Coupled Surface and Saturated/Unsaturated

Ground Water Flow in Heterogeneous Media

Heiko Berninger∗, Ralf Kornhuber, and Oliver Sander

Université de Genève∗, Freie Universität Berlin and Matheon

Multiscale Simulation & Analysis in Energy and the Environment,

Radon Special Semester 2011

Outline

Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10

• solver friendly finite element discretization: linear efficiency and robustness

Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,...

• nonlinear domain decomposition: linear efficiency and robustness

Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11

• continuous of mass flow and discontinuous pressure (clogging)

• mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09, ...

• Steklov–Poincaré formulation and substructuring

• numerical experiments

All computations made with Dune Bastian, Gräser, Sander, ...

Outline

Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10

• solver friendly finite element discretization: linear efficiency and robustness

Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,...

• nonlinear domain decomposition: linear efficiency and robustness

Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11

• continuous of mass flow and discontinuous pressure (clogging)

• mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09

• Steklov–Poincaré formulation and substructuring

• numerical experiments

All computations made with Dune Bastian, Gräser, Sander, ...

Runoff Generation for Lowland Areas

γE

γSP

saturated

γD

h

γSP

γE

vadose

mathematical challenges:

• saturated/unsaturated ground water flow: non-smooth degenerate pdes l Signorini-type bc (seepage face)

• coupling subsurface and surface water: heterogeneous domain decomposition

• uncertain parameters (permeability, ...): stochastic pdes Forster & Kh. 10, Forster 11

Saturated/Unsaturated Groundwater Flow: Richards Equation

∂

∂t θ(p) + div v(x, p) = 0 , v(x, p) = −K(x) kr(θ(p))∇(p− ̺gz)

equations of state: (Brooks & Corey, Burdine)

θ(p) =

{

θm + (θM − θm) (

p pb

)−ε

(p ≤ pb)

θM (p ≥ pb) kr(θ) =

(

θ − θm θM − θm

)3+2ε

−5 −4 −3 −2 −1 0 1 2 3 4 5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

saturation vs. pressure: p 7→ θ(p)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

relative permeability vs. saturation: θ 7→ kr(θ)

Saturated/Unsaturated Groundwater Flow: Richards Equation

∂

∂t θ(p) + div v(x, p) = 0 , v(x, p) = −K(x) kr(θ(p))∇(p− ̺gz)

equations of state: (Brooks & Corey, Burdine)

θ(p) =

{

θm + (θM − θm) (

p pb

)−ε

(p ≤ pb)

θM (p ≥ pb) kr(θ) =

(

θ − θm θM − θm

)3+2ε

−5 −4 −3 −2 −1 0 1 2 3 4 5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

saturation vs. pressure: p 7→ θ(p)

pb, ε → 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

relative permeability vs. saturation: θ 7→ kr(θ)

Homogeneous Equations of State Alt & Luckhaus 83, Otto 97

Kirchhoff Transformation: κ(p) :=

∫ p

0

kr(θ(q)) dq =⇒ ∇κ(p) = kr(θ(p))∇p

−3 −1 0 3 −4/3

−1

0

3

generalized pressure: u := κ(p)

−2 −4/3 −1 0 2 0

0.21

0.95

1

M(u) := θ(κ−1(u))

separation of ill–conditioning and numerical solution: semilinear variational equation

u(t) ∈ H10(Ω) :

∫

Ω

M(u)t v dx+

∫

Ω

(

K∇u−kr(M(u))̺gez )

∇v dx = 0 ∀v ∈ H10(Ω)

Solver-Friendly Discretization

lumped implicit/explicit-upwind discretization in time, finite elements Sj ⊂ H 1 0(Ω):

un+1j ∈ Sj :

∫

Ω

ISj(M(u n+1 j ) v) dx+

∫

Ω

τK∇un+1j ∇v dx = ℓunj (v) ∀v ∈ Sj

equivalent convex minimization problem

uj ∈ Sj : J (uj) + φj(uj) ≤ J (v) + φj(v) ∀v ∈ Sj

quadratic energy J (v) = 12 (τK∇v,∇v) − ℓunj (v)

convex, l.s.c., proper functional φj(v) = ∑

Φ(v(p)) hp = ∫

Ω ISj(Φ(v)) dx

nonlinear convex function Φ : R → R ∪ {+∞} with ∂Φ = M

Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gräser & Kh. 07

• given iterate uνj

• fine grid smoothing: − successive 1D minimization of J + φj in direction of nodal basis functions of Sj:

1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ūνj

• coarse grid correction:

− Newton linearization of M(u) at ūνj − constrain corrections to smooth regime of M

1 step of damped MMG → new iterate uν+1j

=⇒ (J + φj)(u ν+1 j ) ≤ (J + φj)(u

ν j )

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

� � �

� � �

ūνj (p) u

M(u)

Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ −pb, ε ≥ 0 (robustness).

Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gräser & Kh. 07

• given iterate uνj

• fine grid smoothing: − successive 1D minimization of J + φj in direction of nodal basis functions of Sj:

1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ūνj

• coarse grid correction:

− Newton linearization of M(u) at ūνj − constrain corrections to smooth regime of M

1 step of damped MMG → new iterate uν+1j

=⇒ (J + φj)(u ν+1 j ) ≤ (J + φj)(u

ν j )

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

� � �

� � �

ūνj (p) u

M(u)

Signorini condition

0

Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ −pb, ε ≥ 0 (robustness).

Solver–Friendly Discretization

• Kirchhoff transformation

• discretization: discrete minimization problem for uj

• algebraic solution by monotone multigrid (descent method!)

• inverse discrete Kirchhoff transformation

Solver–Friendly Discretization

• Kirchhoff transformation: u = κ(p)

• discretization: discrete minimization problem for uj

• algebraic solution by monotone multigrid (descent method!)

• inverse discrete Kirchhoff transformation

Solver–Friendly Discretization

• Kirchhoff transformation: u = κ(p)

• discretization: discrete minimization problem for uj

• algebraic solution by monotone multigrid (descent method!)

• inverse discrete Kirchhoff transformation

Solver–Friendly Discretization

• Kirchhoff transformation: u = κ(p)

• discretization: discrete minimization problem for uj

• algebraic solution by monotone multigrid

• inverse discrete Kirchhoff transformation

Solver–Friendly Discretization

• Kirchhoff transformation: u = κ(p)

• discretization: discrete minimization problem for uj

• algebraic solution by monotone multigrid

• discrete inverse Kirchhoff transformation: pj = Ij(κ −1(uj))

Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10

reinterpretation in terms of physical variables:

inexact finite element discretization with special quadrature points

convergence properties:

generalized variables: uj → u and M(uj) → M(u) in H 1(Ω)

physical variables pj → p and θj(pj) = Ij (M(uj)) → θ(p) in L 2(Ω)

Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10

reinterpretation in terms of physical variables:

inexact finite element discretization with special quadrature points

convergence properties:

generalized variables: uj → u and M(uj) → M(u) in H 1(Ω)

physical variables pj → p and θj(pj) = Ij (M(uj)) → θ(p) in L 2(Ω)

Experimental Order of L2-Convergence

model problem: time discretized Richards equation without g