Graphical Interpretations for Gas Material Balances

Abstract

A materials balance for a volumetric gas reservoir leads to the well known conclusion that p/z should be a linear function of the cumulative gas produced. Data needed to prepare this balance usually take the form of tabulations of mean reservoir pressure and corresponding cumulative gas production in chronological sequence. It is then necessary to find corresponding gas law deviation factors, z's, compute p/z, and make the plot of p/z vs Gp. If the data form a straight line, it is possible to forecast pressures after any future cumulative gas production by finding the p/z for some future value of Gp. Thus performance p/z for some future value of Gp. Thus performance matching and production forecasting require finding p/z corresponding to some p, or finding the value of p p/z corresponding to some p, or finding the value of p that corresponds to a specific p/z. To aid these calculations, a large-scale plot of p/z vs p is often constructed. But if a plot of z vs p is available (as it usually is), this step is not necessary, and it is also not necessary to calculate p/z. This can be seen with the aid of Fig. 1. Fig. 1 is a conventional plot of z vs p that has been rotated 90 degrees to illustrate the basis of a simple graphical interpretation. The heavy line presents the relationship between z and p for a particular reservoir temperature and gas composition. First, assume that we wish to know the value of pressure that corresponds to a specific value of p/z, say 5,000 psia. The statement p/z = 5,000, or p = 5,000z represents the equation of a straight line on a graph of p vs z. The line passes through the point (0, 0), and also through the point where p = 5,000 and z = 1. This line is shown as the light line on Fig. 1. The intersection of the light line and the heavy line provides both the p and z that correspond to a p/z value provides both the p and z that correspond to a p/z value of 5,000. Now let us assume we want the p/z value corresponding to a given pressure p. Locate the pressure on the heavy line. Pass a straight line from the pressure on the heavy line. Pass a straight line from the origin (0, 0) through the point, and read p/z from the pressure scale where the line intersects the line z = 1. Thus it is not necessary to determine z to find p from p/z, or p/z from p - if a graph such as Fig. 1 p from p/z, or p/z from p - if a graph such as Fig. 1 is available. To summarize, any straight line through the origin on Fig. 1 will represent some constant value of p/z. The intersection of such a line with an appropriate z curve provides a point (p, z) corresponding to the particular value of p/z. This is a graphical solution particular value of p/z. This is a graphical solution of two simultaneous equations. This method could be used to find an empirical pressure coordinate scale for a plot of p vs Gp that would yield a straight line satisfying a gas materials balance. Often, z vs p plots prepared for field work have limited z and p scales such that the graphs can be read with good accuracy. An example is given in Fig. 2. Conventional alignment of the coordinate axes is used in this case. The point at the origin (0, 0) is not on the graph in this case. The graph can still be used readily to find values of p corresponding to p/z values. Again, assume we wish the pressure p corresponding to a p/z value of 5,000 psia. Plot the point p = 5,000 on the line z = 1. P. 837