Porous Fluid

Porous-fluid theory describes fluid transport through the connected pore space of a permeable solid. At the continuum scale, the pore geometry is represented by porosity and permeability rather than by resolving each pore. The formulation below is for a saturated, single-phase Newtonian fluid; it is a general porous-media model and does not imply that every coupling term is active in every OpenFDEM analysis.

Darcy flux

For slow viscous flow through a porous medium, the volumetric Darcy flux is

\[\mathbf{q}=-\frac{\mathbf{k}}{\mu} \left(\nabla p-\rho\mathbf{g}\right),\]

where \(\mathbf{q}\) is the volume of fluid crossing unit bulk area per unit time, \(\mathbf{k}\) is the intrinsic permeability tensor, \(\mu\) is dynamic viscosity, \(p\) is pore pressure, \(\rho\) is fluid density, and \(\mathbf{g}\) is gravitational acceleration. The minus sign makes flow proceed down the hydraulic-potential gradient. For isotropic material, \(\mathbf{k}=k\mathbf{I}\). Darcy’s law assumes that inertial effects are small; high-speed flow may require a non-Darcy relation.

Mass conservation and pressure diffusion

The local fluid mass balance is

\[\frac{\partial (\phi\rho)}{\partial t} +\nabla\cdot(\rho\mathbf{q})=m_f,\]

where \(\phi\) is porosity and \(m_f\) is a mass source per unit bulk volume. For slightly compressible fluid and solid constituents, a commonly used pressure form is

\[S_p\frac{\partial p}{\partial t} +\alpha_B\frac{\partial\varepsilon_v}{\partial t} +\nabla\cdot\mathbf{q}=Q_f,\]

where \(S_p\) is the pressure-storage coefficient, \(\alpha_B\) is the Biot coefficient, \(\varepsilon_v\) is volumetric strain, and \(Q_f\) is a volumetric fluid source. The strain term accounts for the change in pore volume caused by deformation; its sign follows the tension-positive strain convention. If deformation is not coupled, this term is omitted. Combining this balance with Darcy’s law gives the pressure diffusion equation

\[S_p\frac{\partial p}{\partial t} +\alpha_B\frac{\partial\varepsilon_v}{\partial t} -\nabla\cdot\left[ \frac{\mathbf{k}}{\mu}(\nabla p-\rho\mathbf{g})\right]=Q_f.\]

For constant properties, no gravity, and no deformation coupling, this reduces to a diffusion equation for pore pressure. The permeability tensor controls the ease and direction of flow, while storage controls the rate at which pressure responds to fluid injection, production, or boundary loading.

Pressure initial and boundary conditions

A transient problem requires an initial pressure field. Typical boundary conditions prescribe pore pressure, normal fluid flux, a source or sink, or no-flow on an impermeable boundary. Prescribed pressure represents connection to a pressure-controlled reservoir; prescribed flux represents a specified injection or discharge rate. These conditions should be applied consistently with the sign convention used for \(\mathbf{q}\) and \(Q_f\).

The equations above assume a continuum representative volume and single-phase flow. When two immiscible fluids occupy the pore space, saturation-dependent relative permeability and capillary pressure are needed; see Two-Phase Fluid.