Deflated Conjugate Gradient Method for modeling Groundwater Flow

Deflated Conjugate Gradient Method for modeling Groundwater Flow Near Faults Lennart Ros Deltares & TU Delft datum 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)

Outline Deflated CG method 11/23/2020 2

Outline § Introduction Deltares Subsurface, Geohydrology & Faults MODFLOW IBRAHYM & problem § Equation, Discretization & Method § Testcase & Observations § Deflation Techniques & First Results § Further Research & Goals Deflated CG method 11/23/2020 3

Introduction Deflated CG method 11/23/2020 4

Introduction Deltares January 1 st 2008 Deflated CG method 11/23/2020 5

Introduction Subsurface § Subsurface is schematized in layers. § Successive sand clay (aquifers and aquitards) § Assumption: • Horizontal flow in aquifer • Vertical flow in aquitard Deflated CG method 11/23/2020 6

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]. Deflated CG method 11/23/2020 7

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. Deflated CG method 11/23/2020 8

Introduction All Faults in the IBRAHYM model Deflated CG method 11/23/2020 9

Introduction MODFLOW: § MODFLOW is a software package which calculates hydraulic heads. § Developed by the U. S. Geological Survey. § Open-source code: everyone can use and improve this program § Rectangular grid and uses cellcentered variables. § Quasi-3 D model. Deflated CG method 11/23/2020 10

Introduction IBRAHYM: § groundwater model developed for several waterboards in Limburg. § large variety of faults in subsoil. § faults cause model to suffer from bad convergence behavior of solver. § 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. Deflated CG method 11/23/2020 11

Equation, Discretization & Method Deflated CG method 11/23/2020 12

Equation, Discretization & Method Governing Equation: hydraulic conductivities along x, y, and z coordinate axes [LT-1], Where: Deflated CG method h potentiometric head [L], W volumetric flux per unit volume representing sources and sinks of water [T-1], Ss specific storage of porous material [L-1], t Time [T] 11/23/2020 13

Equation, Discretization & Method Finite Volume Discretization: Deflated CG method 11/23/2020 14

Equation, Discretization & Method Finite Volume Discretization: External Sources: Time Discretization: Euler Backwards Deflated CG method 11/23/2020 15

Equation, Discretization & Method Discretized Equation Using Finite Volume Method: Where: Deflated CG method 11/23/2020 16

Equation, Discretization & Method Faults in MODFLOW : When we model a fault in the subsoil we update the hydraulic conductance. Deflated CG method 11/23/2020 17

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 Deflated CG method 11/23/2020 18

Testcase & Observations Deflated CG method 11/23/2020 19

Testcase & Observations Simple Testcase: § 15 rows, 15 colums, 1 layer § 1 fault on 1/3 th of the domain § Cells represent an area of 25 x 25 meters Deflated CG method 11/23/2020 20

Testcase & Observations for simple testcase in Matlab: Preconditioning: Incomplete Cholesky Deflated CG method 11/23/2020 21

Testcase & Observations for simple testcase in Matlab: Deflated CG method 11/23/2020 22

Testcase & Observations for simple testcase in Matlab: Smallest eigenvalue: 0. 00010283296716 Next eigenvalue: Deflated CG method 0. 04870854847951 11/23/2020 23

Testcase & Observations § Due to the small eigenvalue we have a slow converging model. § Want to get rid of this eigenvalue § IDEA: USE DEFLATION Deflated CG method 11/23/2020 24

Deflation Techniques Deflated CG method 11/23/2020 25

Deflation Techniques Basic Idea of Deflation: General linear system of equations: Define: , where: and assume A to be SPD So: and Deflated CG method 11/23/2020 26

Deflation Techniques Basic Idea of Deflation: Note we can write: But since: we only need to compute Since Deflated CG method we solve the deflated system: 11/23/2020 27

Deflation Techniques Deflation using Eigenvectors: Assume that A has eigenvalues: and we choose the corresponding eigenvectors If we now define Then: Deflated CG method 11/23/2020 28 such that

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. Deflated CG method 11/23/2020 29

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 11/23/2020 30

Deflation Techniques Results for the test problem: § Less iterates are needed § Result looks positive Deflated CG method 11/23/2020 31

Further Research Deflated CG method 11/23/2020 32

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. Deflated CG method 11/23/2020 33