  We study measures of the amount of time required for transient flow in
heterogeneous porous media to effectively reach steady state, also known as the
response time. Here, we develop a new approach that extends the concept of mean
action time. Previous applications of the theory of mean action time to
estimate the response time use the first two central moments of the probability
density function associated with the transition from the initial condition, at
$t=0$, to the steady state condition that arises in the long time limit, as $t
\to \infty$. This previous approach leads to a computationally convenient
estimation of the response time, but the accuracy can be poor. Here, we outline
a powerful extension using the first $k$ raw moments, showing how to produce an
extremely accurate estimate by making use of asymptotic properties of the
cumulative distribution function. Results are validated using an existing
laboratory-scale data set describing flow in a homogeneous porous medium. In
addition, we demonstrate how the results also apply to flow in heterogeneous
porous media. Overall, the new method is: (i) extremely accurate; and (ii)
computationally inexpensive. In fact, the computational cost of the new method
is orders of magnitude less than the computational effort required to study the
response time by solving the transient flow equation. Furthermore, the approach
provides a rigorous mathematical connection with the heuristic argument that
the response time for flow in a homogeneous porous medium is proportional to
$L^2/D$, where $L$ is a relevant length scale, and $D$ is the aquifer
diffusivity. Here, we extend such heuristic arguments by providing a clear
mathematical definition of the proportionality constant.
