obtained in this example represent the average k and ß values for the whole porous material, not the local values of k and ß at a selected location along the core.
Note that the errors caused by not using representative core lengths are not negligible. If, for example, a core of 2.54 cm length instead of the representative length of 10 cm had been used, there would have been an error of ×100=21% in the ß value and ×100=-15% in the k value according to the data presented in Fig. 1.