Deflated Conjugate Gradient Method for modeling Groundwater Flow Near Faults

1. Deflated Conjugate Gradient Method for modeling Groundwater Flow Near Faults Lennart Ros Deltares & TU Delft Delft January 11 2008: 13.00 www.deltares.com Supervisors: Prof. Dr. Ir. C. Vuik (TU Delft) Dr. M. Genseberger (Deltares) Ir. J. Verkaik (Deltares)

2. Deflated CG method Outline

3. Deflated CG method Outline • Introduction • Deltares • Subsurface, Geohydrology & Faults • MODFLOW • IBRAHYM & problem • Equation, Discretization & Method • Testcase & Observations • Deflation Techniques & First Results • Further Research & Goals

4. Deflated CG method Introduction

5. Introduction Deltares January1st 2008 Deflated CG method

6. Introduction Subsurface • Subsurface is schematized in layers . • Successive sand and clay (aquifers and aquitards) • Assumption: • Horizontal flow in aquifer • Vertical flow in aquitard Deflated CG method

7. Deflated CG method Introduction Geohydrology • Connected pores give a rock permeability. • The driving force for groundwater flow is the difference in height and pressure. • To represent this difference we introduce the concept of hydraulic heads, h [L].

8. Deflated CG method Introduction Faults • Medium Faults are vertical barriers inside aquifers. • Faults do not usually consist of a single, clean • fracture  fault zone. • Different types of faults. • Main property: low permeability. • Large contrasts in parameters.

9. Deflated CG method Introduction All Faults in the IBRAHYM model

10. Deflated CG method Introduction MODFLOW: • MODFLOW is a software package which calculates hydraulic heads. • Developed by theU.S. Geological Survey. • Open-source code: everyone can use and improvethis program • Rectangular grid and uses cell-centered variables. • Quasi-3D model.

11. Deflated CG method Introduction IBRAHYM: • groundwater model developed for several waterboards inLimburg. • large variety of faults insubsoil. • faults cause model to suffer from bad convergence behavior ofsolver. • uses at most 19 layers to model groundwater flow area. • uses grid cells of 25 times 25 meter to get detailed information. • most famous fault is ”de Peelrandbreuk” in Limburg.

12. Deflated CG method Equation, Discretization & Method

13. Deflated CG method Equation, Discretization & Method Governing Equation: Where:

14. Deflated CG method Equation, Discretization & Method Finite Volume Discretization:

15. Deflated CG method Equation, Discretization & Method Finite Volume Discretization: External Sources: Time Discretization: Euler Backwards

16. Deflated CG method Equation, Discretization & Method Discretized Equation Using Finite Volume Method: Where:

17. Equation, Discretization & Method Faults in MODFLOW : When we model a fault in the subsoil we update the hydraulic conductance. Deflated CG method

18. Deflated CG method Equation, Discretization & Method Solution Method: • MODFLOW use stress, time and inner iteration loops • We look at inner iteration loop: • solves a linear system of equations • matrix is symmertic negative definite • Preconditioned Conjugate Gradient Method: • Incomplete Cholesky Decomposition • also: SOR

19. Deflated CG method Testcase & Observations

20. Deflated CG method Testcase & Observations Simple Testcase: • 15 rows, 15 colums, 1 layer • 1 fault on 1/3th of the domain • Cells represent an area of • 25 x 25 meters

21. Deflated CG method Testcase & Observations Observations for simple testcase in Matlab: Preconditioning: Incomplete Cholesky

22. Deflated CG method Testcase & Observations Observations for simple testcase in Matlab:

23. Deflated CG method Testcase & Observations Observations for simple testcase in Matlab: Smallest eigenvalue: 0.00010283296716 Next eigenvalue: 0.04870854847951

24. Deflated CG method Testcase & Observations • Due to the small eigenvalue we have a • slow converging model. • Want to get rid of this eigenvalue • IDEA: USE DEFLATION

25. Deflated CG method Deflation Techniques

26. Deflated CG method Deflation Techniques Basic Idea of Deflation: General linear system of equations: Define: , where: and assume A to be SPD So: and

27. Deflated CG method Deflation Techniques Basic Idea of Deflation: Note we can write: But since: we only need to compute Since we solve the deflated system:

28. Deflated CG method Deflation Techniques Deflation using Eigenvectors: Assume that A has eigenvalues: and we choose the corresponding eigenvectors such that If we now define Then:

29. Deflated CG method Deflation Techniques Alternative Deflation Techniques: • Random Subdomain Deflation • Deflation based on Physics: • Use faults as boundary of domain • Define vectors such that an element next to a fault has value 1 and otherwise 0.

30. Deflation Techniques Results for the test problem: • Deflation using subdomain deflation • 1 domain left of fault • 1 domain right of fault • The eigenvector corresponding to the smallest eigenvalue is in the span of these two vectors. • Eigenvalues of and are almost the same, but the smallest is cancelled now. Deflated CG method

31. Deflated CG method Deflation Techniques Results for the test problem: • Less iterates are needed • Result looks positive

32. Deflated CG method Further Research

33. Deflated CG method Further Research & Goals • Future Research: • How representive is the Matlab model? • Can faults in IBRAHYM be seen as the sum of local faults? • Is deflation always faster, even if we do not have faults? • Future Goals: • Implementing deflation in MODFLOW. • Choose suitable deflation vectors such that: • vectors are easy to construct, • a priori information is used to construct vectors, • choice of vectors is generetic and not problem dependent. • Reduce number of iterations in PCG solver and gain wall-clock times. • Find criterion for when to use deflation for a general problem.