Darcys Law Calculator - Flow Rate, Gradient, Flux

The darcy's law calculator estimates the volumetric flow rate of water through a saturated soil or rock from the medium's hydraulic conductivity K, the cross-sectional area A, the heads h1 and h2, and the flow path length L. It returns the head difference, the gradient, Q, the Darcy flux q, a porous-media Reynolds number, and a flow-direction label from the same six inputs. Henry Darcy published the relation in 1856 after sand-column experiments for the Dijon water supply, and the same proportionality still drives groundwater, soil-drainage, and aquifer-test work today.

• • Aquifer and well screening: Estimate how much water a sandy aquifer delivers to a well or drain before committing to a pumping test.
• • Soil drainage and septic design: Check how fast water moves through a soil layer under a known head drop, so a leach field or French drain can be sized.
• • Hydrogeology classwork and labs: Solve the Darcy equation for an assignment and compare the result to the calculator's output as a sanity check.
• • Filter and earthwork capacity checks: Predict the flow through a sand filter, gravel blanket, or compacted earthwork by treating it as a 1D porous medium.

In hydrogeology the law is usually written Q = K A i, where i is the dimensionless hydraulic gradient. The form used here replaces i with (h1 - h2) / L so the user can type heads in meters and a flow length directly, and the same expression also returns q = Q / A, the apparent velocity of water through the bulk medium, not the true pore velocity.

The two "Darcy" equations are easy to confuse: this page covers flow through a porous medium, while pipe friction in a closed conduit is covered by the Darcy Weisbach Calculator.

The form reads K, A, h1, h2, and L and solves the head form of Darcy's law in one pass. The same inputs feed a porous-media Reynolds number so you can check whether the law still holds at the computed flux.

• K: Saturated hydraulic conductivity of the medium in m/s.
• A: Cross-sectional area perpendicular to flow in m^2.
• h1: Total hydraulic head at the inlet in meters of water above a common datum.
• h2: Total hydraulic head at the outlet in meters above the same datum.
• L: Flow path length between the two head measurements in meters.

The sign of the result follows the head difference: positive gives forward, negative gives reverse, equal gives static. The Reynolds number uses d10 = 1e-4 m and nu = 1.004e-6 m^2/s, matching the constant-head permeameter convention. Values above about 1 to 10 flag flow that has left the strictly Darcian regime.

According to Wikipedia (Darcy's law), the integral form of the law in a homogeneously permeable medium is Q = (k * A) / (mu * L) * dP, which after substituting the static pressure relation p = rho g h becomes Q = (K * A * dH) / L, where K is the saturated hydraulic conductivity.

When the same head difference is part of a larger pipe or open-channel network, the Bernoulli Equation Calculator ties the head-driven result into the broader energy balance.

Four ideas decide whether the calculator's answer matches a real aquifer or sand column.

Once K, head, and gradient are kept straight, the same expression covers a constant-head permeameter test, a slug test in a monitoring well, and a regional groundwater flow net. Q scales linearly with K and A, so doubling either doubles Q.

For a closed pipe that runs full, head loss follows the Darcy-Weisbach relation, and the Friction Factor Calculator covers that friction-factor branch on its own page.

The form is laid out so medium properties sit on the first row, heads in the middle, and geometry on the bottom row.

1. 1 Pick a material preset or type a custom K value: Choose Gravel, Sand, Silt, or Clay to load a typical saturated K, or leave the preset on Custom and type a measured value.
2. 2 Enter the upstream and downstream hydraulic heads: Type h1 and h2 in meters of water above a common datum.
3. 3 Enter the cross-sectional area and the flow path length: Set A as the area perpendicular to flow in m^2, and L as the distance between the two head points in meters.
4. 4 Read Q, dH, i, q, and the Reynolds number: The primary card shows Q; the rows below give dH, i, q, the porous-media Re, and the flow direction label.
5. 5 Convert the result to the units your project uses: Pipe Q into a flow-rate unit converter to get L/s, GPM, or MGD without re-entering the inputs.

A municipal well taps a sand aquifer with K = 1e-4 m/s. Two observation points 100 m apart read h1 = 12 m and h2 = 10 m, and the section is 10 m^2. The calculator reports dH = 2 m, i = 0.02, Q = 0.00002 m^3/s, q = 0.000002 m/s, and Re about 0.0002, well inside the laminar Darcy regime.

When the project needs the same flow rate in L/s, GPM, or million gallons per day, the Flow Rate Converter rewrites Q in the working units without re-entering the inputs.

Darcy's law is one of the most cited equations in hydrogeology, and a quick numerical answer is usually faster and more accurate than working through the formula by hand.

• • Head form and flux form from the same inputs: Get Q, i, and q from one set of values, so the head-driven answer and the apparent velocity stay consistent.
• • Material preset for common soils: Load a typical saturated K for gravel, sand, silt, or clay with a single click.
• • Built-in Reynolds number check: The porous-media Re is shown alongside Q so you can see whether the flow is still in the laminar Darcy regime.
• • Reverse-flow and static-flow labels: Returns Forward, Reverse, or Static depending on the sign of the head difference.
• • Same form for screening and teaching: Use it to size a leach field, check a slug-test back-calculation, or work a homework problem without switching tools.

Combining Q, the Darcy flux, the porous-media Re, and a flow-direction label in one panel removes the need to keep four separate spreadsheets in sync, and a negative Q is not a bug, it is a reverse gradient.

To convert the volumetric flow rate into a residence time for a basin or aquifer cell, the Hydraulic Retention Time Calculator pairs the same Q with a volume to give a holding time in hours or days.

Several real-world variables decide where the computed flow rate lands and whether the law is even the right model.

• • Darcy's law assumes laminar, single-phase flow in a saturated, homogeneous medium. Fractured rock, karst, and unsaturated flow above the water table need a different model.
• • K is treated as a single scalar, but real aquifers are anisotropic: horizontal K is often 5 to 20 times larger than vertical K, and a 1D calculation overstates flow in any direction with a vertical component.

The K preset is a starting point, not a measured value. For a real project, K should come from a constant-head or falling-head permeameter test, a pumping test with observation wells, or a slug test, with the test method reported so the uncertainty is visible.

According to Wikipedia (Hydraulic conductivity), typical saturated K values for fresh groundwater at 20 degrees C are about 1e-2 m/s for well-sorted gravel, 1e-3 to 1e-5 m/s for sand and silty sand, 1e-5 to 1e-7 m/s for silt, and 1e-7 to 1e-11 m/s for unweathered clay (values adapted from the Bear, 1972 table reproduced in the article).

When the Reynolds number estimate here flags non-Darcy flow, the Reynolds Number Calculator takes a more general pipe or channel Reynolds number and explains the laminar-turbulent cutoff in detail.