Fluid flow in a nice, straight pipe is a piece of cake (well, relatively speaking) – you can solve the Navier–Stokes equations and get a precise result.
But flow in porous rock?
That’s a whole different nightmare. The pore network inside reservoir rock is a convoluted maze of tiny channels that we do not have a complete map for, so we can’t practically solve Navier–Stokes there.
Enter Darcy’s law, the brilliant black-box simplification.
Henry Darcy (way back in the 1800s) basically said: “I can’t see inside the sandbox, but I can measure what goes in and out.” He defined permeability, K, as the proportionality constant linking pressure drop to flow rate, normalized by fluid viscosity and geometry.
This empirical law was originally derived from experiments with single-phase water flow through sand – a clean, elegant solution for one fluid moving through a porous medium under steady conditions.
Darcy’s law was (and still is) a stroke of genius for single-phase flow. It treats the porous rock like a homogenized continuum, skipping all the messy pore-scale complexity. You get a permeability constant K that encapsulates the rock’s ability to conduct one fluid.
In essence, Darcy turned a hopelessly complex pore-scale flow problem into a simple equation:
The reality: oil reservoirs rarely have the decency to contain only one fluid. We’ve got oil, water, gas, all competing to occupy the same pore space.
Darcy’s neat law didn’t directly apply anymore, so what did engineers do? They Frankensteined Darcy’s elegant law to cover multiphase flow.
Essentially, we insisted on using Darcy’s framework for two or three simultaneous fluids by sneaking in a correction multiplier for each fluid. This magical fudge factor is what we call relative permeability.
In other words, we keep Darcy’s equation form for each phase, and let this new kr paremeter carry the burden of all the messy physics of fluids sharing pores.
The second equation is almost identical to the first, except for the introduction of the Kro term. Seeing them side by side makes it clear that Kro acts as a simple correction factor.
This was an ingeniously lazy hack—and it worked. We couldn’t derive a new fundamental law from first principles for multiphase porous flow (too hard), so we tuned the old law with empirical curves.
Each fluid phase (water, oil, gas) gets its own relative permeability that ranges between 0 and 1, depending on its saturation. At 100% saturation, kr=1 (the rock is as good as its absolute permeability for that fluid). At 0% saturation, kr=0 (zero conducting ability for that phase). Between 0 and 1, all bets are off – we fill in that curve from experiments.
It’s the industry’s favorite fudge factor: not fundamentally derived, but calibrated from lab experiments so that Darcy’s law appears to hold for each phase.
Clever, right? Perhaps too clever.
Ask a reservoir engineer and a lab specialist what “relative permeability” means, and you might get two different stories. In the laboratory, relative permeability is defined at the core plug scale under carefully controlled conditions: it’s the functional relationship between point saturation and the fluid’s conduction capacity (flow ability) when the displacement is strictly dominated by viscous forces.
In plainer terms, lab techs measure how much a fluid can flow at a given local saturation, assuming things like capillary forces don’t mess with the flow distribution. They treat the rock sample like a little black box where at any given saturation, you have a well-defined conductivity for each fluid.
In the field/reservoir simulation context, however, relative permeability curves are more about the relationship between average saturation in a region and the productive flow capacity of that phase from that region.
It’s a bulk, effective concept. We assume our simulator grid block (or larger scale region) has some average oil/water/gas saturation, and from that we assign a relative perm to calculate flow. It’s a huge leap of faith that this average saturation meaningfully translates to an equivalent flow capacity as if the block were uniformly at that saturation.
The result? A mismatch between lab and model—and plenty of confusion. We often talk past each other – the lab is talking about pointwise flow capacity under strict conditions, while the simulator just uses a single curve per phase to summarize an entire block’s behavior, regardless of the fluid chaos inside.
One of the most eye-opening reads in my early years was Crotti (2003), who made a subtle but powerful distinction: reservoir engineers care about production—what comes out of a region given an average saturation—while lab measurements focus on conduction—how fluid moves through a small, idealized sample under tightly controlled conditions. That simple framing—production vs. conduction—brilliantly captures the disconnect between lab and field, yet it’s a nuance that often slips under the radar.
