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The evolution of strong transients of temperature and pressure in two adjacent fluid‐saturated porous rocks is described by a Burgers equation in an early model of Natale and Salusti (1996). We here consider the effect of a realistic intermediate region between the two media and infer how transient processes can also happen, such as chemical reactions, diffusion of fine particles, and filter cake formations. This suggests enlarging our analysis and taking into account not only punctual quantities but also “time averaged” quantities. These boundary effects are here analyzed by using a “memory formalism”; that is, we replace the ordinary punctual time‐derivatives with Caputo fractional time‐derivatives. We therefore obtain a nonlinear fractional model, whose explicit solution is shown, and finally discuss its geological importance.