def setUp(self):
mesh = self.get_mesh()
params = Richards.Empirical.HaverkampParams().celia1990
k_fun, theta_fun = Richards.Empirical.haverkamp(mesh, **params)
self.setup_maps(mesh, k_fun, theta_fun)
bc, h = self.get_conditions(mesh)
prob = Richards.RichardsProblem(mesh,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
root_finder_tol=1e-6,
debug=False,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method='mixed')
prob.timeSteps = [(40, 3), (60, 3)]
prob.Solver = Solver
rx_list = self.get_rx_list(prob)
survey = Richards.RichardsSurvey(rx_list)
prob.pair(survey)
self.h0 = h
self.mesh = mesh
self.Ks = params['Ks'] * np.ones(self.mesh.nC)
self.A = params['A'] * np.ones(self.mesh.nC)
self.theta_s = params['theta_s'] * np.ones(self.mesh.nC)
self.prob = prob
self.survey = survey
self.setup_model()
def setUp(self):
M = Mesh.TensorMesh([np.ones(20)])
M.setCellGradBC('dirichlet')
params = Richards.Empirical.HaverkampParams().celia1990
params['Ks'] = np.log(params['Ks'])
E = Richards.Empirical.Haverkamp(M, **params)
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,
mapping=E,
timeSteps=[(40, 3), (60, 3)],
tolRootFinder=1e-6,
debug=False,
boundaryConditions=bc,
initialConditions=h,
doNewton=False,
method='mixed')
prob.Solver = Solver
locs = np.r_[5., 10, 15]
times = prob.times[3:5]
rxSat = Richards.RichardsRx(locs, times, 'saturation')
rxPre = Richards.RichardsRx(locs, times, 'pressureHead')
survey = Richards.RichardsSurvey([rxSat, rxPre])
prob.pair(survey)
self.h0 = h
self.M = M
self.Ks = params['Ks']
self.prob = prob
self.survey = survey
def setUp(self):
M = Mesh.TensorMesh([np.ones(8),np.ones(20),np.ones(10)])
M.setCellGradBC(['neumann','neumann','dirichlet'])
params = Richards.Empirical.HaverkampParams().celia1990
params['Ks'] = np.log(params['Ks'])
E = Richards.Empirical.Haverkamp(M, **params)
bc = np.array([-61.5,-20.7])
bc = np.r_[np.zeros(M.nCy*M.nCz*2),np.zeros(M.nCx*M.nCz*2),np.ones(M.nCx*M.nCy)*bc[0],np.ones(M.nCx*M.nCy)*bc[1]]
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,E, timeSteps=[(40,3),(60,3)], boundaryConditions=bc, initialConditions=h, doNewton=False, method='mixed', tolRootFinder=1e-6, debug=False)
prob.Solver = Solver
locs = Utils.ndgrid(np.r_[5,7.],np.r_[5,15.],np.r_[6,8.])
times = prob.times[3:5]
rxSat = Richards.RichardsRx(locs, times, 'saturation')
rxPre = Richards.RichardsRx(locs, times, 'pressureHead')
survey = Richards.RichardsSurvey([rxSat, rxPre])
prob.pair(survey)
self.h0 = h
self.M = M
self.Ks = params['Ks']
self.prob = prob
self.survey = survey
def run(plotIt=True):
M = Mesh.TensorMesh([np.ones(40)], x0='N')
M.setCellGradBC('dirichlet')
# We will use the haverkamp empirical model with parameters from Celia1990
k_fun, theta_fun = Richards.Empirical.haverkamp(M,
A=1.1750e+06,
gamma=4.74,
alpha=1.6110e+06,
theta_s=0.287,
theta_r=0.075,
beta=3.96)
# Here we are making saturated hydraulic conductivity
# an exponential mapping to the model (defined below)
k_fun.KsMap = Maps.ExpMap(nP=M.nC)
# Setup the boundary and initial conditions
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method='mixed',
debug=False)
prob.timeSteps = [(5, 25, 1.1), (60, 40)]
# Create the survey
locs = -np.arange(2, 38, 4.)
