• INL MOOSE, porous flow module
  • Governing equations
    • Fluid flow (continuity equation)
    • Mass density
    • Flux
    • Solid mechanics
• What’s next

INL MOOSE, porous flow module

I adapt the equations and remove the heat flow term, for we don’t need to history match this one. (Why? 1. we don’t have DTS data; 2. final goal: get the geometry of the frac, perm, and k of the frac. Mechanics is taking control.)

Governing equations

Fluid flow (continuity equation)

\[0=\frac{\partial M^\kappa}{\partial t} + M^\kappa \nabla \cdot \mathbf{v}_s + \nabla \cdot \mathbf{F}^\kappa\]

• dt term: rate of change of fluid.
• vs term: movement of skeleton. If $\nabla\cdot\mathbf v_s$ is positive, the skeleton is expanding. It will suck water from other places.
• fk term: flow term — the fluid flow in vs the fluid flow out.

Variable	Description	Unit
$\mathbf M$	Mass density	$\mathrm{kg/m^3}$
$\mathbf v_s$	Velocity of skeleton	$\mathrm{m/s}$
$\mathbf F$	Flux	$\mathrm{kg\cdot s^{-1}\cdot m^{-2}}$

Mass density

The mass of species $\kappa$ per volume of rock is written as a sum over all phases present in the system:

\[M^\kappa = \phi \sum_\beta S_\beta\rho_\beta\chi_\beta^\kappa + (1-\phi) C^\kappa\]

where:

\[\sum_\beta S_\beta = 1, \qquad \sum_\kappa \chi_\beta^\kappa = 1 \;\; \forall \beta\]

Flux

Flux is the sum of the advective flux and the diffusive-and-dispersive flux:

\[\mathbf F^\kappa = \sum_\beta \chi_\beta^\kappa \mathbf F_\beta^{\mathrm{advective}} +\mathbf F ^\kappa _{\mathrm{diffusion+dispersion}}\]

The advective flux is controlled by Darcy’s law:

\[\mathbf F_{\beta}^{\mathrm{advective}} = \rho_\beta \mathbf v_\beta = -\rho_\beta\frac{kk_{r,\beta}}{\mu_\beta}(\nabla P_\beta -\rho_\beta \mathbf g)\]

In this equation,

Variable	Description	Unit
$\mathbf v_\beta$	Darcy velocity in phase $\beta$	$\mathrm{m/s}$
$k$	absolute permeability (tensor)	$\mathrm{m^2}$
$k_{r,\beta}$	relative permeability	N/A

$k_{r,\beta}$ is always a function of the saturation(s), but with Klinkenberg effects it may also be a function of the gas pressure. Relative permeability can also be hysteretic, so that it depends on the history of saturation.

In reality, relative permeability is actually a tensor (for example it’s usually different in lateral and vertical directions) but is most often treated as a scalar, since it’s hard to get parameters for the tensorial case. In the PorousFlow module it is treated as a scalar.

For diffusion and hydrodynamic dispersion, we have:

\[\mathbf F_{\mathrm{diffusion+dispersion}}^\kappa =-\sum_\beta\rho_\beta \mathcal D_\beta^\kappa\nabla\chi_\beta^\kappa\]

The hydrodynamic dispersion factor is defined as

\[\mathcal D_\beta^\kappa = D_{\beta, T}^\kappa \mathcal I + \frac{D_{\beta, L}^\kappa - D_{\beta, T}^\kappa}{\mathbf v_\beta^2}\mathbf v_\beta \mathbf v_\beta\]

where,

\[D_{\beta, L}^\kappa = \phi\tau_0\tau_\beta d_\beta^\kappa + \alpha_{\beta,L} |\mathbf v|_\beta, \qquad D_{\beta, T}^\kappa = \phi\tau_0\tau_\beta d_\beta^\kappa + \alpha_{\beta,T} |\mathbf v|_\beta\]

In these equations,

Variable	Description	Unit
$d_\beta^\kappa$	molecular diffusion coefficient for component $\kappa$ in phase $\beta$	$\mathrm{m^2/s}$
$\chi_\beta^\kappa$	mass fraction of component $\kappa$ in phase $\beta$	N/A
$\tau_0\tau_\beta$	phase tortuosity	N/A
$\phi$	porosity of the solid	N/A
$\alpha_{\beta, L}$	longitudinal dispersivity of phase $\beta$	$\mathrm{m}$
$\alpha_{\beta, T}$	transverse dispersivity of phase $\beta$	$\mathrm{m}$

It is common to set the dispersion to zero by setting $\alpha_{\beta, T}=\alpha_{\beta, L}=0$.

In PorousFlow, tortuosity is defined as the ratio of the shortest path to the effective path.

Solid mechanics

This term contributes to $M^\kappa \nabla \cdot \mathbf{v}_s$. Based on the conservation of momentum:

\[\rho_{mat} \frac{\partial v_s^j}{\partial t} = \nabla_i \sigma_{ij}^{tot} + \rho_{mat}b_j =\nabla_i\sigma_{ij}^{eff}-\alpha_B\nabla_j P_f+\rho_{mat}b_j\]

And the constitutive law:

\[\sigma_{ij}^{eff} = \sigma_{ij}^{tot}+\alpha_B\delta_{ij}P_f=E_{ijkl}(\epsilon_{kl}^{elastic}-\delta_{kl}\alpha_TT)\]

The elastic tensor $E_{ijkl}$ has an inverse $C_{ijkl}$ called the compliance tensor.

Equilibrium equation:

\[\nabla \cdot (\sigma_{ij}^{eff} - \alpha_B\delta_{ij}P_f)=0\]

What’s next

MOOSE’s design philosophy does not explicitly use PDEs, but assembles the physical processes from kernels. How these processes map to a complete MOOSE kernel configuration is covered in my history matching notes.