Certainly! Let’s address **Problem 1** step-by-step as requested, leading to the final expression: --- ## **Step 1: Derivation of the Phase Mobility Tensors** For three-phase flow (say, water, oil, gas) in an anisotropic, heterogeneous porous medium, the phase mobility tensor for phase α (where α = w, o, g) is defined as: \[ \mathbf{\Lambda}_\alpha = \mathbf{K} \frac{k_{r\alpha}}{\mu_\alpha} \] where: - \(\mathbf{K}\) is the absolute permeability tensor (anisotropic, 2nd order) - \(k_{r\alpha}\) is the relative permeability to phase α (function of saturations) - \(\mu_\alpha\) is the viscosity of phase α --- ## **Step 2: General Pressure Equation (Including Gravity and Capillary Forces)** For incompressible, immiscible three-phase flow, the conservation of mass for each phase α is: \[ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \mathbf{u}_\alpha = \] where \(\mathbf{u}_\alpha\) is the Darcy velocity of phase α: \[ \mathbf{u}_\alpha = -\mathbf{\Lambda}_\alpha \left( \nabla p_\alpha - \rho_\alpha \mathbf{g} \right) \] Capillary pressure between phases (say, oil and water): \[ p_c^{ow} = p_o - p_w, \quad p_c^{go} = p_g - p_o \] The general pressure equation (total flow equation), summing over all phases, is: \[ \sum_{\alpha} \left[ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \mathbf{u}_\alpha \right] = \] Using saturation constraint (\(\sum_\alpha S_\alpha = 1\)), and expressing all phase pressures in terms of a common pressure (e.g., oil phase) and capillary pressures, we get: \[ \nabla \cdot \left( \sum_\alpha \mathbf{\Lambda}_\alpha \nabla p_o + \sum_\alpha \mathbf{\Lambda}_\alpha \nabla p_c^\alpha - \sum_\alpha \mathbf{\Lambda}_\alpha \rho_\alpha \mathbf{g} \right) = \] where \(p_c^\alpha\) are the capillary pressure differences. --- ## **Step 3: Saturation Evolution Equations** For each phase α (e.g., water, oil, gas): \[ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_\alpha (\nabla p_\alpha - \rho_\alpha \mathbf{g}) \right] = \] Expressing \(p_\alpha\) in terms of the reference pressure and capillary pressures: \[ p_w = p_o - p_c^{ow}, \quad p_g = p_o + p_c^{go} \] So, for water: \[ \frac{\partial S_w}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_w (\nabla (p_o - p_c^{ow}) - \rho_w \mathbf{g}) \right] = \] And similarly for oil and gas. --- ## **Step 4: Effect of Relative Permeability Hysteresis** Relative permeability hysteresis means \(k_{r\alpha}\) depends not just on saturation, but on the saturation history, so: \[ k_{r\alpha} = k_{r\alpha}(S_\alpha, \text{history}) \] This makes the system path-dependent, and can be included via empirical or semi-empirical models for \(k_{r\alpha}\). --- ## **Final Expression** The complete set of macroscopic flow equations for three-phase flow in an anisotropic, heterogeneous medium with gravity and capillary pressure is: \[ \boxed{ \begin{aligned} &\nabla \cdot \left( \sum_\alpha \mathbf{\Lambda}_\alpha \nabla p_o + \sum_\alpha \mathbf{\Lambda}_\alpha \nabla p_c^\alpha - \sum_\alpha \mathbf{\Lambda}_\alpha \rho_\alpha \mathbf{g} \right) = \\ &\frac{\partial S_\alpha}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_\alpha (\nabla p_\alpha - \rho_\alpha \mathbf{g}) \right] = \quad \forall\, \alpha \\ &p_w = p_o - p_c^{ow}(S_w, S_o, \text{history}) \\ &p_g = p_o + p_c^{go}(S_g, S_o, \text{history}) \\ &k_{r\alpha} = k_{r\alpha}(S_\alpha, \text{history}),\quad \sum_\alpha S_\alpha = 1 \end{aligned} } \] This system fully describes the macroscopic three-phase flow with anisotropic permeability, capillary pressure, and hysteresis in relative permeability.

