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Richards equation couples conservation of water with a conductivity law for variably saturated soil. Its symbols can look compact, yet a solution depends on water-retention and conductivity functions, coordinate signs, initial water content, upper and lower boundary conditions, and any root-water sink. A profile balance is the quickest way to discover whether those choices tell one consistent story.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Begin with conservation, not the solver
For a one-dimensional soil profile, change in stored water equals inflow minus outflow minus extraction, with any runoff or evaporation placed at the correct boundary. Richards equation expresses this local conservation continuously through depth and time.
A numerical solution that violates the integrated water balance is not rescued by a smooth moisture plot. Always retain a separate ledger of boundary fluxes, sinks and storage change.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. State one sign convention completely
Let vertical coordinate z increase upward, pressure head be h and total head be H=h+z. Darcy–Buckingham flux is q=−K(h)∂H/∂z=−K(h)(∂h/∂z+1).
With a root-water sink S, conservation gives ∂θ/∂t=−∂q/∂z−S. Other texts may take depth positive downward. Their signs can differ without changing the physics, provided every term and boundary follows one convention.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a profile water-balance ledger
A 0.20 m profile rises from average volumetric water content 0.20 to 0.28. Its storage increase is 0.08×200 mm=16 mm. During the interval, 30 mm enters, 8 mm drains from the bottom and roots remove 4 mm.
| Term | Depth equivalent | Balance sign |
|---|---|---|
| Upper-boundary entry | 30 mm | + |
| Bottom drainage | 8 mm | − |
| Root uptake | 4 mm | − |
| Expected storage change | 18 mm | = |
| Observed storage change | 16 mm | Measured |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. See why material functions make the equation nonlinear
Water content θ depends on pressure head through a retention relation, while unsaturated conductivity K can change by orders of magnitude as pores drain. Both functions enter the same transient problem.
Parameters fitted from one moisture range may be weakly identified outside it. A mathematically converged solution can still be a poor physical prediction when retention or conductivity curves are extrapolated beyond measured conditions.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change only the lower boundary
Suppose daily upper entry is 12 mm and root extraction is 2 mm. With 5 mm of bottom drainage, storage rises by 5 mm. If a fixed-head lower boundary instead produces 15 mm of upward-positive outflow magnitude, storage falls by 5 mm.
| Upper entry | Bottom outflow | Sink | Storage change |
|---|---|---|---|
| 12 mm | 5 mm | 2 mm | +5 mm |
| 12 mm | 10 mm | 2 mm | 0 mm |
| 12 mm | 15 mm | 2 mm | −5 mm |
| 12 mm | Unknown | 2 mm | Not identifiable |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Choose upper and lower boundaries deliberately
The upper boundary may prescribe flux while rainfall remains below entry capacity, switch to a pressure head during ponding, and return excess water as runoff or surface storage. A fixed flux that exceeds what the profile can transmit may be physically impossible without such switching.
The lower boundary may be free drainage, a fixed or time-varying water-table head, a measured flux or a seepage-face condition. Each represents a different physical setting and can dominate long simulations.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the existing soil-water routes
Use the hydraulic conductivity and unit-gradient guide for the flux law and the soil-water potential guide for head components and root limits.
The guides to preferential flow and matrix bypass and infiltration, storage and drainage identify processes that a single-domain profile model can blur. The infiltration-curve reading guide separates surface entry rate, cumulative intake and ponding time.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Audit discretisation and solver behaviour
Richards equation is commonly solved numerically because the coefficients are nonlinear and boundary conditions change. Time step, grid spacing, convergence tolerance and interpolation of hydraulic functions can alter the balance and wetting-front shape.
Report cumulative mass-balance error as well as local residuals. Repeat the calculation with finer steps and stricter tolerances until the decision-relevant result is stable, not merely until the program stops warning.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A one-dimensional equilibrium-based Richards model does not automatically represent macropore bypass, hysteresis, entrapped air, swelling, water repellency or lateral redistribution. Extra domains or constitutive rules require data and validation.
Free drainage is a boundary assumption, not a measured fact. A fitted parameter set is not unique merely because one moisture series is reproduced, and matching water content does not prove that simulated fluxes are correct.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record coordinate direction, equation form, initial θ or h profile, retention and conductivity functions, parameter sources, layer geometry, upper and lower boundaries, switching rules, sinks, runoff, solver, grid, time step, tolerances, observations and cumulative balance error.
The USDA Agricultural Research Service HYDRUS-1D technical manual documents variably saturated Richards-equation flow, sink terms and boundary choices. The USDA ARS paper Generalized Richards’ Equation to Simulate Water Transport in Unsaturated Soils explains the classical scaling expectation and evidence that motivates broader models.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is Richards equation just Darcy’s law?
No. It combines a Darcy–Buckingham flux relation with water conservation and a storage relation.
Can rainfall always be imposed as a flux?
Only while the surface can accept it under the model. Ponding or runoff needs a switching boundary or surface-storage treatment.
Does a small solver residual guarantee accurate physics?
No. It shows numerical consistency with the chosen model, not that the functions and boundaries represent the field.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a layered-profile transfer check
Place a fine layer above a coarse layer, impose 20 mm of rainfall and compare a free-drainage lower boundary with a shallow fixed water table. Predict the sign of storage change before solving, then use the integrated balance to check the computed result.
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