Porous Media and Plasticity - Homogenization for Equations ... · PDF fileModelling...

Click here to load reader

  • date post

    21-Jun-2018
  • Category

    Documents

  • view

    221
  • download

    1

Embed Size (px)

Transcript of Porous Media and Plasticity - Homogenization for Equations ... · PDF fileModelling...

  • 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: