Problem Description

This example considers two-phase porous-media flow. In all that follows, we assume that the flow is immiscible and incompressible with no mass transfer between phases. A detailed description of the mathematical model and parameters can be found [DKKN18].

[1]:
try:
    import dune.femdg
except ImportError:
    print("This example needs 'dune.femdg' - skipping")
    import sys
    sys.exit(0)

from matplotlib import pyplot

import numpy
from ufl import *
from dune.ufl import Space, Constant

import dune.fem as fem
import dune.create as create
from dune.generator import algorithm
from dune.common import FieldVector
from dune.grid import cartesianDomain, Marker
from dune.fem.function import levelFunction, gridFunction
from dune.fem import integrate
from dune.plotting import plotPointData as plot

from limit import createOrderRedcution, createLimiter

fem.threading.use = 4

Some parameters

[2]:
maxLevel  = 3
maxOrder  = 3
dt        = 5.
endTime   = 800.
coupled   = False
tolerance = 3e-2
penalty   = 5 * (maxOrder * ( maxOrder + 1 ))
newtonParameters = {"tolerance": tolerance,
                    "verbose": "false", "linear.verbose": "false",
                    "linabstol": 1e-8, "reduction": 1e-8}

Defining the model

using a Brooks Corey pressure law

[3]:
def brooksCorey(P,s_n):
    s_w = 1-s_n
    s_we = (s_w-P.s_wr)/(1.-P.s_wr-P.s_nr)
    s_ne = (s_n-P.s_nr)/(1.-P.s_wr-P.s_nr)
    if P.useCutOff:
        cutOff = lambda a: min_value(max_value(a,0.00001),0.99999)
        s_we = cutOff(s_we)
        s_ne = cutOff(s_ne)
    kr_w = s_we**((2.+3.*P.theta)/P.theta)
    kr_n = s_ne**2*(1.-s_we**((2.+P.theta)/P.theta))
    p_c  = P.pd*s_we**(-1./P.theta)
    dp_c = P.pd * (-1./P.theta) * s_we**(-1./P.theta-1.) * (-1./(1.-P.s_wr-P.s_nr))
    l_n  = kr_n / P.mu_n
    l_w  = kr_w / P.mu_w
    return p_c,dp_c,l_n,l_w

Constants and domain description for anisotropic lens test

[4]:
class AnisotropicLens:
    dimWorld = 2
    domain   = cartesianDomain([0,0.39],[0.9,0.65],[15,4])
    x        = SpatialCoordinate(triangle)

    g     = [0,]*dimWorld ; g[dimWorld-1] = -9.810 # [m/s^2]
    g     = as_vector(g)
    r_w   = 1000.  # [Kg/m^3]
    mu_w  = 1.e-3  # [Kg/m s]
    r_n   = 1460.  # [Kg/m^3]
    mu_n  = 9.e-4  # [Kg/m s]

    lens = lambda x,a,b: (a-b)*                (conditional(abs(x[1]-0.49)<0.03,1.,0.)*                 conditional(abs(x[0]-0.45)<0.11,1.,0.))                + b

    p_c = brooksCorey

    Kdiag = lens(x, 6.64*1e-14, 1e-10) # [m^2]
    Koff  = lens(x, 0,-5e-11)          # [m^2]
    K     = as_matrix( [[Kdiag,Koff],[Koff,Kdiag]] )

    Phi   = lens(x, 0.39, 0.40)             # [-]
    s_wr  = lens(x, 0.10, 0.12)             # [-]
    s_nr  = lens(x, 0.00, 0.00)             # [-]
    theta = lens(x, 2.00, 2.70)             # [-]
    pd    = lens(x, 5000., 755.)            # [Pa]

    #### initial conditions
    p_w0 = (0.65-x[1])*9810.       # hydrostatic pressure
    s_n0 = 0                       # fully saturated
    # boundary conditions
    inflow = conditional(abs(x[0]-0.45)<0.06,1.,0.)*             conditional(abs(x[1]-0.65)<1e-8,1.,0.)
    J_n  = -5.137*1e-5
    J_w  = 1e-20
    dirichlet = conditional(abs(x[0])<1e-8,1.,0.) +                conditional(abs(x[0]-0.9)<1e-8,1.,0.)
    p_wD = p_w0
    s_nD = s_n0

    q_n  = 0
    q_w  = 0

    useCutOff = False

P = AnisotropicLens()

