Modelling Homogenization Mathematical tools

Porous Media and Plasticity - Homogenization for

Equations with Hysteresis

Ben Schweizer

TU Dortmund

Banff, 30.8.2010

Modelling Homogenization Mathematical tools

Modelling subsurface flow

Describe the flow of water in unsaturated porous media

Variables

domain RN

saturation s : (0, T ) Rpressure p : (0, T ) Rvelocity v : (0, T ) RN

Equations

Darcy law v = k(s)pconservation law ts+ v = 0some relation between p and s

We combine these to the evolution equation

ts = (k(s)p).

Modelling Homogenization Mathematical tools

Microscopic Analysis I

relation p s depends on pores

Tube-Model

2d

R

Fluid 1 Fluid 2

d radius of the tube contact angle

H = R1 curvature surface tension

p = H = F (d)

Variable radius

At a given saturation s, pores ofradius d0(s) must be filled.

This needs the pressure

p = pc(s)

Richards equation:

ts = (k(s)[pc(s)])

Modelling Homogenization Mathematical tools

Microscopic Analysis II: Play-type capillary hysteresis

In reality, the radius of the tubesis oscillatory.

This implies that an interval ofpressures is allowed for onesaturation,

p [p1, p2] =: pc,0(s) + [, ].

with the rule: upper/lower valuefor increasing/decreasingsaturation

Resulting model

p pc,0(s) + sign(ts)

with the multi-valued sign-function,sign(0) = [1, 1].

+1

1

S., A stochastic model for fronts in porous

media, Ann. Mat. Pura Appl. 2005

S., Laws for the capillary pressure ...,

SIAM J. Math. Analysis, 2005

Modelling Homogenization Mathematical tools

Existence results for the hysteresis model

p pc,0(s) + sign(ts)

Hysteresis relation p to s

s

p (s)c,0p

Theorem

Let T > 0 and let initial data s0 becompatible. Then there exists a weak

solution of the differential inclusion

ts = p withp as+ b+ sign(ts)

Method of proof: Regularization.

Discretize h > 0

Approximate = sign1 > 0

(.)

tsh, = ([p

h, ash, b]/)

(h)ph, = ([p

h, ash, b]/)

Modelling Homogenization Mathematical tools

Plasticity equations

Variables

domain RN

displacement u : (0, T ) RN

strain : (0, T ) RNN

stress : (0, T ) RNN

Equations

conservation law 2t u = + fstrain relation = 12 (u+ (u)

)a relation between and

Linear elasticity

One uses the simplest choice,

= A .

The observation in plasticity is: the

material flowing beyond some stress

Melan-Prager

One-dimensional relation

sign(t t).

Modelling Homogenization Mathematical tools

Hysteresis problems of plasticity and hydrology

Wave equation with hysteresis

2t u = x + f

xu = +

sign(t).

, , , are parameters.

Richards equation with hysteresis

ts = p

p as+ b+ sign(ts)

a, b, are parameters.

Energy estimate, Plasticity. Testing with tu gives, for f = 0

t1

2

|tu|2 =

txu =

t(+ )

t1

2

||2

[

+

sign(t)

]

t

= t1

2

||2 t1

2

||2

|t|

Modelling Homogenization Mathematical tools

Hysteresis problems of plasticity and hydrology

Wave equation with hysteresis

2t u = x + f

xu = +

sign(t).

, , , are parameters.

Richards equation with hysteresis

ts = p

p as+ b+ sign(ts)

a, b, are parameters.

Energy estimate, Richards. Multiplication of the first equation with p andintegration over gives

|p|2 =

p ts

[as+ b+ sign(ts)]ts

= t

{a

2|s|2 + bs

}

+

|ts|.

Modelling Homogenization Mathematical tools

The fundamental question

We are interested in composite materials

(periodic or stochastic).

The material parameters are constant in

each cell, chosen randomly in each cell.

The grid-spacing is > 0.

Fundamental question of homogenization

If u are solutions to the -problems, and u u.What is the equation for u?

Two-scale convergence (Allaire, ...)

Energy method (Tartar, ...)

B.S. Homogenization of the Prager model in one-dimensional plasticity. Continuum Mechanics andThermodynamics 20(8), 2009.

B.S. Averaging of flows with capillary hysteresis in stochastic porous media. European Journal of AppliedMathematics 18, 2007.

Modelling Homogenization Mathematical tools

Main result on plasticity

Let u be a solution to the problem with oscillating parameters,

2t u = x

+ f

xu = +

sign(t).

