The laws of water part I : How water moves through soil

Darcy's law and Richard's law

The world of water has emerged a number of laws that describe how it behaves. In this essay we will explore two of these laws of water.

When it rains in a bioregion, we want to know: how does that affect the soil moisture? How long does the water stick around in the ground, and how does it seep through to the aquifers below?

To find out, we use watershed modeling. These models rely on what you might call water laws, equations for how the water behaves. Two of the main equations for how water moves through the soil and ground are Darcy’s equation and Richards’ equation. The results from the watershed models can be visualized on what’s essentially an ‘internet of maps’ called GIS (Geographic Information Systems). Through software like ArcGIS or QGIS, you can zoom into your own region on your laptop and see exactly what the soil moisture levels look like.

Imagine the soil is a sponge under a dripping tap. Richards’ Equation is the story of the first few minutes: the dry sponge greedily sucking up droplets through capillary action (suction). Darcy’s Law is the story of what happens an hour later: the sponge is heavy and dripping, and the water just flows through the saturated holes because gravity is pushing it.

Now imagine we’re engaging in ecological and water restoration across a bioregion, like what is happening with the Alvelal project in Spain, the Maharashtra Paani Cup project in India, the zaï pit projects in Burkina Faso, or the half-moon crescent projects in Chad. All of these interventions change how water filters downward through soil. How is that water moving? How much stays in the soil over time? This is where Darcy’s and Richards’ equations earn their keep. We can actually calculate what changes in soil structure and flow rates mean for water movement.

I could write this essay without any math, but for an essay about water laws, the equations themselves can give a little oompf. Feel free to skip the math part, but you might also want to give it a go. Once you stop being intimidated by the math part, you start seeing how cool the math is. For instance, the equation E=mc² packs a punch. When you see an equation, you can translate each symbol into English, then turn that sentence into something even more intuitive. Take E=mc²: E is energy, m is mass, c is speed of light. Translation: “energy equals mass times the speed of light squared.” More intuitive translation: “energy and mass are two forms of the same thing and can convert into each other.” Do this with any equation in any field, and suddenly they’re not so scary. You start to see them for what they really are: elegant abbreviations that pack entire ideas into a few symbols.

At the end of this essay I will show my initial attempt at turning ‘Slow it, spread it, sink it” into math, which will help us begin to calculate how much we need to slow, spread, and sink to restore a watershed.

Water underground

When water arrives as rain you can see and measure, but what happens next, whether it soaks into the soil or sheets away as runoff, how long moisture lingers in the root zone, whether aquifers slowly recharge or remain depleted, at what rate water moves through different soil layers, all of this occurs in darkness, beneath your feet, operating on timescales ranging from hours to decades. Yet these hidden dynamics are key. A permaculture designer planning swale placement needs to know where water naturally accumulates and how long it will persist after storms. An ecorestoration team revegetating a degraded watershed needs to understand whether their site can actually retain enough moisture through dry seasons to keep seedlings alive. A land manager trying to restore groundwater levels needs to predict recharge rates under different scenarios: current compacted conditions versus improved soil structure, native grasses versus bare ground, various rainfall patterns playing out over years. The questions are practical, where to intervene, what techniques will actually work, how to sequence restoration phases, but the answers require peering into processes you cannot directly observe.

This is not a new problem, the fundamental physics governing water’s movement through soil was discovered well over a century ago. Henry Darcy discovered in 1856 that saturated flow through porous media follows a simple law relating flow rate to pressure gradient and the material’s permeability. Lorenzo Richards extended this in 1931 to unsaturated conditions (the vastly more complicated zone where most plants live and die), capturing how water moves through partially wet soil where conductivity itself changes with moisture content, creating nonlinear feedbacks that can lock landscapes into either virtuous or vicious hydrological cycles. These equations (Darcy’s law and Richards’ equation) are the operating instructions for the underground water economy, describing with mathematical precision exactly the processes restoration practitioners need to understand: infiltration capacity, moisture redistribution, the difference between soil that welcomes rain and soil that sheds it like a broken roof. With modern computational power and Geographic Information Systems, these equations can now be used to model entire watersheds and visualize predicted outcomes for restoration interventions.

