In this paper, we consider a fractional differential filtration equation with a transient (non-stationary) filtration law. The essence of the generalization of the fractional differential model of fluid flow in heterogeneous porous media is the assumption that the orders of fractional derivatives are not constant, but are functions of a time and space variable, and, in particular, functions of the desired solution. The generalization hypothesis is to study the fluid flow when the order of the derivative exhibits a monotonic transition from the initial order to the final one, or when the diffusion regime changes at some point in time. In this paper, a fractional-differential generalization of the filtration equation with a transient (non-stationary) filtration law is considered. For the numerical solution, an approximation is constructed using the finite difference method for the fractional time derivative and the finite element method for a spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a second-order formula in time. A priori estimates of the stability of the numerical method are obtained from the initial data and from the right side of the equation