Relative Permeability

The phase-specific, dimensionless constitutive factor that reduces or modifies a porous medium's absolute permeability under multiphase occupancy, conditional on saturation, wettability, displacement history, and normalization convention.

Version

v2 · 2026-09-06 · History

Domain-specific #

2651

Origin domain

porous-media flow

Subdomain

multiphase constitutive modeling

Aliases

Phase relative permeability, Relative-permeability function

Core Idea

Relative permeability is the phase-specific constitutive factor used to represent how the simultaneous occupancy of a porous medium by multiple fluid phases changes the medium's capacity to transmit a selected phase. In the conventional multiphase extension of Darcy's law, the absolute permeability of the solid pore network is multiplied by a dimensionless relative permeability for each phase. The resulting effective or phase permeability is therefore not determined by the dry rock or soil alone: it also depends on which phase is being followed, how the pore volume is partitioned among phases, and how those phases occupy connected pores, throats, films, layers, and trapped clusters.

For phase \(\alpha\), a common continuum form is

\[ \mathbf q_\alpha=-\frac{k_{r\alpha}}{\mu_\alpha}\,\mathbf K \left(\nabla p_\alpha-\rho_\alpha\mathbf g\right), \]

where \(\mathbf q_\alpha\) is Darcy flux, \(\mathbf K\) is the single-phase absolute-permeability tensor, \(\mu_\alpha\) is phase viscosity, \(p_\alpha\) and \(\rho_\alpha\) are phase pressure and density, and \(k_{r\alpha}\) is relative permeability. In the scalar isotropic convention, the phase permeability is \(k_\alpha=k_{r\alpha}k\), so \(k_{r\alpha}=k_\alpha/k\). Muskat and Meres formulated the foundational heterogeneous-fluid extension of porous-flow equations in 1936; the form remains a standard macroscopic closure even though modern pore-scale work exposes its approximations.^[1][2]

The abstraction is not one empirical curve. It is the reusable role-and-relation package that makes a family of curves meaningful: a selected phase; a porous medium with a declared reference permeability; phase saturations constrained to sum to the pore volume; a constitutive relation assigning phase transport capacity over that saturation state; and stated dependencies such as wettability, drainage versus imbibition, capillary number, interfacial topology, and scale. Corey power laws, van Genuchten–Mualem functions, LET correlations, measured tables, hysteretic scanning rules, and pore-network predictions are alternative realizations of this package.^[3][4][5][6]

This node is autonomous because the package recurs as a single decision-bearing object across reservoir simulation, unsaturated-soil and groundwater modeling, geological carbon storage, geothermal flow, contaminant remediation, and multiphase laboratory analysis. It determines phase mobility, fractional flow, displacement fronts, breakthrough, trapping, injectivity, and recovery predictions. The live Permeability prime captures the general capacity of a bounded medium to transmit a carrier. It does not entail the multiphase normalization, competing phase pathways, saturation endpoints, or path-dependent closure family carried here.

Structural Signature

The recurring signature is:

porous medium with absolute permeability + two or more fluid phases + selected phase and saturation state + reference-permeability convention + phase-occupancy and history rule → dimensionless phase relative permeability → phase mobility and multiphase flux

The mandatory roles are:

• Porous-medium reference. A rock, soil, packed bed, membrane, electrode, or other pore network supplies an absolute permeability \(\mathbf K\), or another explicitly declared normalization basis.
• Phase system. At least two phases occupy the pore space. Phase labels, compositions, viscosities, densities, interfacial tensions, and the wetting order must be stated sufficiently for the closure to be interpreted.
• Saturation state. The phase fractions \(S_\alpha\) satisfy \(\sum_\alpha S_\alpha=1\) for the pore space under the adopted convention. Residual or irreducible saturations bound the interval in which phases are mobile.
• Phase-specific capacity. Each selected phase has an effective permeability or Darcy response. Dividing by the declared reference yields \(k_{r\alpha}\), or an equivalent dimensionless function is introduced directly in the phase flux law.
• Constitutive dependence. Relative permeability is assigned as a function of saturation and, when material, of wettability, saturation path, trapping state, capillary number, interfacial topology, composition, rate, or scale.
• Endpoint and curve convention. The normalization basis, endpoint values, saturation scaling, curve direction, and interpolation or correlation family travel with the data. A table without these controls is not a transportable relative-permeability specification.
• Model validity domain. The extended-Darcy assumptions, representative volume, phase continuity, force balance, coupling treatment, and laboratory-to-field scale delimit where the function can be used.

