Homogenization and numerical Upscaling. Unsaturated flow and two-phase flow

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1 Homogenization and numerical Upscaling Unsaturated flow and two-phase flow Insa Neuweiler Institute of Hydromechanics, University of Stuttgart

2 Outline Block 1: Introduction and Repetition Homogenization of the Richards equation Multiscale methods for the Richards equation Block 2: Homogenization of the two-phase flow equations Multiscale methods for the two-phase flow problem

3 Block 2 Homogenization of Two-phase flow equation

4 Homogenization Theory: Two-phase flow Remark: The two-phase flow problem is very complex and cannot be treated in general. Upscaling can only be done for a specific problem. E.g.: Displacement Homogeneous model Heterogeneous model Upscaled homogeneous model?

5 Homogenization Theory: Two-phase flow Two-phase flow equations Continuity Incompressibility Darcy Capillary pressure saturation Fractional flow function: Lambda function:

6 Homogenization Theory: Two-phase flow Two-phase flow equations

7 Homogenization Theory: Two-phase flow Typical scales:

8 Homogenization Theory: Two-phase flow Dimensionless continuity equation: Variables and parameters are dimensionless Gravity number Capillary number Flow dominated by viscous forces:

9 Homogenization Theory: Two-phase flow Fractional flow function: advective velocity (S):

10 Homogenization Theory: Two-phase flow Maximal velocity at d 2 f / ds 2 =0 -> Selfsharpening fronts Front velocity is larger than u t S where d 2 f / ds 2 =0 x

11 Homogenization Theory: Two-phase flow Lambda-function:

12 Homogenization Theory: Two-phase flow Dimensionless continuity equation: Nonlinear advection-diffusion -equation -> Upscaled equation: (Approach of Mauri, PRE, 03, for solute transport)

13 Homogenization Theory: Two-phase flow Equation of order Simplest possible solution: For the case that the capillary pressure is not explicitely space dependent. Capillary equilibrium Otherwise: Equation has to be solved numerically, problem: unknown total flow velocity

14 Homogenization Theory: Two-phase flow Remark on the side about selfaveraging processes: E.g. Solute transport: Diffusion distributes over different streamlines -> real mixing in one single realization Diffusion Initial time Intermediate time Longtime (t>l 2 /D) Selfaveraging for large time scales

15 Homogenization Theory: Two-phase flow Two-phase flow: Selfsharpening interfaces heterogeneous entry pressure Averaging by measurements? Initial time Intermediate time Longtime Well Probability statement

16 Homogenization Theory: Two-phase flow By assuming a saturation dependent but spatially homogeneous capillary pressure function we forced a diffusive process onto the system, which makes the saturation homogeneous on the small scale. Therefore we can assume capillary equilibrium. In reality we don t have that!! Argumentation: With the upscaled problem we predict a stochastically averaged saturation, which is smooth. We therefore introduce and artificial mixing process to realize the smooth shape. End of remark on the side.

17 Homogenization Theory: Two-phase flow Average of zeroth order equation -> first part of upscaled equation Viscous term with averaged velocity field

18 Homogenization Theory: Two-phase flow Average of first order equation -> second part of upscaled equation Undisturbed problem + Gravity and capillary term with averaged permeability field + fluctuations Solution for required

19 Homogenization Theory: Two-phase flow Solution for Linear equation, constant parameter part and fluctuating parameter part

20 Homogenization Theory: Two-phase flow Gaussian Green s function with Equivalent diffusion coefficient: Equivalent velocity: Inhomogeneous part: Infinite sum in exceeding order of the fluctuations

21 Homogenization Theory: Two-phase flow Upscaled equation: In 2 d to second order of the fluctuations: Same type of equation with effective intrinsic permeability and additional dispersive term (Langlo and Espedal, AWR, 1994, Neuweiler et al., Transp. Porous Media, 2004)

22 Homogenization Theory: Two-phase flow Growth of the front of the displacing fluid is determined by the dispersion term Stability Selfaveraging behaviour? ASSUMING a stationary velocity field without trends and a finite integral scale Dispersive term with dimensions:

23 Homogenization Theory: Two-phase flow Example where a dispersive approach is problematic: Modelling of solution of CO 2 in deep brine formations Injection well Caprock Brine CO 2 Flow Dissolution (Bielinski, Class and coworkers, IWS Stuttgart)

