Direct Numerical Simulation of Transport Phenomena on Pore-space Images

1. Direct Numerical Simulation of Transport Phenomena on Pore-space Images Peyman Mostaghimi, Martin Blunt, Branko Bijeljic 11th January 2010, Pore-scale project meeting

2. Flow at pore scale • In petroleum science and engineering, scales of interest may vary from molecular level to a mega level. • The pore scale is of the order of a typical pore which is in the range few microns. Modelling fluid flow at the pore scale can provide a predictive tool for estimating rock and flow properties at larger scales. • One of the most used ways to capture the • morphology of a porous medium as the • main input for pore scale modelling is • micro-CT imaging.

3. Motivation • Network modelling – the representation of the pore space by an equivalent representation of pores and throats – has been successful: we now understand trends in recovery with wettability and can predict single and multi-phase properties. • However…….the extraction of networks involves ambiguities and there are some cases where the method does not work so well. • Now have direct three-dimensional imaging of pore spaces. • Why not simulate multiphase flow directly on these images?

4. Micro-CT imaging AND DIRECT SIMULATION Post processing Micro-CT images, a matrix can be generated for a core which shows whether there is a solid inside the voxel or a pore. Zero means that voxel is a pore and one means it is a solid phase. • Two methods to simulate fluid flow in porous media directly without the need for simplified geometries: • - the lattice Boltzmann method (Edo) • - conventional computational fluid dynamics algorithms based on the relevant flow and conservation equations.

5. Governing equations Conservation of mass: Navier-Stokes equation: Steady-state and incompressible flow:

6. Dimensionless analysis The dimensionless steady-state Navier-Stokes equation: Reynolds number for flow in porous media: Stokes Equation :

7. FORMULATION u Equation (u: velocity in x-direction) v Equation (v: velocity in y-direction) w Equation (w: velocity in z-direction) p Equation

8. Gridding • Marker-and-cell grid: • Existence of solid phase in each grid causes six velocity components be zero in the 3 dimensional models.

9. Discretized Form The momentum equations can be rewritten as:

10. SIMPLE algorithm • The SIMPLE (Semi-Implicit Method for pressure-Linked Equations) Algorithm: • 1. Guess the pressure field p* • 2. Solve the momentum equations to obtain u*,v*,w* by algebraic mutigrid solver • 3. Solve the p’ equation (The pressure-correction equation) by algebraic mutigrid solver • 4. p=p*+p’ • 5. Calculate u, v, w from their starred values using the • velocity-correction equations • 6. Solve the discretization equation for other variables, such as • temperature, concentration, and turbulence quantities. • 7. Treat the corrected pressure p as a new guessed pressure p*, • return to step 2, and repeat the whole procedure until a • converged solution is obtained. Storing matrices in CRS and AMG for solving all linear systems of equations.

11. Boundary Condition

12. . . . Boundary Condition

13. COMPARISON OF THE Three METHODS FOR BC FOR FLOW BETWEEN TWO INFINITE PARALEL PLATES • first method second method third method

14. VELOCITY PROFILE FOR FLOW BETWEEN TWO INFINITE PARALEL PLATES • We see non-zero velocity even for one block within the channel and for more than one we see agreement to within machine accuracy with the analytical solution.

15. Lid-driven Cavity

16. DISPERSION MODELLING • When a miscible fluid is injected in a flowing fluid in a saturated porous media, it will spread by various mechanisms including advection and diffusion. • In brief, dispersion is the spread or mixing of flowing fluids due to all these mechanisms. • To model advection term we use stream tracing algorithm and for diffusion we apply random walk method.

17. STREAMLINE TRACING • Interpolation to estimate the velocity vectors at a point within the grid block The time of flight: • The coordinates of exit location:

18. Diffusion • Random walking method for both advection and diffusion: advection diffusion Random walking method just for diffusion part of flow :

19. . . . Diffusion • Gridding: • Resolution:

20. . . . Particle tracking • Gridding: • Resolution: • Sandpack LV60B

21. . . . Particle tracking Gridding: Resolution: Sandpack LV60B

22. . . . Particle tracking • Gridding: • Resolution: • Sandpack LV60B

23. DISPERSION COEFFIEICENT • The average of positions of particles: • Variance of X can be calculated: • And the longitudinal dispersion coefficient: • For showing the importance of diffusion, dispersion is modelled for a range Peclet number: (Bijeljic et al. 2004)

24. Multiphase flow at the pore scale • Having the interface at different saturations, the flow of each phase can be modelled by the Stokes solver and the relative permeability can be predicted. • Also for reactive transport (Branko), the code can be used to simulate the flow at each time step. Courtesy of MasaProdanovic

25. Many thanks for your attention