For a two-phase oil–water imbibition convention, an effective water saturation is often defined by

\[ S_w^*=\frac{S_w-S_{wc}}{1-S_{wc}-S_{or}}, \qquad 0\le S_w^*\le 1, \]

where \(S_{wc}\) is connate or irreducible water saturation and \(S_{or}\) is residual oil saturation for the stated process. A generic Corey-type pair is

\[ k_{rw}=k_{rw,e}(S_w^*)^{n_w}, \qquad k_{ro}=k_{ro,e}(1-S_w^*)^{n_o}. \]

The endpoints and exponents are fitted or otherwise selected; they are not universal constants. Even the common expectation \(0\le k_{r\alpha}\le1\) is a modeling convention rather than an inviolable theorem. Alternative reference permeabilities, slip-like behavior, unresolved momentum coupling, and unusual phase configurations can produce apparent endpoint values above one.^[7][8]

What It Is Not

Relative permeability is not absolute or intrinsic permeability. Absolute permeability characterizes single-phase transmission through the pore network under the governing continuum convention. Relative permeability is the phase-conditioned dimensionless factor used when other phases occupy and reorganize that network.

It is not phase mobility. The mobility of phase \(\alpha\) is commonly \(\lambda_\alpha=k_{r\alpha}/\mu_\alpha\). Relative permeability omits viscosity; two phases with different \(k_r\) values can nevertheless have comparable mobilities.

It is not fractional flow. With a simplified two-phase model, \(f_\alpha=\lambda_\alpha/\sum_\beta\lambda_\beta\). Fractional flow is derived from all phase mobilities and therefore from viscosities as well as relative permeabilities.

It is not capillary pressure, although simulators usually pair the two closures. Capillary pressure is a pressure difference between phases, such as \(p_n-p_w\), whereas relative permeability multiplies transport capacity. Both can depend on saturation and history without being aliases.

It is not saturation, porosity, wettability, or residual saturation. These are state, geometry, interfacial, or endpoint variables that help determine a relative-permeability function. None alone supplies the function.

It is not a coreflood trace, correlation name, or software table. Pressure, production, and saturation measurements are evidence from which curves are inferred; Corey, Stone, and LET are model families; a simulator keyword is an encoding. The abstraction is the constitutive role that these artifacts estimate or implement.

It is not electromagnetic relative permeability \(\mu_r\), the ratio of a material's magnetic permeability to vacuum permeability. That is a lexical collision in a different physical domain with different variables, units, and governing equations.

Scope of Application

The home scope is continuum modeling and measurement of multiphase flow in porous media. In petroleum and geothermal engineering, oil, water, gas, condensate, steam, and injected fluids compete for pore pathways. In vadose-zone hydrology and soil physics, liquid water and air share pore space, and relative hydraulic conductivity functions play the corresponding water-phase role when fluid properties and the saturated-conductivity reference are held consistent. Mualem's pore-size distribution model and van Genuchten's closed-form retention–conductivity pairing are canonical realizations in that tradition.^[4][5]

The same abstraction supports CO2–brine displacement in geological storage, hydrogen or natural-gas storage, NAPL transport and remediation, water infiltration, and multiphase heat-and-mass simulators. Reynolds and Krevor, for example, measure CO2–brine and nitrogen–water behavior while explicitly using the multiphase Darcy extension and investigating the impact of heterogeneity.^[9] Lawrence Berkeley National Laboratory's TOUGH family implements relative-permeability and capillary-pressure functions across geothermal, waste-isolation, environmental, and coupled heat-flow applications.^[10]

The scope is not every use of a dimensionless material ratio. Literal membership requires phase competition inside a pore network and a phase-specific constitutive transport factor. A wet fabric becoming harder to blow air through may instantiate the abstraction if phase saturation, pressure response, and normalization are modeled. Calling organizational access “relatively permeable” does not.

Clarity

Use four questions to identify a valid instance.

1. Which phase is being transported? A symbol such as \(k_{rw}\) is uninterpretable unless “water,” its companions, and their phase convention are known.
2. Relative to what? State whether the denominator is absolute permeability measured with a designated single phase, permeability at an endpoint condition, saturated hydraulic conductivity under fixed fluid properties, or another reference.
3. At what state and path? Give saturation, drainage or imbibition history, wettability condition, and any rate or capillary-number dependence needed by the model.
4. What does it control? The value must enter a phase flux, mobility, transmissibility, or equivalent constitutive relation—not merely label a sample.

