Find your next learning step
An equilibrium water-retention curve assumes that capillary pressure and saturation have had time to settle onto a quasi-static relation. During rapid drainage or wetting, the measured pressure can lag or overshoot that relation. Dynamic-capillarity models represent the departure with a rate term, but the fitted coefficient can also absorb sample-scale heterogeneity, pressure gradients and sensor response. Rate dependence must therefore be demonstrated rather than assumed.
Open the chapter contents · Soil Science root · Gold Standard of Education · How X Works
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. Separate equilibrium state from transient response
Let S denote water saturation and Pc an explicitly signed capillary pressure. An equilibrium retention relation gives Pc,eq(S) after adequate equilibration. A transient experiment gives Pc,meas(t) while S is changing.
The dynamic residual is the difference between measured and equilibrium pressure under the chosen sign convention. It is not the whole capillary pressure, and its interpretation depends on where and how both pressure and saturation were measured.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Write a rate-dependent closure carefully
A common macroscopic closure relates the pressure residual to the saturation rate through a coefficient τ. Depending on sign definitions, authors write the term with opposite signs. The invariant instruction is to define ΔPc, dS/dt and τ together.
The coefficient may vary with saturation, drainage versus wetting, fluid pair, temperature, pore structure and measurement scale. Treating τ as a universal soil constant usually overstates what the experiment identifies.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a transient pressure ledger
At saturation 0.60, suppose the quasi-static curve gives Pc=12 kPa. Three illustrative drainage runs reach the same saturation at increasing rates and record 13, 15 and 18 kPa. The residuals are 1, 3 and 6 kPa.
| Run | Saturation | |dS/dt| | Measured Pc | Residual above equilibrium |
|---|---|---|---|---|
| Quasi-static | 0.60 | Near zero | 12 kPa | 0 kPa |
| Slow drainage | 0.60 | 0.002 min⁻¹ | 13 kPa | 1 kPa |
| Medium drainage | 0.60 | 0.006 min⁻¹ | 15 kPa | 3 kPa |
| Fast drainage | 0.60 | 0.012 min⁻¹ | 18 kPa | 6 kPa |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Estimate a coefficient without hiding the sign
For the illustrative drainage rows, residual divided by rate gives about 500 kPa·min. That arithmetic is meaningful only after the sign, units, saturation definition and equilibrium reference have been fixed.
Fit the relation over a saturation range and examine residual structure. A constant τ can fit one interval yet fail near saturation or residual water content, where phase connectivity and sensor behaviour change.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change sample size and challenge the mechanism
Repeat the same nominal rate in a small well-characterised cell and a tall heterogeneous column. The larger sample can show a much larger apparent residual because spatial pressure gradients and packing variability are folded into bulk averages.
| Observation | True local dynamics | Gradient or heterogeneity | Sensor lag | Diagnostic move |
|---|---|---|---|---|
| Residual grows with rate | Possible | Possible | Possible | Vary rate and scale |
| Residual grows with column length | Not required | Likely contributor | Possible | Add internal sensors |
| Residual follows sensor time constant | Not required | Not required | Likely | Step-response test |
| Residual persists at matched local state | Supported | Needs model | Must be corrected | Replicate at small volume |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Distinguish dynamic lag from hysteresis
Hysteresis gives different quasi-static paths after drainage and wetting reversals. Dynamic capillarity gives a departure related to how rapidly the state changes. Both can occur together, and a fast loop can confound them.
Measure slow boundary curves first, then run multiple rates with the same path and reversal history. Allow relaxation holds. If pressure returns toward the quasi-static branch at fixed saturation, that relaxation supports a non-equilibrium interpretation.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the established soil-water routes
The hysteresis guide separates path memory from rate. The Richards-equation guide shows how a modified constitutive law changes the transient balance.
Use the hydraulic-conductivity guide for flux sensitivity and the inverse-modelling guide when τ competes with retention, conductivity and boundary parameters.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Design an experiment that can identify rate effects
Use several controlled rates, repeated drainage and wetting paths, equilibrium holds, short sensor spacing and synchronised pressure and saturation observations. Characterise sensor step response before fitting a soil coefficient.
Report specimen dimensions, packing, porosity, temperature, fluid properties, boundary control and representative averaging volume. Validate the fitted closure on a rate and path withheld from estimation.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A loop in Pc–S data does not by itself separate hysteresis, dynamics, gradients and sensor delay. A large fitted τ can be a scale-dependent effective parameter rather than a pore-scale material property.
Dynamic correction is not automatically beneficial. Adding a weakly identified term can trade off against hydraulic conductivity and boundary resistance, fit one experiment and worsen prediction under a changed geometry.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record pressure and saturation definitions, equilibrium curve, path and reversal history, rates, relaxation holds, sample size, internal gradients, sensor response, temperature, fluid properties, τ form and units, fitting interval, parameter covariance, validation rate and uncertainty.
Hou and colleagues used a small-volume unsaturated system to test apparent rate dependence and cautioned about scale and packing effects. Li and colleagues experimentally examined saturation, rate and timing factors in dynamic non-equilibrium effects.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is τ a fixed soil property?
Not generally. It can depend on saturation, path, fluids, structure, scale and the measurement model.
Can fast measurements replace equilibrium retention data?
No. The equilibrium reference is needed to define the transient residual and to separate rate effects from the base curve.
Does a better fit prove the dynamic model?
No. Prediction at changed rates, paths and specimen scales is the stronger test.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a relaxation transfer check
Create three runs that reach the same saturation at different rates, add a hold at that saturation and plot pressure through relaxation. Fit one constant τ, then test it on a rate twice the estimation range and identify whether the failure suggests nonlinearity, gradients or sensor delay.
Continue through the Soil Science root, the Gold Standard of Education and the How X Works hub.
Top · Contents · Previous · How X Works