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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).
Quite a number of problems that cannot be bandied economically with the methods usually applied in reservoir engineering can be solved by the theory of integral equations. The main purpose of this paper is to draw attention once more to the usefulness of integral equations. Their benefit is shown by considering a typical example; namely, the behavior of a horizontal multilayered system of oil reservoirs bounded by aquifers.
A number of reservoir engineering problems involving the flow of compressible fluids in porous media are so complex that they cannot be solved analytically, yet the usual numerical techniques require excessive computer time. An example of such a problem is simultaneous production from a horizontal multilayered system of production from a horizontal multilayered system of separate oil reservoirs, each reservoir being bounded by its own aquifer but sharing production wells with the other reservoirs. The difficulties in obtaining an analytical solution to this problem are insuperable; consequently, Tempelaar-Lietz and Lefkovits et al have restricted their studies to particular aspects under simplified conditions. The modern approach is to simulate the flow problem numerically. This implies that the differential equations that describe the problem are approximated by finite-difference equations, problem are approximated by finite-difference equations, which are subsequently solved by a computer. As both the aquifers and the reservoirs must be simulated, the numerical calculations require considerable computer time and storage capacity. In this paper it is shown that problems involving the influx of fluids (e.g., fluid influx from pay zones into wellbores, or water influx from aquifers into reservoirs) can often be described by a system of intergral equations that require much less computer time for solution. These integral equations can be derived only if the motion of the inflowing fluids can be described by linear and homogeneous differential equations, so that the superposition principle applies. The fluid influx can then be principle applies. The fluid influx can then be calculated from integral expressions that contain standard functions, This obviates the need for numerical simulation of those parts of the reservoir from which the influx originates. The standard functions are the cumulative influx as a function of time for a unit pressure drop at the influx boundary, and the pressure drop at the influx boundary as a function of time for a unit influx rate. These functions are interrelated so that if one is known the other can be calculated. For homogeneous and isotropic flow regions of linear or radial geometry, the standard functions have been evaluated in dimensionless form by van Everdingen and Hurst. For other flow regions, they can be evaluated with an auxiliary computer program. The integral equations presented here are by no means new. They are special cases of Duhamel's theorem (see also Carslaw and Jaeger, p. 30), and one of them is explicitly formulated by Samara in his analysis of pressure decline. The simplest form of our integral equations is
t y (t) . y (t) = (t) + K(t,0) . y (0) . d0, 0
. . . . . . . . . . (1)
where the functions v(t), (t) and K (t, 0) are known, and y(t) must be determined. The equation is termed "integral" because y(t), the function to be determined, also appears under the integral sign. The known function K(t, 0), which depends not only on the current time variable t, but also on the auxiliary variable 0 in the interval O 0 t, is known as the kernel of the integral equation. In many of our problems it is a function of t - 0 only; i.e., of the Faltung type. Moreover, it often becomes infinite if 0 approaches t. However, the improper integrals appearing in this paper can be made proper by transformation with a Dirichlet formula, thus avoiding the evaluation of improper integrals.
An examination is made of the subject matter of the science of reservoirengineering. It is seen that of the several remaining unsolved aspects, theestimation of reserves and recovery presents a problem of practical importance.This problem, as well as many others encountered in the study of reservoirengineering, is of such complexity that rigorous analytical solution is notpossible. In special cases approximate solutions are possible but these usuallygive information of only qualitative significance. The conclusion is thus madethat study of these complex reservoir engineering problems by models might bean acceptable substitute procedure.
Therefore, the basic postulates of dimensional analysis are reviewed, andthe theory of model construction is described. Model types are classified asphysical and analog models, scaled and unscaled models, and exact andverification models. Examples of these types are given, followed by a generaldiscussion of the application of models for the solution of reservoir problems.Reference is also made to the limitations of model work and to the practicaldifficulties encountered in the laboratory construction of models. Finally, anew type of model called the fluid mapper, which heretofore has not beendescribed in the literature of reservoir engineering, is discussed both withregard to the underlying theory and with regard to possible specificapplications.
The general conclusion is drawn that some of the advantages of model studyhave been largely overlooked by reservoir engineers. Specifically, it isconcluded that model study provides the most suitable procedure for evaluationof reserve and recovery factors. It is also concluded that the fluidmapper model technique offers promise of solving certain reservoir engineeringproblems which heretofore have defied analytical description.