Understanding how a reservoir rock transmits fluid, and how that capacity degrades when multiple phases compete for the same pore network, is the single most important rock-physics input to every IPR, Nodal Analysis, and stimulation decision you will make.
Permeability is the property of a porous rock that governs how easily fluids can flow through it under a pressure gradient. In a reservoir context, permeability is never truly absolute: every reservoir of commercial interest contains at least two fluid phases simultaneously, and each phase obstructs the other. The result is that the effective permeability to oil drops the moment free water or gas enters the pore network and it can drop dramatically.
For a production engineer, this is not an academic subtlety. A Gashaka-type reservoir (GK-22) at initial conditions might show a kro of 0.85 at connate water saturation; after a waterflood or aquifer influx drives water saturation from 25% to 60%, that same rock may transmit oil at only 15% of its single-phase capacity. The IPR has changed fundamentally, not because the reservoir pressure has moved, but because the fluid distribution in the pore space has changed. Recognising and quantifying this effect is the purpose of Topic 1.3.
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Lecture: From Darcy to Relative Permeability — Why Single-Phase Models Fail
16:40
Traces the conceptual path from Darcy's single-phase flow law through effective permeability to the relative permeability framework. Uses North Sea and West Africa field examples to show the production impact of saturation changes. Introduces the krw/kro crossover and its engineering significance.
LEARNING OBJECTIVES
After completing Topic 1.3, you will be able to:
1. Define absolute permeability and apply Darcy's Law to quantify single-phase flow capacity. 2. Explain how wetting behaviour controls which fluids occupy which pore sizes, and why this determines relative permeability curve shape. 3. Interpret two-phase kro/krw and kro/krg curves: identify Swc, Soc, crossover points, and end-point values. 4. Distinguish drainage from imbibition processes and predict the direction of hysteresis in both the wetting and non-wetting phase curves. 5. Apply Wyllie–Gardner, Corey, and Pirson correlations to generate synthetic kr data from critical saturation values. 6. Describe three-phase kro estimation using Stone's Models I and II. 7. Connect relative permeability directly to productivity index decline, IPR curvature below bubble point, and future IPR forecasting using the kro/(Boμo) ratio. 8. Identify the engineering levers (water injection profile, drawdown management, stimulation) available to mitigate relative permeability damage.
PREREQUISITE KNOWLEDGE
Topic 1.1 (PVT Behaviour): Fluid phase envelopes, bubble-point pressure, and viscosity/FVF as functions of pressure. Topic 1.2 (Formation Volume Factor): Why reservoir and surface volumes differ. Darcy's Law (introductory): q = kA(ΔP)/(μL), familiarity with the flow-rate/pressure relationship in single-phase flow.
PBL CONNECTION: GASHAKA GK-22 OIL LEG
The permeability work in this topic uses the GK-22 oil leg, Gashaka field, West Africa (k = 8.2 mD, h = 53 ft, φ = 0.19, p̅ = 5,651 psia). The oil leg penetrates a heterogeneous fluvial sandstone with a connate water saturation of 0.22 and critical gas saturation of 0.05. As reservoir pressure drops below the bubble point, solution gas evolves and liquid saturation changes, reducing kro and curving the inflow performance. Understanding relative permeability is essential to diagnosing the productivity decline observed in the GK-22 oil-leg data set and to evaluating whether pressure-maintenance or stimulation can restore deliverability.
Absolute k
The rock's intrinsic flow capacity when 100% saturated with a single fluid. Foundation of all flow equations.
Effective k
The rock's flow capacity to one specific phase when multiple phases are present. Always ≤ absolute k.
Relative k
Effective k normalised by absolute k. A dimensionless fraction (0–1) that depends only on saturation and wettability.
Section 1
Absolute Permeability: The Rock's Flow Capacity
Before introducing multi-phase complexity, we must anchor the concept of absolute permeability: what it measures, how it is obtained, and what geological factors control it.
1.1 Darcy's Law: The Defining Equation
Absolute permeability k is defined through Darcy's Law (1856), which describes single-phase, laminar, steady-state flow through a porous medium. In one-dimensional horizontal form:
q = kA(p₁ − p₂) / (μL)
Where:
q = volumetric flow rate [cm³/s or bbl/day]
k = absolute permeability [Darcy or mD]
A = cross-sectional area [cm² or ft²]
p₁−p₂ = pressure differential [atm or psi]
μ = fluid viscosity [cP]
L = sample length [cm or ft]
UNIT DEFINITIONThe Darcy
One Darcy is the permeability that allows 1 cm³/s of a 1 cP fluid to flow through 1 cm² cross-section under a pressure gradient of 1 atm/cm. Most reservoir rocks are in the millidarcy (mD) range. Tight gas sands: 0.001–0.1 mD. Good quality sandstones: 50–500 mD. Vugy carbonates: >1,000 mD (1 Darcy).
1.2 Conditions Required for Darcy's Law to Hold
Darcy's Law is not universally valid. The following conditions must prevail:
✅ Darcy's Law Applies When:
• Laminar (viscous) flow, low Reynolds number (<1) • Incompressible, homogeneous fluid • Isothermal conditions • No electrokinetic or adsorption effects • Rock pore geometry stable (no fines migration, no swelling clays) • Single fluid phase present
⚠️ Darcy's Law Breaks Down When:
• High flow velocity near wellbore, turbulent/inertial flow (Forchheimer effects in gas wells) • Multiple fluid phases present (requires relative permeability modification) • Klinkenberg gas-slippage at very low pressures • Non-Newtonian polymer fluids • Reactive fluids altering pore geometry
1.3 Geological Controls on Absolute Permeability
Permeability is a rock property determined by pore geometry, not composition. The key controls are:
Coarser grains → larger pores → higher permeability. A poorly sorted grain mixture packs tightly because small grains fill the interstices between large grains, reducing pore throat size and connectivity. Well-sorted sands (uniform grain size) preserve large, open pore throats. In fluvial channel sands (analogous to Gashaka), grain size variation is the primary driver of permeability heterogeneity: point bars and channel axes have coarser, better-sorted material than overbank or levee deposits.
As a rough rule, permeability scales with the square of mean grain diameter: k ∝ d²grain
High porosity alone does not guarantee high permeability. Permeability depends on how well the pores are connected, specifically on the pore throat radius, not the pore body radius. Chalks, for example, have porosities of 30–40% but permeabilities of only 0.1–1 mD because the inter-grain pore throats are microscopically narrow. By contrast, a coarse fluvial sandstone at 22% porosity may have 300 mD permeability because its pore throats are wide and well-connected.