## Problem 1: Macroscopic Fractional Flow Equation for Three-Phase Immiscible Fluid Transport ### Step 1: Derivation of the Phase Mobility Tensors For three-phase flow in an anisotropic porous medium, the phase mobility tensor for each phase \( \alpha \) (where \( \alpha = w, o, g \)) is given by: \[ \mathbf{\Lambda}_\alpha = \mathbf{K} \frac{k_{r\alpha}}{\mu_\alpha} \] Here, \( \mathbf{K} \) is the absolute permeability tensor, \( k_{r\alpha} \) is the relative permeability to phase \( \alpha \), and \( \mu_\alpha \) is the viscosity of phase \( \alpha \). ### Step 2: General Pressure Equation (Including Gravity and Capillary Forces) The conservation of mass for each phase \( \alpha \) is expressed as: \[ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \mathbf{u}_\alpha = 0 \] where the Darcy velocity of phase \( \alpha \) is defined as: \[ \mathbf{u}_\alpha = -\mathbf{\Lambda}_\alpha \left( \nabla p_\alpha - \rho_\alpha \mathbf{g} \right) \] The capillary pressure between phases is represented as: \[ p_c^{ow} = p_o - p_w, \quad p_c^{go} = p_g - p_o \] Thus, the total flow equation, summing over all phases, becomes: \[ \nabla \cdot \left( \sum_{\alpha} \mathbf{\Lambda}_\alpha \nabla p_\alpha + \sum_{\alpha} \mathbf{\Lambda}_\alpha \nabla p_c^\alpha - \sum_{\alpha} \mathbf{\Lambda}_\alpha \rho_\alpha \mathbf{g} \right) = 0 \] ### Step 3: Saturation Evolution Equations For each phase \( \alpha \), the saturation evolution equation can be described as: \[ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_\alpha (\nabla p_\alpha - \rho_\alpha \mathbf{g}) \right] = 0 \] Expressing the pressures in terms of capillary pressures gives: \[ p_w = p_o - p_c^{ow}, \quad p_g = p_o + p_c^{go} \] Thus, for water, we have: \[ \frac{\partial S_w}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_w \left( \nabla (p_o - p_c^{ow}) - \rho_w \mathbf{g} \right) \right] = 0 \] Similar equations hold for oil and gas. ### Step 4: Effect of Relative Permeability Hysteresis The relative permeability hysteresis indicates that \( k_{r\alpha} \) is dependent not only on saturation but also on the history of saturation changes. Therefore, we express: \[ k_{r\alpha} = k_{r\alpha}(S_\alpha, \text{history}) \] This introduces path dependency into the flow equations. ### Final Expression The comprehensive equations governing the macroscopic three-phase flow in an anisotropic, heterogeneous medium with gravity and capillary pressure effects are summarized as follows: \[ \begin{aligned} \nabla \cdot \left( \sum_{\alpha} \mathbf{\Lambda}_\alpha \nabla p_o + \sum_{\alpha} \mathbf{\Lambda}_\alpha \nabla p_c^\alpha - \sum_{\alpha} \mathbf{\Lambda}_\alpha \rho_\alpha \mathbf{g} \right) &= 0 \\ \frac{\partial S_\alpha}{\partial t} + \nabla \cdot \left[ -\mathbf{\Lambda}_\alpha (\nabla p_\alpha - \rho_\alpha \mathbf{g}) \right] &= 0 \quad \forall \, \alpha \\ p_w &= p_o - p_c^{ow}(S_w, S_o, \text{history}) \\ p_g &= p_o + p_c^{go}(S_g, S_o, \text{history}) \\ k_{r\alpha} &= k_{r\alpha}(S_\alpha, \text{history}), \quad \sum_\alpha S_\alpha = 1 \end{aligned} \] This set of equations effectively captures the dynamics of three-phase flow in the specified conditions.