Plume Migration in Weakly Heterogeneous Media
with Pore-Scale Dispersion
A side-by-side IGW-NET simulation comparing two ways to model plume spreading in an aquifer with weak heterogeneity (ln K variance = 0.5). On the left, the plume migrates through a fully resolved heterogeneous conductivity field with pore-scale dispersion. On the right, Gelhar's asymptotic macrodispersion model approximates the same system with a homogeneous medium and a single macrodispersivity coefficient. Watch how well — and where — the approximation holds.
Top panel: detailed heterogeneous field with pore-scale dispersion. Bottom panel: equivalent homogeneous field with Gelhar's macrodispersion coefficient. Both initial plumes are identical; both head boundaries drive flow left-to-right.
Run this experiment yourself
Two model files are available — one for the detailed heterogeneous field, one for the macrodispersion equivalent. Open them in IGW-NET to vary parameters, change boundary conditions, and re-run.
Parameters & Legend
| Parameter | Case 1 (heterogeneous) | Case 2 (macrodispersion) |
|---|---|---|
| Geometric mean hydraulic conductivity, Kg (m/day) | 10 | 10 |
| ln K variance | 0.5 | 0.0 |
| Correlation scale, λ (m) | 1 | N/A |
| Porosity, n | 0.3 | 0.3 |
| Longitudinal pore-scale dispersivity (m) | 0.01 | N/A |
| Transverse pore-scale dispersivity (m) | 0.001 | N/A |
| Longitudinal macrodispersivity (m) | N/A | 0.5 |
| Transverse macrodispersivity (m) | N/A | 0.0008125 |
| Head difference (m) | 1 | 1 |
| Domain size (m) | 200 × 50 | 200 × 50 |
| Covariance function | Exponential | Exponential |
| Initial plume size (m) | ~10 | ~10 |
| Grid | 801 × 201 | 801 × 201 |
| Cell size, Δx (m) | 0.25 × 0.25 | 0.25 × 0.25 |
| Time step, Δt (days) | 1 | 1 |
- The ln K field is normally distributed — the plume has equal probability of encountering low-K or high-K zones along its path.
- The heterogeneous conductivity field is uniquely characterized by four statistical parameters: geometric mean (Kg), correlation scale, variance, and covariance.
- Geometric mean, ln K variance, and correlation scale are held constant across both cases. Only the dispersion treatment differs.
- Cell size Δx is set so the correlation scale λ is well resolved — typically Δx is at least 3 to 4 times smaller than λ.
- Time step Δt is chosen so a plume particle travels less than one λ per step (Courant-style transport stability).
Problem Statement
This simulation compares two modeling techniques for predicting macrodispersion of a conservative solute plume in a weakly heterogeneous medium. The first technique uses a detailed description of the spatial variability of conductivity — characterized by statistical parameters — together with a pore-scale dispersion coefficient. The second technique uses Gelhar's asymptotic macrodispersion model (Gelhar and Axness, 1983), which represents the same system as a homogeneous medium with a single macrodispersion coefficient. The simulation demonstrates the effectiveness of Gelhar's model in predicting plume spreading.
The modelling domain has constant-head boundaries on the left and right, no-flow boundaries on top and bottom, and an initial plume significantly larger than the correlation scale of heterogeneity. Full parameters are listed in the table above.
Key Observations
- Approximate match. The macrodispersion model approximately predicts both plume location and the amount of spreading in the longitudinal and transverse directions.
- Shape is lost. The macrodispersion model cannot reproduce the irregular shape of the plume. As a consequence, it cannot predict the maximum concentration or the degree of dilution.
Subsequent cases — Case 2, Case 3, and Case 4 — repeat this experiment with progressively larger ln K variance to test where the asymptotic approximation breaks down.
Additional Observations
Gelhar's macrodispersion model attempts to predict the large-scale spreading that occurs in a heterogeneous field. The underlying assumption is that the conductivity perturbations driving plume spreading are small. The model predicts that longitudinal spreading is directly proportional to the correlation scale of heterogeneity and to the ln K variance, and that transverse spreading is directly proportional to the pore-scale dispersivity and the ln K variance. Because correlation scale is at least two orders of magnitude larger than pore-scale dispersivity, longitudinal spreading is significantly larger than its transverse counterpart — and the macrodispersion model captures this overall behaviour well.
However, the macrodispersion model represents the medium as homogeneous, so the resulting plume is always smooth and regular. It does not reproduce the irregular shape that emerges in a real heterogeneous field. That irregularity translates into irregular variations of concentration — which are likewise not captured. The model therefore cannot accurately predict maximum concentration or the actual degree of dilution achieved by the plume.
Mathematical Interpretation
The relationship between pore-scale dispersion and macrodispersion is captured by the following equations:
(3.1.1)
(3.1.2)
where C = concentration; ui = seepage velocity; Dij = dispersion coefficient.
It can be shown [Gelhar and Axness, 1983] that:
(3.1.3a)
The coefficient Aii is the macrodispersivity, given by [Gelhar, 1993]:
(3.1.3b)
where σ2ln K = ln K variance; λ = correlation scale; A11 = longitudinal macrodispersivity; A22 = transverse macrodispersivity; αL = longitudinal pore-scale dispersivity; αT = transverse pore-scale dispersivity.
Equation (3.1.3b) shows that longitudinal macrodispersion is unaffected by pore-scale dispersion, whereas transverse macrodispersion is affected. As a result, the magnitude of longitudinal macrodispersion is many orders of magnitude larger than its transverse counterpart. Substituting (3.1.3a) into (3.1.2):
(3.1.4)
(3.1.5)
The pore-scale dispersivity αij is typically on the order of 10−2 m, while the macrodispersivity Aij is on the order of 1 m. Macrodispersion is therefore the dominant process and controls the total spreading of the plume.
Equation (3.1.5) simplifies the complex heterogeneous-field equation (3.1.2). A simple mean macrodispersion model can be obtained by replacing the dispersion coefficient with a macrodispersivity value — and that mean model then predicts the average spreading of the plume.
References
- Gelhar, L. W., & Axness, C. L. (1983). Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resources Research, 19(1), 161–180.
- Gelhar, L. W. (1993). Stochastic Subsurface Hydrology. Prentice-Hall, Englewood Cliffs, NJ.