Using Parameter Estimation Techniques To Convert Centrifuge Data Into a Capillary-Pressure Curve

Abstract

Abstract Previously reported techniques for converting basic centrifuge data into a capillary-pressure curve have one serious drawback: they all involve the graphical or numerical differentiation of experimental data. The problems associated with the differentiation of experimental data can be avoided by using the parameter estimation techniques proposed here. The parameter estimation techniques proposed here. The purpose of this paper is to demonstrate The advantages purpose of this paper is to demonstrate The advantages of using parameter estimation techniques for obtaining a capillary-pressure curve from centrifuge data. Two parameter estimation methods for handling centrifuge data were investigated and compared with a modified form of Hassler's technique for interpreting such data. This investigation indicates that, while Hassler's method and the parameter estimation techniques were equally able parameter estimation techniques were equally able to generate the capillary-pressure curve from centrifuge data, the latter procedures are preferable since they use various integration preferable since they use various integration schemes and, hence, avoid the differentiation problems associated with previously reported problems associated with previously reported methods of data interpretation. Moreover, if the parameter estimation techniques are used, the data can be smoothed and the irreducible water saturation, displacement pressure, and capillary-pressure normalizing parameter can be estimated. Introduction The theory for converting experimental data obtained with a centrifuge into a capillary-pressure curve was developed by Hassler and Brunner. The basic equation used in the conversion is P2 a ci Sn(x) dxP S = cos ---- .....(1)ci n 2 × 21 - ----- sin aPci wherep 2P = ----- w (re2 − ri2), ci 2 p 2x = ------ w (re2 − r2), 2 and ricos a = ----- .re Hassler and Branner were unable to find an analytical solution to Eq. 1, but they demonstrated bow it could be solved by the method of successive approximations. Since this method is very tedious in application, Hassler and Brunner preferred using a simplifying assumption that amounts to setting ri equal to re. This assumption, which the authors considered to be reasonable provided the ratio ri/re was greater than 0.7, resulted in the equation P ciPci Sn = Sn(x)dx, 0 from which it follows that d Sn (Pci) = ------ (Pci Sn)................(2)dP Setting ri equal to re assumes that the length of the core is negligible compared with the radius of rotation of the core. Hoffman has shown that this assumption is unnecessary, and that Eq. 3 should be used to solve for the saturation at each speed level. 2 cos adSnS (P) = -------- (S + P ------)....(3)n ci 1+cos a n ci dPci Eqs. 2 and 3 can be solved by taking slopes of graphs of Sn and Pci Sn vs Pci. SPEJ P. 57