Darcy fingering equations

Governing equations for miscible viscous fingering in Darcy flow.

Dimensional equations

\[\begin{split} \begin{align*} &\text{Find} \\ &c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^d, \\ &p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R} \\ &\text{such that} \\ &\mathbb{IBVP}_{\textbf{u},p,c} \begin{cases} \phi\frac{\partial c}{\partial t} + \textbf{u}\cdot\nabla c = \nabla\cdot(\mathsf{D}\cdot\nabla c) & \\ \nabla\cdot\textbf{u} = 0 & \\ \textbf{u}=-\frac{\mathsf{K}}{\mu}\cdot\nabla p & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ c=c_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ c=c_{\text{D}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{D}, c} \times [0,\infty) \\ \textbf{n}\cdot(\mathsf{D}\cdot\nabla c) = c_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}, c} \times [0,\infty)~,~\partial\Omega_{\text{N}, c}=\partial\Omega/\partial\Omega_{\text{D}, c} \\ \textbf{n}\cdot\textbf{u} = u_{\text{E}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{E}} \times [0,\infty) \\ p = p_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}}\times [0,\infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\textbf{u},p,c}= \begin{cases} \Omega & \text{domain}\\ c_0(\textbf{x}) & \text{concentration initial condition}\\ c_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D},c} & \text{concentration Dirichlet boundary condition} \\ c_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N},c} & \text{concentration Neumann boundary condition} \\ u_{\text{E}}(\textbf{x}, t)~,~\partial\Omega_{\text{E}} & \text{normal velocity essential boundary condition} \\ p_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}} & \text{pressure natural boundary condition} \\ \phi(\textbf{x}) & \text{porosity}\\ \mathsf{K}(\phi) & \text{permeability}\\ \mathsf{D}(\phi, \textbf{u}) & \text{solutal dispersion}\\ \mu(c) & \text{viscosity}\\ \end{cases} \end{align*} \end{split}\]

Non-dimensionalization

Scalings

\[\begin{split} \begin{align*} \textbf{u}&\to\textbf{u}-\textbf{u}_{\text{in}} \\ \textbf{x}&\to\textbf{x}-t\,\textbf{u}_{\text{in}} \\ \end{align*} \end{split}\]

Quantity	\(\vert\textbf{x}\vert\)	\(\vert\textbf{u}\vert\)	\(t\)	\(c\)	\(p\)	\(\psi\)	\(\mu\)	\(\phi\)	\(K\)	\(\vert\mathsf{D}\vert\)
Scaling	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(\Delta c\)	\(\mu_{\text{ref}}\,\mathcal{U}\mathcal{L}/K_{\text{ref}}\)	\(\mathcal{U}\mathcal{L}\)	\(\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\)	\(K_{\text{ref}}\)	\(D_{\text{ref}}\)

Generic dimensionless numbers

\[ Ad=\frac{\mathcal{U}\mathcal{T}}{\phi_{\text{ref}}\mathcal{L}}~,~ Di=\frac{D_{\text{ref}}\mathcal{T}}{\phi_{\text{ref}}\mathcal{L}^2}~,~ In=\frac{\vert\textbf{u}_{\text{in}}\vert}{\mathcal{U}}~,~ X=\frac{\mathcal{L}_\Omega}{\mathcal{L}} \]

Physical dimensionless numbers

Definition	Name	Physical interpretation
\(Pe=\frac{\vert\textbf{u}_{\text{in}}\vert\mathcal{L}_\Omega}{D_{\text{ref}}}\)	Peclet number	Ratio of injection to diffusive speeds.
\(P_\Delta=\frac{K_{\text{ref}}\Delta p}{D_{\text{ref}}\mu_{\text{ref}}}\)	pressure Peclet number	Peclet number for pressure-driven flow.

Scaling choice

Name	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(Ad\)	\(Di\)	\(In\)	\(X\)
advective	\(\mathcal{L}_\Omega\)	\(\lvert\mathbf{u}_{\text{in}}\rvert\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1/Pe\)	\(1\)	\(1\)
diffusive	\(\mathcal{L}_\Omega\)	\(D_{\text{ref}}/\mathcal{L}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1\)	\(Pe\)	\(1\)
pressure-driven	\(\mathcal{L}_\Omega\)	\(K_{\text{ref}}\Delta p/\mathcal{L}\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1/P_\Delta\)	\(0\)	\(1\)

Non-dimensional time-discretized equations

Strong form

\[\begin{split} \begin{align*} &\text{Find}~c^{n+1}, \theta^{n+1},~\textbf{u}^n,~p^n~\text{such that}~\forall n\geq0 \\ &\begin{cases} \phi\frac{c^{n+1}-c^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u}, c}(\textbf{u}\cdot\nabla c) = Di\nabla\cdot\mathcal{D}_{\mathsf{D},c}(\mathsf{D}(\phi, \textbf{u})\cdot\nabla c) & \\ \nabla\cdot\textbf{u}^n = 0 & \\ \textbf{u}^n + In\,\textbf{e}_{\text{in}} =-\frac{\mathsf{K}}{\mu^n}\cdot\nabla p^n \iff \textbf{u}^n = -\frac{\mathsf{K}}{\mu^n}\cdot\nabla(p^n + In\,\mu^n\mathsf{K}^{-1}\cdot\textbf{e}_{\text{in}} ) \\ c^0=c_0 \\ c^n\vert_{\partial\Omega_{\text{D}, c}}=c^n_{\text{D}} \\ \left(\textbf{n}\cdot(\mathsf{D}^n\cdot\nabla c^n)\right)\vert\partial\Omega_{\text{N}, c} = c_{\text{N}}^n \\ (\textbf{n}\cdot\textbf{u}^n)\vert_{\partial\Omega_{\text{E}}} = u^n_{\text{E}}\\ p^n\vert_{\partial\Omega_{\text{N}}} = p^n_{\text{N}} \\ \end{cases}~. \end{align*} \end{split}\]