MAT254 Flow in Porous Media

MAT254

Flow in Porous Media

Florin A. Radu

Department of Mathematics, University of Bergen, P. O. Box 7800, N-5020 Bergen, Norway

mailto:[email protected]

Applications

• Water and soil pollution.

• Concrete carbonation.

• Collagen implants.

• CO₂ capture and sequestration.

• Enhanced oil recovery.

• . . .

Motivation

Motivation

Concentration [mg/l] profiles after T = 5 [years]

Xylene Oxygen Biomass

Motivation

Physical Problem

System of ODE's and/or PDE's

Sequence of Nonlinear Systems

Sequence of Linear Systems

Visualization, Prognose

Mathematical Modeling

Domain Definition, BC, Upscaling

Discretization (time and space)

Linearization Scheme

Linear Solvers

A(x) = b

Jx = b

Motivation

What can go wrong?

• Mathematical Model

•• Wrong model

•• Wrong parameters (Upscalling)

Motivation

What can go wrong?

• Numerics

•• Wrong discretization: method, elements (higher order?)

•• Are the time step and the mesh size small enough? Too small?

•• The programm is too slowly? Wrong (linear or nonlinear) solvers?

•• No convergence? Why? What to do?

AIM of MAT 254

To give an overview on mathematical models and

numerical methods for flow and transport in porous media.

Content

• From modelling to prognose: one example:

•• Contaminant transport in porous media.

• Criteria to choose a simulation software.

• Lecture outgoing.

Solute transport in porous media

Solute transport in porous media

Film: A civil action, with John Travolta, 1988.

To describe

• Water flow, including the unsaturated zone near the subsurface.

• Advective and dispersive transport of multiple contaminants.

• Non-equilibrium and equilibrium sorption.

• Biodegradation.

(Un-)saturated Groundwater Flow

An appropiate model for the water flow in porous media is Richards’ equation (here in the pressure formulation):

∂_tΘ(ψ) − ∇ · K(ψ)∇(ψ + z) = 0 in J × Ω

Water content: θ(ψ) ∈ [0,1]

Pressure head: ψ

Unsaturated hydraulic conductivity: K(ψ) Height against the gravitational direction: z Time interval: J = (0,T)

Domain: Ω in R^d (d = 1,2 or 3)

(Un-)saturated Groundwater Flow

The equation results from

• mass conservation

∂_tΘ(ψ) + ∇ · q = 0

• Darcy’s law

q = −K(ψ)∇(ψ + z)

(Un-)saturated Groundwater Flow

• the soil-water retention Θ(ψ),

• the unsaturated hydraulic conductivity K(Θ),

Gardner Θ_exp(ψ) = Θ_r + (Θ_s − Θ_r)e^αψ Kexp(ψ) = Kse^αψ

Haverkamp ΘHav(ψ) = Θr + ^(Θ_1+(αψ)^s−Θrn⁾

K_Hav(ψ) = _1+(βψ)^Ks _p

van Genuchten- Θ_vG(ψ) = Θ_r + (Θ_s − Θ_r)Φ(ψ) Mualem Φ(ψ) = _(1+(αψ)¹ n)^m , m = 1 − _n¹

K_vG(ψ) = K_sp

Φ(ψ)(1 − (1 − Φ(ψ)^m1 )^m)²

(Un-)saturated Groundwater Flow

Discretization

• Space: mixed hybrid finite element method.

– Lowest order finite elements of Raviart-Thomas type for flux approximation.

– Piecewise constant elements on triangles for pressure.

– Piecewise constant elements on edges for Lagrange multipliers.

• Time: backward Euler.

• Newton method for the nonlinear systems

• Multigrid for the linear systems

Solute transport

General model with multicomponent organic transport and biodegradation

N mobile species, M immobile species

∂t(Θci) + ρb∂tsi − ∇ · (Di∇c_i − qci) = −R_i ,

∂tsi = ki(φ(ci) − si) or si = φ(ci), i ∈ 1, ..., N

∂tci + k_dici =

1 − γi c_i c_imax

Ri, i ∈ N + 1, ..., N + M.

c_i concentration of the species, s_i concentration of the absorbed species , D_i diffusion coefficient, ρ_b bulk density, R_i degradation rate, φ sorption isotherm, k_di death rate, cimax a maximal realistic concentration, γi ∈ {0,1}.

Solute transport

Boundary Conditions

c_i = g_Di on J × Γ_Di,

−D_i∇c_i · n = g_{N i} on J × Γ_{N i},

−D_i∇c_i · n + c_iq · n

| {z }

q_i·n

= g_{F i} on J × Γ_{F i},

Remark. Γ_Di,Γ_{N i},Γ_{F i} are species depending.

Solute transport

Discretization

• Space: mixed hybrid finite element method.

– Lowest order finite elements of Raviart-Thomas type for the flux approximations.

– Piecewise constant elements on triangles for the concentrations.

– Piecewise constant elements on edges for the Lagrange multipliers.

• Time: backward Euler.

• Newton method for the nonlinear systems

• Multigrid for the linear systems

Schematic of the solution algorithm at time t = t_n

The Richards’

equation

Newton solver

Θⁿ_h, qⁿ_h

?

