Abstract
It is easy to assume that the soil with more empty space passes more water. This paper counts, across four soils, that the width of the spaces matters more than their amount. No new theorem or law is claimed.
First, hydraulic conductivity is set by grain size and porosity: by the Kozeny--Carman equation it is $5.530621\times10^{-3}$ for coarse sand, $3.875556\times10^{-4}$ for fine sand, $6.567025\times10^{-6}$ for silt and $1.090000\times10^{-7}$ m/s for clay (Section 2).
Second, and this is the core. Clay, with more empty space, passes less water: its porosity is $\mathbf{1.428571}$ times larger, yet its conductivity is $5.073964\times10^{4}$ times, or $\mathbf{4.705347}$ orders of magnitude, lower (Section 3).
Third, amount favours clay and width favours sand: the porosity term favours clay by $\mathbf{4.927114}$, but the squared grain size favours coarse sand by $250000$, and the ratio of the two matches the conductivity ratio within a relative difference of $10^{-12}$ (Section 3).
Fourth, travel times over $100$ m differ even more than conductivity: at a gradient of $0.01$ they are $2.005350\times10^{-2}$ years for coarse sand and $1.453582\times10^{3}$ years for clay, a ratio of $72485.207101$ ($\mathbf{4.860249}$ orders of magnitude). The porosity ratio $1.428571$ enters once more (Section 4).
Fifth, water moves faster than the Darcy flux: water travels only through the pores, so its speed is the Darcy flux times the reciprocal of porosity; in coarse sand the time from the Darcy flux, $0.057296$ years, is $\mathbf{2.857143}$ times the actual $0.020053$ years (Section 4).
Sixth, the separator is whether one asks how much pore space there is or how wide the pores are: the amount sets how many paths there are, the width sets the resistance of each path, and conductivity follows the latter (Section 5).
Keywords: Darcy’s law, hydraulic conductivity, porosity, Kozeny-Carman equation, groundwater, model.
Scope (what is not claimed)
No new theorem or law is claimed: Darcy’s law, the Kozeny--Carman equation, and the fact that the water’s speed is the Darcy flux divided by porosity, are all known. No real strata are claimed: grain sizes and porosities are model values matching no actual ground. The flow is not widened: flow in soil not filled with water, direction-dependence and layering are not included. The coefficient is not discussed: the constant $180$ is one representative choice for near-spherical grains and moves with grain shape. Nothing is designed: how to build wells or drainage is not stated. Relation to earlier papers: Paper 599 showed that around a well the water table falls only as the logarithm of distance and keeps falling as the logarithm of time. This paper stands beside it. There conductivity was set to a representative $10^{-4}$ m/s to count water levels. Paper 574 showed that lowering the water table by $1$ m raises the bottom of the fresh water by $40$ m. This paper stands beside it too. There conductivity was set to $20$ m/day. Here we count how far such a set value moves with grain size. What is added is laying out conductivity and $100$ m travel time for four soils, measuring that the soil with more porosity passes $4.705347$ orders of magnitude less, splitting that ratio into the porosity term and the squared grain size and matching them, and placing the separator on amount of pore space versus width.
1. Notation and the Question
Setting: a hydraulic gradient $i$ is applied to soil with grain size $d$ and porosity $n$. Hydraulic conductivity $K$, Darcy flux $q$ (discharge divided by the whole cross-section) and the water’s speed $v$ (speed through the pores) are taken as follows.
$$K=\frac{\rho g}{\mu}\cdot\frac{d^{2}n^{3}}{180(1-n)^{2}},\qquad q=Ki,\qquad v=\frac{q}{n}$$Water density $\rho=1000$ kg/m$^3$, gravitational acceleration $g=9.81$ m/s$^2$, viscosity $\mu=1.0\times10^{-3}$ Pa s and gradient $i=0.01$ were used.
Question: does soil with more empty space pass more water?
2. Four Soils
Computation 2.1: conductivity was computed while varying grain size and porosity.
| soil | grain | porosity | conductivity (m/s) |
|---|---|---|---|
| coarse sand | $1$ mm | $0.35$ | $5.530621\times10^{-3}$ |
| fine sand | $0.2$ mm | $0.40$ | $3.875556\times10^{-4}$ |
| silt | $0.02$ mm | $0.45$ | $6.567025\times10^{-6}$ |
| clay | $0.002$ mm | $0.50$ | $1.090000\times10^{-7}$ |
Observation 2.2 as porosity rises, conductivity falls. Of the four soils, coarse sand has the smallest porosity and the largest conductivity: the order by amount of space and the order by water passed point opposite ways.
3. Amount and Width
Computation 3.1: coarse sand and clay were compared term by term.
| compared | value |
|---|---|
| porosity ratio (clay $\div$ coarse sand) | $1.428571$ |
| porosity term $n^{3}/(1-n)^{2}$: coarse sand | $0.101479$ |
| porosity term $n^{3}/(1-n)^{2}$: clay | $0.500000$ |
| porosity-term ratio (favours clay) | $4.927114$ |
| squared grain-size ratio (favours coarse sand) | $250000.000000$ |
| conductivity ratio (coarse sand $\div$ clay) | $5.073964\times10^{4}$ |
| orders of magnitude of that ratio | $4.705347$ |
Observation 3.2 this is the core. Clay, with more empty space, passes less water. The porosity term alone favours clay by $4.927114$, but the squared grain size favours coarse sand by $250000$, leaving coarse sand ahead by $4.705347$ orders of magnitude: dividing the squared grain-size ratio by the porosity-term ratio gives the conductivity ratio within a relative difference of $10^{-12}$.