times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60)
rxSat = Richards.SaturationRx(locs, times)
survey = Richards.RichardsSurvey([rxSat])
survey.pair(prob)
# Create a simple model for Ks
Ks = 1e-3
mtrue = np.ones(M.nC) * np.log(Ks)
mtrue[15:20] = np.log(5e-2)
mtrue[20:35] = np.log(3e-3)
mtrue[35:40] = np.log(1e-2)
# Create some synthetic data and fields
Hs = prob.fields(mtrue)
data = survey.makeSyntheticData(mtrue, std=0, f=Hs, force=True)
if plotIt:
plt.figure(figsize=(14, 9))
plt.subplot(221)
plt.plot(np.log10(np.exp(mtrue)), M.gridCC)
plt.title('(a) True model and data locations')
plt.ylabel('Depth, cm')
plt.xlabel('Hydraulic conductivity, $log_{10}(K_s)$')
plt.plot([-3.25] * len(locs), locs, 'ro')
plt.legend(('True model', 'Data locations'))
plt.subplot(222)
plt.plot(times / 60, data.reshape((-1, len(locs))))
plt.title('(b) True data over time at all depths')
plt.xlabel('Time, minutes')
plt.ylabel('Saturation')
ax = plt.subplot(212)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('(c) Saturation over time')
plt.tight_layout()
def run(plotIt=True):
M = Mesh.TensorMesh([np.ones(40)], x0='N')
M.setCellGradBC('dirichlet')
# We will use the haverkamp empirical model with parameters from Celia1990
k_fun, theta_fun = Richards.Empirical.haverkamp(M,
A=1.1750e+06,
gamma=4.74,
alpha=1.6110e+06,
theta_s=0.287,
theta_r=0.075,
beta=3.96)
# Here we are making saturated hydraulic conductivity
# an exponential mapping to the model (defined below)
k_fun.KsMap = Maps.ExpMap(nP=M.nC)
# Setup the boundary and initial conditions
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method='mixed',
debug=False)
prob.timeSteps = [(5, 25, 1.1), (60, 40)]
# Create the survey
locs = -np.arange(2, 38, 4.)
times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60)
rxSat = Richards.SaturationRx(locs, times)
survey = Richards.RichardsSurvey([rxSat])
survey.pair(prob)
# Create a simple model for Ks
Ks = 1e-3
mtrue = np.ones(M.nC) * np.log(Ks)
mtrue[15:20] = np.log(5e-2)
mtrue[20:35] = np.log(3e-3)
mtrue[35:40] = np.log(1e-2)
m0 = np.ones(M.nC) * np.log(Ks)
# Create some synthetic data and fields
stdev = 0.02 # The standard deviation for the noise
Hs = prob.fields(mtrue)
survey.makeSyntheticData(mtrue, std=stdev, f=Hs, force=True)
# Setup a pretty standard inversion
reg = Regularization.Tikhonov(M, alpha_s=1e-1)
dmis = DataMisfit.l2_DataMisfit(survey)
opt = Optimization.InexactGaussNewton(maxIter=20, maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
beta = Directives.BetaSchedule(coolingFactor=4)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e2)
target = Directives.TargetMisfit()
dir_list = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=dir_list)
mopt = inv.run(m0)
Hs_opt = prob.fields(mopt)
if plotIt:
plt.figure(figsize=(14, 9))
ax = plt.subplot(121)
plt.semilogx(np.exp(np.c_[mopt, mtrue]), M.gridCC)
plt.xlabel('Saturated Hydraulic Conductivity, $K_s$')
plt.ylabel('Depth, cm')
plt.semilogx([10**-3.9] * len(locs), locs, 'ro')
plt.legend(('$m_{rec}$', '$m_{true}$', 'Data locations'), loc=4)
ax = plt.subplot(222)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('True saturation over time')
ax = plt.subplot(224)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs_opt]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('Recovered saturation over time')
plt.tight_layout()
def run(plotIt=True):
"""
FLOW: Richards: 1D: Inversion
=============================
The example shows an inversion of Richards equation in 1D with a
heterogeneous hydraulic conductivity function.
