Homogenization and numerical Upscaling Unsaturated flow and two-phase flow Insa Neuweiler Institute of Hydromechanics, University of Stuttgart
Outline Block 1: Introduction and Repetition Homogenization of the Richards equation Multiscale methods for the Richards equation Block 2: Homogenization of the one-dimensional two-phase flow equation Multiscale methods for the two-phase flow problem
Literature Homogenization of the Richards equation: J. Lewandowska and J.-P. Laurent. Moisture transfer in an unsaturated heterogeneous porous medium. Transport in Porous Media, 45:321 345, 2001. J. Lewandowska, A. Szymkiewicz, K. Burzynski, and M. Vauclin. Modeling of unsaturated water flow in double-porosity soils by the homogenization approach. Advances in Water Resources, 27:283 296, 2004. I. Neuweiler and O. Cirpka. Homogenization of richards equation in permeability fields with different connectivities. Water Resources Research, 41(2): doi:10.1029/2004wr003329, 2005. I. Neuweiler and H. Eichel. Effective parameter functions for richards equation in layered porous media. Vadose Zone Journal, accepted for publication, 2006.
Literature Numerical Homogenization of the Richards equation: Y. Efendiev, T. Hou and V. Ginting: Multiscale finite element methods for nonlinear problems and their application, Comm. Math. Sci. 2(4), 553-589, 2004. Homogenization of viscous dominated flow: R. Mauri. Heat and mass transport in nonhomogeneous random velocity fields. Physical Review E, 68:doi: 10.1103/PhysRevE.68.066306, 2003. C. J. VanDuijn, A. Mikelic, and I. S. Pop. Effective equations for twophase flow with trapping on the micro scale. SIAM Journal of Applied Mathematics, 62(5):1531 1568, 2002.
Literature Multiscale methods for viscous dominated two-phase flow: P. Jenny, S.H. Lee and H. A. Tchelepi: Adaptive multiscale finite/volume method for multiphase flow and transport in porous media, Multiscale Model. Simul. 3(1), 50-64, 2004. P. Jenny, S.H. Lee and H. A. Tchelepi: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, Journal of Computational Physics 187, 47-67, 2003. I. Lunati and P. Jenny, Multiscale finite-volume method for compressible multiphase flow in porous media, Journal of Computational Physics 216, 616-636, 2006. L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Research 27(5), 699-708, 1991. Y. Efendiev and L. J. Durlofsky: Numerical modeling of subgrid heterogeneity in two-phase flow simulations, Water Resources Research 38(8), doi: 10.1029/2000WR000190, 2002.
Block 1 Introduction
Introduction and Repetition Homogenization Theory: Starting point: Model with space-dependent parameters Derivation of a model for the spatially averaged parameters Requirements: Space scales in the medium have to be separated Assumptions: Solution has two separated scales, solution is stationary on the small scale Homogenized model is valid for: Fixed typical time scale (usually fast time scales are neglected), fixed dimensionless numbers (problematic for nonlinear problems, where they change), fixed parameter contrasts (also problematic for nonlinear problems) Treated for the elliptic problem
Introduction and Repetition Fields with scale separation: Counter example: Length scale L REV Length scale l Unit cell l????
Introduction and Repetition Derivation of an upscaled, homogeneous model, where the heterogeneity is no longer seen Consider the limit...
Introduction and Repetition Hypothesis: local periodicity l
Procedure: Introduction and Repetition Adaption of a dimensionless form of the equation Scaling of the dimensionless numbers and parameter contrasts with ε Choice of a reference system Separating the space variables into small scale and large scale Expansion of the variables in orders of ε Solving for different orders of ε separately Limit ε >0, lowest order equation (space averaged over the unit cell) is the upscaled equation Effective parameters of the model are defined from cell problems Proof of convergence, existence, (-> mathematicians)
Introduction and Repetition Credo: Homogenization is a very useful method and improves our understanding of processes in heterogeneous media. It can help to develop models for processes in heterogeneous media. However, we make a lot of inherent assumptions and approximations. As long as we are aware of them: FINE!
