# 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*)

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