The haverkamp model is used with the same parameters as Celia1990_
the boundary and initial conditions are also the same. The simulation
domain is 40cm deep and is run for an hour with an exponentially
increasing time step that has a maximum of one minute. The general
setup of the experiment is an infiltration front that advances
downward through the model over time.
The model chosen is the saturated hydraulic conductivity inside
the hydraulic conductivity function (using haverkamp). The initial
model is chosen to be the background (1e-3 cm/s). The saturation data
has 2% random Gaussian noise added.
The figure shows the recovered saturated hydraulic conductivity
next to the true model. The other two figures show the saturation
field for the entire simulation for the true and recovered models.
Rowan Cockett - 21/12/2016
.. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf
"""
M = Mesh.TensorMesh([np.ones(40)], x0='N')
M.setCellGradBC('dirichlet')
# We will use the haverkamp empirical model with parameters from Celia1990
k_fun, theta_fun = Richards.Empirical.haverkamp(M,
A=1.1750e+06,
gamma=4.74,
alpha=1.6110e+06,
theta_s=0.287,
theta_r=0.075,
beta=3.96)
# Here we are making saturated hydraulic conductivity
# an exponential mapping to the model (defined below)
k_fun.KsMap = Maps.ExpMap(nP=M.nC)
# Setup the boundary and initial conditions
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method='mixed',
debug=False)
prob.timeSteps = [(5, 25, 1.1), (60, 40)]
# Create the survey
locs = -np.arange(2, 38, 4.)
times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60)
rxSat = Richards.SaturationRx(locs, times)
survey = Richards.RichardsSurvey([rxSat])
survey.pair(prob)
# Create a simple model for Ks
Ks = 1e-3
mtrue = np.ones(M.nC) * np.log(Ks)
mtrue[15:20] = np.log(5e-2)
mtrue[20:35] = np.log(3e-3)
mtrue[35:40] = np.log(1e-2)
m0 = np.ones(M.nC) * np.log(Ks)
# Create some synthetic data and fields
stdev = 0.02 # The standard deviation for the noise
Hs = prob.fields(mtrue)
survey.makeSyntheticData(mtrue, std=stdev, f=Hs, force=True)
# Setup a pretty standard inversion
reg = Regularization.Tikhonov(M, alpha_s=1e-1)
dmis = DataMisfit.l2_DataMisfit(survey)
opt = Optimization.InexactGaussNewton(maxIter=20, maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
beta = Directives.BetaSchedule(coolingFactor=4)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e2)
target = Directives.TargetMisfit()
dir_list = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=dir_list)
mopt = inv.run(m0)
Hs_opt = prob.fields(mopt)
if plotIt:
plt.figure(figsize=(14, 9))
ax = plt.subplot(121)
plt.semilogx(np.exp(np.c_[mopt, mtrue]), M.gridCC)
plt.xlabel('Saturated Hydraulic Conductivity, $K_s$')
plt.ylabel('Depth, cm')
plt.semilogx([10**-3.9] * len(locs), locs, 'ro')
plt.legend(('$m_{rec}$', '$m_{true}$', 'Data locations'), loc=4)
ax = plt.subplot(222)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('True saturation over time')
ax = plt.subplot(224)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs_opt]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('Recovered saturation over time')
plt.tight_layout()
def run(plotIt=True):
"""
FLOW: Richards: 1D: Forward Simulation
======================================
The example shows simulation of Richards equation in 1D with a
heterogeneous hydraulic conductivity function.