A useful diagnostic boundary is this: if replacing the other phase with empty pore space would leave the reported “relative permeability” unchanged and no normalization or phase-competition rule is present, the object is probably ordinary permeability or a generic dimensionless coefficient. Conversely, if the same rock has different phase transport factors at different saturations or histories and those factors close a multiphase flow law, the relative-permeability identity is present.

Curves also require convention checks. An “oil endpoint of one” may mean that oil permeability at connate water saturation was chosen as the denominator rather than that the core transmits oil as freely as a fully oil-saturated, independently measured sample. Comparing endpoints without checking the denominator creates a false physical difference.

Manages Complexity

Pore-scale multiphase flow contains moving interfaces, contact-angle effects, snap-off, film and corner flow, ganglion trapping, connectivity changes, and an enormous number of possible fluid configurations. A reservoir- or aquifer-scale model cannot resolve each meniscus. Relative permeability compresses those effects into a small family of phase-conditioned constitutive functions that can be evaluated in every grid cell.

This compression separates three modeling levers. Absolute permeability supplies the persistent medium-scale pathway capacity. Relative permeability supplies the reduction or modification associated with phase occupancy and history. Viscosity supplies phase resistance. Because these levers appear multiplicatively in the Darcy closure, analysts can ask whether a low flux arises from tight rock, a disconnected phase, or a viscous fluid rather than treating all three as one opaque transmissibility.

The abstraction also standardizes experiments and calibration. Steady-state corefloods infer phase permeabilities from stabilized rates, pressure drops, and saturation states. Unsteady-state procedures infer curves from displacement histories; the Johnson–Bossler–Naumann method is a foundational example.^[11] Empirical families reduce noisy curves to endpoints and shape parameters. Moghadasi and colleagues show that model selection remains substantive: Corey, Chierici, LET, and model averaging can differ in fit and parameter burden.^[6]

Compression has a price. Two samples with similar fitted curves need not share the same pore-scale mechanism, and one curve measured on a core may not remain valid at field scale, different wettability, or different capillary number. Relative permeability manages complexity by declaring a closure, not by abolishing the unresolved physics.

Abstract Reasoning

Several inferences follow from the structure.

Phase-flux inference. Holding absolute permeability, gradient, and viscosity fixed, a lower \(k_{r\alpha}\) predicts a proportionally lower Darcy flux for phase \(\alpha\) within the extended-Darcy regime. This is a constitutive-model inference, not a universal pore-scale law.

Mobility-ratio inference. A phase with lower relative permeability can still dominate fractional flow if it is much less viscous. Therefore curve comparison without viscosities cannot determine displacement mobility or stability.

Connectivity inference. A nonzero phase saturation does not guarantee nonzero relative permeability. A phase can occupy isolated ganglia below its connectivity threshold. Conversely, thin wetting films can sustain nonzero transport at low bulk saturation.

History inference. If drainage and imbibition bounding curves differ, saturation alone is not a sufficient state variable. A simulator must remember a turning point, trapped saturation, scanning branch, or another internal history variable. Killough's history-dependent saturation functions are a canonical reservoir-simulation construction, and LBNL's hysteretic functions make the turning-point and trapped-gas state explicit.^[12][13]

Normalization inference. Scaling all effective phase permeabilities by a different reference changes reported \(k_r\) values even when the underlying measured fluxes are identical. Cross-dataset comparison therefore requires denominator harmonization before physical interpretation.

Three-phase inference. Two two-phase curve sets do not uniquely determine three-phase behavior. Stone's probability model estimates a three-phase oil relative permeability from water–oil and gas–oil data under explicit assumptions, illustrating that the bridge is a model rather than an identity.^[14] Three-phase measurements show sensitivity to saturation history, wettability, spreading, and fluid arrangement.^[15]

Knowledge Transfer

Literal transfer occurs within porous-media sciences when the mandatory roles remain intact. An oil–water reservoir curve and an air–water soil conductivity function use different nomenclature and empirical families, yet both normalize phase-specific transmission, index it by saturation, and insert it into a continuum flux law. A CO2–brine storage model can import experimental logic from petroleum core analysis only after matching rock type, wettability, fluid system, capillary number, temperature, pressure, and displacement history.

The transferable workflow is: declare the reference permeability; identify phases and wetting order; choose independent saturation variables; measure or infer phase response; normalize it; select a curve or table representation; attach endpoints, history, and validity domain; and propagate the functions through mobility and conservation equations. This workflow transfers across laboratory, pore-network, and field simulators even when their numerical methods differ.

