An Analysis of a Volumetric Balance Equation for Calculation of Oil in Place and Water Influx
Authors: Tehrani D.H. 
Published: December 2007

Summary
When fluids are withdrawn from a petroleum reservoir, the space left behind is filled partly by the expansion of the remaining fluids and rock and partly by the influx of water from a contiguous aquifer, if it exists. The volumetric balance equation (VBE) is an expression of this same statement. Its simplified form isEquation 1
When sufficient historical data on X and Z are available, various functions for Y can be tried; using the technique of least squares, a set of values can be calculated for Na and ewc.
It has been convenient to write Eq. 1 in the following form:Equation 2
The advantages of this form are that it has only two variables, so least-squares calculations are easier for it, and that the values of Y/X and Z/Xcan be plotted graphically, so that a linear trend can be visually examined. The disadvantage is that the equation has a low resolving power and can produce erroneous answers. Nevertheless, most authors use the form of Eq. 2.
Eq. 1 may be written in many different forms, all of which are algebraically equivalent to each other; however, when the method of least squares is applied to them, they will produce different results.
This paper shows that the best form of the VBE for calculation of the original active oil-in-place (OIP) and water influx constant is the form of Eq. 1. It is also shown that the least-squares calculation based on minimizing the sum of the squared deviations of the calculated oil pressures from the observed pressures is equivalent to carrying out the least-squares method on Eq. 1.

Introduction

The Volumetric Balance Equation
The VBE was first introduced in general form by Schilthuis in 1935.1 It can be written as follows:Equation 3
The left side is the reservoir volume of the total withdrawals, shown here by Z. The first term on the right side (NBti) is the original OIP in reservoir barrels multiplied by the unit expansion [(Bt/Bt-1)+. . .]. The former is shown here by Na and the latter by X. The second term on the right side is the water influx, which can be represented by an influx constant, ewc, multiplied by a water influx function, Y. Eq. 3 can thus be written in the simple form of Eq. 1:Equation
Dependent on the shape, type, and flow characteristics of the aquifer, various equations have been introduced for the Y term in Eq. 1.1–5

Least-Squares Method
Let us indicate one set of data points by Xi, Yi, and Zi, in which the subscript i varies from 1 to n. If we move all terms of Eq. 1 to the right side and replace X, Y, and Z by Xi, Yi, and Zi, we getEquation 4
The least-squares method defines the best set of values of Na and ewc as the set that corresponds to the minimum of sum square of z - i.e., the minimum ofEquation 5
where SSD is the sum squared deviation.

The Volumetric Balance Equation
The VBE was first introduced in general form by Schilthuis in 1935.1 It can be written as follows:Equation 3
The left side is the reservoir volume of the total withdrawals, shown here by Z. The first term on the right side (NBti) is the original OIP in reservoir barrels multiplied by the unit expansion [(Bt/Bt-1)+. . .]. The former is shown here by Na and the latter by X. The second term on the right side is the water influx, which can be represented by an influx constant, ewc, multiplied by a water influx function, Y. Eq. 3 can thus be written in the simple form of Eq. 1:Equation
Dependent on the shape, type, and flow characteristics of the aquifer, various equations have been introduced for the Y term in Eq. 1.1–5

Least-Squares Method
Let us indicate one set of data points by Xi, Yi, and Zi, in which the subscript i varies from 1 to n. If we move all terms of Eq. 1 to the right side and replace X, Y, and Z by Xi, Yi, and Zi, we getEquation 4
The least-squares method defines the best set of values of Na and ewc as the set that corresponds to the minimum of sum square of z - i.e., the minimum ofEquation 5
where SSD is the sum squared deviation.