The Effect of the Tin1e Rate of Change of Momentun1 On Bottom-Hole Pressure In Flo,ving Gas Wells
Authors: Dranchuk P.M. , McFarland J.D. 
Published: March 2010

Abstract
A technique is presented for numerically integrating the differential form of the force-momentum balance in order to calculate pressure in flowing gas wells for which the temperature profiles are known. The method is similar to that suggested by Young, except that integration is effected in a manner such that both production and injection cases may be treated with one equation which has no singularities. The formulation is such that allowance may be made for the variation with depth of the compressibility factor and temperature in the kinetic energy term. Furthermore, the method employs the Ranger Kutta-Merson variable-step size procedure which reduces the number of steps required while increasing accuracy.

Introduction
IF THE TEMPERATURE DISTRIBUTION in a flowing gas stream is known. the pressure distribution in the stream may be determined by integrating the differential form of the force-momentum balance" The integration may be effected by a variety of methods! most of which neglect what is referred to as the time rate of change of momentum, kinetic energy or ulu term.
In 1967, Young(l) developed a method of calculation which included the kinetic energy term. In his work, he concluded that the effect of this term on the bottom-hole pressure was insignificant for normal field situations, but that it could be important under certain conditions. However, an examination of his paper indicated that the differential equation he had employed might be incomplete or incorrect, or both, because it did not disclose the manner in which the variations with depth of compressibility factor and temperature were handled. Furthermore, his method of integration was such that in the case of injection the integral being evaluated could, under certain conditions, become singular so that special treatment was necessary Consequently, the object of the study described herein was to derive the necessary differential equation, examine the nature of the udu term in detail and examine the behaviour of the equation for the injection case in order to remove the difficulty caused by the integral becoming singular.

The Force-Momentum Balance and Young's Method
One may arrive at an integral similar to that suggested by Young by starting with the force-momentum balance in the form:
(Equation Available In Full Paper)
If one chooses the usual values for P" T" and goo expresses pipe diameter as d in inches and multiplies both sides of the equation by the term (Equation Available In Full Paper), Equation 9 becomes
(Equation Available In Full Paper)
If a producing- well is to be examined, L is considered to increase in the direction of flow and S is positive so that
(Equation Available In Full Paper)
This is the same as Young's Equation 2, except that a negative rather than a positive sign precedes the term
(Equation Available In Full Paper)
On the other hand, when an injection well is considered, one must either let L increase in the direction of flow or require that Qo be signed and use QoQo rather than Qo2 in the friction loss term.