Burial diagenesis systematically destroys permeability. Quartz overgrowths cement pore throats. Compaction reduces pore volume. The most damaging agents are authigenic clays: kaolinite, illite, and chlorite fill pore bodies and line pore throats. Illite in particular forms hair-like filaments that dramatically reduce pore throat radius, a 5% illite content in an otherwise excellent sandstone can reduce k from 500 mD to <10 mD. Clay swelling on contact with fresh water (smectite) can completely block pore throats, making formation water salinity compatibility critical in any well operation.
Natural fractures can enhance matrix permeability by orders of magnitude. In tight carbonates and basement reservoirs, fractures provide the principal flow paths. The permeability of a fracture is proportional to the cube of aperture: kfracture ∝ w³. This means fracture permeability is exquisitely sensitive to closure stress, as effective stress increases with depletion, fractures close and permeability plummets. The distinction between matrix k and fracture k is fundamental to stimulation design and recovery factor prediction in carbonate reservoirs.
1.4 Laboratory Measurement of k
Absolute permeability is measured on core plugs using a steady-state flow apparatus. A fluid (gas or brine) is flowed through a cylindrical plug at known rate, and the pressure drop is measured. Darcy's Law then gives k directly.
Measurement Method
Fluid Used
Notes / Correction Required
Typical Application
Liquid permeameter
Brine or oil
Directly gives kliquid ≈ kabs. Most representative.
Conventional core analysis
Gas permeameter (Klinkenberg)
N₂ or He
Gas slippage at low mean free path → overestimates k. Klinkenberg correction: kabs = kgas/(1 + b/p̄)
Routine core analysis, tight sands
Pressure pulse decay
Gas
For ultra-tight samples (k < 0.001 mD). Measures diffusion of pressure pulse.
Tight gas, shale
NMR (T2 relaxation)
Brine in situ
Empirical correlation: k = C·φ⁴·T²₂,lm. Useful log-based estimate.
Log analysis when no core
COMMON PITFALL
Core plug k ≠ reservoir k. Routine core plugs (1–2 inch diameter) capture grain-scale heterogeneity but miss fractures, laminations, and cross-bedding. Full-diameter core analysis, pressure-transient analysis (PTA), and production data analysis are required to upscale plug-scale k to reservoir scale. In the GK-22 well, plug k ranged 50–2,400 mD across 8 metres of pay, the production engineer must use thickness-weighted average k for flow calculations.
WORKED EXAMPLEKlinkenberg Correction, GK-22 Plug
Gas permeameter reading at mean pressure p̄ = 2.5 atm → kgas = 145 mD. Klinkenberg constant b = 0.40 atm.
k_abs = k_gas / (1 + b/p̄)
= 145 / (1 + 0.40/2.5)
= 145 / (1 + 0.16)
= 145 / 1.16
= 125 mD
Without correction: 16% overestimate of absolute permeability.
This matters when calibrating k to PTA-derived kh.
1.5 Permeability Anisotropy: kH vs kV
Reservoirs are almost always anisotropic: horizontal permeability (kH) is typically much higher than vertical permeability (kV) due to layering and compaction. The anisotropy ratio kV/kH typically ranges from 0.01 to 0.5 in sedimentary reservoirs. This ratio is critical for:
Horizontal Wells
Iani = √(kH/kV) appears in Joshi's equation. Low kV/kH penalises horizontal well performance significantly.
Coning Control
Low kV/kH retards water and gas coning, allowing higher critical rates before cone breakthrough.
Gravity Drainage
Gas-cap drainage and waterflooding efficiency both depend on kV to enable vertical fluid segregation and sweep.
Walkthrough of a steady-state permeability measurement on a sandstone plug, including Klinkenberg correction procedure and end-effect considerations. Demonstrates how confining pressure affects measured k, relevant to overburden correction for reservoir conditions.
Section 2
Wettability & Pore Fluid Distribution
Wettability, the preference of a rock surface to be in contact with one fluid over another, controls which fluid occupies which pore sizes, and therefore determines the entire shape of the relative permeability curve.
2.1 What is Wettability?
Wettability is the tendency of a fluid to spread on a solid surface in the presence of another immiscible fluid. It is characterised by the contact angle θ measured through the denser (water) phase:
Contact angle θ measured through the water (denser) phase against the rock surface. Water-wet systems (θ <90°) are most common in fresh, uncontaminated sandstone reservoirs.
The critical insight is this: the wetting phase always occupies the smallest pores and corners of pore space first. This has profound consequences for which fluid loses mobility fastest as its saturation changes.
Water (wetting phase) fills the smallest pores and coats grain surfaces. These small pores contribute very little to flow because permeability is proportional to the square of pore radius (Poiseuille flow in a tube: q ∝ r⁴). Therefore, even at relatively high water saturations, the wetting phase has low krw because it is confined to the smallest, least conductive pores.
Oil (non-wetting phase) occupies the larger, more conductive pores and pore centres. At connate water saturation, the oil fills essentially all the large pore space and kro is at its maximum. As water saturation rises (imbibition, waterflood), oil is displaced from the large pores first, causing kro to fall rapidly. The non-wetting oil phase becomes discontinuous at the critical oil saturation Soc, leaving isolated oil ganglia trapped in pore bodies.
Crossover point: In a water-wet system, the krw/kro crossover occurs at water saturations typically above 50%. At the crossover, both phases are flowing but neither is dominant. This is often near the economic limit for oil production.
In an oil-wet system, the roles are reversed. Oil now coats grain surfaces and fills small pores. Water occupies the large pore centres. The practical consequence is that kro remains relatively high even at high water saturations (because oil still lines the grain surfaces and contributes to flow through surface films), while krw at residual oil saturation can be surprisingly high (because water has access to large pores).
Oil-wet systems tend to produce higher residual oil saturations (Sor) after waterflooding, oil trapped in small pores is harder to displace. This makes oil-wet reservoirs more difficult to water-flood efficiently. Carbonates with adsorbed asphaltenes are the most common oil-wet systems in the industry.
Wettability alteration by surfactants (injected to make oil-wet surfaces water-wet) is a key enhanced recovery mechanism because it lowers Sor and improves kro/krw crossover, making waterflood more efficient.
Most producing reservoirs are neither uniformly water-wet nor oil-wet, they have mixed wettability. The Salathiel (1973) model explains this: when oil migrated into a previously water-wet sandstone, it contacted and altered the wettability of grain surfaces that were in direct contact with oil (i.e., the large pore surfaces). Grain surfaces coated by water films (the smallest pore contacts) remained water-wet.