Okay, so we’ve accepted that we need these fudgy relative permeability curves. How do we get them? Through painstaking lab experiments on core samples – which is easier said than done. There are two classic approaches:
This approach recreates Darcy’s ideal conditions in a core for each fixed saturation. You inject two fluids (say oil and water) at constant rates until the production rates level out to equal the injection – a steady state for both phases.
At that point, you assume the core has a uniform saturation distribution, and you can compute relative perms from the stabilized flow rates and pressure drop. Then you change the ratio of injected fluids and repeat for a new saturation. It’s laborious and time-consuming (each point on the curve can take days), but conceptually straightforward. You are essentially holding the black box in place until inflow=outflow for each phase, ensuring a single saturation throughout the core, just like Darcy’s single-phase experiment — albeit with two fluids now. This method gives you one pair of kr,o and kr,w at each equilibrium saturation. Drawback: It assumes you can reach a uniform saturation. In practice, capillary forces can cause one end of the core to have more of one fluid even at steady state, etc. But in theory, steady-state gives “true” relative perm under ideal conditions.
Rather than many separate steady runs, this method (thanks to Buckley-Leverett theory and later refinements by Welge and Johnson, Bossler, Naumann) uses one continuous displacement experiment. For example, take a core initially full of oil, then start injecting water at one end. As water floods the core, oil is produced at the other end. The saturations in the core are changing with time (hence “unsteady”). By measuring the produced fluids over time and knowing injection rates, you can use the Buckley-Leverett fractional flow theory to back-calculate the relative permeability curves. Essentially, Buckley and Leverett gave us a mathematical recipe to interpret the changing production data – the Welge method lets us derive a fractional flow curve, and from that we deduce rel perm values.
This is much faster than steady-state (one experiment to get the whole curve), but it comes with assumptions: the core is homogeneous, the flow is 1D, and crucially, the displacement is dominated by viscous forces (we usually crank up the flow rate or use viscous oils to satisfy that). Capillary forces and gravity are assumed negligible in the analysis, so the math works out nicely on paper.
Both methods – if done perfectly – should in theory give the same relative perm curves for a given rock and fluid system. And indeed, in a perfect homogeneous core under strictly controlled conditions, they often do. But here’s the kicker: those conditions hardly reflect a real reservoir. Both lab techniques basically engineer a situation where viscous force is king.
Here's a fundamental question: are the curves measured under these special conditions actually valid for the unsteady, heterogeneous, capillary-and-gravity-driven reality of reservoirs? It’s a question we conveniently keep sweeping under the lab rug.
Also worth noting: even the lab measurements themselves can vary. Steady-state vs. unsteady-state often yield slightly different curves on the same rock, especially if there are capillary end-effects (e.g. water piling up at the outlet face due to capillary pressure). Labs try to minimize those, but small differences in procedure or wettability can lead to different “official” rel perm curves. So if you’ve ever wondered why literature has multiple oil-water rel perm curves for what seems like the same rock type – well, it might depend on how the poor grad student or technician ran the experiment.
Here’s the ugly truth: the beautiful curves we measure in the lab often have little to do with what happens in an actual reservoir. There are several reasons for this disconnect:
Heterogeneity: Lab experiments typically use a small, presumably homogeneous core sample. Real reservoirs are anything but homogeneous – they have layers, fractures, channel sands, you name it. In a stratified reservoir, for example, water might preferentially flow through high-permeability streaks and bypass tight rock. The effective relative permeability for the field (the relationship between average saturation and flow) ends up vastly different from the lab core which had uniform flow. A lab core can’t capture the fact that in the field, some regions flood out (100% water local saturation) while others remain mostly oil – yet our simulator still assigns one rel perm curve to the whole mix. The result? Using lab curves in a heterogeneous model often mis-predicts recovery, because the model doesn’t account for the uneven sweep. So, engineers often have to generate pseudo-relative-permeabilities for heterogeneous reservoirs – essentially fudged curves that incorporate the effect of heterogeneity by tuning the rel perm.
