Demystifying Saturation Functions: From Theory to Simulation

Reservoir engineering is as much an art of interpretation as it is a science of precision. At its core, it requires a delicate balance between physics-based modeling, empirical data, and numerical approximation—yet even subtle changes in input parameters can trigger significant deviations in predicted reservoir performance. Over time, I’ve seen how engineers, despite their best efforts, can unknowingly misinterpret recovery forecasts simply due to the way saturation functions are defined and applied in simulation models.

Among all the inputs in a reservoir simulation, saturation functions are perhaps one of the most sensitive yet least intuitively understood. These functions—governing the relationship between fluid saturations, relative permeability, and capillary pressure—dictate the flow and distribution of oil, water, and gas in the porous media. However, their application is often oversimplified or mis-calibrated, leading to inaccurate fractional flow predictions, poor displacement efficiency, and even numerical instability in simulations.

There is a widespread assumption that lab-derived relative permeability curves are absolute truths—when in reality, they are fitting parameters that require careful adjustment to match real reservoir conditions.

One of the most striking examples of this, as L.P. Dake pointed out:

“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.”

Dake brilliantly highlights a fundamental challenge in reservoir simulation—scale disparity. The relative permeability curves we measure on tiny 1?-inch core plugs simply don’t translate directly to the vast 100 × 100 × 1.0 m grid cells used in field-scale models. While engineers use upscaling techniques like piston-like and thickness-averaged curves to bridge this gap, these methods often introduce discontinuities that can disrupt simulation stability and convergence.

This is why raw laboratory-measured curves are rarely used as-is in numerical models. Instead, they must be carefully modified, upscaled, and calibrated against history-matched data to ensure they reflect actual reservoir behavior. A common approach is to fit a Corey-type curve to the lab data, smoothing out inconsistencies and providing a more realistic, numerically stable input for simulation. Without these adjustments, we risk basing critical field development decisions on misleading or overly rigid interpretations of lab-derived data.

Corey built upon the foundational work of Purcell and Burdine, whose models were widely accepted for their simplicity and practicality. While Corey's original equations were developed for the drainage process in water-wet sandstones, they have since been applied to carbonate formations as well. His approach provides a mathematically convenient way to model relative permeability, making it a standard in reservoir simulation.

There are several forms of these equations, with the most common normalizing the saturation over the mobile hydrocarbon phase, as depicted in equations:

Where:

krow= maximum oil relative permeability at Swc,

krw= maximum water relative permeability at Sorw,

Swc=critical water saturation, and

Sorw = residual oil saturation.

Bridging the Gap Between Theory and Application

Despite the fundamental role of saturation functions in reservoir simulation, many engineers today lack an intuitive understanding of how these functions influence fluid flow. This is largely due to the fact that workflows are often centered on numerical curve fitting and simulation tuning, rather than a physical understanding of fractional flow dynamics.

This gap in understanding has significant business implications:

? Overestimated recovery predictions can lead to misguided investment decisions.

? Poorly tuned relative permeability functions can result in inefficient recovery strategies.

? Over-reliance on lab measurements without context can introduce significant uncertainties in field development planning.

To help engineers go beyond equations and truly visualize the impact of saturation functions, I teamed up with Alan Mourgues at CrowdField - Res. Eng. Hub . With his guidance and support, I developed a Streamlit-based app designed to make relative permeability and saturation function analysis more intuitive and interactive. This tool lets users adjust Corey exponents and endpoint saturations in real-time, instantly revealing how these parameters influence fluid flow and reservoir simulation outcomes.

From the start, we envisioned an app that would be simple, interactive, and practical—a tool that not only helps engineers experiment with relative permeability but also integrates seamlessly into reservoir simulation workflows. We set out to build an interface that would:

? Support multiple fluid systems (oil-water, oil-gas, gas-water) via an easy-to-use dropdown list.

? Allow quick parameter adjustments (saturation and relative permeability end-points, Corey exponents) using intuitive drag-and-slide horizontal scroll-bars.

? Provide real-time visualization of Kr and Pc curves, dynamically updating as users modify parameters.

? Enable seamless data export, including Eclipse-formatted INCLUDE files for direct integration into reservoir simulators.

? Offer an option to save results in Excel, making it easy for engineers to revisit and refine their analysis.

Below is a snippet of the app in action, showcasing its interactive features and real-time updates.

The Impact

This project taught me several valuable lessons:

1. Modern engineering tools don't have to be complicated to be useful.

2. Python and Streamlit enable the creation of professional applications without requiring extensive software development expertise.

3. Hosting calculations in a web app simplifies sharing and collaboration with colleagues.

This project wasn’t just about coding—it was about making complex reservoir engineering concepts more intuitive and accessible. If you're curious about how this tool came to life, check out our blog post:

?? Case Study: Collaborating with a Young Engineer to Develop a Streamlit Relative Permeability App.

For those eager to build their own interactive tools, we've put together something that could help you get started, without a big investment of time or big bucks.

?? Beginner's Guide to Developing a Streamlit App for Relative Permeability and Capillary Pressure Curves.

Inside, you’ll find:

? A web landing page with an easy-to-follow tutorial, complete with animated GIFs and a video walk-through to guide you step-by-step.

? A detailed PDF guide, packed with relevant resources to help you navigate the process.

? The full source code, thoughtfully commented so you can follow along and learn as you go.

For a small amount, you can unlock everything you need to start building. When you’re ready, you can access it all right here:

?? Start Your App Development Journey!