$Homogenization Theory: Two-phase flow Two-phase two-component process, assuming thermal and chemical equilibrium Mass fraction of dissolved CO 2 in water depends on$

24 Homogenization Theory: Two-phase flow Two-phase two-component process, assuming thermal and chemical equilibrium Mass fraction of dissolved CO 2 in water depends on pressure and temperature, not on the phase saturation CO 2 equilibrium 0 Saturation Concentration of CO 2 in the water On off switch, depending on presence of the second phase

25 Homogenization Theory: Two-phase flow Heterogeneous model Homogenized model Saturation Mass fraction of CO 2 in water

$Saturation Mass fraction of$

26 Homogenization Theory: Two-phase flow Heterogeneous model Homogenized model Saturation Mass fraction of CO 2 in water

27 Block 2 Multiscale methods for the two-phase flow problem (pressure equation)

28 Multiscale methods for two-phase flow Multiscale Finite Volume Method for visc. dominated flow Jenny et al., J Comp Phys, 2003, Jenny et al., Multiscale Model. Simul., 2004, Lunati and Jenny, J Comp Phys, 2006 Pressure equation: Saturation equation:

29 Multiscale methods for two-phase flow Fractional flow function:

30 Multiscale methods for two-phase flow Mobility function:

31 Multiscale methods for two-phase flow Problem compared to the linear elliptic problem Linear problem: Pressure solution on a coarse grid Field properties Pressure and velocity information on the fine grid

32 Multiscale methods for two-phase flow Two-phase problem: Pressure solution on a coarse grid Time i Time i+1 Field properties Pressure and velocity information on the fine grid Propagation of fluid saturation

33 Multiscale methods for two-phase flow Problem compared to the linear elliptic problem In the linear problem the main goal is to have efficient methods to calculate the basis functions, as they are calculated once. In the two-phase problem it is an additional issue that the field changes with time -> one big calculation of the basis functions might be more useful than many fast caculations. The velocity field is crucial for the two-phase flow, as transport of fluid depends on it. Two-phase flow is highly non-linear -> The methods are often quite practical and ad hoc

34 Multiscale methods for two-phase flow Coarse scale: Properties on the coarse scale Reconstruct on the fine scale Velocity on the fine scale Saturation transport on the fine scale

35 Multiscale methods for two-phase flow Solid lines: Coarse mesh Dotted lines: Fine mesh Dashed lines: Dual mesh C D A B 7 8 9

36 Multiscale methods for two-phase flow Definition of basis functions for the dual mesh, the same way as for the multiscale finite element method for the linear elliptic problem (Hou and Wu, J Comp Phys, 1997) Solution of 4 basis functions Dual cell Boundary conditions: on the edges, t is the direction of the edge

37 Multiscale methods for two-phase flow Given the coarse scale solution (pressure on the black nodes), the fine scale pressure field in each dual cell can be constructed with the base functions The fine scale flux field can then be calculated as

38 Multiscale methods for two-phase flow 4 A From the fluxes in the dual cell the fluxes at the interface of the adjacent coarse cells can be calculated. From the averaged fluxes at the interfaces the effective transmissivities for the coarse mesh are calculated.

39 Multiscale methods for two-phase flow With the effective transmissibilities for the coarse mesh: Solve for the coarse scale pressures on the coarse grid. So far: Upscaling As a finite volume approach is used the mass is conserved on the coarse scale.

40 Multiscale methods for two-phase flow Next step: Reconstruction of the small scale velocity field Reconstruction from the dual basis functions is problematic: Flux discontinuous -> divergences give large errors -> Mass conservative reconstruction of the velocity field

41 Multiscale methods for two-phase flow Calculate a set of velocity basis functions K: label of coarse cell, i: index for dual basis functions which reach into k. C D 5 A B Note: The dual cells A, B, C and D are influenced by the surrounding coarse cells (1-9), not only by cell 5. -> We need 9 basis functions for the pressure in cell 5

42 Multiscale methods for two-phase flow Calculation of the basis function as Here: i=5 Calculation of the fine scale fluxes and use them as boundaries for the construction of the basis function -> mass conservative

43 Multiscale methods for two-phase flow Calculation of the fine scale velocity field in coarse cell V5: With the fine scale velocity field -> solution of a time step in the saturation equation With the fine scale velocity field -> solution of a time step in the saturation equation

44 Multiscale methods for two-phase flow Alltogether 9 times number of coarse volumes and 4 times number of dual volumes elliptic problems have to be solved. At initial time the whole number is calculated. After that, the fine scale velocity field and the transmissibilities are only recalculated in the regions of interest (at interfaces etc., where mobilities changed).