Observation 3.3 width acts as a square. With porosity fixed at $0.4$, doubling grain size made conductivity exactly $4$ times larger: within $10^{-12}$. The resistance of each path is set by its width.
4. Travel Time
Computation 4.1: at a gradient of $0.01$, Darcy flux, the water’s speed and the time to travel $100$ m were computed.
| soil | Darcy flux (m/s) | water speed (m/s) | years for $100$ m |
|---|---|---|---|
| coarse sand | $5.530621\times10^{-5}$ | $1.580178\times10^{-4}$ | $2.005350\times10^{-2}$ |
| fine sand | $3.875556\times10^{-6}$ | $9.688889\times10^{-6}$ | $3.270560\times10^{-1}$ |
| silt | $6.567025\times10^{-8}$ | $1.459339\times10^{-7}$ | $2.171400\times10^{1}$ |
| clay | $1.090000\times10^{-9}$ | $2.180000\times10^{-9}$ | $1.453582\times10^{3}$ |
Observation 4.2 the water’s speed is the Darcy flux times the reciprocal of porosity. The Darcy flux is divided by the whole cross-section, but water can pass only through the pores: in coarse sand the time from the Darcy flux is $0.057296$ years, $2.857143$ times the $0.020053$ years from the water’s speed. In all four soils the water’s speed times porosity matched the Darcy flux.
Observation 4.3 at the same flux, water is actually slower in the soil with more pore space. The travel-time ratio of clay to coarse sand is $72485.207101$ ($4.860249$ orders of magnitude), larger than the $4.705347$ orders in conductivity: the time ratio matched the conductivity ratio times the porosity ratio $1.428571$ within a relative difference of $10^{-12}$. Here too, more pore space does not speed the water.
5. The Separator
Observation 5.1 the separator is whether one asks how much pore space there is or how wide the pores are. The amount sets how many paths the water has; the width sets how much water each path carries: clay wins on amount and loses by $250000$ on width. Conductivity follows the width.
Observation 5.2 placed beside Papers 599 and 574. 599 set conductivity to $10^{-4}$ m/s to count water levels around a well, and 574 set it to $20$ m/day to count the bottom of the fresh water. Here we count how far such a set value moves with grain size and porosity: $4.705347$ orders of magnitude across the four soils.
Observation 5.3 what is not said. Real strata, flow in soil not filled with water, direction-dependence, layering, and the design of wells or drainage are not treated. What is shown is that the width of the pores, more than their amount, sets conductivity.
6. Census and Closing
- conductivity is set by grain size and porosity (Section 1): the Kozeny--Carman equation.
- coarse sand $5.530621\times10^{-3}$ m/s (Computation 2.1): porosity $0.35$.
- clay $1.090000\times10^{-7}$ m/s (Computation 2.1): porosity $0.50$.
- clay has $\mathbf{1.428571}$ times the porosity (Computation 3.1): more space in clay.
- conductivity differs by $\mathbf{4.705347}$ orders (Computation 3.1): the core.
- porosity term favours clay by $\mathbf{4.927114}$ (Computation 3.1): the amount side.
- squared grain size favours coarse sand by $250000$ (Computation 3.1): the width side.
- clay: $1.453582\times10^{3}$ years for $100$ m (Computation 4.1): coarse sand $2.005350\times10^{-2}$ years.
- the Darcy flux gives times $\mathbf{2.857143}$ times too long (Observation 4.2): water moves only through pores.
- amount of pore space, or width (Observation 5.1): the separator; Papers 599, 574.
Closing: Clay has more empty space than sand, yet lets less water through. Computing conductivity from grain size and porosity, clay has $1.428571$ times the porosity yet $4.705347$ orders of magnitude less conductivity, because the porosity term favours clay by $4.927114$ while the squared grain size favours coarse sand by $250000$. At a gradient of $0.01$, water takes $2.005350\times10^{-2}$ years to travel $100$ m through coarse sand and $1.453582\times10^{3}$ years through clay: and since water moves only through the pores, times computed from the Darcy flux are $2.857143$ times too long. Placed among the earlier papers: beside the representative conductivities set in Papers 599 and 574, we placed how far such a value moves with grain size and porosity. To be honest: grain sizes and porosities are model values, the coefficient moves with grain shape, and soil not filled with water and layering are not included.
References
- Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier.
- Freeze, R. A., & Cherry, J. A. (1979). Groundwater. Prentice-Hall.
- Yamagishi, Y. (2026). Lower the Water Table by $1$ m and the Bottom of the Fresh Water Rises $40$ m [Paper 574]. YOI: https://www.yamagishiyuuki.com/yoi/574; DOI: 10.5281/zenodo.23014304
- Yamagishi, Y. (2026). Around a Well the Water Table Falls Only as the Logarithm of Distance, and Keeps Falling as the Logarithm of Time [Paper 599]. YOI: https://www.yamagishiyuuki.com/yoi/599; DOI: 10.5281/zenodo.23014427
Reference
Yamagishi, Y. (2026). Clay Has More Empty Space Than Sand, Yet Lets Less Water Through [Paper 863]. YOI: https://www.yamagishiyuuki.com/yoi/863
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 863. This paper is a preprint and has not been peer reviewed.