Some transfer is only analogical. The idea that competing occupants reduce one another's access resembles congestion in networks, but traffic has no phase saturation, interfacial tension, or Darcy normalization. That portable residue belongs to broader Permeability, Competition, Constraint, or Hysteresis patterns. It does not make Relative Permeability a prime.

Examples

Canonical two-phase Corey-type specification. Consider an illustrative waterflood model with \(S_{wc}=0.20\), \(S_{or}=0.20\), \(k_{rw,e}=0.30\), \(k_{ro,e}=0.90\), \(n_w=3\), and \(n_o=2\). At \(S_w=0.50\), the normalized saturation is

\[ S_w^*=\frac{0.50-0.20}{1-0.20-0.20}=0.50. \]

The model then gives \(k_{rw}=0.30(0.5)^3=0.0375\) and \(k_{ro}=0.90(0.5)^2=0.225\). With water viscosity \(1\) cP and oil viscosity \(5\) cP, the mobilities are proportional to \(0.0375\) and \(0.045\) cP\(^{-1}\), so the simplified water fractional flow is \(0.0375/(0.0375+0.045)\approx0.455\). The example is constructed, not a universal rock. It demonstrates why relative permeability, mobility, and fractional flow must remain distinct.

Unsaturated water flow. Mualem predicts unsaturated conductivity from the moisture-retention curve and saturated conductivity; van Genuchten supplies a retention relation that yields a closed-form relative conductivity when combined with Mualem's model.^[4][5] At fixed water properties, this relative conductivity functions as the water-phase relative permeability. The state coordinate is effective water saturation, the dry-end and saturated endpoints are declared, and the curve closes a continuum water-flow equation. A petroleum Corey curve cannot simply be substituted because the pore-size model, retention relation, and history convention differ.

History-dependent gas trapping. During primary drainage, nonwetting gas invades as liquid saturation falls. Later imbibition can strand gas in disconnected clusters. At the same current liquid saturation, the gas relative permeability can therefore differ depending on the maximum gas saturation and turning history. A hysteretic simulator selects or interpolates a scanning rule rather than reading one single-valued table. The abstraction persists—phase capacity normalized to a medium reference—but saturation alone no longer supplies the full state.^[12][13]

Three-phase reservoir prediction. When water, oil, and gas coexist, direct three-phase measurements are difficult. Stone's construction estimates three-phase behavior from two two-phase data sets, while experimental reviews show why wettability, spreading, and saturation history can defeat naive interpolation.^[14][15] The example separates the stable abstraction from one closure model: relative permeability remains required, but the function cannot be identified by name alone.

Structural Tensions

• Macroscopic closure vs. pore-scale mechanism. A compact \(k_r(S)\) function makes field simulation possible, but different interfacial topologies can map to the same saturation. Diagnostic: if pore-scale configuration changes while saturation does not, test whether one curve predicts both fluxes.
• Saturation-only simplicity vs. state memory. Single-valued curves are efficient, while drainage, imbibition, trapping, and wettability alteration create path dependence. Diagnostic: cycle saturation and compare responses at repeated saturation values.
• Normalization portability vs. denominator ambiguity. Dimensionless values invite comparison, but absolute-, endpoint-, and fluid-specific bases can differ. Diagnostic: reconstruct dimensional phase permeability before comparing datasets.
• Laboratory control vs. field representativeness. Corefloods constrain fluids and boundary conditions, while reservoirs contain larger-scale heterogeneity and mixed histories. Diagnostic: test scale, capillary-number, and directional sensitivity before upscaling.
• Empirical fit vs. parameter identifiability. Flexible correlations can match measured points better but may trade endpoint and shape parameters or extrapolate poorly. Diagnostic: compare predictive error and parameter stability, not fit residual alone.
• Decoupled phase Darcy laws vs. interphase coupling. The conventional model assigns each phase its own permeability factor; momentum coupling, non-Darcy behavior, and slip-like effects can produce apparent \(k_r>1\) or rate dependence. Diagnostic: vary rate and model coupling before treating an anomalous endpoint as impossible.^[7][8]
• Distinct domain identity vs. broad Permeability. Every instance uses phase-specific permeability, but the defining residual is the multiphase normalized closure with saturation and history. Diagnostic: remove phase competition and normalization; if the account remains complete, route it to Permeability rather than this node.

