Hubbert's fluid potential concept is indeed foundational in hydrogeology and reservoir engineering, particularly for describing groundwater flow and single-phase fluid flow in porous media. However, its application to variable-density, multiphase flow systems, such as oil reservoirs, introduces complexities that deserve careful consideration.

1. Potential Field and Variable-Density Flow

In single-phase, constant-density flow, the governing equations reduce to Laplace's or Poisson's equations, ensuring that flow lines strictly follow the gradient of a scalar potential function (hydraulic head or pressure). However, in a variable-density system—such as oil-water or gas-oil systems—density variations make the flow field nonlinear and dependent on:

• Fluid density gradients.
• Pressure gradients.
• Gravitational forces acting on fluid densities.

2. Can a Potential Field Still Be Defined?

For variable-density flow, we can define an equivalent potential field, but it is no longer scalar in the classical sense. Instead:

• The hydraulic potential must account for variations in fluid density, particularly due to the pressure dependence of compressible fluids (oil and gas).
• For oil-water systems, pressure gradients combine with capillary and gravitational effects, resulting in deviations from purely irrotational fields.
• The effective potential field might include a pseudo-potential term that incorporates buoyancy effects and density variations.

3. Finite Field of Force and Rotational Components

In multiphase systems, a rotational component can emerge due to:

• Density-driven convection (gravity segregation).
• Capillary pressure gradients across phases.
• Viscous coupling between phases (relative permeability effects).

This implies the flow is no longer purely irrotational, and vorticity (rotational flow) may exist locally. The rotational component does not invalidate the potential field concept but rather suggests it must be augmented with additional vector terms to describe convective and capillary-driven flows.

4. Superposition of Rotational Components on the Potential Field?

Yes, rotational components arising from density-driven flow or capillary effects can be superimposed onto the primary potential field, especially in transition zones like the water-oil contact (WOC) and gas-oil contact (GOC). These regions often exhibit capillary equilibrium, where gradients in capillary pressure balance gravitational forces, creating complex flow patterns.

Such flows are governed by coupled Darcy and continuity equations, including terms for relative permeabilities, saturation gradients, and multiphase interaction forces.

5. Implications for Reservoir Engineering

• For reservoir simulation, the pressure potential approach must be adapted to pseudopotentials or stream functions that capture density variations.
• Numerical models use finite-difference or finite-element approximations to represent rotational effects in the governing equations.
• Convective mixing and fingering instability at WOC and GOC must be explicitly modeled in compositional or black-oil simulators.

6. Final Thoughts

While Hubbert's potential field remains a useful approximation, the introduction of variable density, capillary forces, and multiphase flow challenges its direct application. Instead, augmented potential fields and hybrid formulations incorporating rotational effects provide a more comprehensive framework for reservoir engineering. This approach allows capturing both irrotational flow components (potential gradients) and rotational or convective movements (vorticity effects) in variable-density systems.