Analytical model for tracer dispersion in porous media

In this work, we present a novel analytical model for tracer dispersion in laminar flow through porous media. Based on a straightforward physical argument, it describes the generic behavior of dispersion over a wide range of Peclet numbers (exceeding 8 orders of magnitude). In particular, the model accurately captures the intermediate scaling behavior of longitudinal dispersion, obviating the need to subdivide the dispersional behavior into a number of disjunct regimes or using empirical power law expressions. The analysis also reveals the existence of a new material property, the critical Peclet number, which reflects the mesoscale geometric properties of the microscopic pore structure.

💡 Research Summary

In this paper the authors develop a unified analytical model for tracer dispersion in porous media under laminar flow conditions, addressing both longitudinal (parallel to the mean flow) and transverse (perpendicular) directions. Traditional approaches have treated dispersion as a series of distinct Peclet-number regimes, each described by its own empirical power‑law relationship. The new model replaces this fragmented view with a single continuous expression derived from a simple physical picture: a tracer alternately experiences pure molecular diffusion and “mechanical” dispersion that arises from the heterogeneous velocity field within the pore network.

The key assumption is that the fraction of time a tracer spends in mechanically dominated channels, tL, grows monotonically with the mean flow velocity v, while the complementary time t0 corresponds to pure diffusion. By choosing the simplest functional form f(x)=x for the ratio tL/t0 = f(v/vc), where vc is a critical velocity at which mechanical dispersion begins to dominate, the authors obtain for the longitudinal effective diffusivity

DtL = D0 + βL v (v/vc) / (1 + v/vc) .

Here D0 is the molecular diffusion coefficient, βL is a geometric proportionality constant, and vc marks the transition between diffusion‑dominated and advection‑dominated transport. In dimensionless form, using the Peclet number Pe = v G/Dm (G = characteristic grain or pore length, Dm = molecular diffusivity), the expression becomes

DtL/Dm = D0/Dm + β′L Pe (Pe/Pe_c) / (1 + Pe/Pe_c) ,

where Pe_c = vc G/Dm is the newly introduced “critical Peclet number,” a material property reflecting mesoscopic pore geometry.

For the transverse direction, the same reasoning leads to

DtT/Dm = D0/Dm + β′T Pe (Pe/Pe_c) / (1 + Pe/Pe_c) ,

with β′T < β′L because the mean flow does not directly drive transverse spreading; molecular diffusion is always present, and mechanical dispersion contributes only in channels oriented at modest angles to the flow.

The model predicts three asymptotic regimes: (i) at very low Pe, Dt ≈ D0 (pure diffusion); (ii) at very high Pe, Dt ≈ β v (linear mechanical dispersion); and (iii) an intermediate regime where Dt scales as v²/vc (or Pe²/Pe_c), producing an apparent power‑law exponent greater than one. The intermediate scaling appears only when the dimensionless group D0/(β vc) ≪ 1, i.e., when the transition is well separated from the diffusion limit.

To validate the theory, the authors fit the analytical expressions to high‑resolution numerical simulations of flow through micro‑CT scanned sandstone (references