We start with the following parabolic PDE: $$ \left( c \rho_R + \phi \frac{\partial \rho_R}{\partial p}\right) \frac{\partial p}{\partial t}  \nabla \cdot \left[ \rho_R \frac{\kappa}{\mu} \left( \nabla p + \rho_R g \right) \right] $$ $$ Q_p = 0. $$ where
In order to obtain a unique solution it is necessary to specify conditions on the boundary $\Gamma$ of the domain $\Omega$.
The benchmark at hand should demonstrate the primary variable constraint Dirichlettype boundary condition. Here, the size of the subdomain, the Dirichlettype boundary condition is defined on, is variable and changes according to a condition depending on the value of the primary variable.
$$ \Gamma^\ast_D = { x \in \mathbb{R}^d, x \in \Gamma_D, \text{Condition}(p(x)) } $$
On the left (x=0) and right side (x=1) of the domain $\Omega = [0,1]^3$ the usual Dirichlettype boundary conditions are set, i.e., $$ p = 1, \quad x=0 \qquad \text{and} \qquad p = 1\quad x=1 $$ The initial condition $p_0$ is set to zero. Additionally, a primary variable constraint Dirichlettype boundary condition (PVCDBC) is specified: $$ p = 0.1, \quad \text{for}\quad z = 1 \quad \text{and}\quad p(x,y,1) > 0. $$ At the beginning of the simulation the PVCDBC is inactive. Because of the ’normal’ Dirichlettype boundary conditions the pressure is greater than zero after the first time step and the PVCDBC is activated in the second time step. The effect is depicted in the figure:
This article was written by Thomas Fischer. If you are missing something or you find an error please let us know.
Generated with Hugo 0.117.0
in CI job 364423

Last revision: July 24, 2023
Commit: [PL/PhF] Rename x_dot to x_prev 648b0315
 Edit this page on