R E R I

Hoteit, H., and Firoozabadi, A.: "Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods," SPE J (2006) 11(1), 19-34.

There is a vast difference between the numerical errors in first-order finite difference (FD) and discontinuous Galerkin (DG) methods for the same mesh size. In 2D, the computational speed in about two orders higher in our DG compared to first-order finite difference schemes for compositional flows.

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Mikyška, J., and Firoozabadi, A.: "Implementation of Higher-Order Methods for Robust and Efficient Compositional Simulation," J Comput Phys (2010) 229, 2898–2913.

To take advantage of the powerful features of higher-order methods, we sometimes need to formulate our basic equations in accordance with the numerical scheme. With the appropriate formulation the code runs robustly as seen above from the CO₂ injection from the bottom.

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Hoteit, H., and Firoozabadi, A.: "Numerical Modeling of Two-Phase Flow in Heterogeneous Permeable Media with Different Capillarity Pressures," Adv Water Resour (2008) 31, 56-73.

Capillary pressure contrast between the layers can sharpen the front in layered media. Without the capillary pressure or when an average capillary pressure is used, the results on the left are obtained.

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Hoteit, H., and Firoozabadi, A.: "An Efficient Numerical Model for Incompressible Two-Phase Flow in Fractured Media," Adv Water Resour (2008) 31, 891-905

a) Gridding

Unstructured gridding with any quality is a requirement for accurate modeling of flow in descrete fractured porous media. Our higher-order method provides accurate modeling of species flow and velocity fields in complex fractured media.

b) Flow Paths

Water injection in such a complex fractured media can only be performed with a higher-order method if accurate results are desired.

c) Flow Paths in Matrix

d) Flow Path in Fractures

3D geometry and fractured complexity create challenges that can only be addressed by using a comprehensive numerical scheme.

Flow in the fractures (shown above) is coupled with matrix flow.

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Moortgat, J., Li, Z. and Firoozabadi, A.: "Three-Phase Compositional Modeling of CO₂ Injection by Higher-Order Finite Element Methods with PR and CPA Equations of State," Water Resour Res (2012) 48, W12511:

Viscous fingering during WAG injection in anisotropic 600 m by 60 m domain saturated with oil and 30% connate water. The vertical permeability is 50 times lower than the 221 md horizontal permeability. Alternating slugs of 2.5% pore volume of water and 2.5% slugs of gas are injected uniformly from the left boundary, and production is at constant pressure from the right. The gas is 80 mol% CO₂ and 20mol% methane and has a density similar to the oil in the reservoirs, but a much lower viscosity. Because of the high viscosity ratio, pronounced viscous fingers develop. These fingers are easily resolved by our higher-order finite element methods, while they are often suppressed by numerical dispersion when traditional finite difference methods are used. Mixtures of the injected gas and the 9-(pseudo)component oil are near the critical point, which makes the three-phase compositional modeling and phase behavior computations challenging.

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Moortgat, J., Firoozabadi, A., Li, Z. and Esposito, R.: "CO₂ Injection in Vertical and Horizontal Cores: Measurements and Numerical Simulation," SPE J (2013), 18(2), 331-344.

Numerical modeling of experiments in which supercritical CO₂ is injected in a core saturation with oil and connate water. The supercritical CO₂ at some reservoir temperatures and pressures is denser than the oil, so injection from the top is prone to gravitational instabilities. When Fickian diffusion is neglected, the DG simulations show pronounced gravitational fingering and natural convection (right), while the fingering is suppressed by numerical dispersion when traditional FD methods are used (left) even on this very fine 50 by 250 element grid.

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Moortgat, J., Li, Z. and Firoozabadi, A.: "Three-Phase Compositional Modeling of CO₂ Injection by Higher-Order Finite Element Methods with PR and CPA Equations of State," Water Resour Res (2012) 48, W12511.

When CO₂ dissolves in water, it increases the water density in the top of the aquifer, which is gravitationally unstable. As a result, gravitational fingers may develop that transport the injected and dissolved CO₂ throughout the aquifer by convection. The gravito-convective mixing is much faster than the diffusive time-scale, which makes the sequestration process more efficient. The on-set time, critical wavelength of the fingers, and propagation speed through the porous medium all scale with the formation permeability. The higher the permeability, the more efficient the process. High permeability aquifers should therefore be considered in sequestration pilot projects.

Our simulations take into account Fickian diffusion, use the cubic-plus-association equation-of-state for the CO₂-water mixtures, consider rock compressibility, and use impermeable boundary conditions to allow a study of the associated pressure build-up.

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Moortgat, J., and Firoozabadi, A.: "Three-Phase Compositional Modeling with Capillarity in Heterogeneous and Fractured Media," SPE J (2013), 18(6), 1150-1168.

a) Mesh and location of discrete fractures, b) gas saturation for DG simulation with capillarity, c) gas saturation for DG simulation without capillarity.

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