The result is a reservoir where large pores are oil-wet and small pores remain water-wet, the opposite of uniform water-wet behaviour. Mixed wettability systems show distinctive kr curves: very low irreducible water saturation, very low residual oil saturation after waterflooding, and near-continuous oil production to very low kro values. This explains why many mature fields continue to produce oil at very high water cuts, the mixed wettability allows oil to trickle through the oil-wet large pore network even when globally waterlogged.
2.3 Pore-Scale Fluid Distribution and Flow
Understanding permeability requires thinking at the pore scale. Flow through a porous rock occurs through a network of pore bodies connected by pore throats. Permeability is controlled primarily by the smallest constriction, the pore throat.
Water-wet sandstone at intermediate Sw ≈ 45%. Water coats grain surfaces and occupies small pore throats; oil fills large pore bodies. Both phases flow simultaneously, the essence of two-phase relative permeability.
KEY INSIGHTWhy the Non-Wetting Phase Hurts kro so Much
The non-wetting phase (oil in a water-wet system) occupies the largest, most connected pores. These are the pores that dominate flow because permeability scales with r⁴. When even a small amount of water (non-wetting to oil's perspective as it enters large pores first during imbibition) enters these large pores, kro drops precipitously. A 10% increase in water saturation from Swc can cut kro in half. This is why water breakthrough has such a devastating and immediate impact on oil productivity, even before water reaches the surface, the relative permeability effect has already begun reducing the IPR.
Section 3
Two-Phase Relative Permeability Curves
The relative permeability curve is the most important petrophysical input to flow modelling. Understanding its anatomy, every critical saturation point and curve shape feature, is a prerequisite for interpreting well performance.
3.1 Definition and Normalisation
Relative permeability to phase j is defined as the effective permeability to that phase normalised by the absolute (or base) permeability:
k_rj = k_j(S_w) / k_base
Where:
k_rj = relative permeability to phase j (dimensionless, 0 ≤ k_rj ≤ 1)
k_j(S_w) = effective permeability to phase j at water saturation S_w
k_base = absolute permeability (most common) OR k_o at S_wc (used in some correlations)
Note: Using k at S_wc as k_base means (k_ro)_Swc = 1.0 by definition.
Using k_abs as k_base means (k_ro)_Swc < 1.0 (typically 0.7–0.95).
3.2 Anatomy of the Water–Oil kr Curve
The figure below shows a canonical water-wet system. Learn each critical point, they each have engineering meaning.
Figure 3.1: Canonical water-wet two-phase relative permeability curves. Green curve = k_ro (oil). Blue curve = k_rw (water). Regions A, B, C defined by saturation endpoints. All four critical points annotated.
FOUR CRITICAL SATURATION POINTS
Point 1 — Connate Water Saturation (Swc) — LEFT BOUNDARY:
The minimum irreducible water saturation. Water is present in all reservoir pores but is immobile: held by capillary forces in the smallest pores and surface films. kro is at its maximum value here. krw = 0 (water cannot flow). This is the initial reservoir condition for an oil reservoir above bubble point.
Point 2 — Critical Oil Saturation (Soc) — RIGHT BOUNDARY:
The minimum oil saturation at which oil can still flow. Below Soc, oil exists as isolated, disconnected ganglia trapped in pore bodies, it cannot form a continuous flow path. kro = 0 here. This is the residual oil saturation after waterflooding, and sets the maximum oil recovery from a waterflood.
Point 3 — End-Point kro at Swc [(kro)Swc]:
The maximum relative permeability to oil. Typically 0.70–0.95 depending on wettability and lithology. This is the starting point for IPR at initial conditions.
Point 4 — End-Point krw at Soc [(krw)Soc]:
The relative permeability to water when oil saturation is reduced to Soc (residual). In a water-wet system, krw at endpoint is typically 0.3–0.8. This determines the WOR at breakthrough and during the declining oil phase.
3.3 Gas–Oil Relative Permeability Curves
Gas–oil systems have a different shape. In the presence of connate water, oil acts as the wetting phase with respect to gas. The critical gas saturation Sgc is typically very small (0.01–0.08), gas begins to flow almost as soon as it forms a continuous phase. The oil relative permeability curve shape is concave upward (wetting phase shape) rather than S-shaped.
Figure 3.2: Gas–oil relative permeability curves. Red = k_rg. Amber = k_ro (oil acts as wetting phase, giving concave-upward shape). Critical gas saturation is very small. This system applies when reservoir pressure drops below bubble point.
Parameter
Water-Oil System
Gas-Oil System
Engineering Implication
Wetting phase
Water
Oil
Determines which phase occupies small pores
Non-wetting phase
Oil
Gas
Occupies large pores; faster mobility onset
kro curve shape
S-shaped (concave)
Concave upward
Oil recovers more slowly in gas-oil systems below BP
Critical saturation
Soc = 0.15–0.40
Sgc = 0.01–0.08
Gas flows almost immediately; oil needs more saturation to flow
Irreducible wetting S
Swc = 0.15–0.35
Swc in rock (immobile)
Sets left boundary; wetting phase always present
(kr)end-point
kro at Swc = 0.7–0.95
krg at Swc = 0.8–1.0
Starting IPR condition
FIELD EXAMPLEReading the kr Crossover: Dunbeath Field
The Dunbeath DB-07 producer shows a water-oil kr crossover at Sw = 0.58. Above this saturation, krw > kro, water has higher relative permeability than oil. At the prevailing viscosity ratio (μo/μw = 3.2), the mobility ratio M = (krw/μw) / (kro/μo) exceeds unity well before the crossover. This means that once water starts flowing preferentially, it will accelerate its own movement, a positive feedback causing rapid water cut increase after breakthrough. The crossover saturation is therefore not just a petrophysical curiosity, it is the inflection point in the production decline curve.
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Animation: How kr Curves Change Along the Waterflood Front
10:30
Animated visualisation showing how the local saturation state, and therefore the local kr values, evolve as a waterflood front progresses through a reservoir. Shows the connection between the Buckley-Leverett displacement theory and the kr curve shape. Illustrates why a favourable mobility ratio (M < 1) is needed for efficient sweep.
Section 4
Saturation History, Drainage & Imbibition
Relative permeability is not just a function of current saturation. It depends on how that saturation was reached. The path matters. This path-dependence is called hysteresis and it has direct consequences for waterflood design and recovery prediction.
4.1 The Two Fundamental Processes
Every kr measurement and every reservoir process follows one of two saturation-change directions. Getting this right matters enormously for which kr dataset you use in your model.
⬇ DRAINAGE: Wetting Phase Decreasing
Definition: The wetting phase saturation decreases (the non-wetting phase displaces the wetting phase).
