Darcy convection equations

Governing equations for thermosolutal convection-reaction coupled to Darcy flow.

Dimensional equations

\[\begin{split} \begin{align*} &\text{Find} \\ &c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &\theta(\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} \begin{cases} \phi\frac{\partial c}{\partial t} + \textbf{u}\cdot\nabla c = \nabla\cdot(\mathsf{D}\cdot\nabla c) + R & \\ \phi\frac{\partial\theta}{\partial t} + \textbf{u}\cdot\nabla\theta = \nabla\cdot(\mathsf{G}\cdot\nabla\theta) + H& \\ \nabla\cdot\textbf{u} = 0 & \\ \textbf{u}=-\frac{\mathsf{K}}{\mu}\cdot(\nabla p - \rho g\,\textbf{e}_g) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ c=c_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ \theta=\theta_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} \\ \theta=\theta_{\text{D}} & \forall (\textbf{x}, t)\in\partial\Omega_{\text{D}, \theta} \times [0,\infty] \\ \textbf{n}\cdot(\mathsf{G}\cdot\nabla \theta) = \theta_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}, \theta} \times [0,\infty]~,~\partial\Omega_{\text{N}, \theta}=\partial\Omega/\partial\Omega_{\text{D}, \theta} \\ \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} \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain}\\ c_0(\textbf{x}) & \text{concentration initial condition}\\ \theta_0(\textbf{x}) & \text{temperature initial condition}\\ c_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D},c} & \text{concentration Dirichlet boundary condition} \\ \theta_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D},\theta} & \text{temperature Dirichlet boundary condition} \\ c_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N},c} & \text{concentration Neumann boundary condition} \\ \theta_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}, \theta} & \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}\\ \mathsf{G}(\phi, \textbf{u}) & \text{thermal dispersion}\\ \rho(c, \theta) & \text{density}\\ \mu(c, \theta) & \text{viscosity}\\ R(c,\theta, \phi) & \text{solutal reaction}\\ H(c,\theta, \phi) & \text{thermal reaction}\\ \end{cases} \end{align*} \end{split}\]

Non-dimensionalization

Scalings

Quantity	\(\vert\textbf{x}\vert\)	\(\vert\textbf{u}\vert\)	\(t\)	\(c\)	\(\theta\)	\(\rho g\)	\(p\)	\(\psi\)
Scaling	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(\Delta c\)	\(\Delta\theta\)	\(g \Delta\rho\)	\(\mu_{\text{ref}}\,\mathcal{U}\mathcal{L}/K_{\text{ref}}\)	\(\mathcal{U}\mathcal{L}\)

\(\mu\)	\(\phi\)	\(K\)	\(\vert\mathsf{D}\vert\)	\(\vert\mathsf{G}\vert\)	\(R\)	\(H\)
\(\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\)	\(K_{\text{ref}}\)	\(D_{\text{ref}}\)	\(G_{\text{ref}}\)	\(\Delta R\)	\(\Delta H\)

Abstract dimensionless numbers

\[ Ad=\frac{\mathcal{U}\mathcal{T}}{\phi_{\text{ref}}\mathcal{L}}~,~ Di=\frac{D_{\text{ref}}\mathcal{T}}{\phi_{\text{ref}}\mathcal{L}^2}~,~ Ki=\frac{\mathcal{T}\Delta R}{\phi_{\text{ref}}\Delta c}~,~ Bu=\frac{K_{\text{ref}}\,g\Delta\rho}{\mu_{\text{ref}}\,\mathcal{U}}~,~ X=\frac{\mathcal{L}_\Omega}{\mathcal{L}} \]

Physical dimensionless numbers

Definition	Name	Physical interpretation
\(Ra=\frac{\mathcal{L}_\Omega K_{\text{ref}}g\Delta\rho}{\mu_{\text{ref}}D_{\text{ref}}}=\underbrace{\frac{K_{\text{ref}}\,g\Delta\rho}{\mu_{\text{ref}}}}_{\text{convective speed}} \big/ \underbrace{\frac{D_{\text{ref}}}{\mathcal{L}_\Omega}}_{\text{diffusive speed}}\)	Rayleigh	Ratio of convective to diffusive speeds, defined with respect to the transport of \(c\) and domain length scale.
\(Da=\frac{\mathcal{L}_\Omega \mu_{\text{ref}}\,\Delta R}{K_{\text{ref}}\,g\Delta\rho\Delta c} = \underbrace{\frac{\Delta R}{\Delta c}}_{\text{reaction rate}} \big/ \underbrace{\frac{K_{\text{ref}}\,g\Delta\rho}{\mathcal{L}_\Omega \mu_{\text{ref}}}}_{\text{convection rate}}\)	Damköhler	Ratio of reaction to convection rates, defined with respect to the transport of \(c\) and domain length scale.
\(Le=\frac{G_{\text{ref}}}{D_{\text{ref}}}\)	Lewis	Ratio of thermal to solutal diffusivities.
\(Lr=\frac{\Delta H\Delta c}{\Delta\theta \Delta R} = \underbrace{\frac{\Delta H}{\Delta\theta}}_{\text{thermal reaction rate}} \big/ \underbrace{\frac{\Delta R}{\Delta c}}_{\text{solutal reaction rate}}\)	Lerwis	Ratio of thermal to solutal reaction rates.