45 Block 2 Numerical upscaling for viscous dominated twophase flow (saturation equation)

46 Multiscale methods for two-phase flow II Numerical upscaling of the viscous dominated twophase flow problem (Efendiev and Durlofsky, WRR 02, Durlofsky, WRR, 2001) Viscous dominated flow problem without gravity: Saturation equation: Pressure equation:

47 Multiscale methods for two-phase flow II Comparison to before: Pressure solution on a coarse grid Time i Time i+1 Field properties Pressure and velocity information on the fine grid Propagation of fluid saturation

48 Multiscale methods for two-phase flow II This time: Information about fluctuations Pressure solution on a coarse grid Time i Propagation of fluid saturation Time i+1 Field properties

49 Multiscale methods for two-phase flow II With homogenization theory we have derived an upscaled form of the equation (setting the gravity number and the capillary number to zero): Derivation by spatial averaging (no homogenization) yields: Angular brackets: space average

50 Multiscale methods for two-phase flow II Approximation (tested numerically): Coarse grid problem to solve:

51 Multiscale methods for two-phase flow II Numerical scheme to solve for Coarse grid: Fine grid:

52 Multiscale methods for two-phase flow II Discretized from (Finite Volume Scheme): -> need parametrization for Best: without solving a small scale problem

53 Multiscale methods for two-phase flow II We had: However: Approximation for a periodic field or for an average over an REV. Here: Any field should be possible. First step: Approximate solution for

54 Multiscale methods for two-phase flow II Perturbation expansion

55 Multiscale methods for two-phase flow II Calculate the function along coarse scale trajectories: Trajectories have to be updated after each time step!

56 Multiscale methods for two-phase flow II Approximate solution to second order in fluctuations, using lots of tedious algebra: Coarse scale function:

57 Multiscale methods for two-phase flow II Involves the two-point correlation function Computationally very expensive, requires the fine scale velocity field. As the approach should be applicable to fields with long-range correlations: The fine scale solution could not be decoupled into cell problems. Could be done if the term is needed only in a small area, but even then requires keeping track of the trajectories -> Parametrisation of the term

58 Multiscale methods for two-phase flow II Approach Efendiev and Durlofski, WRR 02: Field with correlation length in the range of the grid size: Compare: Averaging over many correlation lengths Note: not dependend on the correlation length <-> dispersion effect Variance of log saturated conductivity

59 Multiscale methods for two-phase flow II Approximation: Change of averaged saturation with time is small Approximation: Expansion in to eliminate the local effects: Length of the trajectory

60 Multiscale methods for two-phase flow II Problem: Goal: Estimation without solving the global fine scale model. Use the fine scale velocities from the local cell problem which has to be solved to calculate the coarse grid permeability for the pressure equation (time independent). Rescale, such that the gradient of the local problem is equal to that of the global problem (also time independent). Use these rescaled velocities as For the coarse scale saturation equation use

61 Multiscale methods for two-phase flow II Estimation of the length of the trajectory: What is needed after the initial local fine grid solution: Pressure solution on the coarse grid Saturation solution on the coarse grid Trajectories on the coarse grid

62 Multiscale methods for two-phase flow II Procedure Calculate the local single phase flow fields Solve the coarse scale pressure equation using effective permeabilities after Durlofsky, WRR, 2001 Calculate the length of the trajectories and the fluctuations of the fluxes Estimate the coarse scale dispersion coefficient Calculate one time step in the coarse scale saturation equation

Toward reservoir simulation on geological grid models

1 Toward reservoir simulation on geological grid models JØRG E. AARNES and KNUT ANDREAS LIE SINTEF ICT, Dept. of Applied Mathematics, P.O. Box 124 Blindern, NO-0314 Oslo, Norway Abstract We present a reservoir