Reservoir examples:
• Oil charging a water-saturated formation (geological migration)
• Gas expanding below bubble point (solution gas drive)
• Gas cap expanding into oil column
• Gas injection: gas displacing oil and water
Laboratory setup: Core initially saturated with water; oil is injected to displace water down to connate water saturation (Swc).
⬆ IMBIBITION: Wetting Phase Increasing
Definition: The wetting phase saturation increases (the wetting phase displaces the non-wetting phase).
Reservoir examples:
• Waterflooding: water displacing oil
• Aquifer encroachment
• Water influx from adjacent high-pressure water bearing zones
• Gas condensate systems: condensate accumulating as pressure drops
Laboratory setup: Core at Swc and residual oil condition; water is injected to displace oil toward residual oil saturation (Soc).
CRITICAL RULE FOR ENGINEERS
Always match the reservoir saturation history to the correct kr curve. Using drainage kr data to model a waterflood (imbibition process) will overestimate oil productivity and underestimate water cut, potentially by a large margin. Similarly, using imbibition kr for a gas cap drainage calculation underestimates gas production. Most oil field kr datasets are measured by imbibition (waterflood direction) because that is the most common production mechanism.
4.2 Hysteresis: Why Drainage and Imbibition kr Curves Differ
For the same saturation value Sw = 0.50, the relative permeability to oil differs depending on whether you arrived there by drainage (oil charging in) or imbibition (water flooding in). This is hysteresis, fundamentally driven by pore-scale trapping mechanisms.
Figure 4.1: Hysteresis in relative permeability. Left: k_ro, imbibition reaches zero at lower S_w (more oil trapping). Right: k_rw — drainage curve lies slightly above imbibition. The non-wetting phase (oil) shows stronger hysteresis than the wetting phase (water).
4.3 Mechanisms Behind Hysteresis
Why oil (non-wetting) shows stronger hysteresis: During imbibition (water flooding in), the invading water film advances along grain surfaces. When the water film reaches a pore throat, it can snap-off the non-wetting oil, pinching it into an isolated blob trapped in the pore body. This trapped oil blob cannot flow and does not contribute to kro. The snap-off mechanism depends on pore geometry (the ratio of pore body to pore throat radius, the aspect ratio). High aspect ratio pores (narrow throats, large bodies) promote snap-off and lead to high residual oil saturation Sor during imbibition.
The consequence is that imbibition Soc > drainage Soc, imbibition traps more oil. For waterflood design, this means the kr measured by drainage will overestimate oil recovery. Always use imbibition kr for waterflood calculations.
The wetting phase (water) shows less pronounced hysteresis because it preferentially occupies the smallest pores regardless of saturation direction, its pore occupancy is more deterministic. However, some hysteresis exists: during drainage, water is forced out of progressively larger pores (a continuous film remains on surfaces), while during imbibition, water re-enters those pores at slightly different conditions.
The practical consequence is that krw hysteresis is small enough that most engineering workflows use a single krw curve. The more critical hysteresis to capture correctly is kro.
Because Swc and Soc vary between core samples (and between drainage and imbibition), the effective (normalised) saturation S* is used to correlate kr data consistently across samples:
These effective saturations range from 0 to 1 and are used as the x-axis in most analytical kr correlations (Wyllie-Gardner, Corey, etc.). Normalising removes the effect of different connate water saturations so that kr from multiple core samples can be averaged onto a common scale.
4.4 Core Preparation and Its Effect on Measured kr
The kr values measured in the laboratory depend critically on how the core was prepared. Using the wrong preparation method can yield data that is completely unrepresentative of the reservoir.
Core State
Wettability Preserved?
Best For
Risk
Native State
Yes, if handled correctly
Most representative kr measurement
Expensive; wettability can alter during recovery if pressure drops
Restored State
Partially, after cleaning and reconditioning with crude oil
As a drainage-versus-imbibition analogue, consider the GK-22 gas-condensate interval (the fluid studied in Topics 1.1–1.2, distinct from the oil leg used for the permeability worked examples here). Its core kr data was measured using the restored-state imbibition method. The original reservoir pressure at discovery was above the dew point (6,200 psia) so that interval was single-phase gas-condensate. As pressure declined below the dew point, retrograde condensate began forming, this is a drainage process for gas (gas saturation decreasing as condensate accumulates) and an imbibition process for condensate (liquid saturation increasing). The correct kr to use for gas productivity prediction is the drainage krg versus liquid saturation, with the imbibition kro for condensate recovery if cycling is planned.
Section 5
Analytical kr Correlations
When core kr data is absent or unreliable, analytical correlations allow synthetic kr curves to be generated from critical saturation values alone. These correlations are essential for sensitivity analysis, early-field development planning, and numerical simulation input preparation.
5.1 The Wyllie–Gardner Correlations (1958)
The most widely used first-generation correlations. Based on capillary tube bundle models, they express kr as a power function of normalised saturation. Different exponents apply to different rock types.
DRAINAGE: Oil-Water System
Rock Type
kro = f(S*w)
krw = f(S*w)
Unconsolidated sand, well-sorted
(1 − S*w)³
(S*w)³
Unconsolidated sand, poorly sorted
(1 − S*w)³·⁵
(S*w)²(1 − S*w)
Cemented sandstone / oolitic limestone
(1 − S*w)²(1 − S*w²)
(S*w)⁴
DRAINAGE: Gas-Oil System
Rock Type
kro = f(S*o)
krg = f(S*o)
Unconsolidated sand, well-sorted
(S*o)³
(1 − S*o)³
Unconsolidated sand, poorly sorted
(S*o)³·⁵
(1 − S*o)²(1 − S*o²)
Cemented sandstone / oolitic limestone
(S*o)⁴
(1 − S*o)²(1 − S*o²)
KEY FORMULANormalised Saturation (used by all correlations)
S*_w = (S_w − S_wc) / (1 − S_wc)
S*_o = S_o / (1 − S_wc) = (1 − S_w) / (1 − S_wc)
S*_g = S_g / (1 − S_wc)
These range from 0 (at residual/connate condition) to 1 (at maximum mobile saturation).
WORKED EXAMPLE 5.1Wyllie-Gardner: Oil-Water Drainage Table (training values)
Given: Well-sorted sandstone reservoir. Swc = 0.25, Soc = 0.30, Sgc = 0.05. (Generic training endpoints: note the GK-22 oil leg itself carries Swc = 0.22; the method is identical.)
Generate the drainage oil-water relative permeability table.
Corey proposed a simple power-law model specifically for gas-oil drainage systems. It uses the effective gas saturation S*g and generates both kro and krg from critical saturation values alone.
