01 lnK Var=0.5, w/ Pore-Scale Dispersion
When an aquifer is only mildly variable, can a simple effective model stand in for it?
What you’re watching Top: real transport through a mildly heterogeneous aquifer (lnK variance 0.5), the plume gently fingered. Bottom: the macrodispersion model — a uniform aquifer with one large effective dispersivity — producing a smooth band of about the same extent.
The mechanism The macrodispersion model lumps all the conductivity variability into a single effective (Fickian) dispersivity in a homogeneous aquifer. At low variance the real plume is only mildly fingered, so the smooth effective band matches its overall spread fairly well.
Why it matters It marks the regime where the convenient textbook approach is defensible: weakly heterogeneous aquifers.
IGW-NET Running the true field and its effective twin together shows at a glance how close the approximation is — the model-validity test made visual.
02 lnK Var=1.0, w/ Pore-Scale Dispersion
How does the match hold up as variability doubles?
What you’re watching At lnK variance 1.0 the real plume (top) is more fingered and channelized; the effective model (bottom) is still a smooth band, but now noticeably smoother than reality.
The mechanism As variance rises, channeling concentrates solute along fast paths that a single-dispersivity model cannot reproduce. The effective band still gets the rough extent but begins to lose the structure and the peak.
Why it matters Variance near 1 is common in real aquifers — right where the effective approximation begins to strain.
IGW-NET The growing gap between the two panels is the approximation breaking down in front of you.
03 lnK Var=2.0, w/ Pore-Scale Dispersion
At strong variability, is the effective model still trustworthy?
What you’re watching At variance 2.0 the real plume is strongly channelized and irregular; the macrodispersion band is much smoother and more diluted than the actual plume.
The mechanism Strong heterogeneity produces pronounced channeling and tailing — non-Fickian behavior. The effective model, Fickian by construction, captures neither; it overstates dilution and misses the concentrated core.
Why it matters It depends on what you care about. For plume extent, the effective model may suffice; for exceeding a concentration limit (an MCL), it is dangerously optimistic — the fingers keep peaks high even where the average looks safe.
IGW-NET Side by side, the smooth model versus the channeled reality shows why ‘effective’ can be dangerously optimistic.
04 lnK Var=4.0, w/ Pore-Scale Dispersion
In a severely heterogeneous aquifer, what does the effective model miss entirely?
What you’re watching At variance 4.0 the real plume is a tight, channelized streak with a diffuse halo (top); the macrodispersion model is a broad, smooth, symmetric ellipse (bottom) — they barely resemble each other.
The mechanism Extreme variance drives strong preferential channeling: most mass races along a few connected high-K paths. The effective model reproduces the plume’s overall spread (its spatial moments) by smearing that mass evenly — but real mixing happens only at the thin finger edges, so the true effective water volume is far smaller and the concentration far higher than the smooth band implies.
Why it matters If the question is ‘how big is the plume,’ the effective model may answer it; if the question is ‘does it exceed the MCL at the well,’ it can understate the peak by a wide margin — the core reason effective models poorly represent field transport where concentration matters.
IGW-NET The dramatic mismatch between the two panels is the clearest possible demonstration of the effective model’s limits — visible because both run together.
05 lnK Var=2.0, w/ No Pore-Scale Dispersion
Does adding local dispersion rescue the effective model?
What you’re watching The strong-heterogeneity case run without pore-scale dispersion; compared with the with-dispersion case, the real plume stays sharper and more fingered — and still does not match the smooth effective band.
The mechanism Local dispersion mixes within fingers and smooths the real plume a little, nudging it toward the effective model — but the channeling from heterogeneity dominates, so the mismatch with the Fickian model remains.
Why it matters It rules out the hope that ‘enough mixing’ makes the effective model valid: local dispersion is exactly the dilution mechanism, and even with it on, the fingers keep peak concentrations high. Spreading averages out long before dilution does.
IGW-NET Toggling pore-scale dispersion on and off shows how little it closes the gap — isolating heterogeneity as the real culprit.
06 Multiple Scales of Heterogeneity
What happens when an aquifer varies at several scales at once?
What you’re watching Transport through an aquifer with heterogeneity at multiple scales — small patches nested within larger structures — producing spreading that grows as the plume samples ever-larger features.
The mechanism Real aquifers vary across a hierarchy of scales. As a plume grows it ‘sees’ progressively larger heterogeneity, so its apparent dispersion keeps increasing with travel distance — scale-dependent dispersivity, which no single constant can capture.
Why it matters Scale-dependent dispersivity is one of the most cited puzzles in field hydrogeology — why lab and field dispersivities differ by orders of magnitude.
IGW-NET Watching the plume’s spreading rate climb as it meets larger structures makes scale-dependence — usually just a curve in a paper — something you see happen.
