Theoretical derivation of the conservation equations for single phase flow in porous media: a continuum approach

Abstract

This paper deals with the theoretical derivation of the conservation equations for single phase flow in a porous medium. The derivation is obtained within the framework of the continuum mechanics and classical thermodynamics. The adopted procedure provides the conservation equations of mass, momentum, mechanical energy, total energy, internal energy, entropy, temperature, enthalpy, Gibbs free energy and Helmholtz free energy. The obtained results highlight the connection between the basic equations of fluid mechanics and of fluid flow in porous media, as well as the restrictions and the limitations of Darcy’s law and Richards’ equation.

Appendices

Appendix 1

Equation (8) can be obtained by using the following classical thermodynamic relations [33]:

$$ \frac{\delta \beta }{\alpha } = \frac{1}{p} $$

(89)

$$ T{\text{d}}s = {\text{d}}u_{i} - \frac{p}{{\rho^{2} }}{\text{d}}\rho = \left( {\frac{{\partial u_{i} }}{\partial T}} \right)_{\rho } {\text{d}}T + \left( {\frac{{\partial u_{i} }}{\partial \rho }} \right)_{T} {\text{d}}\rho - \frac{p}{{\rho^{2} }}{\text{d}}\rho = c_{w} {\text{d}}T + \left[ {\left( {\frac{{\partial u_{i} }}{\partial \rho }} \right)_{T} - \frac{p}{{\rho^{2} }}} \right]{\text{d}}\rho $$

(90)

$$ \left[ {\left( {\frac{{\partial u_{i} }}{\partial \rho }} \right)_{T} - \frac{p}{{\rho^{2} }}} \right] = - \frac{T}{{\rho^{2} }}\left( {\frac{\partial p}{\partial T}} \right)_{\rho } = - \frac{\delta Tp}{{\rho^{2} }} = - \frac{\alpha T}{{\beta \rho^{2} }} $$

(91)

$$ T{\text{d}}s = {\text{d}}h - \frac{1}{\rho }{\text{d}}p = \left( {\frac{\partial h}{\partial T}} \right)_{p} {\text{d}}T + \left( {\frac{\partial h}{\partial p}} \right)_{T} {\text{d}}p - \frac{1}{\rho }{\text{d}}p = c_{p} {\text{d}}T + \left[ {\left( {\frac{\partial h}{\partial \rho }} \right)_{T} - \frac{1}{\rho }} \right]{\text{d}}p $$

(92)

$$ \left[ {\left( {\frac{\partial h}{\partial p}} \right)_{T} - \frac{1}{\rho }} \right] = - \frac{T}{{\rho^{2} }}\left( {\frac{\partial \rho }{\partial T}} \right)_{p} = - \frac{\alpha T}{\rho } $$

(93)

$$ c_{p} - c_{w} = \frac{\alpha T\delta p}{\rho } = \frac{{T\alpha^{2} \varepsilon }}{\rho } $$

(94)

In the above equations, w = 1/ρ; α = − (∂w/∂T) _p /w is the coefficient of thermal expansion; β = − (∂w/∂p) _T /w the coefficient of compressibility; ε = 1/β the modulus of elasticity; δ = (∂p/∂T) _w /p the thermal coefficient of pressure; c _w  = (∂u _i /∂T) _w  = T(∂s/∂T) _w the specific heat at constant volume; c _p  = (∂h/∂T) _p  = T(∂s/∂T) _p the specific heat at constant pressure.

With Eq. (91), Eq. (90) may be rewritten as:

$$ \rho T\frac{{{\text{D}}s}}{{{\text{D}}t}} = \rho c_{w} \frac{{{\text{D}}T}}{{{\text{D}}t}} - \frac{\alpha T}{\beta \rho }\frac{{{\text{D}}\rho }}{{{\text{D}}t}} = \rho c_{w} \frac{{{\text{D}}T}}{{{\text{D}}t}} + \alpha T\varepsilon \nabla \cdot \varvec{v} $$

(95)

while, with Eq. (93), Eq. (92) becomes:

$$ \rho T\frac{{{\text{D}}s}}{{{\text{D}}t}} = \rho c_{p} \frac{{{\text{D}}T}}{{{\text{D}}t}} - \alpha T\frac{{{\text{D}}p}}{{{\text{D}}t}} $$

(96)

By comparing Eq. (95) with Eq. (96), it follows that:

$$ \rho \left( {c_{p} - c_{w} } \right)\frac{{{\text{D}}T}}{{{\text{D}}t}} = \alpha T\left( {\varepsilon \nabla \cdot \varvec{v} + \frac{{{\text{D}}p}}{{{\text{D}}t}}} \right) $$

(97)

Finally, by using Eq. (94), Eq. (97) reduces to Eq. (8).

Appendix 2

In accordance with the constitutive theory of continuum mechanics, the term ∇·T _r can be expressed as:

$$ \nabla \cdot \varvec{T}_{r} = \left( {\nabla \cdot \varvec{T}_{r} } \right)_{s} + \left( {\nabla \cdot \varvec{T}_{r} } \right)_{b} $$

(98)

where the vectors (∇·T _r ) _s and (∇·T _r ) _b are the friction forces arising from the shear and the bulk viscosities, respectively [18, 34, 35]. Eq. (31) implicitly assumes that the bulk viscosity vanishes (∇·T _r ) _b  = 0 and, consequently, the fictitious fluid is elastic (no energy dissipation occurs during the compression or expansion).