Balance of Linear Momentum

Similar to the mass balance, we derive the balance of linear momentum for the pore water phase as well as for the mixture as a whole, starting from a general form of the balance of linear momentum.

General form of the balance of linear momentum

For each phase \(\alpha\) of the porous medium, the balance of linear momentum considers the equilibrium of forces acting on an arbitrary control volume. These forces are:

Surface forces (tractions), denoted by the vector \(\mathbf{t}^\alpha\), acting on the boundary surface element \(\mathrm{d}a\) of the control volume. According to Cauchy's theorem, the surface traction is related to the Cauchy stress tensor \(\boldsymbol{\sigma}^\alpha\) and the outward unit normal vector \(\mathbf{n}\) as follows:

\[ \mathbf{t}^\alpha = \boldsymbol{\sigma}^\alpha \mathbf{n}. \]
Body forces per unit volume, denoted by \(\rho^\alpha \mathbf{b}^\alpha\). These forces act uniformly throughout the volume element \(\mathrm{d}v\) and commonly represent gravitational forces.
Interaction forces, represented by \(\mathbf{h}^\alpha\), which quantify internal forces per unit volume exchanged between phase \(\alpha\) and the other phases of the mixture. These arise due to relative movements and interactions between phases (e.g. drag forces due to fluid flow).

Considering these forces, the local form of the balance of linear momentum for phase \(\alpha\) becomes:

\[ \begin{equation} \label{eq:linear_momentum_alpha} \rho^\alpha \mathbf{b}^\alpha + \nabla \cdot \boldsymbol{\sigma}^\alpha + \mathbf{h}^\alpha = \rho^\alpha \bar{\mathbf{a}}^\alpha. \end{equation} \]

In this equation:

\(\rho^\alpha \bar{\mathbf{a}}^\alpha\) represents the inertia force per unit volume (rate of change of linear momentum).
\(\rho^\alpha \mathbf{b}^\alpha\) denotes external body forces per unit volume, such as gravitational forces.
\(\nabla \cdot \boldsymbol{\sigma}^\alpha\) is the divergence of the stress tensor, representing net internal mechanical forces per unit volume due to spatial variation of stress.
\(\mathbf{h}^\alpha\) captures the forces per unit volume exerted by other phases onto phase \(\alpha\), reflecting interactions within the multiphase system.

This formulation provides the fundamental basis for deriving specific linear momentum equations for the pore water, solid skeleton, and the entire mixture.

Balance of linear momentum of pore water

In geotechnical engineering, accurately describing fluid flow in porous media requires establishing a momentum balance equation specifically for the pore water. The momentum balance for pore water accounts for the interaction forces between the fluid and the solid skeleton, as well as body forces such as gravity, fluid pressure gradients, and viscous resistance to flow. Starting from the general form of linear momentum balance (Equation \(\ref{eq:linear_momentum_alpha}\)), we specify the relevant terms for the pore water phase.

Under isothermal conditions, the interaction force per unit volume exerted by the pore water on the solid skeleton can be expressed as:

\[ \begin{equation}\label{eq:interaction_force} \mathbf{h}^w = \bar{p}^w \nabla \varphi^w - \varphi^w \left(\mathbf{K}^w\right)^{-1} \bar{\gamma}^w \mathbf{w}^w, \end{equation} \]

where:

\(\bar{p}^w\) is the intrinsic pore fluid pressure,
\(\varphi^w\) is the volume fraction of water (equal to porosity \(n\) for saturated soils),
\(\mathbf{K}^w\) is the hydraulic conductivity tensor of the porous medium (w.r.t water),
\(\bar{\gamma}^w = \bar{\rho}^w \mathbf{g}\) is the intrinsic unit weight vector of the pore water (where \(\mathbf{g}\) is gravitational acceleration),
\(\mathbf{w}^w\) is the Darcy velocity, representing the volumetric discharge of water per unit cross-sectional area relative to the solid skeleton.

The partial Cauchy stress tensor of the pore water is isotropic, given by:

\[ \boldsymbol{\sigma}^w = \varphi^w \bar{p}^w \boldsymbol{\delta}, \]

where \(\boldsymbol{\delta}\) is the identity tensor. This form reflects that pore fluid exerts isotropic (equal in all directions) pressure within the porous medium.

Considering that all phases experience the same external body force \(\mathbf{b}^w = \mathbf{b}\) (typically gravity), the balance of linear momentum for the pore water is given as:

\[ \begin{equation} \label{eq:linear_momentum_fluid} \varphi^w \bar{\rho}^w \mathbf{b} + \varphi^w \nabla \bar{p}^w + \bar{p}^w \nabla \varphi^w - \varphi^w \left(\mathbf{K}^w\right)^{-1} \bar{\gamma}^w \mathbf{w}^w = \varphi^w \bar{\rho}^w \mathbf{a}^w. \end{equation} \]

Under common practical conditions, it can be assumed that the spatial gradient of the water volume fraction (or porosity) is negligible, i.e. a homogeneous porosity field (\(\nabla \varphi^w = 0\)). Furthermore, the hydraulic conductivity tensor \(\mathbf{K}^w\) is usually related to the intrinsic permeability tensor of the soil skeleton \(\mathbf{K}^s\) through:

\[ \begin{equation}\label{eq:hydraulic_conductivity} \mathbf{K}^w = \frac{\gamma^w}{\bar{\mu}^w}\mathbf{K}^s, \end{equation} \]

where \(\bar{\mu}^w\) is the dynamic viscosity of the pore water. Under these conditions, the momentum balance reduces to the generalized Darcy law:

\[ \begin{equation} \label{eq:darcy_general} \mathbf{w}^w = \frac{k^w}{\bar{\mu}^w}\mathbf{K}^s \left(-\nabla \bar{p}^w + \bar{\rho}^w(\mathbf{b} - \mathbf{a}^w)\right), \end{equation} \]

or in index notation:

\[ \begin{equation} w_i^w = \frac{k^w}{\bar{\mu}^w}K^s_{ij}\left(-\bar{p}_{,j}^w + \bar{\rho}^w(b_j - a_j^w)\right). \end{equation} \]

