[Paper Review] A Fully Equivalent Global Pressure Formulation for Three-Phase Compressible Flow

Guy Chavent|ArXiv.org|Jan 11, 2009

Computational Fluid Dynamics and Aerodynamics2 references3 citations

TL;DR

This paper presents a fully equivalent global pressure formulation for three-phase compressible flow in porous media, enabling the total volumetric flow to follow a classical Darcy law via a new global pressure variable. The formulation is valid only for Total Differential (TD)-three-phase data—defined by global capillary pressure and mobility functions—enabling accurate interpolation from two-phase data that satisfy a TD-compatibility condition.

ABSTRACT

We introduce a new global pressure formulation for immiscible three-phase compressible flows in porous media which is fully equivalent to the original equations, unlike the one introduced in \cite{CJ86}. In this formulation, the total volumetric flow of the three fluids and the global pressure follow a classical Darcy law, which simplifies the resolution of the pressure equation. However, this global pressure formulation exists only for Total Differential (TD) three-phase data, which depend only on two functions of saturations and global pressure: the global capillary pressure and the global mobility. Hence we introduce a class of interpolation which constructs such TD-three-phase data from any set of three two-phase data (for each pair of fluids) which satisfy a TD-compatibility condition.

Motivation & Objective

• To develop a fully equivalent global pressure formulation for three-phase compressible flow that avoids approximations present in prior work.
• To identify the necessary and sufficient conditions (TD-condition) under which such a global pressure formulation exists.
• To enable interpolation of three-phase relative permeabilities and capillary pressures from two-phase experimental data by introducing a new class of TD-interpolations.
• To ensure the interpolated three-phase data honor the original two-phase data on the ternary diagram boundary.
• To simplify numerical simulation by reducing the three-phase flow system to a single global pressure equation with classical Darcy behavior.

Proposed method

• Introduces a global pressure variable $ P = P_2 + P_{cg}(S, P) $, where $ P_{cg} $ is the global capillary pressure function.
• Derives a global Darcy law for total volumetric flow $ q $, ensuring full equivalence to the original three-phase equations under the TD-condition.
• Defines TD-three-phase data as those where relative permeabilities and capillary pressures depend only on two functions: global capillary pressure $ P_{cg} $ and total mobility $ d $.
• Establishes a TD-compatibility condition on three two-phase data sets (on $ \ ilde{\mathbb{T}} $) that must be satisfied for consistent interpolation.
• Proposes a two-step interpolation: first, construct $ P_{cg} $ and $ d $ on $ \mathbb{T} $ satisfying Dirichlet and Neumann boundary conditions derived from two-phase data.
• Uses PDEs (biharmonic for $ P_{cg} $, Laplace for $ d $) with boundary conditions to ensure smooth, consistent interpolation over the ternary diagram.

Experimental results

Research questions

• RQ1 Under what conditions can a fully equivalent global pressure formulation be derived for three-phase compressible flow in porous media?
• RQ2 How can three-phase relative permeability and capillary pressure data be consistently interpolated from two-phase experimental data sets?
• RQ3 What mathematical structure (TD-condition) must three-phase data satisfy to allow a global pressure formulation with classical Darcy behavior?
• RQ4 Can the global pressure formulation be constructed without approximating volume factors, ensuring full equivalence to the original equations?
• RQ5 What boundary conditions must be imposed on the global capillary pressure and mobility functions to match given two-phase data on the ternary diagram edges?

Key findings

• A fully equivalent global pressure formulation exists for three-phase compressible flow if and only if the three-phase data satisfy the Total Differential (TD)-condition.
• The total volumetric flow $ q $ of the three phases follows a classical Darcy law with respect to the global pressure $ P $, simplifying numerical solution.
• The global capillary pressure $ P_{cg}(s,p) $ and total mobility $ d(s,p) $ must be chosen such that their boundary values on $ \partial\mathbb{T} $ match the three given two-phase data sets.
• The TD-compatibility condition ensures that three two-phase data sets (on the edges of the ternary diagram) are consistent with a single set of TD-three-phase data.
• The global pressure formulation satisfies the stability condition $ 1 - \partial P_{cg}/\partial p > 0 $, and under realistic compressibility assumptions, $ 1 > 1 - \partial P_{cg}/\partial p > 0 $, ensuring well-posedness.
• Smooth interpolation of TD-three-phase data is achieved by solving biharmonic and Laplace equations for $ P_{cg} $ and $ d $, respectively, with boundary conditions derived from two-phase data.