Section 3.1 — The Pattern
A Number That Knows the Rock
Here's something surprising: when scientists pumped thousands of wells and measured both the specific capacity (Q/s) and independently determined the effective porosity of those aquifers, they found a pattern. Wells in tight, low-porosity rock had low specific capacities. Wells in open, porous rock had high specific capacities. Plot them on a graph and they follow a curved line — almost every time. The teal curve in the animation below is the Wilkinson equation fitted to that data. The colored dots are individual well measurements.
That means if you know the specific capacity of a well, you can predict the effective porosity of the aquifer it sits in. Two numbers you already measure during a routine pumping test. No lab required. No core samples. No tracer chemicals. Just Q and s.
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The physical intuition behind the Q/s — Ne relationship: a permeable medium with many open, connected pores tends to have both high transmissivity (T) and high effective porosity. The Wilkinson equation captures this covariance empirically across a very wide range of hydrogeologic settings.
Crucially, the relationship is power-law (not linear), reflecting that the gain in effective porosity per unit gain in specific capacity diminishes at high Q/s values — tight rocks vary enormously in Q/s, while high-permeability aquifers cluster in a narrower Ne range. The exponent 0.0826 is close to zero mathematically, which means the function is remarkably stable: small percentage errors in Q/s produce even smaller percentage errors in the predicted Ne. This is a desirable property for a field-applied predictive tool.
Section 3.2 — Validation
Five Thousand Wells Can't Be Wrong
To know if a pattern is real, you have to test it on a lot of data — not just a few wells you already know about. Dr. Wilkinson gathered data from nearly 5,000 wells across Texas, New Mexico, and South Sudan in Africa. He compared what the equation predicted for effective porosity against values that were independently measured or confirmed. The equation held up every time.
That consistency — across deserts, plains, and different continents — is what makes this equation trustworthy. It wasn't built from one region and applied to another. It was tested across every major aquifer type that exists.
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The validation dataset comprises approximately 5,000 pumping tests spanning diverse hydrogeologic provinces: sedimentary aquifers of Texas, alluvial and basin-fill aquifers of New Mexico, and fractured/weathered basement and sedimentary aquifer systems in South Sudan. This geographic and lithologic diversity was deliberate — the goal was to test whether the Q/s–Ne relationship held across aquifer types, climatic regimes, and well construction practices.
The coefficient of determination (R²) ranged from 0.987 to 0.991 across all tested datasets from inception to present — no exclusions, no subsets. This level of predictive power is exceptional for an empirical relationship in hydrogeology, where subsurface heterogeneity typically limits R² values to 0.6–0.8 for transmissivity–lithology correlations.
Section 3.3 — The Equation
Meet the Equation
Here it is — the equation at the heart of this tool. You put in two numbers you can measure at any well: Q (how much water you are pumping) and s (how much the water level dropped during the pumping test). The calculator accepts SI, US customary, Imperial UK, and CGS units — enter your values in whichever unit system your well data is reported in. Divide them to get Q/s. Raise that to the power 0.0826, then multiply by 0.15108. The result, Ne, is the effective porosity — a number between 0 and 1, where 0.25 means 25% of the rock is usable pore space.
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Coefficient 0.15108: The scaling coefficient was derived using SI units (Q in m³/day, s in meters), which is how the equation is published. The calculator handles all unit conversions internally, so you can work in SI, US customary (gpm/ft), Imperial UK, or CGS units without any manual conversion. The predicted Ne is always dimensionless and independent of the unit system used.
Exponent 0.0826: The power-law exponent governs the sensitivity of Ne to changes in Q/s. Because 0.0826 is much less than 1, the relationship is highly compressed — increasing Q/s by an order of magnitude raises Ne by only about 21%. This is mathematically elegant: it means small field measurement errors in Q or s propagate to even smaller percentage errors in the predicted Ne.
Independence from aquifer type: The equation was developed and validated across confined, unconfined, karst, and fractured-rock systems with no limits on applicability. Traditional methods (laboratory core analysis, tracer tests) require different protocols for each aquifer type and are expensive and time-consuming. This equation requires only a standard pumping test.
