Semi-analytical solutions for multilayer reservoirs

Abstract

This work provides the development, validation, and application of new approximate (semi-analytical) solutions for the wellbore pressure and fractional flowrate responses of commingled layered reservoirs - without interlayer crossflow (crossflow is only permitted in the wellbore, not in the reservoir). For simplicity, our solutions are presented and compared in terms of dimensionless variables (these variables are specifically formulated for the multilayer reservoir case). Our developments utilize the traditional approach of working in the Laplace domain (and for one specific application, in the real domain). When working in the Laplace domain, we require the computation of the pressure and rate responses (total response as well as individual layers) via inversion of the Laplace domain solutions. In general, the Laplace domain solutions for both the single and multilayer reservoir cases cannot be inverted analytically, and are typically inverted via numerical means. In this study, we develop, validate, and present five new approximate solutions for the case of a multilayer reservoir system - these solutions are: [ Solution p[wDj(tD)] Description 1 a[j] Constant p[wDj(tD)] Case 2 a[j tD] Linear p[wDj(tD)] Zero Intercept Case 3 aj + bj tD Linear p[wDj(tD)] Case 4 a[j] exp(bj tD) Exponential p[wDj(tD)] Case 5 General formulation: p[wsD(tD)] = [] Total Pressure/Rate Averaging} Our work illustrates the validity of each approximate multilayer solution. In particular, the solutions based on the assumption of a constant p[wDj(tD)], linear p[wDj(tD)] zero intercept, and exponential p[wDj(tD)] work well for the prediction of the wellbore pressure response. However, the solutions for the "constant p[wDj(tD)]" case, the "linear p[wDj(tD)]" (zero intercept) case, and the "exponential p[wDj(tD)]" case may not accurately predict the layer rate responses during the transition period from transient to boundary-dominated flow conditions. The solutions based on the "linear p[wDj(tD)]" case tend to work very well for the prediction of the total pressure response and the layer rate responses - for all flow regimes. The "total pressure/rate averaging" (i.e., the TPRA) solution generally works well - except during the transition period. In particular, the TPRA method appears to work well even for cases where the pressure transient in each layer reaches the external boundary for each layer at significantly different times.