Hydraulic Gradient Calculator - i = (h1 - h2) / L

A hydraulic gradient calculator turns two head measurements and the distance between them into the dimensionless ratio i, which tells you both the magnitude and the direction of subsurface water flow. Hydraulic gradient is the slope of the hydraulic head line between two wells, piezometers, or reference points, and it is the driving term in Darcy's law.

• • Two-well groundwater flow: Compute the gradient between two piezometers or wells to flag which way groundwater is moving.
• • Drainage and French system design: Estimate the gradient across a foundation drain or French drain so the design flow can be checked against the soil K.
• • Contaminant tracking: Combine the gradient with K and effective porosity to estimate seepage velocity and travel time for a contaminant plume.
• • Classroom and field work: Pair a field-measured i with a sieve-derived or lab K to estimate Darcy velocity during a hydrogeology lab.

Hydraulic gradient is dimensionless but usually expressed as meters per meter. A gradient of 0.01 m/m means the hydraulic head drops 1 cm for every meter between the two points.

Once the gradient is in hand, the next step is to multiply it by K from the Hydraulic Conductivity Calculator to estimate Darcy velocity and discharge.

The calculator chains i from heads and distance, then layers in Darcy velocity, seepage velocity, and flow rate when K, ne, and A are provided.

• h1, h2: Hydraulic heads at the two points in meters of water, read from a piezometer, water-table elevation, or total head value on the same reference datum.
• L: Straight-line distance between the two head measurements in meters. Must be greater than zero.
• K: Saturated hydraulic conductivity in m/day from a sieve analysis, lab test, or pump test.
• ne: Effective porosity as a decimal between 0 and 1. Excludes dead-end and bound water that does not contribute to flow.
• A: Cross-sectional area perpendicular to the flow direction in m^2, such as saturated thickness times width.

The sign of i tells you which point the flow runs toward: positive means flow goes from point 1 to point 2, negative means flow goes from point 2 to point 1.

According to U.S. Geological Survey, hydraulic gradient is the head loss per unit distance between two points, written as i = (h1 - h2) / L, and the sign of i indicates which way groundwater moves between the heads.

According to U.S. Geological Survey, Darcy's law in volumetric form gives Q = K * i * A and defines the Darcy velocity v = K * i and seepage velocity vs = K * i / ne.

Once Q is known, the volume-over-flow step that follows naturally lands on the Hydraulic Retention Time Calculator for detention basins, clarifiers, and recharge trenches.

Four ideas explain what the gradient actually means and why the calculator also returns Darcy velocity, seepage velocity, and flow rate instead of just one number.

These four ideas form the chain the calculator walks through: head sets i, i drives Darcy velocity, and ne converts Darcy velocity into the seepage velocity that moves solutes.

When the question is soil-water energy rather than saturated flow, the Water Potential Calculator keeps the same head vocabulary while moving into matric and osmotic potentials.

The form updates the gradient from the three primary inputs, and the optional K, A, and ne inputs layer in the Darcy-law outputs only when they are needed.

1. 1 Enter the head at point 1 and the head at point 2: Use piezometer readings, water-table elevations, or total-head values in meters of water. The two values must share the same reference datum so the difference is meaningful.
2. 2 Enter the distance between the two points: Use the straight-line distance between the two measurement locations in meters. A zero or negative distance is treated as zero gradient.
3. 3 Add hydraulic conductivity K if you want velocities: K in m/day turns the gradient into a Darcy velocity. Pull K from a sieve analysis, a constant head test, a falling head test, or a published K table for the soil type.
4. 4 Add cross-sectional area A and effective porosity ne as needed: A turns Darcy velocity into a volumetric flow rate Q. ne converts Darcy velocity into seepage velocity. Leave either at 0 to skip the matching output.
5. 5 Read the gradient, direction, and derived outputs: The primary output is i in m/m with a direction flag. Below it the calculator shows Darcy velocity in m/day and m/s, seepage velocity in m/day, and flow rate in m^3/day.

With h1 = 12 m, h2 = 8 m, L = 20 m, K = 5 m/day, A = 2 m^2, ne = 0.2, the gradient reads 0.2 m/m, the Darcy velocity is 1 m/day, the seepage velocity is 5 m/day, and Q is 2 m^3/day. Set K to 0 if you only need the gradient.

When the same head loss shows up in a pipe rather than an aquifer, the Friction Factor Calculator carries the same i concept over into pipe-flow Darcy-Weisbach head loss.

Four reasons a hydraulic gradient calculator is worth keeping next to field notes, lab sheets, and design spreadsheets.

• • Head-to-gradient math in one step: Subtracts h2 from h1 and divides by L, so the user never hand-divides head by distance.
• • Direction-aware sign handling: Returns a positive or negative i with a plain-language direction flag, so the user does not need to remember which way the formula points.
• • Chains into Darcy velocity and flow rate: Pulls K from a sieve, lab test, or pump test, then multiplies by i to expose Darcy velocity in m/day and m/s.
• • Adds seepage velocity when ne is given: Divides Darcy velocity by effective porosity so the right speed for tracer studies is one entry away.
• • Useful for coursework, fieldwork, and design review: Lets a student check a homework problem, a hydrogeologist sanity-check a two-well reading, and a reviewer verify a third-party drainage report.

These benefits compound: a single form that returns i, direction, Darcy velocity, seepage velocity, and flow rate saves the unit-conversion and sign-tracing work in a typical groundwater check.

Where the gradient feeds an underdrain, leach field, or recirculation loop, the Wastewater Calculator picks up the same Q plus the biological side of the design.

Five physical factors decide how the calculator output should be read. They explain why the same i can produce very different Q values in two adjacent sites.

• • The calculator assumes steady, saturated, single-phase flow with the same K and ne everywhere between the two points. Layered aquifers, transient pumping, and density-driven flow need a different model.
• • Sign conventions for hydraulic gradient vary by textbook. The calculator reports a positive value when h1 is greater than h2, but a downstream convention may report the opposite sign for the same physical setup.

Treat the result as a screening value, then move to a pumping test or numerical model once the design depends on the number.

According to U.S. Geological Survey, groundwater moves from areas of higher hydraulic head to areas of lower hydraulic head, so the hydraulic gradient controls both the magnitude and the direction of subsurface flow.

When the resulting Darcy velocity implies non-laminar pore flow, the Reynolds Number & Flow Regime Calculator gives the same regime check that Reynolds number gives in pipe flow.