Laplace Transforms of Solutions for a Radial Advection–Diffusion Equation to Simulate Aquifer Thermal Energy Storage

Abstract The radial advection–diffusion equation with thermal conductivity plays a role at the injection of warm water in aquifers. In this paper we generalize the well-known solutions of solute transport at a fully screened injection well by taking into account molecular diffusion. We translate the results into the terms appropriate for heat transport. Because the initial value for the extraction depends on the results of the injection attention is also given to solutions with arbitrary initial values. So, we present the semi-analytical solutions (Laplace transforms) for arbitrary initial data F ( r ) and several (Dirichlet, Neumann, Robin/Steklov) inhomogeneous type boundary conditions G ( t ) at the well radius $$r=r_{0}$$ r = r 0 during injection or extraction. The attention given for arbitrary initial values is new in this setting. In the given representation Kummer functions turn up. We find the existing solutions for pure dispersion back if our dimensionless parameter S in our expressions goes to 0. S measures the ratio between the diffusive transport by the advective transport and is equal to $$1/P_{e}$$ 1 / P e , where $$P_{e}$$ P e is the Péclet number. We also allow a rest period after the injection and present the appropriate solution. In the periods of rest with no extraction or injection the Péclet number is 0 and we find for $$S\\rightarrow \\infty $$ S → ∞ in the limit the solutions for the radial Laplace equation. For various cases we use a Matlab script to invert the Laplace transforms numerically to find the solution in the time domain. We exhibit some graphs for various values of S , and a real world case for heat injection, rest and extraction. For all examples we compare the results with those for the pure dispersion case. In the whole paper we use dimensionless variables. The mathematical derivation of the results and some extra examples can be found in the Supplementary Material.