A Generalized Water Displacement Curve of Forecasting Oilfield Development Indexes
Authors: Qitai Yu 
Published: December 2007

Abstract
The water displacement curve (WDC) method is the most common way for China's water-flood fields to predict their development indexes. Four WDCs presented by the former Soviet Union scholars are now in extensive use in China: Maksimov curve, Sazonov curve, Cipachev curve and Nazalov curve. These curves are all common water displacement curves, each reflecting only one rule of water-cut rise. Based on our theory of generalized water displacement curve (GWDC), where the curve can reflect various rules of water-cut rise, we have presented a new GWDC expression:
lgN p=a-blg(Lp/Wp) (Np for cumulative oil production, Wp for cumulative water production, and Lp for cumulative liquid production) We took three old oilfields as examples that represent different rules of water-cut rise. These examples are Renqiu Oilfield, Yangsanmu Oilfield and Shayi Pool of Pucheng Oilfield. We used the new GWDC and the four common WDCs respectively for calculation. As the comparison of these results demonstrates, the new GWDC has many advantages: It has a simple expression, its parameters are easy to calculate, the prediction with it is accurate, it has a good applicability, and its property is far better than that of the common WDCs.

Preface
A water displacement curve (WDC) is a statistical empirical formula for relation between the cumulative oil production, cumulative water production and cumulative liquid production of a waterflood field. The WDC method is the main way for China's water-flood fields to predict their development indexes. It was founded and presented by the former Soviet Union scholar Maksimov (M, , M,a K C II M O B) in his study of the water-displacement dynamic data from the October Oilfield. Its expression is as follows: (1)Np=a+blgWp
Because of the use of cumulants in it, Formula (1) is called the integral form of WDC.
In 1971, the U.S. scholar E.H. Timmerman founded the relation between Np and OWR (i.e. (1-fw)/fw)2. He said, this relation was derived from the water-flood fields in Illinois, and it could match the dynamic conditions of most water-flood fields in Oklahoma, Kansas, Texas, Wyoming and Montana. This relation is as follows:Equation (2)
Formula (2) can be called the differential form of WDC, because differentiating Formula (1) with the time t can lead to it. Please see Appendix A for the derivation.
Mainly due to the researches conducted by Chinese, former Soviet Union and Russian scholars, 40 WDC expressions have been presented up till now3,4,5,6,7. In 1990, we screened out four WDCs from the literature8, namely Maksimov curve 1, Sazonov curve9, Cipachev curve10 and Nazalov curve11; and recommended the crude oil viscosity limits for the use of these curves in oilfields. Since then, these curves have been in extensive use for China's water-flood fields in their development index predictions. Unlike in China and Russia, the integral form of WDC is seldom used in U.S., where the extensively used, instead, are the differential form, i.e. Formula (2), and the following relation deduced and presented in the literature 12.Equation (3)
The fw~R* relation can typically express the water-cut rise rule of a water-flood field. Different WDCs reflect different of water-cut rise, respectively. In order to compare them for the difference of water-cut rise rules, you can set the same initial condition: if R*=0, fw=0 or a number approaching 0 (e.g. fw=0.02), as well as the same ending condition: if R*=1.0, fw=0.98 (In China, Np which corresponds to fw =0.98 is defined as NR, the technical recoverable reserve). From WDCs, you can derive their corresponding fw~R* relations. For example, the following fw~R* relation can be derived from Formula (1): (Please see Appendix B)