Capillary and Gravity Forces: Lab measurements usually assume these forces are negligible (especially unsteady-state tests done at high speed). In the quiet corners of a reservoir, however, capillary forces can trap or redistribute fluids independent of flow, and gravity can segregate fluids by density. For instance, in the field, water might imbibe into small pores (due to capillarity) that it never reaches in a fast lab flood, effectively lowering oil relative permeability at a given average saturation (because some oil is stranded in micro-pores). Gravity can cause water to sink and oil to rise in a reservoir over time, leading to vertical saturation gradients that no lab core oriented horizontally could mimic. The net effect: the reservoir’s apparent relative permeability behavior (including end-point saturations like residual oil) can differ from lab results, which were obtained under a very different force balance. The lab yields a curve for the special case of viscous-dominated flow, which “only applies to conditions... under predominance of viscous forces”– clearly not the full story in real reservoirs.
Scale and Saturation Distribution: In a core flood, when we say “the core has 50% water saturation,” we often (in steady-state) mean uniformly 50% throughout, or (in unsteady) an average 50% with a certain saturation profile along the core that we interpret via theory. In a reservoir sector, “50% water saturation” might mean one half of the volume is at 100% water and the other half is at 0%.
If half a reservoir rock is thoroughly water-flooded and half is virgin oil, the average is 50% water saturation – but what’s the effective conductivity to water? Is it the average of fully conducting in one part and zero in the other? In reality, until the water-connected pathways span from injector to producer, you get injection without production (water can enter rock that isn’t producing yet) – a phenomenon the single-valued rel perm curve struggles to capture. In short, average saturation is not a sufficient descriptor of flow when saturation isn’t uniformly distributed. Field models often require adjusting rel perms to account for this saturation shock or “end-point bypass” effect.
Wettability and Fluid Differences: Reservoirs often have mixed-wet or oil-wet behavior, while many lab tests (especially older ones) assumed water-wet conditions or used refined oils and brines that don’t perfectly represent the actual fluids. A curve measured on a water-wet core may be very different from what the reservoir (which might be mixed-wet due to crude oil contact for ages) would have. This can lead to lab-measured residual oil saturations that are overly optimistic or pessimistic compared to field reality. Although nowadays labs do special core analysis trying to honor reservoir wettability, it’s another source of potential mismatch.
Given all these factors, it’s no surprise that the lab’s “intrinsic” relative permeability curves often fail to predict field performance. As Dake pointed out rather bluntly, reservoir engineers have historically placed almost religious faith in full-core relative permeability curves – treating them with “great veneration” – yet this is a “questionable attitude” because such lab curves are never used directly in a real reservoir study without modification. Unless your reservoir happens to literally behave like a tiny core plug flooded with 17 centipoise oil, you will have to tweak those curves. The disconnect is so bad that one could argue the lab measurements serve only as an initial guess to get the ball rolling, that’s all.
Even if Mother Nature were kind enough to make your reservoir homogeneous and perfectly characterized by those lab curves (she won’t, but imagine), the numerical simulation of flow in reservoirs introduces its own bag of problems. Reservoir simulators solve flow equations on a grid, and their limited resolution and numerical algorithms can distort how multiphase flow is represented:
Numerical Dispersion: In theory (Buckley-Leverett), a waterflood might have a sharp saturation front moving through the reservoir. In practice, simulators with finite grid spacing smear that front because of numerical diffusion. Water saturation transitions become gradual over several grid blocks. This “smearing” means the water cut (fractional flow) prediction comes out more gradual than reality. To compensate, engineers often adjust the relative permeability curves – essentially steepen them or use so-called pseudo-relative-perms – so that in the coarse grid model the water front is sharper and matches observed well behavior. It’s a bit ironic: we tweak the fudge factor to counteract another fudge (numerical diffusion). But such is life in reservoir simulation.