Setup grid, discrete spaces and functions

[5]:
grid = create.view("adaptive", "ALUCube", P.domain, dimgrid=2)

if coupled:
    spc          = create.space("dglegendrehp", grid, dimRange=2, order=maxOrder)
else:
    spc1         = create.space("dglegendrehp", grid, dimRange=1, order=maxOrder)
    spc          = create.space("product", spc1,spc1, components=["p","s"] )

solution     = spc.interpolate([0,0], name="solution")
solution_old = spc.interpolate([0,0], name="solution_old")
sol_pm1      = spc.interpolate([0,0], name="sol_pm1")
intermediate = spc.interpolate([0,0], name="iterate")
persistentDF = [solution,solution_old,intermediate]

fvspc        = create.space("finitevolume", grid, dimRange=1, storage="numpy")
estimate     = fvspc.interpolate([0], name="estimate")
estimate_pm1 = fvspc.interpolate([0], name="estimate-pm1")
[6]:
uflSpace = Space((P.dimWorld,P.dimWorld),2)
u        = TrialFunction(uflSpace)
v        = TestFunction(uflSpace)
cell     = uflSpace.cell()
x        = SpatialCoordinate(cell)
tau      = Constant(dt, name="timeStep")
beta     = Constant(penalty, name="penalty")

p_w  = u[0]
s_n  = u[1]

p_c,dp_c,l_n,l_w = P.p_c(s_n=intermediate[1])

Bulk terms

[7]:
dBulk_p  = P.K*( (l_n+l_w)*grad(p_w) + l_n*dp_c*grad(s_n) )
dBulk_p += -P.K*( (P.r_n*l_n+P.r_w*l_w)*P.g )
bulk_p   = -(P.q_w+P.q_n)
dBulk_s  = P.K*l_n*dp_c*grad(s_n)
dBulk_s += P.K*l_n*(grad(p_w)-P.r_n*P.g)
bulk_s   = -P.q_n

Boundary and initial conditions

[8]:
p_D, s_D = P.p_wD, P.s_nD,
p_N, s_N = P.J_w+P.J_n, P.J_n
p_0, s_0 = P.p_w0, P.s_n0

Bulk integrals

[9]:
form_p = ( inner(dBulk_p,grad(v[0])) + bulk_p*v[0] ) * dx
form_s = ( inner(dBulk_s,grad(v[1])) + bulk_s*v[1] ) * dx

Boundary fluxes

[10]:
form_p += p_N * v[0] * P.inflow * ds
form_s += s_N * v[1] * P.inflow * ds

DG terms

[11]:
def sMax(a): return max_value(a('+'), a('-'))
n         = FacetNormal(cell)
hT        = MaxCellEdgeLength(cell)
he        = avg( CellVolume(cell) ) / FacetArea(cell)
heBnd     = CellVolume(cell) / FacetArea(cell)
k         = dot(P.K*n,n)
lambdaMax = k('+')*k('-')/avg(k)
def wavg(z): return (k('-')*z('+')+k('+')*z('-'))/(k('+')+k('-'))

Penalty terms (including dirichlet boundary treatment)

[12]:
p_c0,dp_c0,l_n0,l_w0 = P.p_c(0.5)
penalty_p = [beta*lambdaMax*sMax(l_n0+l_w0),
             beta*k*(l_n0+l_w0)]
penalty_s = [beta*lambdaMax*sMax(l_n0*dp_c0),
             beta*k*(l_n0*dp_c0)]
form_p += penalty_p[0]/he * jump(u[0])*jump(v[0]) * dS
form_s += penalty_s[0]/he * jump(u[1])*jump(v[1]) * dS
form_p += penalty_p[1]/heBnd * (u[0]-p_D) * v[0] * P.dirichlet * ds
form_s += penalty_s[1]/heBnd * (u[1]-s_D) * v[1] * P.dirichlet * ds

Consistency terms

[13]:
form_p -= inner(wavg(dBulk_p),n('+')) * jump(v[0]) * dS
form_s -= inner(wavg(dBulk_s),n('+')) * jump(v[1]) * dS
form_p -= inner(dBulk_p,n) * v[0] * P.dirichlet * ds
form_s -= inner(dBulk_s,n) * v[1] * P.dirichlet * ds

