Normalised Plot - A Novel Technique for Reservoir Characterisation and Reserves Estimation
Authors: Kabir Mohammad I. 
Published: December 2007

Abstract
A novel normalised plot technique is developed for reservoir characterisation and reserves estimation. This method is based on the Buckley-Leverett and Welge's fluid displacement theories. The theories suggest that under ideal conditions a normalised plot of oil recovery as a function of water or gas (displacing phase) throughput is independent of rate, drainage volume, and geometry of the system. This implies that in a homogeneous reservoir, well by well normalised recovery plots should collapse to one curve. In reality, well performance is influenced by reservoir heterogeneity, drive mechanism (bottom vs edge drive), gravity, and well locations. Therefore, the normalised curves do not often match and that leads to reservoir dynamic characterisation. The normalising factor, sweep volume, is related to the recoverable reserves (RR) that we seek to estimate. This paper presents the theoretical background of the technique and illustrates its application by presenting two field examples.

Introduction
Use of decline curve analysis for reserves estimation is a common practice in the petroleum industry. Reference 3 presents a summary discussion of the decline methods. However, these techniques are not based on any physical theories. Furthermore, decline analysis assumes constant operating conditions (choke size, artificial lift, etc), which is hardly ever met in field production operations. Experience has shown that analysis by exponential decline generally yields a conservative forecast (also known as a 'buyer's forecast'). On the other hand, hyperbolic extrapolation often results in an optimistic estimate ('vendor's forecast'). The applicability of the above techniques is basically a matter of convenience. The x-cut plot analysis, a second method, is based on fractional flow theory, but the assumption that a semi-log plot of permeability versus saturation data would be a straight line is not valid in all reservoirs and fluid flow systems. A third approach used in the industry is cumulative oil versus cumulative liquid (representing water influx in a water drive reservoir) production plot. Since drainage volumes and consequently recoverable reserves generally vary from well to well, such plots cannot be compared on a one to one basis. A meaningful comparison is achieved if the above conventional cumulative plots are normalised by their respective reservoir drained volumes (hence the name normalised plot).

Mathematical Basis
In 1942 Buckley and Leverett presented the basic equation for describing immiscible displacement in one dimension (also known as Frontal Advance Equation). In 1952 in another paper of fundamental importance, Welge enlarged upon Buckley and Leverett's work and derived an equation that relates the average displacing fluid saturation to that saturation at the producing end of the system. Welge also determined that pore volumes of cumulative injected fluid is equal to the inverse of the slope of the tangent on fractional flow curve. The above mathematical developments have become the basis of performance predictions of incompressible fluids in one- dimensional systems.
In reality, reservoir fluids (particularly gas) are compressible and flow dynamics hardly occur in a linear system. Kern has shown application of the Buckley and Leverett equation to reservoir geometry to predict gas flooding performance. A generalised form (variable cross-section) of Frontal Advance displacement developed by Latil shows that the Buckley-Leverett theory is applicable in a reservoir as long as the displacement takes place in a porous medium whose geometry leads to the equipotentials and isosaturations being superimposed. The pore volume traversed by iso-saturation in a variable geometry reduces to a distance travelled in a linear system.
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