Equation (\(\ref{eq:darcy_general}\)) clearly highlights how various forces drive or resist fluid motion in a porous medium:

\(-\nabla \bar{p}^w\): The driving force due to the pore water pressure gradient, which causes fluid to flow from regions of higher pressure to regions of lower pressure.
\(\bar{\rho}^w \mathbf{b}\): Body force due to gravity, causing downward movement of the fluid.
\(\bar{\rho}^w \mathbf{a}^w\): Fluid inertia term, generally small but potentially significant under dynamic loading conditions (such as earthquakes).
\(\frac{k^w}{\bar{\mu}^w}\mathbf{K}^s\): A tensorial coefficient representing the ease with which fluid moves through the pore structure, capturing soil permeability and fluid viscosity. A higher permeability or lower viscosity facilitates greater fluid mobility.

Thus, Equation (\(\ref{eq:darcy_general}\)) provides a physically consistent and mathematically rigorous description of fluid movement through saturated soils, forming the basis for coupling hydraulic processes with mechanical deformation in finite element analyses.

Balance of Linear Momentum of the Mixture

In the following, we derive the balance of linear momentum for the mixture as a whole by summing the momentum equations of each phase. This approach significantly simplifies the analysis, particularly when modelling the coupled deformation and fluid flow in saturated soils.

Starting from the general balance equation of linear momentum (Equation \(\ref{eq:linear_momentum_alpha}\)), we consider a porous medium composed of solid and fluid phases and sum the contributions from each phase. To form a physically consistent mixture formulation, the following assumptions are typically made:

All phases have the same acceleration: \(\bar{\mathbf{a}}^\alpha = \mathbf{a}\),
All phases experience the same external body force: \(\mathbf{b}^\alpha = \mathbf{b}\),
Interaction forces between phases cancel out: \(\sum_\alpha \mathbf{h}^\alpha = \mathbf{0}\),
The total stress is the sum of individual partial stresses:

\[ \boldsymbol{\sigma}^{tot} = \sum_\alpha \boldsymbol{\sigma}^\alpha. \]

Under these assumptions, the balance equation of linear momentum for the mixture simplifies to:

\[ \begin{equation}\label{eq:linear_momentum_TPM} \rho^{tot} \mathbf{b} + (\nabla \cdot \boldsymbol{\sigma}^{tot}) - \rho^{tot}\mathbf{a} = \mathbf{0}. \end{equation} \]

In this equation:

\(\rho^{tot}\) is the total density of the mixture, defined as:

\[ \rho^{tot} = \rho^s + \rho^w = \varphi^s\bar{\rho}^s + \varphi^w\bar{\rho}^w. \]
\(\boldsymbol{\sigma}^{tot}\) is the total stress tensor, representing both mechanical and fluid pressures.
\(\mathbf{b}\) represents body forces per unit mass, typically gravitational forces.
\(\mathbf{a}\) is the acceleration vector shared by all phases.

Introducing the concept of effective stress allows us to separate mechanical stresses carried by the solid skeleton from pore fluid pressures:

\[ \boldsymbol{\sigma}^{tot} = \boldsymbol{\sigma} - \bar{p}^w\boldsymbol{\delta}, \]

where:

\(\boldsymbol{\sigma}\) is the effective stress tensor of the solid skeleton, responsible for soil deformation,
\(\bar{p}^w\) is the intrinsic pore water pressure,
\(\boldsymbol{\delta}\) is the identity tensor.

With this substitution, the linear momentum balance of the mixture becomes explicitly coupled to deformation and fluid pressure:

\[ \begin{equation}\label{eq:linear_momentum} \rho^{tot}\mathbf{b} + \nabla\cdot\left(\boldsymbol{\sigma} - \bar{p}^w\boldsymbol{\delta}\right) - \rho^s\bar{\mathbf{a}}^s - \rho^w\bar{\mathbf{a}}^w = \mathbf{0}, \end{equation} \]

or equivalently expressed in index notation:

\[ \begin{equation} \rho^{tot}b_i + \left(\sigma_{ij} - \bar{p}^w\delta_{ij}\right)_{,j} - \rho^s\bar{a}_i^s - \rho^w\bar{a}_i^w = 0_i. \end{equation} \]

Equation (\(\ref{eq:linear_momentum}\)) provides a fundamental framework for analysing saturated soils subjected to coupled mechanical and hydraulic loading conditions, including consolidation processes, groundwater flow, and soil-structure interactions. It explicitly accounts for the following terms:

\(\rho^{tot}\mathbf{b}\): Total external body forces per unit volume (such as gravitational forces) acting uniformly throughout the porous medium.
\(\nabla \cdot \boldsymbol{\sigma}\): Internal mechanical forces arising from spatial variations in effective stress within the solid skeleton. These forces govern deformation of the soil structure.
\(\nabla \bar{p}^w\): Forces induced by pore water pressure gradients. These gradients influence fluid flow and oppose compression by exerting pressure against the solid skeleton.
\(\rho^s\bar{\mathbf{a}}^s + \rho^w\bar{\mathbf{a}}^w\): Combined inertial response of the solid and fluid phases. While often negligible in quasi-static conditions, these inertia terms become crucial under dynamic loading conditions, such as earthquakes, capturing the dynamic interaction between fluid and solid phases.