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Section 3.4 — Interpretation
What Does The Number Mean?
Once you have your Ne value, what does it actually tell you? Think of it this way: an Ne of 0.30 means 30 out of every 100 "units" of rock are usable water-holding, water-moving space — like a really good sponge. An Ne of 0.05 means only 5 out of 100 units count — like granite or clay. Knowing this tells engineers how much water is actually available, and how fast a contaminant could travel underground.
The table below shows typical Ne ranges for different geological materials. Notice that many ranges overlap — this is intentional. Nature doesn't follow clean boundaries. Interpreting these results correctly requires reading a well log, understanding the local geology, and bringing the judgment of an experienced hydrogeologist.
| Ne Range | Material | Notes |
|---|---|---|
| 0.35 – 0.45 | Karst limestone, karst dolomite | Conduit and cave systems — like punching a hole into an underground river or lake. Extremely productive. |
| 0.30 – 0.45 | Vesicular basalt | Gas-bubble voids formed during lava cooling. Highly productive where well-connected. |
| 0.28 – 0.38 | Clean gravel, coarse gravel | Classic high-yield alluvial aquifers. Grain size and sorting are the key controls. |
| 0.22 – 0.33 | Clean coarse sand, sand & gravel mix | High productivity. Overlap with gravel zone — sorting is critical to distinguish. |
| 0.18 – 0.28 | Medium sand, poorly sorted sand & gravel | Overlap zone — sorting and fine content control the result. Well log interpretation essential. |
| 0.12 – 0.22 | Fine sand, well-cemented sandstone | Common in sedimentary basin aquifers worldwide. Cementation degree is the key variable. |
| 0.08 – 0.18 | Silty sand, weathered / vuggy limestone | Transition zone. Vug connectivity in limestone makes well log reading critical. |
| 0.04 – 0.12 | Silt, fine sandstone, fractured limestone | Heterogeneous results. Fracture vs. matrix contribution varies greatly between wells. |
| 0.01 – 0.06 | Fractured crystalline rock, weathered granite | Fracture aperture and connectivity dominate. Matrix porosity is near-zero. |
| < 0.02 | Dense granite, unfractured basalt, tight shale | Near-zero flow. Structural geology controls any water movement that does occur. |
Overlapping ranges are by design. Effective porosity is interpretive science — the correct material assignment requires geological judgment from a seasoned geologist reading a well log, not mechanical table lookup.
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1. Groundwater velocity estimation:
Average linear groundwater velocity: v = (K × i) / Ne, where K is hydraulic conductivity (m/day) and i is the hydraulic gradient (m/m). Ne appears in the denominator — a lower Ne produces faster groundwater velocities for the same hydraulic conditions. This counterintuitive but important result is critical for contaminant transport assessments.
2. Aquifer storage volume:
Ne reflects the potential storage capability of the aquifer material — the fraction of the rock or sediment volume that is available to hold and transmit water. Whether that capacity is realized depends on the degree of saturation, which is independent of whether the aquifer is confined or unconfined. Unconfined aquifers are often partially saturated, though they can be fully saturated and will discharge at the surface or produce swampy conditions if they are. Confined aquifers are usually fully saturated, but exceptions exist. The key point is that saturation state does not change what Ne tells you — it describes the material, not the current water inventory.
In practice, this is a useful tool for volumetric estimation. For a hypothetical basin aquifer, the geometry (length × width × thickness) gives the total rock volume. Multiplying by Ne of the sediments yields the potential storage volume of that basin — a straightforward and powerful application of the Wilkinson equation results.
3. Particle tracking and transport modeling:
In 3D numerical models (e.g., MODFLOW/MODPATH), Ne is assigned on a cell-by-cell basis. The Wilkinson equation enables spatially distributed Ne estimation from existing specific capacity data in well databases and driller reports, replacing uniform textbook assignments and substantially improving transport model accuracy. This is the primary professional application for which the equation was developed.
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