Time discretization

[14]:
form_s = P.Phi*(u[1]-solution_old[1])*v[1] * dx + tau*form_s

Stabilization (Limiter)

[15]:
limiter = createLimiter( spc, limiter="scaling" )
tmp = solution.copy()
def limit(target):
    tmp.assign(target)
    limiter(tmp,target)

Time stepping Converting UFL forms to scheme

[16]:
if coupled:
    form = form_s + form_p
    tpModel = create.model( "integrands", grid, form == 0)
    # tpModel.penalty  = penalty
    # tpModel.timeStep = dt
    scheme = create.scheme("galerkin", tpModel, spc, ("suitesparse","umfpack"),
                     parameters={"nonlinear." + k: v for k, v in newtonParameters.items()})
else:
    uflSpace1 = Space((P.dimWorld,P.dimWorld),1)
    u1        = TrialFunction(uflSpace1)
    v1        = TestFunction(uflSpace1)
    form_p = replace(form_p, { u:as_vector([u1[0],intermediate.s[0]]),
                             v:as_vector([v1[0],0.]) } )
    form_s = replace(form_s, { u:as_vector([solution[0],u1[0]]),
                             intermediate:as_vector([solution[0],intermediate[1]]),
                             v:as_vector([0.,v1[0]]) } )
    form = [form_p,form_s]
    tpModel = [create.model( "integrands", grid, form[0] == 0),
               create.model( "integrands", grid, form[1] == 0)]
    # tpModel[0].penalty  = penalty
    # tpModel[1].penalty  = penalty
    # tpModel[1].timeStep = dt
    scheme = [create.scheme("galerkin", m, s, ("suitesparse","umfpack"),
                     parameters={"nonlinear." + k: v for k, v in newtonParameters.items()})
                for m,s in zip(tpModel,spc.components)]

Stopping condition for iterative approaches

[17]:
def errorMeasure(w,dw):
    rel = integrate([w[1]**2,dw[1]**2], order=5, gridView=grid)
    return numpy.sqrt(rel[1]) < tolerance * numpy.sqrt(rel[0])

Iterative schemes (iterative or impes-iterative)

[18]:
def step():
    n = 0
    solution_old.assign(solution)
    while True:
        intermediate.assign(solution)
        if coupled:
            scheme.solve(target=solution)
        else:
            scheme[0].solve(target=solution.p)
            scheme[1].solve(target=solution.s)
        limit(solution)
        n += 1
        # print("step",n,flush=True)
        if errorMeasure(solution,solution-intermediate) or n==20:
            break

HP Adpativity

Setting up residual indicator

[19]:
uflSpace0 = Space((P.dimWorld,P.dimWorld),1)
v0        = TestFunction(uflSpace0)

Rvol = P.Phi*(u[1]-solution_old[1])/tau - div(dBulk_s) - bulk_s
estimator = hT**2 * Rvol**2 * v0[0] * dx +      he * inner(jump(dBulk_s), n('+'))**2 * avg(v0[0]) * dS +      heBnd * (s_N + inner(dBulk_s,n))**2 * v0[0] * P.inflow * ds +      penalty_s[0]**2/he * jump(u[1])**2 * avg(v0[0]) * dS +      penalty_s[1]**2/heBnd * (s_D - u[1])**2 * v0[0] * P.dirichlet * ds
estimator = replace(estimator, {intermediate:u})

estimatorModel = create.model("integrands", grid, estimator == 0)
# estimatorModel.timeStep = dt
# estimatorModel.penalty  = penalty
estimator = create.operator("galerkin", estimatorModel, spc, fvspc)

Marker for grid adaptivity (h)

[20]:
hTol = 1e-16                           # changed later
def markh(element):
    center = element.geometry.referenceElement.center
    eta    = estimate.localFunction(element).evaluate(center)[0]
    if eta > hTol and element.level < maxLevel:
      return Marker.refine
    elif eta < 0.01*hTol:
      return Marker.coarsen
    else:
      return Marker.keep

Marker for space adaptivity (p)

[21]:
pTol = 1e-16
def markp(element):
    center = element.geometry.referenceElement.center
    r      = estimate.localFunction(element).evaluate(center)[0]
    r_p1   = estimate_pm1.localFunction(element).evaluate(center)[0]
    eta = abs(r-r_p1)
    polorder = spc.localOrder(element)
    if eta < pTol:
        return polorder-1 if polorder > 1 else polorder
    elif eta > 100.*pTol:
        return polorder+1 if polorder < maxOrder else polorder
    else:
        return polorder

