A time-convolution approach for modeling heat exchange between a wellbore and surrounding formation

In oil, gas, and geothermal energy production, as well as geological CO2 storage, the target formation is typically deeper than 1000 meters. As a result, associated wellbores have a large heat exchange area with the surrounding formation. Large gradients and temporal variations in temperature induced by the injection and production of fluids require accurate and efficient ways to calculate the heat exchange between fluids in the wellbore and the formation. One way to calculate this heat exchange is to fully discretize and numerically model the formation that surrounds the wellbore. However, because only the energy equation needs to be solved (i.e., there is no fluid exchange between the cased wellbore and the formation), this approach is computationally inefficient. In this work, we propose a time-convolution method, where only the wellbore is fully discretized, and heat exchange between fluids in the wellbore and the formation is calculated using semi-analytical solutions of radial conductive heat flow. The time-dependent temperature evolution in the wellbore is calculated numerically using a wellbore simulator for non-isothermal, multiphase fluid mixtures. At each time step, radial heat transfer with the formation is calculated by superposition of analytical solutions of heat flow that are dependent on the temperature differences between subsequent time steps. This coupling scheme is implemented in the TOUGH2 suite of reservoir simulators. To verify the proposed semi-analytical method and demonstrate its applicability, we present examples and compare them to full numerical solutions.