Temperature Transient Tests: Modeling, Interpretation, and Nonlinear Parameter Estimation

Abstract

This study presents semi-analytical and analytical solutions based on a coupled transient wellbore/reservoir thermal model to predict temperature transient measurements made under constant rate and bottom-hole pressure production as well as variable rate and bottom-hole pressure production histories in a vertical or an inclined wellbore across from the producing horizon or at a gauge depth above it. Slightly compressible, single-phase, and homogeneous infinite-acting single-layer geothermal reservoir system is considered. The models account for Joule-Thomson heating/cooling, adiabatic fluid expansion, conduction and convection effects both in the reservoir and wellbore. The transient wellbore model accounts for friction and gravity effects. The solutions of the analytical and semi-analytical reservoir models are verified by use of a general-purpose thermal simulator. Wellbore temperatures at a certain gauge depth are evaluated through a wellbore thermal energy equation coupling the reservoir temperature equation. It is shown that unlike the “sandface” temperature measurements made close to the producing zone, the temperature measurements made at locations significantly above the producing horizon are dependent upon the geothermal gradient and radial heat losses from the wellbore fluid to the formation on the way to gauge and hence more difficult to interpret for well productivity evaluation and reservoir characterization. The solutions are used as forward models for estimating the parameters of interest by nonlinear regression built on a gradient-based maximum likelihood estimation (MLE) method. A methodology is proposed to obtain good initial guesses of parameters which derive the MLE objective function to have reliable optimized estimates. It is based on straight line analyses of flow regimes (as derived from analytical solutions) identified on log-log diagnostic plots of bottom-hole pressure- and temperature-derivative data. The results show that the rock and fluid flow/thermal properties of a nearby skin zone and reservoir beyond skin zone can be reliably estimated by regressing on temperature transient data jointly with pressure transient data in presence of noise.