Grid Size and Scale-Up: A fine-grid model (tiny blocks) can retain more of the true physics (and less numerical dispersion) than a coarse model. But we usually can’t simulate millions of cells for full fields – we upscale to larger grid blocks. When we do that, the effective relative permeability for that larger block is not the same as the lab curve for a small core. Instead, it’s influenced by sub-grid saturation distribution. The industry approach is to derive upscaled or pseudo relative permeability curves for the coarse blocks that try to mimic what a finer model (or the real reservoir) would do. The outcome is that the rel perm values become grid-dependent. Change your grid resolution or even the orientation, and you may need a different rel perm curve to history match. This isn’t because physics changed – it’s our numerical representation that’s changing.
It’s a dirty little secret that your simulator’s results can depend on grid orientation for processes like waterflooding. If you rotate the grid or refine it, the water breakthrough might change unless you recalibrate rel perms. Ideally, a property of the rock should be independent of grid discretization, but relative permeability in simulation is as much a numerical tuning knob as a rock property.
Endpoint Adjustments and Saturation Functions Tuning: Simulators also require you to input endpoint parameters like residual oil saturation, critical water saturation, etc., along with rel perm curves. The shape of the rel perm curve, combined with those endpoints, can greatly affect how quickly a simulator cell goes from 0 to 100% water (or oil). Often, history matching involves tweaking these endpoints (e.g. increasing the residual oil saturation or lowering endpoint relative permeabilities) to get the correct recovery and production rate. That effectively changes the rel perm behavior from the lab-derived one. It’s not uncommon that the “best match” rel perm curves in a reservoir model look flatter or more one-sided than any lab core test suggested.
Let's walk through some of these issues using a few simple schematics. To keep things clear, we'll focus only on the water relative permeability.
You can think of relative permeability as a measure of a phase's "capacity to flow." Since we're interested in reservoir-scale behavior, we'll observe a simulation cell, with the x-axis representing the average water saturation in the cell.
First, let’s consider two possible types of water-oil displacement:
We’re interested in deriving the relative permeability curve for the outlet cell face on the right.
In the first case, water does not flow through the outlet face until the plug becomes fully water-saturated. As a result, water relative permeability jumps from zero when the vertical interface reaches the outlet.
In the second case, water begins to flow through the outlet face as soon as the bottom of the inclined interface reaches it. This happens at a lower average water saturation within the cell. From that point on, the ability to flow (i.e., the relative permeability) increases steadily until full water saturation is achieved.
Thus, depending on the angle of the water-oil interface, we end up with different relative permeability curves for this type of displacement.
Next, let’s consider the exact same physical displacement, but change the size of the observation window — in other words, the size of the simulation cell we’re looking into.
The first case is the same as before. However, when the cell is smaller (we halve it in this example), given the angle of the interface, the water front reaches the cell outlet at a lower average cell water saturation (for example, 80% in the top case and 50% in the bottom case). As a result, the relative permeability curve starts rising from zero earlier.
Let’s now examine a bottom-up displacement with a water-oil interface, keeping it flat for simplicity, and think about the flow through the top and side faces.
What we observe is that water can flow through the side faces as soon as the flat interface enters the cells. However, the top cell — just like in our first piston-like horizontal displacement example — won’t experience any water flow-through until it becomes fully water-saturated.
Wait —
so the shape of the relative permeability curves doesn’t just depend on rock wettability and displacement type (which are purely physical phenomena), but also on grid-related parameters like cell size (grid resolution), grid orientation (relative to the front advancement), and even which specific cell face we observe.
In summary, even if we had perfect lab relative permeability curves, the act of putting them into a reservoir simulator forces us to confront numerical artifacts. We end up modifying the curves or using pseudo-functions to ensure the simulation doesn’t stray from reality. This brings us to a somewhat uncomfortable truth that many reservoir engineers know deep down: the lab-measured rel perms are almost never the ones we finally use in our models.
At the end of the day, what do reservoir engineers trust the most? Not the lab report, but the field performance data. We adjust the model until it reproduces the past behavior of the reservoir – a process aptly named history matching. And guess which parameters are the first to get bent to our will? Yes, the relative permeability curves. Those sacred lab curves often get stretched, squeezed, and tweaked beyond recognition to fit the observed production profiles. Dake wryly noted that we treat full-core rel perm curves as sacrosanct in theory, yet in practice we never hesitate to mangle them because the reservoir is not a core plug. If the water breakthrough happened sooner in the field than the lab curve predicts, we might increase the kr,w at intermediate saturations or reduce the residual oil saturation. If the water cut climbed slower in reality, we might flatten the curves to introduce more mixing. The lab data provides a starting template, but the truth is what history match tells us it must be (within the limits of model assumptions).
