Part 1: The Ahmed Miscibility Function

Every CO₂ EOR project depends on the minimum miscibility pressure.

Above the MMP, CO₂ and oil become miscible — interfacial tension vanishes, residual oil drops sharply, and recovery improves significantly. Below it, the fluids remain as separate phases and sweep efficiency suffers.

In a recent project I helped evaluate CO₂ injection potential in a giant Middle East carbonate reservoir. The fluid data remain confidential. The workflow can be reproduced using public data.

This two-part series does exactly that — using open-source Python and published slimtube experiments to calculate CO₂ MMP and compare analytical predictions against laboratory measurements and compositional simulation.

Objectives

1. Implement Ahmed's (1994) EOS-based miscibility function in Python and verify it against two published slimtube datasets.
2. Build and run an E300 compositional slimtube simulation on the same fluids and compare the result against both the analytical method and the physical experiment.
3. Establish a complete, reproducible CO₂ MMP screening workflow that any engineer can run without a commercial PVT package.

Part 1 covers the analytical method. Part 2 covers the E300 simulation. All code is available on GitHub.

Results First

Both fluids are from Rathmell, Stalkup and Hassinger (1971) SPE 3483 — one of the few published datasets that releases full fluid compositions, EOS parameters, and slimtube results together. Ahmed used the same fluids to validate his method in CIPC 2000-01, giving a genuine three-way comparison: physical experiment, Ahmed's published prediction, and this Python implementation.

The Fluid A overprediction is real but bounded. Rathmell's data bracket the MMP between 1200 psig (immiscible) and 1500 psig (miscible), so 1549 psia sits just outside the experimental window. For Fluid B, where the long-core data are conclusive at 2000–2015 psig, the 2013 psia prediction is excellent.

EOS Setup

The fluids were characterised using a Peng–Robinson EOS with Peneloux volume correction. Standard EOS parameters were generated from critical properties and acentric factors, with C7+ critical temperature regressed against the experimental bubble point in PVTi 2023.1. The Python implementation reproduces the PVT package bubble point within 0.1% for both fluids.

Ahmed's contribution is surprisingly simple. He made two modifications to standard PR:

1. Fix α_CO₂ = 1. Standard PR uses a temperature-dependent alpha function calibrated to CO₂ vapour pressure below its critical temperature of 88°F. Above this temperature — which covers essentially all reservoir conditions — the standard function extrapolates poorly, underestimating CO₂ density and producing optimistic MMP predictions. Setting α_CO₂ = 1 and recalibrating the attraction and co-volume parameters against Vukalvich and Altunin's (1968) CO₂ density data corrects this. At 103°F, the modified parameters give CO₂ density approximately 8% higher than standard PR.

2. Revised CO₂–hydrocarbon BIPs. Standard BIPs between CO₂ and hydrocarbons default to 0.10 — reasonable, but not calibrated for miscibility prediction. Ahmed optimised these against slimtube data:

Each modification independently increases the predicted MMP:

Everything else is standard PR EOS machinery.

The Ahmed Miscibility Function

The workflow follows five steps:

As CO₂ is added incrementally to the oil, the bubble point of the enriched mixture rises. K-values at the bubble point measure how far each component is from the critical condition where K_i = 1 for all components simultaneously. Ahmed's miscibility function (equation 19, SPE 27032) quantifies that distance:

At low CO₂ fractions, K-values are far from unity and F_M is large and positive. As CO₂ content increases and pressure rises, K-values converge toward unity and F_M approaches zero. The bubble point pressure at which F_M = 0 is the MMP.

The parallel with the physical slimtube is direct: the slimtube finds miscibility through recovery vs pressure; the F_M function finds it through K-value convergence vs pressure. Both identify the same condition — the pressure at which the CO₂–oil system reaches its critical point.

The C7+ Exclusion

With a single lumped C7+ pseudocomponent, K_C7+ at bubble point conditions is of order 10⁻⁵ — five orders of magnitude below unity. Including this term drives F_M to order 10,000 and the function never approaches zero at any achievable pressure.

Vaporising-gas-drive miscibility is controlled by intermediate hydrocarbons (C₂–C₆), not heavy ends. CO₂ extracts the intermediates into the vapour phase; the C7+ fraction remains in the liquid throughout. Excluding components with K < 10⁻⁴ from the F_M sum is physically justified and produces monotonically decreasing F_M curves with a well-defined zero crossing.

Results

Fluid A — T = 109°F. The F_M curve decreases smoothly from approximately 0.85 at zero CO₂ addition to zero at 1549 psia. The zero crossing is sharp and well-defined. Rathmell's data bracket the MMP between 1200 psig (immiscible) and 1500 psig (miscible). The prediction of 1549 psia sits just above the upper bound, consistent with the known behaviour of the method at lower temperatures and lighter C7+ fractions.

Fluid B — T = 103°F. The F_M curve reaches zero at 2013 psia and is within Rathmell's long-core experimental range of 2000–2015 psig.

Fluid B (103°F). F_M and bubble point pressure vs CO₂ addition fraction. Zero crossing at 2013 psia, within Rathmell's experimental window of 2000–2015 psig.

Full Python implementation and notebook: github.com/eskoantg/CO2_MPP

Caveats

Ahmed's method answers one question: at what pressure does the CO₂–oil system reach its critical point under equilibrium conditions? Three things it does not capture:

Numerical dispersion. In a finite-difference simulator, grid blocks mix fluids over finite volumes, spreading the transition zone and delaying the development of a miscible front. A 100-cell slimtube model may predict a different MMP than a 1500-cell model of the same system. The Ahmed function has no grid — it is inherently dispersion-free.

The recovery curve shape. Ahmed's method gives a single pressure. The slimtube gives a full recovery curve — the rising slope below MMP, the kink at MMP, the plateau above it. The shape of this curve contains information about how readily miscibility develops in a real reservoir. Part 2 addresses this directly.

Drive mechanism competition. For fluids where both condensing and vaporising mechanisms contribute, the pure vaporising assumption may not hold. Both Rathmell fluids are unambiguously vaporising — confirmed by the FCMP composition (78.6 mol% CO₂ at 6908 psia for Fluid B) and a ratio FCMP/MCMP ≈ 3.4 consistent with a vaporising drive system. For other fluids, caution is warranted.