07 Small Source & Small λ Case 1
Does a small contaminant source ever behave like the ‘average’ effective model?
What you’re watching A small source in an aquifer with a small correlation scale, compared with the macrodispersion model at two variances; because the plume still spans many small patches, it tracks the effective band reasonably.
The mechanism The macrodispersion model is an ensemble-average concept. A real plume matches it only when it samples many correlation scales — the ‘ergodic’ limit. A small source in a small-λ field still crosses many patches, approaching ergodic behavior.
Why it matters Whether the effective model is valid depends not just on variance but on the size of your plume relative to the aquifer’s correlation scale.
IGW-NET Comparing the real small-source plume with the effective model shows ergodicity as a visible criterion, not an abstract assumption.
08 Small Source & Small λ Case 2
Is that agreement repeatable, or a lucky realization?
What you’re watching A second realization of the small-source, small-λ case; the plume again roughly tracks the effective model, confirming the agreement is not a fluke.
The mechanism Near the ergodic limit, different realizations behave similarly and each is close to the ensemble effective model — low realization-to-realization scatter.
Why it matters Low scatter between realizations is what makes a single effective prediction trustworthy.
IGW-NET Re-running the realization and seeing the same agreement is the visual test of ergodicity.
09 Small Source & Larger λ Case 1
What if the aquifer’s patches are large compared with the plume?
What you’re watching A small source in a large-λ field; the plume now spans only a few big patches, and its shape departs sharply from the smooth effective model.
The mechanism When the correlation scale is large relative to the source, the plume samples too few patches to average out — non-ergodic. Its path is dominated by the particular big features it happens to hit, so it can run fast or slow, left or right, unlike the symmetric effective band.
Why it matters A small spill in a coarsely heterogeneous aquifer is the worst case for the effective model — and a common real situation.
IGW-NET The mismatch between the erratic real plume and the smooth model makes non-ergodicity unmistakable.
10 Small Source & Larger λ Case 2
How different can two small-source plumes be in the same large-λ aquifer?
What you’re watching A second realization of the small-source, large-λ case; the plume takes a markedly different path and shape than Case 1 — large realization-to-realization scatter.
The mechanism Because each plume is steered by a few large features, equally-likely realizations diverge widely. No single effective curve represents them; only a distribution does.
Why it matters This is the formal reason a small-source prediction in a coarse aquifer needs Monte Carlo, not a single effective model.
IGW-NET Watching two realizations diverge so much from identical statistics is the case for probabilistic prediction in one image.
11 Larger Source & Small λ Case 1
Does a big source average out the heterogeneity?
What you’re watching A large source in a small-λ field; the broad plume spans very many patches and tracks the smooth effective model closely.
The mechanism A large plume over a small correlation scale samples a huge number of patches, so heterogeneity averages out — the strongly ergodic limit, where the effective model is most accurate.
Why it matters It defines the best-case conditions for using the convenient effective model with confidence.
IGW-NET The close match between the broad real plume and the effective band confirms when averaging is justified.
12 Larger Source & Small λ Case 2
Is the large-source agreement consistent across realizations?
What you’re watching A second realization of the large-source, small-λ case; the plume again closely tracks the effective model — minimal scatter.
The mechanism In the strongly ergodic regime, realizations are nearly indistinguishable and all match the ensemble effective model.
Why it matters When these conditions hold, expensive Monte Carlo is not needed — the effective model is enough.
IGW-NET Seeing realizations nearly coincide justifies the simpler model, visually.
13 Larger Source & Larger λ Case 1
Does a large source still help when the patches are also large?
What you’re watching A large source in a large-λ field; the plume spans only a moderate number of patches, so it tracks the effective model better than a small source would, but with more deviation than the small-λ case.
The mechanism Ergodicity depends on the ratio of plume size to correlation scale. Scaling up both source and λ together keeps that ratio modest, so agreement is partial — between the strongly ergodic and non-ergodic extremes.
Why it matters It clarifies the real criterion for effective-model validity: how many correlation scales the plume spans, not how big it is in meters.
IGW-NET Comparing this with the small-λ large-source case isolates the ratio as the controlling factor.
14 Larger Source & Larger λ Case 2
How much scatter remains in this intermediate regime?
What you’re watching A second realization of the large-source, large-λ case; moderate realization-to-realization differences — less than the small-source case, more than the small-λ case.
The mechanism Intermediate source-to-λ ratios give intermediate scatter: the effective model is a fair central estimate, but a useful uncertainty band remains around it.
Why it matters Most real sites live in this middle ground — the effective model is useful but should be reported with an uncertainty range.
IGW-NET The moderate spread across realizations argues for pairing an effective estimate with a probabilistic band — which the digital laboratory produces directly.