COREY GAS-OIL DRAINAGE:
S*_g = S_g / (1 - S_wc) [effective gas saturation]
k_ro = (1 - S*_g)⁴ [oil relative permeability — strong sensitivity]
k_rg = (S*_g)²(1 - (1-S*_g)²) [simplified Corey gas — common approximation]
Note: The exponent 4 on kro makes gas-oil kro extremely sensitive to gas saturation.
Even 5% free gas saturation can cut kro by roughly a quarter (see Worked Example 5.2).
WORKED EXAMPLE 5.2Corey Method: Below Bubble Point in GK-22
Reservoir pressure drops below bubble point. Swc = 0.22. Calculate kro as free gas saturation builds.
k_ro = (1 - S*_g)⁴ where S*_g = S_g / (1 - 0.22) = S_g / 0.78
S_g S*_g k_ro Production Impact
0.00 0.000 1.000 Initial — full oil mobility
0.03 0.038 0.855 3% free gas → 15% drop in k_ro
0.05 0.064 0.767 Critical gas saturation often ~5%
0.10 0.128 0.578 10% Sg → 42% k_ro loss
0.20 0.256 0.306 20% Sg → 69% k_ro loss
0.30 0.385 0.143 30% Sg → 86% k_ro loss
(each k_ro = (1 − S_g/0.78)⁴, e.g. S_g = 0.10: (1 − 0.128)⁴ = 0.872⁴ = 0.578)
KEY INSIGHT: Below bubble point, solution gas drive is initially efficient
but k_ro collapses rapidly as gas saturation builds.
Pressure maintenance (gas injection/cycling) preserves k_ro above ~0.77.
5.3 Pirson's Correlation (1958)
A petrophysical approach that differentiates between imbibition and drainage processes for the non-wetting phase. Useful for capturing hysteresis effects without full dual-curve laboratory data.
PIRSON — Water-Wet Rock:
Wetting phase (water): Eq. 5-13, both drainage and imbibition:
k_rw = S*_w · (S_w)³
Two features worth internalising (forms per Ahmed, Reservoir Engineering Handbook, Eqs. 5-13 to 5-15). The wetting-phase law deliberately mixes the effective saturation S*w — which forces krw = 0 exactly at connate water — with the total saturation Sw³, reflecting Pirson's observation that wetting-phase conductivity scales with the full water-filled pore volume. And the imbibition oil curve reaches zero at Sw = 1 − Snw (trapped residual oil), whereas the drainage curve only reaches zero as S*w → 1: the hysteresis gap between the two is the trapped non-wetting saturation. At the benchmark Sw = 0.40 (Swc = 0.25 → S*w = 0.20): krw = 0.20 × 0.40³ = 0.0128, just above Wyllie-Gardner's (S*w)³ = 0.008, running two correlations and reading the spread as a free uncertainty band is the whole point of using them.
5.4 Generalised Corey (Power-Law) Model for Simulation
Modern reservoir simulators use end-point-scaled power-law models with adjustable exponents. This "generalised Corey" form is the default kr input format in most simulators (and is the model behind the Topic simulator in Section 8). The fully flexible LET model (Lomeland, Ebeltoft & Thomas, 2005) extends it with three shape parameters L, E, T per phase to capture S-shaped laboratory curves that a single exponent cannot, reach for LET only when measured data visibly misfit the power law. The general form for oil-water is:
k_ro = (k_ro)_Swc × [(1 - S_w - S_orw) / (1 - S_wc - S_orw)]^n_o
k_rw = (k_rw)_Soc × [(S_w - S_wc) / (1 - S_wc - S_orw)]^n_w
Where:
(k_ro)_Swc = k_ro at connate water saturation (end-point)
(k_rw)_Soc = k_rw at residual oil saturation (end-point)
S_orw = residual oil saturation in water-oil system
n_o, n_w = Corey exponents (typically 1.5–4 for oil, 1.5–3 for water)
COREY EXPONENT INTERPRETATION
The Corey exponent controls the curvature of the kr curve:
n = 1.0: Linear kr, straight line from endpoint to endpoint. Rare; only in perfect capillary tube bundles. n = 2.0: Mildly curved, typical for unconsolidated, well-sorted sands (Wyllie well-sorted correlation). n = 3–4: Strongly curved, kro drops steeply with initial water saturation increase. Typical for cemented sandstones, carbonates. Implies very rapid productivity loss after water breakthrough. n > 4: Extreme curvature, near-threshold behaviour; productivity collapses at low S_w above S_wc. Indicates significant pore structure heterogeneity (bimodal pore size distribution).
5.5 Normalisation and De-normalisation for Field Application
When averaging kr across multiple core samples with different Swc and Soc, the standard normalisation procedure removes end-point variability before averaging, then de-normalises to reservoir representative endpoints. This prevents arithmetic averaging from yielding physically meaningless intermediate values.
1
Collect kr data from all core plugs. Record measured (S_w, k_ro, k_rw) triples and note S_wc and S_oc for each plug.
2
Normalise each core plug: Calculate S*_w = (S_w − S_wc)/(1 − S_wc − S_oc). Calculate k*_ro = k_ro/(k_ro)_Swc and k*_rw = k_rw/(k_rw)_Soc. Now all curves span [0,1] on both axes.
3
Average normalised curves. At each S*_w value, compute permeability-thickness weighted average: (k*_ro)_avg = Σ[h_i · k_i · (k*_ro)_i] / Σ[h_i · k_i]
4
Establish representative end-points from log analysis, capillary pressure, and correlation plots: (k_ro)_Swc vs S_wc, (k_rw)_Soc vs S_oc.
Walkthrough: Generating kr Tables Using Wyllie-Gardner and Corey Correlations
12:20
Step-by-step generation of oil-water and gas-oil kr tables using all three main correlations. Shows how different rock types produce different curve shapes and end-point values. Compares correlation outputs against measured laboratory data from Gashaka-analogue West Africa fluvial sands.
Section 6
Three-Phase Relative Permeability
When gas, oil, and water coexist simultaneously, as in solution gas drive reservoirs below bubble point, gas-cap reservoirs, or gas injection schemes, three-phase kr estimation is required. Direct measurement is extremely difficult; practical engineering relies on analytical models built from two-phase data.
6.1 Three-Phase Flow Rules
The governing simplification, confirmed by experimental data, is that each phase's relative permeability depends primarily on its own saturation in a predictable way:
Water (Wetting)
krw = f(Sw only) Same function as in two-phase water-oil system. Water occupies smallest pores regardless of other phases present.
Gas (Non-Wetting)
krg = f(Sg only) Same function as in two-phase gas-liquid system. Gas occupies largest pores regardless of oil presence.
Oil (Intermediate)
kro = f(Sw, Sg) Most complex, oil is intermediate-wetting; its kr depends on BOTH water and gas saturations. Requires special models.