Operator for projecting into space with a reduced order on every element

[22]:
orderreduce  = createOrderRedcution( spc )

Main program

Pre adapt the grid

[23]:
hgrid = grid.hierarchicalGrid
hgrid.globalRefine(1)
for i in range(maxLevel):
    print("pre adaptive (",i,"): ",grid.size(0),end="\n")
    solution.interpolate( as_vector([p_0,s_0]) )
    limit(solution)
    step()
    estimator(solution, estimate)
    hgrid.mark(markh)
    fem.adapt(persistentDF)

print("final pre adaptive (",i,"): ",dt,grid.size(0),end="\n")
pre adaptive ( 0 ):  240
pre adaptive ( 1 ):  231
pre adaptive ( 2 ):  363
final pre adaptive ( 2 ):  5.0 273

Define the constant for the h adaptivity

[24]:
solution.interpolate( as_vector([p_0,s_0]) )
limit(solution)
estimator(solution, estimate)
timeTol = sum(estimate.dofVector) / endTime
print('Using timeTol = ',timeTol, end='\n')
Using timeTol =  3.216131221875019e-15

Time loop

[25]:
t = 0
saveStep = 0
while t < endTime:
    step()

    # h adaptivity
    hTol = timeTol * dt / grid.size(0)
    estimator(solution, estimate)
    hgrid.mark(markh)
    fem.adapt(persistentDF)

    # p adaptivity
    estimator(solution, estimate)
    orderreduce(solution,sol_pm1)
    estimator(sol_pm1, estimate_pm1)
    fem.spaceAdapt(spc, markp, persistentDF)
    t += dt

    if t>=saveStep:
        print(t,grid.size(0),sum(estimate.dofVector),hTol,"# timestep",flush=True)
        plot(solution[1],figsize=(15,4))
        saveStep += 100
5.0 240 0.0 5.890350223214321e-17 # timestep
_images/twophaseflow_nb_49_1.jpg
100.0 363 0.0 4.46684891927086e-17 # timestep
_images/twophaseflow_nb_49_3.jpg
200.0 498 0.0 3.248617395833353e-17 # timestep
_images/twophaseflow_nb_49_5.jpg
300.0 582 0.0 2.748830104166683e-17 # timestep
_images/twophaseflow_nb_49_7.jpg
400.0 669 0.0 2.4145129293356e-17 # timestep
_images/twophaseflow_nb_49_9.jpg
500.0 774 0.0 2.0856882113326973e-17 # timestep
_images/twophaseflow_nb_49_11.jpg
600.0 885 0.0 1.8232036405187182e-17 # timestep
_images/twophaseflow_nb_49_13.jpg
700.0 1017 0.0 1.585863521634625e-17 # timestep
_images/twophaseflow_nb_49_15.jpg
800.0 1134 0.0 1.4255900806183596e-17 # timestep
_images/twophaseflow_nb_49_17.jpg

Postprocessing Show solution along a given line

[26]:
x0 = FieldVector([0.25,  0.65])
x1 = FieldVector([0.775, 0.39])
p,v = algorithm.run('sample', 'utility.hh', solution, x0, x1, 1000)

x = numpy.zeros(len(p))
y = numpy.zeros(len(p))
l = (x1-x0).two_norm
for i in range(len(x)):
    x[i] = (p[i]-x0).two_norm / l
    y[i] = v[i][1]
pyplot.plot(x,y)
[26]:
[<matplotlib.lines.Line2D at 0x7faa9fdd1c90>]
_images/twophaseflow_nb_51_1.jpg
[27]:
from dune.fem.function import levelFunction
@gridFunction(gridView=grid,name="polOrder",order=0)
def polOrder(e,x):
    return [spc.localOrder(e)]
plot(solution[0],figsize=(15,4))
plot(solution[1],figsize=(15,4))
plot(polOrder,figsize=(15,4))
plot(levelFunction(grid),figsize=(15,4))
_images/twophaseflow_nb_53_0.jpg
_images/twophaseflow_nb_53_1.jpg
_images/twophaseflow_nb_53_2.jpg
_images/twophaseflow_nb_53_3.jpg

This page was generated from the notebook twophaseflow_nb.ipynb and is part of the tutorial for the dune-fem python bindings DOI