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model_description.md

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Physics

PDEs

One mass balance per component \kappa (water and CO2).

\frac{\partial M^{\kappa}}{\partial t} + \nabla\cdot \mathbf{F}^{\kappa} - q^{\kappa} = 0

where the mass of each species is (summed over each phase \beta)

M^{\kappa} = \phi\sum_{\beta}S_{\beta}\rho_{\beta}\chi_{\beta}^{\kappa}

and the flux of each component \mathbf{F}^{\kappa} is the sum of advective and diffusive fluxes \mathbf{F}^{\kappa} = \mathbf{F}^{\kappa}{\mathrm{advective}}+ \mathbf{F}^{\kappa}{\mathrm{diffusion}}

where the advective flux is given by Darcy's law, and the diffusive flux is the usual Fickian diffusion.

All parameters above are their usual values (e.g., porosity, density etc).

Constitutive relations

Fluid-matrix interaction

  • Capillary pressure: Brooks-Corey p_c(S_{l}) = p_\text{entry}S_{le}^{-1/\lambda}

  • Relative permeability: Brooks-Corey k_{\mathrm{r, w}} = \left(S_{\mathrm{eff}}\right)^{(2 + 3 \lambda)/\lambda} and k_{\mathrm{r, nw}} = (1 - S_{\mathrm{eff}})^2 \left[1 - \left(S_{\mathrm{eff}}\right)^{(2 + \lambda)/\lambda}\right]

Phase composition: Applied equations of state

  • CO2 in liquid phase: Calculated using fugacity formulation of Spycher et al. (2003) and Spycher et al. (2005)

  • Water in gas phase: Calculated using fugacity formulation of Spycher et al. (2003) and Spycher et al. (2005)

Density

  • Liquid phase: Water density calculated using IAPWS Industrial Formulation, density change due to dissolved CO2 calculated using formulation by Garcia, 2001.

  • Gas phase: Approximated as pure CO2, with properties calculated using the Span and Wagner (1996) EOS.

Solubility limit

CO2 solubility at base of tank 1.786 kg/m3 (0.001782 kg/kg)

Temperature

20°C

Domain volume

The volume of the sand region (NOT including water region at top of tank) 0.0865 m3. The pore volume of the model is 0.0383 m3.

Spatial parameters

The parameters k, \phi, p_\text{entry}, \lambda, S_{lr}, S_{gr}, k_{lr}, k_{gr}, for the different facies are given in spatial_parameters.csv.

Numerics

Coupling of flow and transport, temporal and spatial discretization

Fully coupled, fully implicit, cell-centered FV.

Linearization and Solvers

Newton with line search, direct solver (MUMPS) for the linear systems.

Primary Variables

Persistent set of primary variables comprising fluid pressure and total mole fraction of CO2 summed over all phases, with gas saturation calculated using a compositional flash see MOOSE documentation for details.

Computational Grid

Uniform structured grid of 44,284 quadrilateral elements (approx dimensions of each element is 10mm x 10mm).

Performance

During the injection period (0 to 5 hours)

Indicator Average Min Max Median
time step size [s] 1.96 0.0078125 2 2
# nonlinear iterations per time step 3 1 15 3
# linear iterations per solve 1 1 1 1

During the subsequent rest period (5 to 120 hours)

Indicator Average Min Max Median
time step size [s] 35.92 0.15625 60 20
# nonlinear iterations per time step 3 1 15 2
# linear iterations per solve 1 1 1 1