A Simple Technique To Determine the Unknown Constant in y or x for Producing y vs. x as a Straight Line in Reservoir Engineering Problems
- M.C. Aaron Cheng (Marietta College)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- May 1982
- Document Type
- Journal Paper
- 1,140 - 1,142
- 1982. Society of Petroleum Engineers
- 5.6.4 Drillstem/Well Testing, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc)
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It is common practice in some reservoir engineering and well-testing problems to plot a certain function A vs. another function B such that a straight line results. One well-known example is the graphical interpretation of the material-balance equation as a straight line. However, most of the time there is an unknown factor in either A or B. A trial-and-error procedure for estimating this factor until a straight line results is the method usually mentioned in the literature. This paper presents a simple technique for computing, the unknown factor directly.
Using common algebraic notations, a straight line has the equation y = mx + b.
A plot of y vs. x will give a straight line with slope m and intercept b. Also, if three (or more) pairs of data, (x1, y1),(x2,y2), and (x3,y3), are known,
y2 - y1 y3 - y1 --------- - --------- = m x2 - x1 x3 - x1
or y2 - y1 y3 - y1 --------- - --------- = 0. x2 - x1 x3 - x1
If y is comprised of a certain unknown constant, say c, this equation can be expressed as
f(c) = ----------------- - ---------------- = 0.......(1) x2 - x1 x3 - x1
Similar expression holds for x being a function of c.
Eq. 1 is the formulation of the common root-solving problem, where c is the root to be determined such that problem, where c is the root to be determined such that f(c)=0. Depending on the nature of the equation, the solution of f(c)=0 can be obtained with a closed type formula or, if this is not possible. with an efficient iterative root-solving algorithm such as the Newton's method.
Selection of Data Points
The current technique requires the use of three data points for formulating Eq. 1. If more than three points points for formulating Eq. 1. If more than three points are available, the selection of any three points from the given data would be adequate if all the data points are known to be correct. This is valid because any three correct data points would yield the same root c in Eq. 1. However, if it is not known which data points are correct, which three points to choose is a problem. One togical approach is to choose the two endpoints, (x1, y1) and (xn, yn) and the median or central point from the given set of n data points (x1, y1),(x2,y2),....., (xn, yn) to span the whole data set.
The following examples illustrate the current technique.
Example 1-Determination of Pressure Buildup Correction Factor C
The afterflow analysis of Russell for pressure buildup analysis requires the plot of p/(1-1/C t) vs. log t to be a straight line. The correction factor C is usually an unknown. Russell suggested that C be determined by trial and error until the plot is a straight line. The current technique instead solves for C directly. Dake's afterflow analysis data are used in this example and there are 15 data points available. Select the two endpoints and the central point (Table 1).
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