The modification of the Cullender and Smith equation presented here consists primarily of adding a gas-water ratio term, and a friction factor term as given by the explicit Jain-Swamee correlation. The results of this study show a reduction in the average error in predicting bottom-hole pressure of 3.4 percent for the flowing cases and 1.9 percent for the static case when the modified Cullender and Smith equation was used instead of the original Cullender and Smith equation. This study resulted in two major conclusions. First, modification of the Cullender and Smith equation produces a more accurate method of calculating static and flowing bottom-hole pressures. Second, using an apparent roughness of 0.0023 inches instead of an absolute roughness of 0.0006 inches will further reduce the RMS error for the flowing case.
In order to estimate the absolute open flow potential of a gas well, it is necessary to determine the static and flowing bottom-hole pressures. This is done either by actual measurement with a bottom-hole pressure gauge, or by calculation using wellhead pressure measurements. This paper is only concerned with the calculation of bottom-hole pressure from wellhead measurements. Currently, the Texas Railroad Commission recommends calculating bottom-hole pressure by the average temperature and compressibility method. However, as the name implies, this method assumes a constant gas compressibility factor determined from an assumed average temperature and pressure for the entire flow column. Although these assumptions are fairly accurate in shallow wells, this procedure results in more error as depth. temperature, and pressure increase. Aziz concluded that a more practical approach is that of Cullender and Smith, which treats the gas compressibility factor as a function of depth. However, the Cullender and Smith method was developed for dry gas wells in rough-turbulent flow with a absolute roughness of 0.0006 inches. Peffer, Miller, and Hill used a variation of the Cullender and Smith method to calculate bottom-hole pressure. They modified the gas gravity correlation to take into account condensate and/or water production, and used the Nikuradse friction factor correlation for rough-turbulent flow. The purpose of this study is to modify the Cullender and Smith equation to account for water production and employ a friction factor correlation to take into account smooth-turbulent and rough-turbulent flow at any absolute roughness. The data on 78 wells obtained from the Peffer, Miller, and Hills' paper is reported in Table 1. Unfortunately, information on hydrogen sulfide, carbon-dioxide, and nitrogen concentrations were not provided for wells 29 through 78.
Determination of the Modified Cullender and Smith Flowing Bottom-Hole Pressure Equation
The modification of the Cullender and Smith equation to take into account condensate and/or water production stems from the fundamental mechanical energy equation. This energy balance can be expressed for steady-state flow as shown in Eq. 1. (1)