Structural–Framed Character

Relative Permeability is mixed-structural, leaning structural. Its strongest structural commitment is the phase Darcy decomposition: reference pathway capacity, dimensionless phase factor, viscosity, and driving potential have distinct mathematical roles. Saturation conservation, phase-specificity, and multiplication by the reference permeability survive changes of fluid system and solver.

Its framing dependence is nevertheless substantial. The reference denominator, residual-saturation convention, drainage or imbibition path, relative-permeability model, wettability state, measurement method, and scale determine the reported curve. A value such as \(k_{rw}=0.2\) has little meaning without those declarations. Even bounds often taught as definitional can be violated as apparent parameters when the underlying flow law omits coupling or adopts a different basis.

The proper reading is therefore neither “arbitrary fitted curve” nor “intrinsic rock constant.” It is a controlled constitutive abstraction: structurally fixed in role, empirically and conventionally framed in value.

Structural Core vs. Domain Accent

The portable structural core is a graded transmission capacity for a selected carrier through a persistent medium, factored into a baseline pathway capacity and a context-dependent modifier. That core instantiates the live Permeability prime. History-dependent branches also instantiate Hysteresis when the same saturation supports different responses after different displacement paths.

The domain accent is indispensable: pore volume is partitioned into fluid phases; saturations obey a sum constraint; capillary interfaces and wetting order organize connected pathways; an extended multiphase Darcy law defines the response; endpoint saturations delimit mobility; and laboratory or pore-scale evidence supplies the closure. Remove those commitments and only generic permeability, congestion, normalization, or path dependence remains.

This explains why literal recurrence across petroleum, soil physics, hydrogeology, carbon storage, and geothermal modeling does not justify prime classification. Those practices share porous-media physics. The structure does not transfer unchanged to unrelated organizational or computational substrates.

Instantiates / Related Primes

Permeability is the minimal live parent relation. Relative Permeability uses the same medium–carrier–pathway–gradient structure, but conditions the carrier's capacity on simultaneous phase occupancy and expresses it relative to a reference. The proposed DAG relation is composition/part-of rather than subsumption: the dimensionless factor itself is not the entire dimensional permeability or flux relation.

Hysteresis is instantiated when drainage and imbibition or scanning curves return different \(k_r\) values at the same saturation because internal phase configuration preserves history. Hysteresis is common but not mandatory in every simplified relative-permeability model, so it is related rather than a second parent.

Normalization is a portable operation in the definition \(k_{r\alpha}=k_\alpha/k\), but no normalization operation determines phase topology or saturation dependence. Constraint appears in the saturation sum and endpoint interval. Competition appears as phases contend for connected pathways. These primes illuminate roles without jointly composing away the domain-specific abstraction.

The live domain-specific Wettability node is a strong causal neighbor. Wettability affects which phase coats surfaces or occupies pore centers and therefore changes relative-permeability curves, but a contact-angle or wetting-state description does not supply phase transport capacity. Neither node covers the other.

Relationships to Other Abstractions

Current abstraction Relative Permeability Domain-specific

Parents (1) — more general patterns this builds on

• Relative Permeability is part of Permeability Prime

  Permeability is the minimal live parent relation.

Hierarchy path (1) — routes to 1 parentless root

• Relative Permeability → Permeability → Boundary

Neighborhood in Abstraction Space

Relative Permeability sits in a sparse region of the domain-specific corpus (93^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Ergun equation — 0.80
• Subsurface Flow — 0.79
• Confined Liquid — 0.79
• Subsidence — 0.77
• Van der Waals Equation — 0.77

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Permeability: the broad or single-phase pathway capacity; no competing-phase saturation closure is required.
• Effective/phase permeability: the dimensional capacity \(k_\alpha=k_{r\alpha}k\), not the dimensionless factor alone.
• Hydraulic conductivity: a dimensional flux coefficient that includes fluid density and viscosity as well as medium properties; relative conductivity may coincide with a relative-permeability factor only under controlled fluid conventions.
• Mobility and mobility ratio: quantities that divide relative permeability by viscosity and compare phases.
• Fractional flow: a derived share of total flow formed from all phase mobilities.
• Capillary pressure: the phase-pressure difference closure, often paired with but not identical to relative permeability.
• Wettability: the interfacial preference controlling phase arrangement, not the transport curve itself.
• Porosity or saturation: pore-volume measures, not phase transmission capacities.
• Transmissibility: a discretized flow coefficient that also contains geometry and may combine permeability, mobility, and grid connection factors.
• Electromagnetic relative permeability: \(\mu_r\), an unrelated magnetic-material ratio.