Darcy’s Law

Darcy’s Law is the rulebook for understanding how water seeps through sand, soil, and rock (the underground plumbing beneath our feet). Think of drinking a thick milkshake through a narrow straw versus sipping water through a wide one: the effort and flow are completely different. French engineer Henry Darcy cracked the math behind this in 1856 while studying sand filters for Dijon’s water supply. He discovered that water flowing through sand behaves predictably based on three things: how hard you’re pushing it (pressure or slope), how much space it has to flow through (cross-sectional area), and how easily the material lets water pass, called hydraulic conductivity.

[ Water moves at different rates through clay, silt and sand, which each has a different K, conductivity, a measure of how well it passes water through. Picture from this video]

Darcy’s insight mirrors Ohm’s law in electricity: just as electrons flow through conductors in response to voltage, water flows through soil in response to pressure gradients. Soil acts like a resistor network - well-connected pores are low-resistance pathways, while compacted soil creates bottlenecks. Transport through complex media depends on the interaction between the material and the driving force.

What emerged from Dijon’s need for clean drinking water became a window into how underground fluids move. Darcy measured flow through sand columns with patient precision and found that the relationship between pressure, flow, and resistance followed a crisp rule. Push water twice as hard, get twice the flow. Double the area, double the flow. Switch the material, and seepage slows.

Lets look at the math here. You have to unpack it slower than how you approach understanding a normal English sentence. The power of equations lies in their ability to capture relationships. Darcy’s equation does exactly this: Q = -KA(dh/dl). Take your time with it. Slowly translate what each letter means in English. On the left is Q, the rate of volume of water flowing through the soil. On the right, the terms that determine it: K is hydraulic conductivity (how easily water passes through that particular material). Don’t let the fancy term intimidate you. It’s just like electrical conductivity, except instead of measuring how well electrons flow through copper or rubber, it measures how well water flows through sand or clay. A is the cross-sectional area the water flows through, and dh/dl is the hydraulic gradient (the change in pressure or elevation driving the flow). Think of it as the slope of the water table. Steeper slope, faster flow, just like a ball rolling down a hill.

Now read your translated sentence: The rate of volume of water flowing through the soil equals how easily water passes through that particular material, multiplied by the cross-sectional area, multiplied by the change in pressure or elevation driving the flow.

You can translate it into even simpler terms: faster flow = larger area + more permeable + steeper slope.

This general approach can help you feel less intimidated by equations in any field of knowledge. Translate what each symbol means, then write out the sentence. Then translate again into something more intuitive.

The beauty is that K, the measure of how easily water passes through the material, varies wildly depending on what you’re dealing with - coarse beach sand might conduct water 10,000 times faster than dense clay. But once you measure K in the field for your specific soil, you can plug it into the equation alongside your gradient and area, and suddenly you have a quantitative approximation for how fast water is actually moving underground. The equation transforms scattered observations into predictions.

The law’s power lies in what it reveals about the invisible architecture beneath us. Every aquifer is a vast dark river flowing at geological patience through pores smaller than pinheads. Ecologists see how wetlands filter nutrients, their saturated soils acting as natural kidneys for entire watersheds. Foresters see why hillside springs appear where they do, why some slopes stay green through drought while others parch. Every contaminated site is a slow-motion catastrophe: invisible plumes of gasoline or solvents spreading through soil at feet per year, not miles per hour, following Darcy’s arithmetic.

Darcy stumbled onto something cool: an organizing principle connecting how rain soaks into a forest floor to how coastal marshes buffer storm surges, how desert playas trap fleeting moisture to how mountain snowmelt feeds valley streams months later. The equation has held for over 165 years showing the pattern of how viscosity, pressure, and geometry work together to move fluids through the hidden spaces of our world.

Richard’s equation

Richards came along in 1931. Lorenzo Richards, a soil physicist working in the arid landscapes of the American West, confronted a problem that had bedeviled hydrologists for decades: water moving through unsaturated soil refuses to behave politely. Darcy’s law worked beautifully when every pore space was filled with water, when the underground was a saturated sponge. But what about the vast zones above the water table, where soil is only partially wet, where water clings to particles in films and threads, where air pockets interrupt the flow?