Block 1 Homogenization of Richards equation
Contaminant River Infiltration Root uptake Unsaturated zone: air + water Scale: 10 0 m Aquifer
Richards equation Mass balance of the water phase assuming air is always at atmospheric pressure Flux: Darcys law Saturation of water Flux Water pressure head Saturated conductivity Relative permeability
Air pressure is constant: Richards equation Capillary pressure head Capillary pressure saturation relation Relative permeability
Capillary pressure saturation relation in porous media Unique relation between water saturation and capillary pressure: (Pressure on the macroscale, averaged over the pore space) Entry pressure (capillary entry pressure of the largest pore)
Relative permeability
Richards equation But also
z above groundwater [m] Homogenization: Richards equation Typical staedy state profiles: Groundwaterlevel: Constant head 20 18 16 14 12 10 8 6 4 2 0 Water-Content Profile in Silt as Function of Recharge [mm/year] Gravity dominated 0 0,1 0,2 0,3 0,4 0,5 θ [-] 20 18 16 14 12 10 8 6 4 2 0 Capillary-Head Profile in Silt as Function of Recharge [mm/year] Gravity dominated 0 2 4 6 8 10 -h c [m] Constant flux hydrostatic 10 100 1000 10000
1. Step: Dimensionless equation Variables with stars are dimensionless Typical values for: Pressure head (needed only for the parameter functions) Conductivity (e.g. geometric mean in 2d) Length scale (there are two, here: large length scale) Time scale Dimensionless number:
Choice of the time scale Time scale due to large scale pressure gradients: Time scale due gravity, large scale: Time scale due to small scale pressure gradients: Time scale due gravity, small scale: In principle a scale separation of time would be necessary. This is not done, usually a large time scale is chosen.
Problem: estimation of pressure head fluctuations Could be due to large scale boundary conditions Could be due to fluctuations of the solution in a heterogeneous medium Could be due a solution in a saturation regime, where the P c -S curve is steep Difficult to estimate a priori
Example: Capillary-Head Profile in Silt as Function of Recharge [mm/year] 20 18 16 Due to small scale 14 fluctuations 12 10 8 6 4 Due to small large scale solution 2 0 0 2 4 6 8 10 -h c [m] hydrostatic 10 100 1000 10000
Choice of the parameter contrast, e.g in: In a two-material composite, what determines the flow is With typical values: Problem: during a transient flow process this ratio can change a lot, as it depends strongly on the saturation. For the estimation of the dimensionless numbers, time scale and parameter contrast we have in principle to choose the saturation dependent values!! Means: We have to have an idea about the solution already.
2. Step: Two-scale Expansion Reference system: Large scale L Separation of scales -> space variable can be made dimensionless with two different length scales -> Two different variables: Transformation of the gradient:
Expansion of the variables in terms of ε
First case: Homogenization: Richards equation -> moderate heterogeneities E.g. lognormally distributed field -> small variances -> Typical pressure head fluctuations are in the same order of magnitude as the large length scale
Example for such a scenario: Slow changes in a scenario such as: 6 Silt with recharge rate: 10 mm / year Material 1 Material 2 z [m] 0 0 0,5 1 1,5 2 2,5 3 3,5 h C [m] Or horizontal flow
Expansion of everything in terms of ε The whole equation with these parameters:
Solving for different orders of ε separately: Order Boundary conditions: Is -periodic is in average constant on Problem corresponds to gravitational free unsaturated flow with periodic boundary conditions without source term -> only solution is that is constant on the small scale (would otherwise be a perpetuum mobile).
This condition is equivalent to capillary equilibrium The typical time scale is too large to appear in this condition -> quasi-instantaneous The pressure head (equal to negative capillary pressure head) is constant over the unit cell -> not influenced by the heterogeneities Implications for the paramter functions: Not constant on the small scale, but instantaneously in equilibrium -> not dynamics
Unit cell Heterogeneous capillary entry pressure head: Water Saturation: Unsaturated conductivity:
Order Coupling of large and small scale -> decoupling via
Cell problem: Boundaries: is periodic. As in single phase flow, as depends only explicitely on,but not indirectly, as is constant in -> Problem boils down to the single phase flow problem with a fixed value of
Upscaled problem: Order Same problem as the Richards equation with effective retention function and effective unsaturated conductivity function
For an isotropic field:
3. Step: Derivation of the effective retention function Choose a value for the pressure head h (0) Calculate the saturation field at this pressure head according to the local retention curves Average the saturation over the unit cell This yields a point on the effective retention curve Repeat the procedure for more values h (0).