Idea: Material label y I := [0, 1]. The measure M(I) denotes theprobability distribution of materials.

The strain in material y I is w(x, t, y). Problem (P) is

2t u = x

+ f

xu =

I

w(y) d(y) +

(y)w(y) (y) (y)sign(tw(y)) a.e. y I

where is the expected value of .

Theorem (S. 2009, Cont. Mech. Therm.)

Under ergodicity assumptions, the functions u converge to the uniquesolution u almost surely.

Modelling Homogenization Mathematical tools

Main result in hydrology: Expected pressure

The pressure has bounded gradients, hence p without oscillations.The saturation s, instead, oscillates.

A new quantity: The expected capillary pressurew := as + b = pc,0(s

).

1. case: saturation decreases. Then

s = pb+

ais oscillatory

The expected pressure is

w := as + b = p +

y1

w

p

... at places with = y.

2. case: Variable saturation

Expected pressure

w(x, y, t) := as + b

y

p

1

w

w encodes the saturation history of the porous medium!

Modelling Homogenization Mathematical tools

Equations for w

Averaged equations. Conservation law:

ts = (Kp) x ,

where K determined by a cell-problem.The saturation is reconstructed from w via

s(x, t) :=

I

w(x, y, t) b

ady,

where b = b and a = 1/a1

.The hysteresis relation holds point-wise,

p(x) w(x, y) + y sign(tw(x, y)) x , y [0, 1].

Theorem (S. 2004/07, Eur. J. Appl. Math.)

The equations posess a unique weak solution (s, w, p).For solutions (s, p) of the stochastic -problem we have

s s, p p almost surely.

Modelling Homogenization Mathematical tools

Scanning curves for the effective equations.

The presence of the history varia-ble w alters the scanning curves. s

p

Hysteresis-cell-problem andoperator-cell-problem decouple:

Play-type hysteresis averages to

Prandtl-Ishlinskii-hysteresis

Francu and Krejci (1999): 1-dim. deterministic wave-eq.

Visintin (02-), Alber (09): n-dim. deterministic static

S.-Veneroni (09): n-dim. deterministic wave-eq.

The effective model has scanningcurves that are qualitatively as inthe experiment.

Desirable for realistic

modelling!

Modelling Homogenization Mathematical tools

Oscillating test-functions

The correct description of porous media:

1 : Rd I = [0, 1] describes the distribution of material, it is

chosen stochastically, e.g. constant in unit cubes.

Heterogeneous media: : [0, 1], x 7 1(x/)

Parameters depend on the material: a(x) = a0((x)) etc.

Method of oscillating test-functions for plasticity. Knowing the

homogenized solution (u, , w), we may construct

w(t, x) := w(t, x, (x)).

We expect w to be similar to .

E(t) =1

2

|tu tu

|2 +1

2

| w|2 +

1

2

| |2

A direct calculation gives

E(t)

t

{(

I

tw(y) d(y) tw

)

( )t

}

( )

Modelling Homogenization Mathematical tools

Stochastic choice of 1

To describe stochastic media, one chooses 1 stochastically.Let be the distribution of values of 1.

Loose definition of ergodicity

The stochastic process is ergodic, if spatial averages yield the expectedvalues (almost surely).

The ergodicity of the medium implies

Definition (Ergodicity property)

Let g Lq(I, d) for q 1 and let g : R be defined as

g(x) = g((x)).

Then g converges weakly to a constant function,

g(x) g in Lq() almost surely.

Modelling Homogenization Mathematical tools

Two-scale ergodicity

Definition (Two-scale ergodicity property)

We say that the stochastic process and a function g : I R satisfythe two-scale ergodicity property with q [1,) if the following holds.Consider g : R and g : R,

g(x) = g(x, (x)), g (x) =

I

g(x, y) d(y).

Then

g g for 0 in Lq() almost surely.

The pair is two-scale ergodic when is ergodic and

g is continuous or

has finite support (finite number of materials).

This is the case in the discrete approximation!

Modelling Homogenization Mathematical tools

Conclusions

The homogenized system is the one you had guessed in the firstplace ... once you understood the system well.

Method of oscillating test-functions is very powerful for rigorousresults

Technical problems are BV -controls and two-scale ergodicity.

Further steps:

fingering in porous media

improved existence in porous media hysteresis

periodic coefficients for plasticity in Rn ( Marco Veneroni)

Open problem: