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Two soil samples can hold the same water saturation yet contain very different meniscus areas, curvatures and trapped clusters. A single capillary-pressure–saturation curve compresses those configurations into one coordinate and therefore carries hysteresis. Adding specific fluid–fluid interfacial area can recover part of the missing state, but only when measurement scale, wettability and equilibrium assumptions are declared.
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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. Identify the closure problem
Classical two-phase models often prescribe capillary pressure Pᶜ as a function of wetting saturation Sʷ. Drainage and imbibition follow different paths because the same saturation can arise from different pore occupancies, contact-line histories and trapped clusters.
A constitutive closure must supply enough macroscopic variables to predict the required fluxes or pressure relations. If distinct microstates collapse to the same declared state but produce different responses, the closure is incomplete for that purpose.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Add specific interfacial area
Specific fluid–fluid interfacial area aʷⁿ is the fluid–fluid interface area divided by a declared bulk or pore volume. It carries information about how finely the phases are divided and how much interfacial energy is stored.
The extended proposal writes Pᶜ=F(Sʷ,aʷⁿ), producing a surface rather than separate drainage and imbibition curves. The area definition must say whether films, trapped clusters and unresolved interfaces are included.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work an illustrative state-surface ledger
Use the teaching surface Pᶜ=5+30(1−Sʷ)+0.8aʷⁿ kPa, where aʷⁿ is reported in mm⁻¹ for this illustration. At Sʷ=0.60 and aʷⁿ=3.0 mm⁻¹, Pᶜ=5+12+2.4=19.4 kPa.
| State item | Value | Contribution | Interpretation |
|---|---|---|---|
| Wetting saturation Sʷ | 0.60 | 12.0 kPa | Bulk phase amount |
| Interfacial area aʷⁿ | 3.0 mm⁻¹ | 2.4 kPa | Configuration coordinate |
| Offset | — | 5.0 kPa | Illustrative baseline |
| Predicted Pᶜ | — | 19.4 kPa | Point on the surface |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. See hysteresis as a projection
If drainage and imbibition states lie near one surface in (Sʷ,aʷⁿ,Pᶜ) space, projecting that surface onto the (Sʷ,Pᶜ) plane can produce two apparent curves. The omitted area coordinate explains part of the loop.
This does not make every loop disappear. Contact-angle hysteresis, topology, disconnected phases, dynamic pressure and solid deformation can require additional variables or path information.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change area at fixed saturation
On the illustrative surface, keep Sʷ=0.60 but reduce aʷⁿ from 3.0 to 1.5 mm⁻¹. The prediction falls from 19.4 to 18.2 kPa. The water amount is unchanged, but the phase configuration has changed.
| Changed state | Sʷ | aʷⁿ | Illustrative Pᶜ |
|---|---|---|---|
| Finely divided interface | 0.60 | 3.0 mm⁻¹ | 19.4 kPa |
| Coarser interface | 0.60 | 1.5 mm⁻¹ | 18.2 kPa |
| Drier, same area | 0.50 | 3.0 mm⁻¹ | 22.4 kPa |
| Same saturation only | 0.60 | Unreported | Not closed |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Connect area to trapping and mobilisation
The snap-off guide explains how pore geometry creates disconnected interfaces. The ganglion-mobilisation guide shows how forcing changes those trapped configurations.
Two states with the same phase volume can have different cluster counts and total interface. Mobilisation, coalescence and fragmentation therefore move the state through area as well as saturation.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect hysteresis and dynamics
The hysteresis guide tracks drainage, imbibition and scanning paths. The dynamic-capillarity guide adds rate-dependent pressure lag.
Use the air-entrapment guide for intermittent filling and residual gas, and the pressure-overshoot guide for unstable fronts.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Measure area and scale honestly
Micro-computed tomography can segment fluid phases and estimate interface area, while pore-network models and interfacial tracers provide other routes. Resolution, segmentation threshold and unresolved films can shift the estimate substantially.
The averaging volume must be large enough to represent the material but small enough to preserve relevant gradients. Surface smoothing changes area, so the reconstruction protocol belongs with the reported number.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A tight fitted Pᶜ–Sʷ–aʷⁿ surface in one material and flow regime is not a universal state law. Wettability, pore shape, saturation history, rate and phase connectivity can change the relation.
Interfacial area is not the same as mean curvature. Young–Laplace pressure depends on curvature, while area records extent; topology and curvature measures may be needed for stronger geometric closure.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use an evidence and source routine
Record fluids, interfacial tension, contact angles, porosity, saturation path, equilibration rule, pressure convention, area definition, averaging volume, image resolution, segmentation, connected versus trapped interfaces, fitted surface, residuals and out-of-path validation.
Reeves and Celia test a capillary-pressure–saturation–area relationship with a pore-network model. Pyrak-Nolte and colleagues relate capillary pressure to measured interfacial areas. Porter and colleagues characterise the three-variable relation using microtomography.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does interfacial area remove all hysteresis?
No. It can reduce path dependence in some systems, but contact-angle hysteresis, topology and dynamics may remain.
Can area be inferred from saturation alone?
Only through a model with assumptions; the purpose of the extra state variable is that identical saturation can support different areas.
Why divide area by volume?
Specific area provides a scale-aware macroscopic density, but the chosen bulk or pore volume must be declared.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a state-surface transfer check
Use the illustrative surface at Sʷ=0.70 and aʷⁿ=2.0 mm⁻¹. Then hold saturation fixed and double the area. Compute both pressures and explain why this is a configuration change rather than a change in water content.
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