Material Balance Calculations with Water Influx in the Presence of Uncertainty in Pressures
Authors: Mcewen C.R. 
Published: December 2007

Abstract
This paper presents a technique for calculating the original amount of hydrocarbon in place in a petroleum reservoir, and for determining the constants characterizing the aquifer performance, based on pressure-production data. A method for doing this based on a least-squares line-fitting computation was proposed by van Everdingen, Timmerman and McMahon in 1953. We found that their method would not work when there is error in the reservoir pressure dataeven moderate error. The technique presented here appears to give reasonable answers when pressure data are uncertain to the degree expected in reservoir pressure determinations. The major change introduced in the present analysis is to limit the least-squares line-fitting to yield only one constant the amount of hydrocarbon in place. The water-influx constant is then taken as proportional to the oil (or gas) in place. The constant of proportionality can be computed from estimates of effective compressibility and reservoir water saturation. It is also pointed out that the commonly used least-squares analysis assumes all of the uncertainty to be in the dependent variable. The material balance should be arranged so that this condition is fulfilled if correct inferences are to be made from statistical calculations. Examples are shown of the application of the new technique to gas reservoirs both hypothetical and real and to the oil reservoir example of van Everdingen, Timmerman and McMahon.

Introduction
The amount of hydrocarbon originally in place in a petroleum reservoir can be estimated by means of the material-balance calculation. Simultaneous observations of pressure and amounts of produced fluids are required, together with the PVT data applicable to the reservoir fluids. If water encroachment is occurring, it is desirable to try to infer the behavior of the aquifer, as well as the original hydrocarbon in place, from the pressure-production data. This imposes additional demands on the method of calculation, and uncertainty in the data can result in large uncertainty in the answer. In addition, if the size of a gas cap is to be established, the whole problem becomes indeterminate, as pointed out by Woods and Muskat. Brownscombe and Collins simulated a gas reservoir and its aquifer on a reservoir analyzer and derived quantitative information on the effect of uncertainty in pressure and aquifer permeability on computed gas in place. Among the various techniques of estimating the performance of an aquifer, the method of van Everdingen and Hurst, based on compressible flow theory, seems to have been the most generally successful (see Ref. 4, for example). In this paper we shall confine ourselves to their representation of the aquifer. In 1953, van Everdingen, Timmerman and McMahon introduced a statistical technique for deriving the amount of oil originally in place and the parameters which describe the aquifer. (We shall refer to this technique as the "VTM method", as suggested by Mueller.) Their example reservoir had no gas cap. It has been our experience that the VTM method gives a reasonable answer when the data are very accurate, but that inaccuracy (particularly in pressure) can cause the method to break down. The effect was first observed in gas reservoirs, but has since been seen in oil reservoirs also. In this paper we present another statistical method which has been successful in achieving a reasonable answer where the VTM method has failed. In the new method, one less parameter is derived from material-balance computations. It is assumed that values can be established for effective compressibility in the aquifer and reservoir water saturation independently of the material-balance calculation. The water-influx constant can then be obtained from these data and the quantity hydrocarbon in place.
SPEJ
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