A spatial Markov model for the evolution of the joint distribution of groundwater age, arrival time, and velocity in heterogeneous media

Abstract

AbstractThe evolution of the joint distribution of groundwater age, velocity, and arrival times based on a Markov model for the velocities of fluid particles in heterogeneous porous media has been quantified. An explicit evolution equation for the joint distribution of age, arrival time, and particle velocity is derived, which is equivalent to a continuous time random walk for age, velocity, and arrival time. The approach is fully parameterized by the correlation model and the distribution of groundwater flow velocities. The transition probability for subsequent particle velocities along streamlines is implemented by a Copula, which is an efficient method to generate a correlated velocity series with prescribed marginal distribution. We discuss different solution methods based on finite‐differences and random walk particle tracking. The latter is based on continuous time random walks, whose transition times are obtained kinematically from the flow velocities. Specifically, we discuss a renormalization scheme to accelerate the particle‐tracking simulations based on the definition of aggregate particle transitions while at the same time renormalizing velocity correlation. The impact of velocity correlation and velocity distribution on the evolution of age at different distances from the inlet plane is also studied. At distances of the order of the correlation length, persistent particle velocities give the same behavior as stochastic streamtube models. For velocity distributions which give rise to transition times with finite variance, the age distributions evolve toward an inverse Gaussian. For heavy‐tailed weighting times, they evolve toward stable distribution as the distance from the inlet increases.