Simultaneous Estimation of Aquifer Parameters and Original Hydrocarbons in Place From Production Data Using Numerical Inversion of Laplace Transform

A new method is presented for the simultaneous estimation of original hydrocarbons in place OHIP and aquifer parameters for a circular aquifer surrounding a hydrocarbon reservoir. Water influx data are represented by the van-Everdingen and Hurst (VEH) unsteady state solution. In addition to the initial oil and/or gas in place, the aquifer parameters are also estimated. These parameters include the water influx constant B, the relative aquifer size ReD , and the time adjustment factor c which transforms real time t into dimensionless time tD.. The parameters B and c can be used to estimate the storativity hφCt and the transmissibility kh/μ of the aquifer. The method of least squares is applied to the material balance equation to estimate the parameters resulting in a non-linear regression problem. Because the solution in Laplace space is simpler than the solution in the real time domain, numerical inversion of Laplace transform was used to obtain the partial derivatives of the VEH solution with respect to aquifer parameters needed for least squares method. Nonlinear regression analysis using numerical inversion of Laplace transform is applied. The Levenberg method was used for parameter estimation to guarantee convergence when the initial guess is not close enough to the correct solution.. The model is linear with respect to original hydrocarbons in place N , Gi ,and the water influx constant B, but is nonlinear with respect the dimensionless aquifer size ReD and time adjustment factor c which transforms real time t into dimensionless time tD.. Assuming the values for c and ReD enables the calculation of N , Gi ,and B and hence the estimation of the sum of squares of the residuals .Maps of the sum of squares of the residuals are generated and displayed to show regions of maxima and minima in the aquifer parameters space. Maps explain the non-uniqueness of the solution usually encountered in nonlinear least squares. The knowledge of the approximate values of some of the aquifer parameters from geological and engineering data will help choosing the proper initial guess to achieve convergence to the correct solution Introduction The estimation of initial hydrocarbons (oil and/or gas) in place OHIP and the prediction of future reservoir performance are of great importance for the development of these reservoirs. Volumetric methods based on geological and seismic .data can be used to estimate the OHIP. With a reasonable estimates of OHIP, the material balance equation MBE can be used to predict future reservoir performance for volumetric reservoirs (no water influx). For non-volumetric (water drive) reservoirs, the aquifer parameters are needed. If enough production data are available for a given reservoir, the MBE can be used to estimate the OHIP and aquifer parameters. For volumetric reservoirs (no water influx), the MBE is linear the parameters N and G. In this case, the MBE represent an equation of a plane. Havlena and Odeh showed how it can be arranged as an equation of a straight line by grouping production and pressure dependent terms. Tehrani, however, indicated that regression should be performed on the original (non-grouped) MBE to preserve the physical meaning of regression variables. In these cases, linear or multiple regression analysis by the method of least squares is used to estimate the original oil in place N and the gas cap ratio m for oil reservoirs or the original gas in place Gi for gas reservoirs. For non-volumetric (water drive) reservoirs the material balance equation can be used to estimate both OHIP and aquifer parameters. An aquifer model describing water influx from the aquifer into the reservoir is needed. In most field cases, such model is nonlinear. The van-Evedingen and Hurst's (VEH) unsteady state model is an exact analytical solution for circular aquifers with homogeneous properties. The model is linear with respect to the water influx constant B, but nonlinear with respect the dimensionless aquifer size ReD and time adjustment factor c which transforms real time t into dimensionless time tD. It is therefore obvious that linear regression can not be used directly to estimate both OHIP and aquifer parameters B, c, and ReD. To overcome the nonlinearity problem, most investigators used some kind of a trial and error approach. In such cases , values for aquifer parameters are assumed and linear regression is performed to estimate N and m for oil or Gi for gas reservoirs. The standard deviation or the sum of squares of SPE 104603 Simultaneous Estimation of Aquifer Parameters and Original Hydrocarbons in Place From Production Data Using Numerical Inversion of Laplace Transform Noaman El-Khatib, SPE, Sudan U. for Science & Technology