6.2 Stone's Model I (1970)
The most widely implemented three-phase kro model. It uses a probability (channel flow) framework to combine two sets of two-phase data into a three-phase prediction. The model requires a minimum oil saturation Som — the oil saturation below which oil becomes immobile when displaced by both water and gas simultaneously.
Stone simplified his approach to avoid the difficulty of estimating Som:
STONE'S MODEL II:
k_ro = (k_ro)_Swc × [(k_row/(k_ro)_Swc + k_rw) × (k_rog/(k_ro)_Swc + k_rg) − (k_rw + k_rg)]
Where:
(k_ro)_Swc = oil kr at connate water saturation (base value)
k_row = oil kr from water-oil system at S_w
k_rog = oil kr from gas-oil system at S_g
k_rw = water kr at S_w
k_rg = gas kr at S_g
Advantage: No Som required. More tractable for simulation.
Disadvantage: Can give k_ro > 1.0 at some saturations, must cap at (k_ro)_Swc.
WORKED EXAMPLE 6.1Stone's Model II: GK-22 Below Bubble Point
At reservoir conditions: So = 0.40, Sw = 0.30, Sg = 0.30. (kro)Swc = 0.88.
From two-phase data: krow = 0.403 (at Sw=0.30), krw = 0.030, krog = 0.524 (at Sg=0.30), krg = 0.650.
Stone's Model II:
k_ro = (k_ro)_Swc × [(k_row/(k_ro)_Swc + k_rw) × (k_rog/(k_ro)_Swc + k_rg) − (k_rw + k_rg)]
Term A = k_row/(k_ro)_Swc + k_rw = 0.403/0.88 + 0.030 = 0.458 + 0.030 = 0.488
Term B = k_rog/(k_ro)_Swc + k_rg = 0.524/0.88 + 0.650 = 0.595 + 0.650 = 1.245
Term C = k_rw + k_rg = 0.030 + 0.650 = 0.680
k_ro = 0.88 × [(0.488 × 1.245) − 0.680]
= 0.88 × [0.608 − 0.680]
= 0.88 × (−0.072) → NEGATIVE → k_ro = 0
INTERPRETATION: At Sg = 0.30 and Sw = 0.30, oil has lost all mobility.
Stone II predicts k_ro = 0: oil is immobile even though So = 0.40.
This is the combined impact of water and gas competing for pore space.
This reservoir is approaching abandonment conditions for oil production.
ENGINEERING IMPLICATION: THREE-PHASE DAMAGE
The three-phase example above illustrates a critical production engineering principle: when both free water and free gas are present simultaneously, oil relative permeability collapses far faster than either two-phase system would predict. A well that would be viable with only a waterflood (kro manageable) or only a gas drive (kro manageable) can become non-producing when both phases are simultaneously present. This is why maintaining reservoir pressure above bubble point (preventing gas liberation) and managing water injection profiles carefully (avoiding early breakthrough in high-permeability layers) are the two most impactful production engineering interventions.
6.4 Ternary Diagrams for Three-Phase kr Visualisation
Three-phase kr is commonly displayed on a ternary diagram. Each vertex represents 100% saturation of one phase. Contours of constant kro (oil isoperms) show how oil mobility varies across all possible three-phase saturation combinations.
Schematic ternary diagram showing oil isoperms in a water-wet, three-phase system. Oil mobility collapses rapidly as both S_w and S_g increase simultaneously. The working point from Worked Example 6.1 falls near the k_ro = 0 boundary.
Interactive Tool
Relative Permeability Curve Generator
Adjust critical saturation values and Corey exponents to generate and visualise kr curves in real time. Observe how curve shape changes with rock type, wettability, and saturation endpoints.
Scenarios to explore:
• Set Swc=0.20, Sor=0.35, no=4 → Cemented sandstone (high trapping)
• Set Swc=0.15, Sor=0.15, no=2 → Unconsolidated sand (efficient waterflood)
• Set krwe=0.8, nw=1.5 → Oil-wet system (high water mobility at Sor)
• Observe how crossover saturation shifts with exponent changes
kro Impact on Productivity Index (PI) INTERACTIVE
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Understanding the PI decline:
J = k·k_ro·h / (141.2·Bo·μo·[ln(re/rw)−0.75+S])
As S_w rises above S_wc, k_ro drops and J falls proportionally. This simulator shows the direct engineering consequence of water saturation increase on well deliverability, independent of any change in reservoir pressure.
SIMULATOR INTERPRETATION GUIDE
What to look for in Simulator 1:
1. The crossover saturation, where k_ro = k_rw lines cross. Above this S_w, water has higher relative permeability than oil.
2. How increasing n_o (Corey exponent) makes kro collapse faster after the first increase in water saturation: high n_o means poor tolerance to any water influx.
3. How endpoint kro and krw control the maximum producible rates at initial and abandonment conditions.
What to look for in Simulator 2:
1. The PI reduction as S_w rises, even a 10% rise above Swc can cut PI by 30–50% depending on rock type.
2. How high k and h partially compensate for kro loss, a thick, high-k reservoir tolerates water better than a thin, tight one.
3. That high oil viscosity amplifies the impact of kro loss: heavy oil reservoirs are particularly sensitive to water invasion.
Section 8
Relative Permeability in IPR & Production Engineering
Relative permeability is not just a rock characterisation exercise, it directly controls the shape of the IPR, the rate of productivity decline, and the choice of production strategy. This section connects kr to the engineering decisions that define field performance.
8.1 Relative Permeability and the Productivity Index
The productivity index J (or J*) in pseudo-steady-state flow is:
J* = q / (p̄ - pwf) = k·k_ro·h / [141.2·Bo·μo·(ln(re/rw) - 0.75 + S)]
KEY INSIGHT: J* is directly proportional to k_ro.
As water saturation rises above Swc:
• k_ro decreases → J* decreases proportionally
• Bo and μo also change as solution gas evolves below bubble point
• Combined effect: J* can decline to 10-20% of initial value
FUTURE IPR FORECASTING: Vogel Method with kr
The ratio of future to present productivity index (from Guo, Section 3.6.1):
This means that to forecast IPR at a future reservoir pressure, you need:
1. The future k_ro, from the kr curve at the projected future saturation
2. The future Bo and μo, from PVT (Topic 1.1 and 1.2)
The kr contribution is often the dominant driver of productivity decline below bubble point, as k_ro can fall by 50–80% while Bo and μo change by only 20–40%.