The equations for the species:

transport, sorption, biodegradation

fully coupled Newton solver

Multicomponent organic transport and biodegradation

Implementation

The algorithm is implemented in the software package:

UG, version 3.8

P. Bastian et al., UG-a flexible toolbox for solving partial differential equation, Comput. Visualization in Science 1, pp. 27-40, 1997.

Benzene Biodegradation

???

...

Ω₁

Ω₂ Γ₁

(0,0) (2,0)

(0,3) (2,3)

•Water Flow : F3 days rain, 4 days dry

Fvan Genuchten-Mualem Model

•Biodegradation : F2 mobile species, 1 biomass

Fno sorption

FMonod Model

Benzene concentration at T = 30, 60, 90, 120, 150, 160 days

Oxygen concentration at T = 30, 60, 90, 120, 150, 160 days

Biomass concentration at T = 30, 60, 90, 120, 150, 160 days

Real case study: Xylene Degradation

•Water Flow : F stationary flow

F variable permeability

•Biodegradation : F 2 mobile species, 1 biomass

F without sorption

F Monod Model

Xylene degradation: variable permeability

Domain Pressure Profile Flux

Concentration [mg/l] profiles at T = 1 [year]

(without additional delivery of contaminant)

Xylene Oxygen Biomass

Concentration [mg/l] profiles at T = 3 [years]

(without additional delivery of contaminant)

Xylene Oxygen Biomass

Concentration [mg/l] profiles at T = 5 [years]

(without additional delivery of contaminant)

Xylene Oxygen Biomass

Are the results always correct?

• Wrong model (parameters)?

• Enough accuracy?

• Wrong discretization?

Wrong parameters?

A first example

∂

∂tc + v · ∇c = D∆c in Ω_p, D∇c · n = R(c) on Γ_s, c = c₀ on Γⁱ_f,

∇c · n = 0 on Γ^o_f .

where

R(c) = − k_max c K_m + c

Wrong parameters?

Wrong parameters?

• Simulation of contaminant transport using local and upscaled parameters in a reaction limited regime (D >> R).

Wrong parameters?

Wrong parameters?

• Simulation of contaminant transport using local and upscaled parameters in a diffusion limited regime (D << R).

Example 2: not accurate enough?

Accuracy

• Example from M. Bause and P. Knabner, Numerical simulation of contaminant biodegradation by higher order methods and

Example 2: not accurate enough?

• Simulation performed with linear elements on an uniform grid (left) and quadratic elements on a pre-refined grid (right).

• Example from M. Bause and P. Knabner, Numerical simulation of contaminant biodegradation by higher order methods and

adaptive time stepping, Comput. Visualiz. Sci. 7 (2004).

Example 3: Wrong Discretization?

Wrong Discretization?

Richy 1D (left) versus Feflow (right): pressure

Example 3: Wrong Discretization?

Wrong Discretization?

Richy 1D (left) versus Feflow (right): water flux

Summary

Numerical Simulation

• Mathematical Model

• Discretization (Time and Space)

• Solver for nonlinear systems

• Solver for linear systems

• Visualization tool

• Adaptivity

• Parallelization

Summary

Criteria to choose a software tool

• How comprehensive is the implemented mathematical model? How difficult is to implement a problem

(sophisticated geometry, boundary conditions,...)?

• How are the equations discretized?

• What solvers for the nonlinear systems are available?

• What solvers for the linear systems are available?

• Can I compute adaptively?

• Works on parallel computers (clusters)?

Lecture organization

• Lectures: Mo., 12:15 and Th., 8:15, Room: Audi Π i fjerde.

• Exam (oral): 21.11.2013 ?

• No lecture on 30.09.2013 (conference).

• At the beginning we have only lectures, after the first chapter we start also exercises (simulation problems).

Others

• Homepage: http://people.uib.no/fra001/Radu.html

• EMAIL: [email protected]

• Knowledge requirements: Vector Calculus, Linear Algebra.

• Software: Matlab or Octave.

Lecture structure

1. Single phase flow in porous media.

– Darcy’s law. Hydraulic head. Hydraulic conductivity and permeability.

– Conservation laws and governing equations.

– Energy conservation.

– Model simplifications. Analytical solutions. Reduction of dimensionality.

– Numerical methods.

2. Two-phase flow in porous media.

– Two-phase flow.

– Richards’ equation.

– Non-standard models.

– Buckley-Leverett solution.

– Numerical methods.

3. Solute transport in porous media.

– One-component transport.

– Multicomponent reactive transport.

– Numerical methods.

4 Homogenization.

– Formal derivation of Darcy’s law by homogenization.

Related lectures

• MAT 160, MAT 260

• Numerical methods for PDEs: FEM, FV, MPFA

• MAT 255

Bibliography

• Nordbotten, J. M. and M. A. Celia, Geological Storage of CO2:

Modeling Approaches for Large-Scale Simulation, 2011, John Wiley and Sons, Inc.

• Bear, J. F and Y. Bachmat. Introduction in Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers,

Boston, 1991.

• Helmig, R., Multiphase flow and transport processes in the subsurface. Springer.

• Chen, Z., G. Huan and Y. Ma, Computational methods for

multiphase flow in porous media, SIAM, Computational science and engineering.

Master thesis

• Mathematical modeling

• Discretization schemes, implementation or/and analysis

• Topics from envirnomental sciences, biology, CO₂ storage...

Hopefully you will enjoy it!!!