The unsteady fluid flow in a three-layer porous medium is numerically investigated and is an important and topical problem. An analytical solution of the equation for the pressure fluid layer is obtained on the basis of the theory of elastic regime, taking into account the overflow from the coating and the low-permeability layer into the low-permeability bulkhead and external sources that greatly affect the liquid level change. In the main bounded aquifer only horizontal liquid migrations prevail, and in the cover and low-conductive layers only vertical migrations prevail, allowing for horizontal components of the flow rate to be omitted here. Evaporation from the surface of a liquid in a porous medium is considered. Evaporation from the surface of a liquid in a porous medium significantly affects the distribution of overflows in the layers and the distribution of the liquid level in a porous medium. Therefore, when designing vertical drains for enhanced oil recovery in multilayer reservoirs and designing fluid flows in reservoirs, evaporation must be taken into account. The computational experiments have established that the dynamics of changes in the liquid level in a porous medium depends significantly on the evaporation parameter. With an increase in its value, the liquid level in a porous medium decreases proportionally over time. The accuracy of the numerical solution using the balance equation showed that the error does not exceed 1.3%.