Drilling and Production-Equipment, Methods and Materials - Determining Friction Factors for Measuring Productivity of Gas Wells

The theoretical background for calculating friction factors for flow in gas wells by two methods is presented. The first method, requiring pressures, temperatures and specific volumes of the flowing fluids at various depths in the well bore, shows how the mechanical-energy-balance equation for vertical flow may be graphically integrated over the actual path of the expansion of the fluid in the well. Thus, assumptions regarding the effective temperature and effective compressibility of the fluid in the well are avoided. The second method presents an equation derived on a basis of the assumptions that both the temperature and the compressibility are fixed at constant effective values throughout the flowing column of gas. The second method provides a convenient and practical means of calculating friction factors for gas wells and lends itself readily to the problem of calculating subsurface pressures in a flowing gas well. The application of both methods to actual test data taken on a flowing gas well is illustrated in the paper. INTRODUCTION As friction factors for the producing strings of flowing gas wells cannot be measured directly and must be calculated from flow-test data, study of the methods of arriving at friction factors is a necessary adjunct to understanding the characteristics of flow in gas wells. There are two methods of calculating friction factors for gas wells; they differ from one another mainly in the treatment of the path of expansion of the fluid in the well. In the flowing well, the energy consumed in lifting the fluid from the bottom to the top of the well, overcoming the friction between the moving fluid and the pipe walls, and increasing the velocity of the fluid as it flows up the producing string is supplied by expansion of the flowing fluid. The available energy is determined by the expansion of the fluid that follows a path determined by conditions of temperature, compressibility and phase changes of the fluid during the expansion. A means of evaluating the available energy in a flowing gas well and determining the proportion of the available energy used in lifting the gas, overcoming friction, and increasing the velocity of flow is developed in this report. Knowing how much energy is consumed in overcoming friction makes it possible to calculate friction factors for given flow rates in given sizes of pipes in wells. Friction coefficients, as used in this report, are dimension- less proportionality multipliers used in the flow equations to satisfy the equality between the terms of the equation. The square root of the reciprocal of the friction coefficient is termed the friction factor. It has been known for many years that friction factors may be computed directly from mathematical formulas based on certain assumptions regarding the temperature and compressibility of the moving fluid in a well. A general equation is presented in this report without assumptions for vertical flow of fluids and a conventional-type equation is derived on the basis of assumptions that fix the temperature and compressibility at constant values. Accurate pressures at the sand faces in wells are required in the method of determining the productivity of gas wells, as outlined by Rawlins and Schellhardt1 in Bureau of Mines Monograph 7. Where measurements are not made with subsurface-pressure gauge; or static gas columns are unavailable, flowing pressures customarily are calculated at the sand face in the well by the use of the well-known Weymouth formula1. Natural gas engineers have realized that errors introduced by the use of friction factors as given by the Weymouth formula are relatively unimportant in testing low-capacity gas wells; they also know that such factors are important considerations in testing large-capacity gas wells. Accordingly. present research on the productivity of gas wells at the Petroleum Experiment Station of the Bureau of Mines, Bartlesville, Okla., is being directed toward measurement of the pressure loss due to friction in flowing gas wells. It is beyond the scope of this report to show how friction factors vary with rate of flow and in pipes of different diameters, as it is intended only to develop and illustrate the use of mathematical expressions for calculating friction factors from flow-test data. The equations presented apply only to turbulent flow in circular pipes. ENERGY RELATIONS FOR FLOW OF FLUIDS2 The concept of conservation of energy is usually the basis of any study of fluid flow through vertical pipes as in gas wells, horizontal pipe lines, or orifices. In deriving equations, the following symbols are used: