Modeling and analysis of temperature transient sandface and wellbore temperature data from variable rate well test data

In this paper, we present new semi-analytical and analytical solutions based on coupled transient wellbore/ reservoir thermal models to predict temperature transient measurements made under variable rate production history 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 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 reservoir model includes the effect of heat losses to over-burden and under-burden strata. The transient wellbore model accounts for friction and gravity effects. The solutions of the semi-analytical reservoir model 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. Our results show that unlike pressure transient data, the variable-rate temperature transient data cannot be transformed to an equivalent constant-rate temperature data by use of linear superposition. It is shown that the temperature-derivative data taken with respect to the elapsed time for a given period, plotted against elapsed time better identifies flow regimes exhibited by the transient temperature data recorded under variable rate history. Furthermore, we show that the temperature measurements made at locations above the producing horizon are dependent upon the flow rate, geothermal gradient, and radial heat losses from the wellbore fluid to the formation on the way to gauge. The temperature responses during early times of drawdown and buildup periods are mainly affected by the thermal and wellbore storage effects and do not exhibit the early-, intermediate- and late-time infinite acting flow regimes (IARFs) that would be exhibited by the sandface temperatures unless the gauge distance is not close to the producing horizon. The sensitivity of temperature recorded above the producing zone to reservoir permeability and skin decreases with gauge placed far away above the producing zone. Synthetic test examples are presented to demonstrate the use of the proposed models and methods.