# Front Tracking Using a Hybrid Analytical Finite Element Approach for Two-Phase Flow Applied to Supercritical CO_{2} Replacing Brine in a Heterogeneous Reservoir and Caprock

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## Abstract

Predicting fluid replacement by two-phase flow in heterogeneous porous media is of importance for issues such as supercritical CO_{2} sequestration, the integrity of caprocks and the operation of oil water/brine systems. When considering coupled process modelling, the location of the interface is of importance as most of the significant interaction between processes will be happening there. Modelling two-phase flow using grid based techniques presents a problem as the fluid–fluid interface location is approximated across the scale of the discretisation. Adaptive grid methods allow the discretisation to follow the interface through the model, but are computationally expensive and make coupling to other processes (thermal, mechanical and chemical) complicated due to the constant alteration in grid size and effects thereof. Interface tracking methods have been developed that apply sophisticated reconstruction algorithms based on either the ratio of volumes of a fluid in an element (Volume of Fluid Methods) or the advective velocity of the interface throughout the modelling regime (Level set method). In this article, we present an “Analytical Front Tracking” method where a generic analytical solution for two-phase flow is used to “add information” to a finite element model. The location of the front within individual geometrical elements is predicted using the saturation values in the elements and the velocity field of the element. This removes the necessity for grid adaptation, and reduces the need for assumptions as to the shape of the interface as this is predicted by the analytical solution. The method is verified against a standard benchmark solution and then applied to the case of CO_{2} pooling and forcing its way into a heterogeneous caprock, replacing hot brine and eventually breaking through. Finally the method is applied to simulate supercritical CO_{2} injected into a brine saturated heterogeneous reservoir rock leading to significant viscous fingering and developement of preferential flow paths. The results are compared with to a finite volume simulation.

## Keywords

Two-phase flow Hybrid analytical numerical CO_{2}sequestration Caprock integrity Front tracking

## List of Symbols

*A*Area of element normal to the fluid flux (m

^{2})- \({A_{\rm{2D}}^{\prime}}\)
2D local coordinate system

- \({A_{\rm {3D}}^{\rm g}}\)
3D global coordinate system

*A*_{ne}Area of influence assigned to a node in the element (m

^{2})- \({C_{ij}^{\rm e} }\)
Element storage matrix

*C*_{ij}Global storage matrix

- \({D_{t_{n}}}\)
Distance from an origin at time step n (m)

*D*_{front}Distance of a node normal to the saturation front (m)

*D*_{node}Distance of a node to the saturation front (m)

*D*1_{front}Distance of node with

*S*_{max}normal to the saturation front (m)*D*2_{front}Distance of node with

*S*_{mid}normal to the saturation front (m)*f*_{1}Flow rate of phase 1 into or out of a volume (m

^{3}/s)*f*_{1i}Flow rate of phase 1 into or out of a volume discretised to node i (m

^{3}/s)- \({f_{\alpha i}^{\rm e} }\)
Element contribution to flow rate of phase

*α*discretised node i (m^{3}/s)*g*Acceleration due to gravity (m/s

^{2})*i*,*j*,*k*Iteration integers

*k*_{rα}Relative permeability of phase

*α*(-)**k**Intrinsic permeability tensor (m

^{2})*md*Maximum distance predicted by analytical solution from origin (m)

*MN**β*Shape function for the node,

*β*= 1 for node*S*_{max}to*β*= 3 for node*S*_{min}*n*_{β}Node number

*β**ne*Number of elements

*nn*Number of nodes

*P*_{Smax}Global point coordinates for node with maximum saturation in element

*p*_{α}Fluid pressure of phase

*α*(*Pa*)*p*_{1},*p*_{2}Fluid pressure of phases 1 and 2 (Pa)

*p*_{c}Capillary pressure (Pa)

*p*_{w}Fluid pressure of phase of wetting phase (Pa)

*Q*_{total}Total flow rate of all phases (m

^{3}/s)*Q*_{1}Flux of phase 1 (m/s)

*q*_{total}Total Darcy flow velocity of all phases (m/s)

**q**_{α}Fluid velocity vector (m/s)

**q**′Fluid velocity vector in local coordinates (m/s)

*S*_{1}Saturation of fluid phase 1 (-)

*S*_{2}Saturation of fluid phase 2 (-)

*S*_{α}Saturation of fluid phase

*α*(*α*= 1*or*2)(-)*SF*Scaling factor (-)

*SF*(*t*_{e})Scaling factor dependent on

*t*_{e}(-)*S*_{front}Saturation of phase 2 at the top of the saturation front (-)

*S*_{max}Maximum saturation of phase 2 of the element nodes (-)

*S*_{min}Minimum saturation of phase 2 of the element nodes (-)

*S*_{mid}Middle saturation of phase 2 of the element nodes (-)