It’s almost comedic: we spend tens of thousands of dollars on special core analysis to get “accurate” relative permeability, then we spend months of simulation work effectively throwing those results out in favor of whatever makes the model line up with reality. In the end, the relative permeability used in the reservoir simulation is a field-calibrated, case-specific curve. It may still be loosely based on lab measurements (perhaps honoring endpoints or general shape), but it’s tuned for that particular reservoir’s performance.
Let’s be honest: relative permeability is not a fundamental rock property like porosity or absolute permeability. It’s not even “real” in the physical sense. It’s a mathematical convenience, a workaround, a fudge factor. It exists solely to let us keep using Darcy’s elegant, old-school law in a messy, multiphase world where fluids fight over pore space and the real physics is far too complex to model directly.
If we had a full map of pore geometry, rock-fluid interactions, wettability patterns, capillary pressures, and could solve Navier-Stokes across a billion-cell reservoir model in real-time… we wouldn’t need relative perms at all. But we don’t. So instead, we fake it—with rel perms.
They serve a purpose:
So, yes, the concept of relative permeability is needed.
But here’s the kicker: lab-measured relative perms are not.
They're often based on experiments under idealized conditions—tiny cores, assumed homogeneity, viscous-dominated displacement—that bear little resemblance to real reservoir behavior. Once fed into a simulator, they’re reshaped, reinterpreted, or outright ignored. Because in the end, it’s not about what flows in a lab core—it’s about what flows in your field.
And that brings us to the heart of the matter:
Only if:
But if you have field production data?
You're far better off tuning simulation-specific rel perms that reflect what the reservoir actually did—not what a 3-inch core under lab conditions thinks it might do.
Final verdict:
"Full rock relative permeabilities always seem to have been treated with great veneration throughout the history of reservoir engineering. They are assumed to be intrinsically correct and all theory and practice is geared to accommodate this commonly held view. In fact, as argued throughout the chapter, this is a questionable attitude and full rock curves are never used directly: unless the problem in hand is that of flooding a reservoir which has the dimensions of a core plug and is full of 17 cp oil, which is a condition seldom encountered in practice."
“Fractional flow is the most fundamental concept in the whole subject of waterdrive, much more so than relative permeabilities.”
“The shape of rel perms reveals all about the displacement mechanism, whereas the rel perms do not.”
"It is therefore interesting to note that when you enquire of Service Company laboratory staff why they conduct relative permeability experiments with an artificial high-viscosity oil, the answer you receive is that if they did not adopt this practice they would not obtain 'curves' only 'points' — and that would not satisfy the customer! This has always seemed a perfectly reasonable if somewhat commercial reply."
"Invariably they are constructed using the finite difference analogue and as such the input of step-functions can give the machines indigestion, to the extent that they can come to a grinding halt."
"The most unfortunate aspect concerning relative permeabilities and their use in coarse gridded numerical simulation is not that the models do not respect the concept of fractional flow but rather — that because they ignore it, then so too do reservoir engineers."
L.P. Dake, The Practice of Reservoir Engineering
“In one way or another, and indeed now for almost a Century, the voluminous literature on the subject has substantially been influenced by the evolving understandings about the meanings and applicability of what today still remains to be a perplexing (rather than an ubiquitous) relative permeability concept.”
W. Rose - SPE 57442
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Alan is a Consulting Petroleum Reservoir Engineer with 20+ years of international industry experience. Alan is the founder of CrowdField, a marketplace that connects Oil & Gas and Energy businesses with a global network of niche talent for task-based freelance solutions. His mission is to help skilled individuals monetize their knowledge as the Energy transition unfolds, by bringing their expertise to the open market and creating digital products to sell in CrowdField's Digital Store.
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