Numerical Aspects of Vertical Equilibrium Models for
Simulating CO2 Sequestration
I. S. Ligaarden H. M. Nilsen
Department of Applied Mathematics, SINTEF ICT, Oslo, Norway
Summary/Background
The Johansen Model
Using a vertical equilibrium (VE) assumption, the flow of a layer of CO2 can be approximated in
terms of its thickness to obtain a 2D simulation model. Although this approach reduces the
dimension of the model, important information of the heterogeneities in the underlying 3D
medium is preserved.
In many cases the errors resulting from the VE assumption may be significantly smaller than
the errors introduced by the overly coarse resolution needed to make the 3D simulation model
computationally tractable. Vertical equilibrium simulations may then be attractive to increase
(lateral) resolution while saving computational cost.
Here we consider the Johansen formation, a candidate for CO2 sequestration, for comparing
the use of 3D simulations to simulations with a vertical equilibrium 2D model. We discuss
numerical aspects of using the different methods, and demonstrate that the vertical equilibrium
model provides more accurate results when the vertical grid resolution is low.
The Johansen formation [1] is a deep saline aquifer in the North sea which under consideration to
be used for storage of CO2 in a future pilot project for CCS at Mongstad, Norway.
Vertical Equilibrium Model
The VE formulation for two-phase flow in heterogeneous media can be written in a standard
fractional flow formulation as a system of a pressure equation and a transport equation. When
we consider immiscible incompressible flow without capillary pressure the equations read
Pressure
∇k · ~v = qtot (~x )
i
λw (s, ~x )
~v = −λt (s, ~x ) ∇kpt − f (s, ~x )ρco2 + [1 − f (s, ~x )]ρw ~gk(~x ) +
∇kpc (s, ~x )
λt (s, ~x )
h
The left plot above shows a height-map of the
Johansen sector model with the well indicated. The
right plot shows the z-coordinates of the cells in the
corresponding 2D grid used in the vertical equilibrium simulation (here the main fault is modeled as
an internal no-flow boundary).
The right figure shows gravity lines of the top surface
of Johansen, where the cells that correspond to local
maxima of the surface are colored red and cells that
correspond to local minima are colored green.
Transport
h
i
∇k · f (s, ~x )~v + fg (s, ~x ) ~gk(~x ) + ∇pc (s, ~x ) = qco2(~x )
h: height of CO2
H: height of reservoir
s: s = h/H
pt : pressure at the top surface
k: parallel to the top surface
λα: pseudo mobilities
pc : gravity component perpendicular to the
surface
Comparison of Averaged Permeability and Full Permeability
It is possible to do several approximations in the vertical equilibrium model. Here we compare
how averaging of the permeability in the vertical direction influences the accuracy of the VE
solution. The permeability is included in the pseudo mobility function, the difference between
using averaged permeability and the full 3D permeability is shown in the figures below.
2D Example - Comparison of 3D Simulations and VE Simulation
CO2 is very mobile and the flow is usually confined to thin layers, which put severe requirements
on the vertical grid resolution for 3D simulators. Here we compare simulations of a 2D cut of the
Johansen formation using a 3D simulator with different vertical grid resolutions to a simulation
with a vertical equilibrium model. The focus is on the post injection dynamics, or equivalent, the
behavior of the plume far from the injector.
Averaged pseudo mobility functions.
Vertically integrated pseudo mobility functions
Simulation setup
I 2D model: vertical slice from the Johansen
formation (right figure)
I initial plume: placed in stratigraphic trapping
region with an identified spillpoint (right figure)
I simulation time: 2000 years
I z-resolutions in 3D simulator: 5 (original), 10 and
20 cells
I simplifications: no residual trapping, simple relative
permeability
I boundary: no-flow on all edges
I permeability/porosity: 200 mD/20 %
2D cut of Johansen model with initial CO2 distribution
and spillpoint marked with black.
The figures above show the CO2 saturation for VE simulation with averaged permeability (left)
and VE with nonlinear pseudo mobility (right) 500 years after start of injection, injecting 110
years. The averaging of the permeabilities make the flow go too slow along the upper surface
because the permeability of Johansen in general increases towards the top. Hence, the flow is
underestimated where the layer of CO2 is thin relative to the height of the formation.
Time step Requirements for Different Terms of 3D and VE Equations
Time steps [years] for the VE model
Time steps [years] the 3D model
Time
Advection Convection Segregation Parabolic
injection
1
201
8
6
post injection
164
8
3
Time
Advection Convection Segregation
injection
0.1
10
0.04
post injection
12
0.04
The tables show the maximal time step for an explicit method for different parts of the dynamics
of the transport equation. We see that the VE model enables use of longer time steps.
Conclusions
I for grids with low z-resolution, a VE model is more accurate than a 3D model away from
the well.
I large timescale difference between the lateral and vertical movement of a CO2 plume
makes 3D numerical resolution of vertical dynamics intractable for the post injection period
I it is important to preserve the 3D properties of the permeability field in the vertical
equilibrium formulation
I the VE model has a stronger decoupling between the pressure and transport equations in
the post injection scenario compared with the 3D model
These features of VE models can be be utilized to develop fast simulation methods for large scale
CO2 migration problems.
The above figures show the CO2 distribution after 2000 years of simulation computed with the
3D simulator for different z-resolutions and for the vertical equilibrium model (bottom right). A
rather fine z-resolution is needed to get accurate results for the 3D simulator.
References
[1] Eigestad, G., Dahle, H., Hellevang, B., Riis, F., Johansen, W. and Øian, E. [2009] Geological modeling and simulation of CO2
injection in the Johansen formation. Comput. Geosci., 13(4), 435-450, ISSN 1420-0597, doi:10.1007/s10596-009-9153-y.
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