*S*_{res}Residual saturation (-)

- \({S_{t_{n}}}\)
Saturation at time step n (-)

*S*_{1r},*S*_{2r}Residual saturation of fluid phase 1 and 2 (-)

*sv*_{β}Side vectors of the element in local coordinates

*T*Thickness of an element (m)

_{2D}*T*_{3D}Transformation matrix from 3D global coordinates to coordinates to 2D local coordinates

*t*Time (s)

*t*_{1},*t*_{0}Indicates time step, 0 precedes 1

*t*_{e}Time since the front entered an element (s)

- Δ
*t* Time step length (s)

*V*Integration volume (m

^{3})*V*^{e}Element volume (m

^{3})*V*^{n}Mesh volume mapped to a node (m

^{3})*v*_{α}Advective velocity of fluid

*α*(m/s)*x*Distance (m)

*x*_{n}Normalised distance from origin (-)

*x*_{real}Actual distance from origin (m)

*x*′,*y*′Local coordinate system

*x*_{g},*y*_{g},*z*_{g}Global coordinate system

**u**Displacement tensor (m)

*z*Height above datum (m)

*α*Fluid phase (-), 1 or 2

*β*Number from 1 to 3 unless otherwise stated

- \({\phi}\)
Porosity (-)

*μ*Dynamic viscosity (N/m

^{2}s)*ρ*_{α}Density of phase

*α*(kg/m^{3})- \({\varpi_i}\)
Weighting function of Galerkin finite element scheme (-) for node i

*ω*_{j}Weighting function of Galerkin finite element scheme (-) for node j

*a*_{x},*b*_{x},*c*_{x},*d*_{x},*e*_{x},*f*_{x},*g*_{x}Polynomial coefficients for a function

*y*=*f*(*x*)*a*_{s},*b*_{s},*c*_{s},*d*_{s},*e*_{s},*f*_{s},*g*_{s}Polynomial coefficients for a function

*y*=*f*(*S*)