8.2 IPR Curvature Below Bubble Point: The kr Explanation
Vogel's empirical IPR equation for two-phase (saturated) reservoirs produces a curved, concave-inward IPR. The physical explanation is entirely in relative permeability:
Above Bubble Point: Linear IPR
Single-phase oil. kro = constant at (kro)Swc. J* is constant. q increases linearly with drawdown (p̄ − pwf). Vogel's equation reduces to: q = J* × (p̄ − pwf).
Below Bubble Point: Curved IPR
Free gas forms. krg increases, kro decreases. Near wellbore (lowest pressure), gas saturation is highest → kro lowest. Higher drawdown creates higher gas saturation → further kro reduction. This positive feedback creates the curved, flattening IPR characteristic of Vogel's equation.
IPR curve anatomy. Linear portion (above bubble point): k_ro is fixed, J* is constant. Curved portion (below bubble point): rising gas saturation progressively reduces k_ro, causing the characteristic Vogel concavity. The curvature is driven entirely by relative permeability.
8.3 Permeability Ratio krg/kro: Gas-Oil Ratio Control
The producing gas-oil ratio (GOR) at any point in the reservoir is governed by the mobility ratio of gas to oil:
GOR = R_s + (k_rg/k_ro) · (μ_o·B_o) / (μ_g·B_g)
Where R_s = solution GOR (gas staying in solution at reservoir pressure)
The second term = free gas production contribution
This means:
• When k_rg/k_ro is small (below critical gas saturation): GOR = Rs (solution gas only)
• When k_rg/k_ro rises (free gas above Sgc): GOR rises rapidly
• At high k_rg/k_ro: production becomes predominantly gas — "gas channelling"
log(k_rg/k_ro) = log(a) + b·S_g [semi-log linear relationship]
8.4 Engineering Levers: Managing the kr Constraint
Understanding relative permeability is not just analytical, it defines the engineering options available to manage productivity. The following table maps the kr constraint to the available production engineering response:
kr Problem
Mechanism
Engineering Response
Kr Impact
Water saturation rising → kro falling
Aquifer encroachment or waterflood breakthrough
Water shut-off (cement, mechanical), zonal isolation, selective perforation of low-Sw zones
Maintains kro in producing interval at lower Sw
Gas saturation building → kro falling below BP
Solution gas drive, free gas expansion
Pressure maintenance (gas injection, water injection), gas-lift optimisation to control drawdown
Keeps Sg below Sgc; preserves kro at near-initial values
Near-wellbore condensate banking
Retrograde condensate accumulates → krg collapses
Lean-gas cycling, solvent injection, periodic blow-down, hydraulic fracturing to bypass damage zone
Reduces liquid saturation near wellbore; restores krg
Mixed wettability → low kro
Asphaltene deposition / wettability alteration by crude
Chemical wettability modification (surfactant EOR), acid treatment to restore water-wet state
Shifts kro-krw curves toward water-wet (lower Sor, higher kro)
Preserves effective k; kro unaffected if k maintained
LOOKING AHEAD: kro LOSS IS PI LOSS, ONE-FOR-ONE
Because keff = k·kro multiplies the numerator of every inflow equation, any fractional loss of relative permeability is the same fractional loss of productivity index: halve kro and you have halved J*, exactly as if half the pay had vanished. Course 02 Module 09 builds its entire multiphase-flow treatment on this identity, there written as PIoil = PIinitial × kro/kro* (current end-point over original end-point), and extends it to gas wells losing krg to condensate banking and water loading. Everything in this topic's correlations exists to let you put a number on that ratio before the reservoir does it to you.
PBL TASK PREVIEW: MODULE 07 NODAL ANALYSIS
In the Module 07 Nodal Analysis problem set, you will use the GK-22 oil-leg reservoir data to:
1. Generate kr curves using Wyllie-Gardner correlation from given critical saturations (Swc=0.22, Sgc=0.05)
2. Calculate initial PI using kro at Swc
3. Predict future PI at reservoir pressure 2,800 psia (below the oil-leg bubble point of 3,250 psia) using the (kro/Boμo) ratio method
4. Plot future IPR and overlay VLP to find deliverable rate
5. Evaluate whether stimulation (fracturing to restore near-wellbore kro) can restore initial productivity
▶
Case Study: Diagnosing GK-22 Productivity Decline — A kr-Based Analysis
18:45
Full field case study using the Gashaka GK-22 well. Integrates PVT data, kr correlation outputs, and observed production data to diagnose the productivity decline. Demonstrates the Vogel method future IPR forecast and evaluates three remediation options: cycling, stimulation, and drawdown management. Reference production data and calculations provided in the case study supplement.
Assessment
Knowledge Check
Test your understanding of permeability and relative permeability. 12 questions covering all sections. Immediate feedback and explanation provided.
1. Absolute permeability is defined using Darcy's Law. Which of the following correctly states what "absolute permeability" represents in reservoir engineering?
Absolute permeability is the intrinsic capacity of the rock to transmit any single fluid, measured when the pore space is fully occupied by that one fluid. It is a rock property, not a fluid property, and does not depend on which fluid is used, though gas measurements require a Klinkenberg correction. It is NOT the permeability to oil at Swc (that is the effective permeability to oil at Swc, which equals the absolute k only if oil is non-wetting and fully occupies all connected pore space).
2. In a water-wet sandstone reservoir, which of the following best describes the pore-scale fluid distribution at connate water saturation (Swc)?
In a water-wet system, water (the wetting phase) always preferentially occupies the smallest pore spaces and coats grain surfaces. Oil (the non-wetting phase) is displaced to the larger, more central pore bodies. This is what makes kro at Swc close to its maximum: oil has access to the large, highly conductive pore space. The water at Swc is immobile precisely because it is in the smallest, capillary-dominated pores.
3. A well in a water-wet sandstone shows kro = 0.85 at initial conditions (Sw = Swc = 0.22). After an aquifer influx raises Sw to 0.50, what do you expect to have happened to kro?
For a well-sorted sandstone using the Wyllie-Gardner correlation (kro = (1-S*w)³), with Swc=0.22 and Sor=0.25: at Sw=0.22, S*w=0 → kro=1.0 (normalised). At Sw=0.50, S*w=(0.50-0.22)/(1-0.22)=0.359 → kro=(1-0.359)³=0.263. This is roughly a 74% reduction in kro, a massive productivity loss. A 28% increase in water saturation from connate typically cuts oil productivity by more than half. This is why water management is such a high-value activity.
4. What is the critical oil saturation (Soc) and why is it important for recovery factor estimation?
The critical oil saturation Soc (also called residual oil saturation Sor in an imbibition context) is the saturation at which oil becomes disconnected into isolated, immobile ganglia trapped in pore bodies. Below Soc, kro = 0: no oil can flow regardless of pressure gradient. It directly sets the maximum oil recovery by waterflooding: RF = (1 - Swc - Soc)/(1 - Swc). Lower Soc means higher waterflood recovery. Wettability alteration to more water-wet conditions typically reduces Soc and improves recovery.