Richards saw that this wasn’t just a technical problem but a conceptual challenge. He needed to marry Darcy’s elegant linearity with the fact that water is flowing from places where it isn’t to places where it is, while accounting for the fact that a soil’s ability to conduct water changes continuously as it wets or dries. The math describes how moisture content changes through space and time, how water creeps downward through gravity while simultaneously being sucked sideways and upward by capillary forces, how the hydraulic conductivity itself depends on how wet the soil is at any given moment.

Start by imagining what soil actually is. It’s not a solid block. It’s more like a chaotic sponge made of particles of different sizes jumbled together. Sand grains the size of beach sand. Silt particles smaller than table salt. Clay flakes microscopic and flat like tiny playing cards. Now add in the living stuff: root exudates (sticky sugars that plants ooze out), dead organic matter, microbial goo, all of which glue particles together into clumps called aggregates. The result is a three-dimensional maze of pore spaces (some wide enough to see with a magnifying glass, others so narrow that water molecules barely squeeze through single-file).

Now picture water moving through this. When the soil is completely saturated (every nook and cranny filled), water flows relatively smoothly, like traffic on an open highway. But most of the time, soil isn’t saturated. There’s water and air sharing the pore space. Water doesn’t fill pores evenly; it clings to particle surfaces in thin films, bridges across narrow gaps by surface tension, pools in tiny pockets. Imagine trying to pour water through a sponge that’s only half-wet: some pathways are open rivers, others are dead-ends where water has to creep along grain surfaces, and air bubbles block what would otherwise be shortcuts. Water gets stuck at constrictions, has to detour around air pockets, clings stubbornly to clay surfaces because of electrostatic attraction. The soil’s “willingness” to let water through (its hydraulic conductivity) isn’t a fixed number anymore. It changes depending on how wet the soil is at that exact moment. Dry soil conducts water miserably because there are so few connected pathways; as it wets up, more pores link together and flow accelerates. But even that process isn’t straightforward. Wetting soil behaves differently than drying soil, because water invades pores differently than it retreats from them. It’s like the soil has a memory of where it’s been.

To get to the law, the equation let’s start with the core idea in plain English: A patch of soil gets wetter or drier over time depending on whether more water is flowing into it from above than is flowing out of it below, and how fast water flows depends on how wet the soil already is, plus gravity always pulling downward.

Here’s what Richards did: He took Darcy’s equation (which works beautifully for saturated soil) and modified it in two crucial ways. First, he made the hydraulic conductivity K variable instead of constant, since it changes dramatically as soil wets and dries. Second, he combined it with the idea that water moves from place to place : water flowing into a layer minus water flowing out equals the change in how much water that layer is storing.

Why does this matter? Picture pouring water onto a dry kitchen sponge. Darcy’s equation would treat the sponge’s conductivity as fixed (measure it once, use that number forever). But when you first pour water on the dry sponge, it barely penetrates. The conductivity is terrible because water is clinging to scattered points with hardly any connected pathways. But as more water accumulates, suddenly it starts flowing faster. Pathways link up, the sponge darkens from the top down, and conductivity skyrockets, maybe 100-fold. If you tried to predict how fast water would reach the bottom using Darcy with a single K value, you’d be wildly wrong. Richards’ equation captures this: at every moment, it recalculates how easily water flows based on how wet the sponge is right now, then uses conservation of mass to figure out how the wetness changes in the next instant. The conductivity and the moisture content are locked in a feedback loop, constantly chasing each other.

Now picture a farmer’s field (two versions). In the first, the farmer has tilled for decades, applied synthetic fertilizer, and the soil structure has collapsed into a compacted mass with uniform, tiny pores. Rain hits this field and either pools on the surface or races straight down through a few cracks. Most of it becomes runoff. The hydraulic conductivity K is low and doesn’t vary much because there’s not much structural diversity.

In the second field, the farmer stopped tilling years ago, built soil organic matter, and let earthworms and roots create a chaotic architecture of aggregates (clumps ranging from sand-grain size to golf-ball size, with air pockets scattered throughout). When rain falls here, something different happens. Water first clings to the surfaces of these aggregates, held by surface tension and the stickiness of organic coatings. The small pores between and within aggregates fill first, and K is still relatively low. But as moisture builds, the larger pores between clumps start conducting water, and suddenly K jumps (maybe 10-fold, 100-fold). The soil can hold more water in temporary storage (higher moisture content at any given pressure) and the varying pore sizes mean water moves through the profile more gradually rather than all at once.