## References

- Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media, Hydrology Paper, vol. 3, Band 3, Colorado State University, Fort Collins (1964)Google Scholar
- Buckley S.E., Leverett M.C.: Mechanism of fluid displacements in sands. Trans. Am. Inst. Min. Metall. Eng. (TAIME)
**146**, 107–116 (1941)Google Scholar - Chen Y., Durlofsky L.J., Gerritsen M., Wen X.H.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour.
**26**, 1041–1060 (2003)CrossRefGoogle Scholar - Chen Z.X.: Some invariant solutions to two-phase fluid displacement problems including capillary effect. Soc. Pet. Eng. Reserv. Eng.
**3**(2), 691–700 (1988)Google Scholar - Durlofsky L.J., Efendiev Y., Ginting V.: An adaptive local-global multiscale finite volume element method for two-phase flow simulations. Adv. Water Resour.
**30**, 576–588 (2007)CrossRefGoogle Scholar - Fucik R., Mikyška J., Beneš M., Tissa H.: Semianalytical solution for two-phase flow in porous media with a discontinuity. Vadose Zone.
**7**, 1001–1007 (2008)CrossRefGoogle Scholar - Glimm, J., Grove J.W., Li X.L., Zhao N.: Simple front tracking. In: Chen G.-Q., DiBenedetto E. (eds) Contemporary mathematics, vol. 238, pp. 133–149, American Mathematical Society, Providence (1999)Google Scholar
- Helmig, R.: Multiphase flow and transport processes in the subsurface; a contribution to the modeling of hydrosystems, Environmental Engineering. Springer, Berlin, ISBN 3-540-62703-0 (1997)Google Scholar
- Helmig R., Braun C., Manthey S.: Upscaling of two-phase flow processes in heterogeneous porous media: determination of constitutive relationships. IAHS AISH Publ.
**277**, 28–36 (2002)Google Scholar - Helmig R., Huber R.: Comparison of Galerkin-type discretization techniques for two-phase flow in heterogeneous porous media. Adv. Water Resour.
**21**, 697–711 (1998)CrossRefGoogle Scholar - Hoteit H., Firoozabadi A.: Numerical modeling of two-phase flow in heterogeneous permeable media with different capillarity pressures. Adv. Water Resour.
**31**, 56–73 (2008)CrossRefGoogle Scholar - Huang H., Meakin P.: Three dimensional simulation of liquid drop dynamics within unsaturated vertical Hel-Shaw cells. Water Resour. Res.
**44**, 10 (2008)Google Scholar - Huang H., Meakin P., Moubin L.: Computer simulation of two phase-immiscible fluid motion in unsaturated complex fractures using a volume of fluid method. Water Resour. Res.
**41**, 12 (2005)Google Scholar - Ippisch O., Vogel H.J., Bastian P.: Validity limits for the van Genuchten-Mualem model and implications for parameter estimation and numerical simulation. Adv. Water Resour.
**29**, 1780–1789 (2006)CrossRefGoogle Scholar - Istok, J.: Groundwater modeling by the finite element method, American Geophysical Union, 2000 Florida Avenue, NW, Washington, DC 20009 (1989)Google Scholar
- Juanes R., Patzek T.P.: Multiscale-stabilized finite element methods for miscible and immiscible flow in porous media. J. Hydraul. Res. Extra Issue
**42**, 131–140 (2004)Google Scholar - Klieber W., Riviere B.: Adaptive simulations of two phase flow by discontinous Galerkin methods. Comput. Methods Appl. Mech. Eng.
**196**, 404–419 (2006)CrossRefGoogle Scholar - Kolditz O.: Modelling of flow and heat transfer in fractured rock: conceptual model of a 3-D deterministic fracture network. Geothermics
**24**(3), 451–470 (1995)CrossRefGoogle Scholar - Kolditz O.: Non-linear flow in fractured rock. Int. J. Numer. Methods Heat Fluid Flow
**11**, 547–575 (2001)CrossRefGoogle Scholar - Lewis R.W., Schrefler B.A.: The finite element method in the static and dynamic deformation and consolidation of porous media, 2 edn, pp. 492. Wiley, Chichester (1998)Google Scholar
- McDermott C.I., Lodemann M., Ghergut I., Tenzer H., Sauter M., Kolditz O.: Investigation of coupled hydraulic-geomechanical processes at the KTB site: pressure-dependent characteristics of a long-term pump test and elastic interpretation using a geomechanical facies model. Geofluids
**6**, 67–81 (2006)CrossRefGoogle Scholar - McDermott C.I., Tarafder S.A., Schüth C.: Vacuum assisted removal of volatile to semi volatile organic contaminants from water using hollow fiber membrane contactors II: a hybrid numerical-analytical modeling approach. J. Membr. Sci.
**292**, 17–28 (2007)CrossRefGoogle Scholar - McDermott C.I., Walsh R., Mettier R., Kosakowski G., Kolditz O.: Hybrid analytical and finite element numerical modeling of mass and heat transport in fractured rocks with matrix diffusion. Comput Geosci. (2009). doi: 10.1007/s10596-008-9123-9
- McWhorter D.B., Sunada D.K.: Exact integral solutions for two-phase flow. Water Resour. Res.
**26**, 399–413 (1990)CrossRefGoogle Scholar - Meakin P., Tartakovsky A.M.: Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev. Geophys.
**47**, 47 (2009)CrossRefGoogle Scholar - Niessner J., Hassanizadeh S.M.: A model for two-phase flow in porous media including fluid–fluid interfacial area. Water Resour. Res.
**44**, W08439 (2008). doi: 10.1029/2007WR006721 CrossRefGoogle Scholar - Thorenz, C.: Model adaptive simulation of multiphase and density driven flow in fractured and porous media. PhD thesis, Universitaet Hannover, 2001, ISSN 0177-9028 (2001)Google Scholar
- Thorenz C., Kosakowski G., Kolditz O., Berkowitz B.: An experimental and numerical investigation of saltwater movement in partially saturated systems. Water Resour. Res.
**38**(6), 1001 (2002). doi: 10.1029/2001WR000364 CrossRefGoogle Scholar - Unverdi S.O., Tryggvason G.: A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys.
**100**, 25–37 (1992)CrossRefGoogle Scholar - van Duijn C.J., de Neef M.J.: Similarity solution for capillary redistribution of two phases in a porous medium with a single discontinuity. Adv. Water Resour.
**21**, 451–461 (1998)CrossRefGoogle Scholar - van Genuchten M.T.: A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**44**, 892–898 (1980)CrossRefGoogle Scholar - Versteeg H.K., Malalasekera W.: An introduction to computational fluid dynamics, the finite volume method, 2 edn, pp. 503. Pearson Prentice Hall, Englewood cliffs (2007)Google Scholar
- Wang W., Kolditz O.: Object-oriented finite element analysis of thermo-hydro-mechanical (THM) problems in porous media. Int. J. Numer. Methods Eng.
**69**(1), 162–201 (2007)CrossRefGoogle Scholar - Wang W.Q., Kosakowski G., Kolditz O.: A parallel finite element scheme for thermo-hydro-mechanical (THM) coupled problems in porous media. Comput. Geosci.
**35**(8), 1631–1641 (2009)CrossRefGoogle Scholar - Younes, A., Ackerer, P., Delay, F.: Mixed finite elements for solving 2-D diffusion type equations. Rev. Geophys. 48, RG1004/2010 pp. 26, (2010)Google Scholar
- Zienkiewicz O.C., Taylor R.L.: The finite element method, 6 edn, pp. 752. Butterworth Heinemann, Oxford (2005)Google Scholar