References

[1] Muskat, Morris, and Milan W. Meres. “The Flow of Heterogeneous Fluids Through Porous Media.” Physics 7, no. 9 (1936): 346–363. Foundational multiphase porous-flow formulation in which saturation determines separate phase permeabilities. registry ↩

[2] Blunt, Martin J. “Navier–Stokes Equations, Darcy's Law and Multiphase Flow.” In Multiphase Flow in Permeable Media: A Pore-Scale Perspective. Cambridge University Press, 2017, 219–314. Authoritative derivation, definition, assumptions, pore-scale interpretation, and extensions of multiphase Darcy flow. See also Chapter 7, “Relative Permeability.” registry ↩

[3] Corey, A. T. “The Interrelation Between Gas and Oil Relative Permeabilities.” Producers Monthly 19 (1954): 38–41. Foundational power-law correlation family for phase relative permeabilities. registry ↩

[4] Mualem, Yechezkel. “A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media.” Water Resources Research 12, no. 3 (1976): 513–522. Primary pore-size/retention-based model for unsaturated conductivity. registry ↩a ↩b ↩c

[5] van Genuchten, M. Th. “A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils.” Soil Science Society of America Journal 44, no. 5 (1980): 892–898. Primary closed-form retention and relative-conductivity construction. registry ↩a ↩b ↩c

[6] Moghadasi, Leili, Alberto Guadagnini, Fabio Inzoli, and Martin Bartosek. “Interpretation of Two-Phase Relative Permeability Curves through Multiple Formulations and Model Quality Criteria.” Journal of Petroleum Science and Engineering 135 (2015): 738–749. Experimental and model-selection comparison of Corey, Chierici, LET, and averaging approaches. registry ↩a ↩b

[7] Berg, Steffen, A. W. Cense, J. P. Hofman, and R. M. M. Smits. “Two-Phase Flow in Porous Media with Slip Boundary Condition.” Transport in Porous Media 74, no. 3 (2008): 275–292. Primary investigation of endpoint relative permeabilities above one and a slip-based interpretation. registry ↩a ↩b

[8] Bravo, Maria C., and Mariela Araujo. “Analysis of the Unconventional Behavior of Oil Relative Permeability during Depletion Tests of Gas-Saturated Heavy Oils.” International Journal of Multiphase Flow 34, no. 5 (2008): 447–460. Primary evidence and generalized-coupling interpretation for apparent oil relative permeability above one. registry ↩a ↩b

[9] Reynolds, C. A., and S. Krevor. “Characterizing Flow Behavior for Gas Injection: Relative Permeability of CO2–Brine and N2–Water in Heterogeneous Rocks.” Water Resources Research 51 (2015). Primary application and measurement study addressing the multiphase Darcy model and heterogeneity. registry ↩

[10] Lawrence Berkeley National Laboratory. “TOUGH Software Suite.” Official documentation for multiphase porous-flow simulators in which phase interference is represented by relative-permeability functions. registry ↩

[11] Johnson, E. F., D. P. Bossler, and V. O. Naumann. “Calculation of Relative Permeability from Displacement Experiments.” Transactions of the AIME 216 (1959): 370–372. Foundational method for inferring individual phase curves from displacement experiments. registry ↩

[12] Killough, J. E. “Reservoir Simulation With History-Dependent Saturation Functions.” Society of Petroleum Engineers Journal 16, no. 1 (1976): 37–48. Primary hysteretic drainage, imbibition, and scanning-function construction. registry ↩a ↩b

[13] Doughty, Christine. User's Guide for Hysteretic Capillary Pressure and Relative Permeability Functions in iTOUGH2. Lawrence Berkeley National Laboratory, 2013. Authoritative implementation of drainage, imbibition, turning-point, and trapped-gas state rules. registry ↩a ↩b

[14] Stone, H. L. “Probability Model for Estimating Three-Phase Relative Permeability.” Journal of Petroleum Technology 22, no. 2 (1970): 214–218. Primary model estimating three-phase data from two-phase measurements. registry ↩a ↩b

[15] Alizadeh, Ali H., and Mohammad Piri. “Three-Phase Flow in Porous Media: A Review of Experimental Studies on Relative Permeability.” Reviews of Geophysics 52 (2014). Authoritative synthesis of saturation-history, wettability, spreading, interfacial-tension, and measurement effects. registry ↩a ↩b