Regenerative farming practices (no-till, cover crops, compost additions) are literally reengineering this K(θ) function. You’re changing the shape of the curve that relates moisture content to conductivity, making it so the soil can absorb more water, hold it longer, and release it more gradually.

Now we translate this into mathematical symbols: ∂θ/∂t = ∂/∂z[K(θ)(∂h/∂z + 1)]

Here’s what each symbol means: θ (theta) is the moisture content (the fraction of soil that’s water). The ∂θ/∂t on the left is “how fast moisture content changes over time” at one spot. That’s the conservation of mass part, tracking whether water is accumulating or draining. On the right, ∂/∂z means “how things change with depth going downward.” K(θ) is hydraulic conductivity as a function of moisture. This is Darcy’s law with variable conductivity. The (∂h/∂z + 1) term captures both the pressure gradient (∂h/∂z) and gravity (the +1, always pulling down). The whole right side describes how flow changes with depth. If more water flows in from above than flows out below, that spot gets wetter.

Let’s look at the full equation again and connect it to “slow it, spread it, sink it”:

“Slow it” refers to interventions like swales, zaï pits, and terraces that change the surface conditions and keep water in contact with soil longer. Without these structures, rain runs off immediately and ∂θ/∂t (the rate moisture accumulates) stays near zero. With them, water is held on the surface, giving it time to infiltrate, so ∂θ/∂t becomes positive. The soil actually gets wetter instead of staying dry while water sheets away. In math terms, the earthworks help establish the boundary conditions.

“Spread it” means creating conditions for lateral water movement. When you build soil structure with varying pore sizes and add organic matter unevenly across a field, you create spatial variation in both moisture content and conductivity. This generates lateral pressure gradients. Water doesn’t just plunge straight down; it redistributes horizontally from wetter zones to drier ones. While this specific version of the equation tracks water moving down,, the same logic applies sideways. By creating different soil textures, we create side-to-side 'pulls' that spread water horizontally across the landscape.

“Sink it” is about increasing K(θ), the hydraulic conductivity at different moisture levels. Practices like adding compost, eliminating tillage, and growing deep-rooted cover crops create macropores and improve soil structure, which increases K. Higher K means the flow rate (the K(θ)(∂h/∂z + 1) term on the right side) is larger, so water penetrates faster and deeper into the soil profile. You get positive ∂θ/∂t deeper in the soil column. Water sinks into storage rather than running off the surface or evaporating from the top few inches.

Richards’ equation lets us quantify exactly how much you need to slow, spread, and sink. If you measure K(θ) before and after implementing regenerative practices, you can model how a 2-inch rainstorm will distribute itself through the new soil structure versus the old. Will that rain mostly run off, or will 80% of it infiltrate and be available to plants a week later? The equation gives you numbers. It gives you some idea whether your interventions are enough to prevent erosion on a 5% slope during spring storms, or whether you need more cover crop biomass to hit your infiltration targets.

This is where regenerative practice meets computational power to create a feedback loop for restoration. As communities around the world engage in bioregional regeneration of the water cycle (digging swales and terraces that reshape topography and slow surface flow, building soil organic matter that increases aeration and aggregate stability, planting deep-rooted perennials that create macropore networks), each intervention changes the physical parameters in Richards’ equation. Measure the new K(θ) curves after adding compost. Survey the altered slopes and water table depths after installing earthworks. Plug these updated values into watershed models, and the equations calculate how water will actually move through your transformed landscape during the next monsoon or spring melt.

These predictions can be visualized on GIS platforms: interactive maps showing where moisture will accumulate, where erosion risk drops, where groundwater recharge increases. Farmers, land managers, and communities can see projected outcomes, adjust their designs, implement changes, then measure again and refine the models. The equations become a feedback mechanism: practice informs measurement, measurement refines models, models guide better practice. Richards’ equation becomes a tool for collective learning, a way to track, quantify, and accelerate the healing of landscapes one watershed at a time.

By viewing our soil as a giant, responsive sponge, we can use Richards’ Equation to model the soak and Darcy’s Law to model the seep. When we feed these rules into a GIS, we turn invisible underground flows into a digital ‘weather map’ for soil. This allows us to see exactly how our restoration efforts (like swales or cover crops) are physically changing the way our bioregion holds onto life-giving water.