5. What is the difference between a drainage and an imbibition relative permeability measurement, and which is appropriate for designing a waterflood?
Drainage involves decreasing the wetting-phase (water) saturation, the process that occurred geologically when oil charged the reservoir. Imbibition involves increasing the wetting-phase saturation, exactly what happens during a waterflood when injected water displaces oil. Using drainage kr for waterflood calculations overestimates oil recovery because drainage gives a lower residual oil saturation and higher kro at intermediate saturations than imbibition. The hysteresis (path-dependence) means the two processes follow different kr curves.
6. A gas-condensate well (GK-22 analogue) shows productivity declining significantly as reservoir pressure drops below the dew point. Using relative permeability concepts, what is the primary mechanism driving this decline?
Below the dew point, retrograde condensate liquid accumulates near the wellbore where pressure is lowest. This liquid saturation rise reduces the relative permeability to gas (krg) sharply, a phenomenon called "condensate banking." Because condensate is the non-wetting phase with respect to gas, its critical saturation is small (Soc/Slc ~ 0.05–0.15), meaning it traps easily. Even a 10% liquid saturation above irreducible can cut krg by 50–70%. This near-wellbore damage is severe because the pressure gradient is highest near the wellbore, exactly where the most condensate accumulates. Hydraulic fracturing or cyclic gas injection are the primary remediation strategies.
7. Using the Wyllie-Gardner correlation for a well-sorted sandstone (drainage, water-oil), at Sw = 0.40 with Swc = 0.20, what is kro?
For well-sorted sandstone (drainage): k_ro = (1 − S*_w)³. Always normalise first: S*_w = (S_w − S_wc)/(1 − S_wc) = (0.40 − 0.20)/(1 − 0.20) = 0.25. Then k_ro = (1 − 0.25)³ = (0.75)³ = 0.422. The commonest error is substituting the raw S_w = 0.40 directly, which gives (0.60)³ = 0.216 and understates the oil mobility.
8. In Stone's Model II for three-phase relative permeability, what happens to kro when high water saturation AND high gas saturation coexist simultaneously?
Stone's Model II combines the k_row and k_rog contributions multiplicatively (not additively), with a subtracted term for water and gas kr. The combined effect is synergistically damaging: each phase occupying pore space reduces the available pathways for oil, and the combined loss is greater than either phase alone. In the worked example, at So=0.40, Sw=0.30, Sg=0.30, Stone II predicted k_ro = 0 despite a substantial oil saturation. This three-phase competition for pore space is the mechanism behind the very poor oil recoveries seen in reservoirs that undergo simultaneous water influx and solution gas drive.
9. A reservoir engineer wants to forecast the future IPR at a lower reservoir pressure using Vogel's method. Which of the following kr-related inputs is required?
The Vogel-based future IPR method (Guo, Eq. 3.51) uses: J*_f = J*_p × (k_ro/B_o·μ_o)_f / (k_ro/B_o·μ_o)_p. This requires the future k_ro from the kr curve at the projected future oil saturation (after accounting for gas liberation), and the future Bo and μo from PVT. The k_ro contribution typically dominates the J* change below bubble point. This is why both Topics 1.1 (PVT) and 1.3 (kr) are prerequisites for Module 07 Nodal Analysis.
10. The producing GOR of a well below bubble point is controlled partly by the krg/kro ratio. If the semi-log plot of krg/kro vs Sg gives log(krg/kro) = −2.35 + 11.5·Sg, what is krg/kro at Sg = 0.20?
Using the semi-log relationship directly: log₁₀(k_rg/k_ro) = −2.35 + 11.5×0.20 = −2.35 + 2.30 = −0.05, so k_rg/k_ro = 10^(−0.05) = 0.89. The key concept is that the ratio grows exponentially with gas saturation, so small increases in Sg lead to large GOR increases.
11. What does a Corey exponent no = 4 (instead of no = 2) imply about the reservoir rock's response to water saturation increase?
Higher Corey exponent n_o means stronger curvature: kro stays high initially but then collapses faster as Sw increases. At n_o=2: kro = (1-S*w)². At n_o=4: kro = (1-S*w)⁴. Compare at S*w=0.30: n_o=2 → kro=0.49; n_o=4 → kro=0.24. The n_o=4 rock has already lost 76% of its oil flow capacity at the same saturation where n_o=2 still has 51% capacity. Cemented sandstones and carbonates with tight pore throats typically have n_o=3–5. This is critical for waterflood sensitivity analysis: a high n_o rock is a poor candidate for conventional waterflooding.
12. A production engineer observes that a well's PI has declined by 55% over 18 months despite reservoir pressure remaining essentially constant. Core data shows Swc = 0.20, Soc = 0.30, and the Wyllie-Gardner exponent of 3 applies. Which of the following best explains the observation?
PI ∝ k_ro. A 55% PI decline means k_ro has fallen to 45% of initial. With Swc=0.20, kro_initial = (1-S*w_initial)³ = (1-0)³ = 1.0. Need k_ro = 0.45 → (1-S*w)³ = 0.45 → S*w = 1 - 0.45^(1/3) = 1 - 0.766 = 0.234. Then Sw = 0.20 + 0.234×(1-0.20) = 0.20 + 0.234×0.80 = 0.20 + 0.187 = 0.387. So Sw ≈ 0.39, which rises toward 0.48 if we include the full denominator (1-Swc-Soc). The key answer is D: constant reservoir pressure eliminates depletion; the kr explanation (rising Sw from aquifer influx or water injection breakthrough) accounts for PI decline without any change in absolute permeability or skin. This is a classic production engineering diagnosis question.
TOPIC COMPLETE: NEXT STEPS
You have completed Topic 1.3: Permeability & Relative Permeability.
Topic 1.4 — Drainage Geometry builds directly on this foundation: how boundary shapes, drainage radius, and well placement interact with permeability in the flow equations.
Module 07 PBL Hub applies Topic 1.3 directly: you will generate kr tables, calculate the future PI for GK-22, and evaluate remediation options for the condensate banking problem.
Recommended supplementary reading:
Honarpour M., Mahmood S.M. (1988) Relative-permeability measurements-an overview, Journal of Petroleum Technology 40, 963–966. The definitive reference for all correlations covered in this topic.
Rose, W. 1987. Relative Permeability. In Petroleum Engineering Handbook, Chapter 28. Richardson, Texas: Society of Petroleum Engineers (SPE). A scholarly overview of kr measurement methodologies and their limitations.