Exercise 1: Example: 2d Checkerboardfield Porosity: Unit cell Material 1: Material 2: Calculate the values of the effective retention curve at
Solution: Θ
4. Step: Derivation of the effective unsaturated conductivity Cell problem In each component:
That means: Boundaries: is periodic. This is equivalent to a singlephase pressure problem on the unit cell with a unit pressure gradient in both directions. If the conductivity is isotropic, it is sufficient to solve one problem, the effective unsaturated conductivity is the resulting averaged flux (average pressure gradient is unit). As this is equivalent to single phase flow: Can be estimated with the methods we know for this problem.
Exercise 2: Example: 2d Checkerboardfield Porosity: Unit cell Material 1: Material 2: Calculate the values of the effective conductivity curve at Use the geometric mean as approximation for the effective conductivity at one value
Solution:
Analytic results for log-normally distributed K field and Brooks Corey model assuming Gaussian fields Effective retention curve: (no approximation) From: Neuweiler and Cirpka, WRR, 2005
Analytic results for log-normally distributed K field and Brooks Corey model assuming Gaussian fields Effective permeability curve: (second order in variance of total permeability) From: Neuweiler and Cirpka, WRR, 2005
Second case: Strong parameter contrast in a two-material composite (Lewandowska et al., AWR, 2004) -> Typical pressure head fluctuations are in the same order of magnitude as the large length scale
In material 1: In material 2:
Variable of the upscaled system: Was: Is now: Consider only: Small capacity Large capacity Coupled to the inclusion via the flux boundaries
Order in material 1: -> Capillary equilibrium in material 1 Order in material 2: -> dynamics on the small scale
Order in material 1:. As in the Richards problem before, only this time we consider only material 1. The inclusion sphere of material two is treated as impermeable. -> Effective unsaturated conductivity from the cell problem:
Order in material 1: Before: Averaging over the unit cell cancelled the derivative with respect to the small scale. Now: Leads to coupling to the inclusion
Leads to the averaged equation of order in material 1: Effective parameters as in the usual Richards problem with the cell problem Dynamic effect: Requires a parametrization.
Physical interpretation: Typical time scale of gravity on the large scale is equal to the typical time scale of capillary forces on the small scale. -> Appears as an additional effect
Third case: Gravity dominated flow (Neuweiler and Eichel, Vadose Zone Journal, in press)
Order : No longer capillary equilibrium, equation has to be solved numerically.
Example vertical flow in a layered medium: Local head distribution
Upscaled problem in leading order: -> Effective curves can be approximated reasonably well with the capillary equilibrium approach
Block 1 Numerical homogenization of Richards equation
Numerical Homogenization of Richards equation Aim: Improvement of fine scale FE methods Idea: making use of two scale expansions Richards Equ. -in variational formulation - on the coarse mesh
Numerical Homogenization of Richards equation Macro/micro approach: use a coarse mesh for defining the nodal values of φ( x j ) and a fine mesh for computing the basis functions ϕ j ( x). The problem dimension is that of the coarse mesh Multiscale FEM are defined for non-periodic problems but their quantitative convergence analysis is made in the periodic case. Main references for this example: Hou, Efendiev, Wu, Babuska, Matache, Schwab, Brizzi...
Numerical Homogenization of Richards equation ε [ K u ( φ )( φ + e )] ε = 0 i i z The construction of the basis functions neglects gravity effects!
Comparison of fine scale solution with MsFEM: Numerical Homogenization of Richards equation We follow infiltration fronts moving from top to bottom in an a)isotropic medium b)anisotroic medium