Scaling choice

Name	\(\mathcal{L}\)	\(\mathcal{U}\)	\( \mathcal{T}\)	\(\{Ad, Di, Ki, Bu, X\}\)	Examples
advective	\(\mathcal{L}_\Omega\)	\(K_{\text{ref}}\,g\Delta\rho/\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(\{1, 1/Ra, Da, 1, 1\}\)	Hewitt et al. (2012)
diffusive	\(\mathcal{L}_\Omega\)	\(D_{\text{ref}}/\mathcal{L}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(\{1, 1, RaDa, Ra, 1\}\)	Ritchie & Pritchard (2011)
advective-diffusive	\(D_{\text{ref}}/\mathcal{U}\)	\(K_{\text{ref}}\,g\Delta\rho/\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(\{1, 1, Da/Ra, 1, Ra\}\)	Slim (2014)
reactive	\(\sqrt{D_{\text{ref}}\mathcal{T}/\phi_{\text{ref}}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{T}\)	\(\phi_{\text{ref}}\Delta c/\Delta R\)	\(\{1, 1, 1, \sqrt{Ra/Da}, \sqrt{RaDa}\}\)	Kabbadj et al. (2025)

Non-dimensional time-discretized equations

Strong form

\[\begin{split} \begin{align*} &\text{Find}~c^{n+1}, \theta^{n+1},~\textbf{u}^n,~p^n~\text{such that}~\forall n\geq0 \\ &\begin{cases} \phi\frac{c^{n+1}-c^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},c}(\textbf{u}\cdot\nabla c)=Di\nabla\cdot\mathcal{D}_{\mathsf{D},c}(\mathsf{D}\cdot\nabla c) + Ki\,\mathcal{D}_R(R) \\ \phi\frac{\theta^{n+1}-\theta^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},\theta}(\textbf{u}\cdot\nabla\theta)=Di\nabla\cdot\mathcal{D}_{\mathsf{G},\theta}(\mathsf{G}\cdot\nabla\theta) + LrKi\,\mathcal{D}_H(H) \\ \nabla\cdot\textbf{u}^n=0 \\ \textbf{u}^n=-\frac{\mathsf{K}}{\mu^n}\cdot\left(\nabla p^n - Bu\,\rho^n\,\textbf{e}_g\right) \\ c^0=c_0 \\ \theta^0=\theta_0 \\ c^n\vert_{\partial\Omega_{\text{D}, c}}=c^n_{\text{D}} \\ \left(\textbf{n}\cdot(\mathsf{D}^n\cdot\nabla c^n)\right)\vert_{\partial\Omega_{\text{N}, c}} = c_{\text{N}}^n \\ \theta^n\vert_{\partial\Omega_{\text{D}, \theta}}=\theta^n_{\text{D}} \\ \left(\textbf{n}\cdot(\mathsf{G}^n\cdot\nabla\theta^n)\right)\vert_{\partial\Omega_{\text{N}, \theta}} = \theta_{\text{N}}^n \\ (\textbf{n}\cdot\textbf{u}^n)\vert_{\partial\Omega_{\text{E}}} = u^n_{\text{E}}\\ p^n\vert_{\partial\Omega_{\text{N}}} = p^n_{\text{N}} \\ \end{cases} \end{align*} \end{split}\]

Weak forms

Mixed formulation

\[\begin{split} \begin{align*} &\text{Find} \\ &(\textbf{u}^n, p^n)\in V_\textbf{u}\times V_p \\ &c^{n+1}\in V_c \\ &\theta^{n+1}\in V_\theta \\ &\text{such that} \\ &\mathbb{F}_{\textbf{u},p} \begin{cases} F_{\textbf{u},p}(\textbf{u}^n, p^n, \textbf{v}, q)=0 \quad\forall(\textbf{v}, q)\in V_{\textbf{u}} \times V_p\\ F_c(c^{n+1}, v) = 0\quad\forall v\in V_c \\ F_\theta(\theta^{n+1}, v) = 0\quad\forall v\in V_\theta \\ \end{cases} \end{align*} \end{split}\]

Streamfunction formulation

\[\begin{split} \begin{align*} &\text{Find} \\ &\psi^n\in V_\psi \\ &\textbf{u}^n\in V_\textbf{u} \\ &c^{n+1}\in V_c \\ &\theta^{n+1}\in V_\theta \\ &\text{such that} \\ &\mathbb{F}_\psi \begin{cases} F_\psi(\psi^n, v) = 0\quad\forall v\in V_\psi \\ F_{\textbf{u}}(\textbf{u}^n, \textbf{v}) = 0\quad\forall \textbf{v}\in V_{\textbf{u}} \\ F_c(c^{n+1}, v) = 0\quad\forall v\in V_c \\ F_\theta(\theta^{n+1}, v) = 0\quad\forall v\in V_\theta \\ \end{cases} \end{align*} \end{split}\]