The haverkamp model is used with the same parameters as Celia1990_
the boundary and initial conditions are also the same. The simulation
domain is 40cm deep and is run for an hour with an exponentially
increasing time step that has a maximum of one minute. The general
setup of the experiment is an infiltration front that advances
downward through the model over time.
Figure (a) shows the heterogeneous saturated hydraulic conductivity
parameter and the location of the data collection, which happens every
minute from 30 seconds into the simulation. Note that the simulation
mesh and the data locations are not aligned, and linear interpolation
is used to collect the data. The points are sampled in pressure head
and then transformed to saturation using the haverkamp model for
the water retention curve.
Figure (b) shows the data collected from the simulation. No noise is
added to the data at this time. The various data locations register
the infiltration event through increasing saturation as the front moves
past the receiver. Notice that the slope of the curves are not equal
as the hydraulic conductivity function is heterogeneous.
Figure (c) shows the saturation field over the entire experiment. Here
you can see that the timestep is not constant over time (5 seconds
at the start of the simulation, 60 seconds at the end). You can also
see the effect of the highly conductive layer in the model between
20 and 25 cm depth. The water drains straight through the conductive
unit and piles up on the other side - advancing the fluid front
faster than the other layers.
Rowan Cockett - 21/12/2016
.. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf
"""
M = Mesh.TensorMesh([np.ones(40)], x0='N')
M.setCellGradBC('dirichlet')
# We will use the haverkamp empirical model with parameters from Celia1990
k_fun, theta_fun = Richards.Empirical.haverkamp(M,
A=1.1750e+06,
gamma=4.74,
alpha=1.6110e+06,
theta_s=0.287,
theta_r=0.075,
beta=3.96)
# Here we are making saturated hydraulic conductivity
# an exponential mapping to the model (defined below)
k_fun.KsMap = Maps.ExpMap(nP=M.nC)
# Setup the boundary and initial conditions
bc = np.array([-61.5, -20.7])
h = np.zeros(M.nC) + bc[0]
prob = Richards.RichardsProblem(M,
hydraulic_conductivity=k_fun,
water_retention=theta_fun,
boundary_conditions=bc,
initial_conditions=h,
do_newton=False,
method='mixed',
debug=False)
prob.timeSteps = [(5, 25, 1.1), (60, 40)]
# Create the survey
locs = -np.arange(2, 38, 4.)
times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60)
rxSat = Richards.SaturationRx(locs, times)
survey = Richards.RichardsSurvey([rxSat])
survey.pair(prob)
# Create a simple model for Ks
Ks = 1e-3
mtrue = np.ones(M.nC) * np.log(Ks)
mtrue[15:20] = np.log(5e-2)
mtrue[20:35] = np.log(3e-3)
mtrue[35:40] = np.log(1e-2)
# Create some synthetic data and fields
Hs = prob.fields(mtrue)
data = survey.makeSyntheticData(mtrue, std=0, f=Hs, force=True)
if plotIt:
plt.figure(figsize=(14, 9))
plt.subplot(221)
plt.plot(np.log10(np.exp(mtrue)), M.gridCC)
plt.title('(a) True model and data locations')
plt.ylabel('Depth, cm')
plt.xlabel('Hydraulic conductivity, $log_{10}(K_s)$')
plt.plot([-3.25] * len(locs), locs, 'ro')
plt.legend(('True model', 'Data locations'))
plt.subplot(222)
plt.plot(times / 60, data.reshape((-1, len(locs))))
plt.title('(b) True data over time at all depths')
plt.xlabel('Time, minutes')
plt.ylabel('Saturation')
ax = plt.subplot(212)
mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N')
sats = [theta_fun(_) for _ in Hs]
clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax)
cmap0 = matplotlib.cm.RdYlBu_r
clr[0].set_cmap(cmap0)
c = plt.colorbar(clr[0])
c.set_label('Saturation $\\theta$')
plt.xlabel('Time, minutes')
plt.ylabel('Depth, cm')
plt.title('(c) Saturation over time')
plt.tight_layout()