def setupSecondarySurvey(
self, primaryProblem, primarySurvey, map2meshSecondary
):
print('Setting up Secondary Survey')
nx = 41
ny = nx
rx_x, rx_y = 2*[np.linspace(-2050, 2050, nx)]
self.rxlocs = Utils.ndgrid([rx_x, rx_y, np.r_[-1]])
self.rx_x = self.rxlocs[:, 0].reshape(nx, ny, order='F')
self.rx_y = self.rxlocs[:, 1].reshape(nx, ny, order='F')
rx_ex = FDEM.Rx.Point_e(self.rxlocs, orientation='x', component='real')
rx_ey = FDEM.Rx.Point_e(self.rxlocs, orientation='y', component='real')
RxList = [rx_ex, rx_ey]
sec_src = [
FDEM.Src.PrimSecMappedSigma(
RxList, freq, primaryProblem, primarySurvey,
map2meshSecondary=map2meshSecondary
)
for freq in self.freqs
]
print('... done secondary survey')
return FDEM.Survey(sec_src)
def setupSecondaryProblem(self, mapping=None):
print('Setting up Secondary Problem')
if mapping is None:
mapping = [('sigma', Maps.IdentityMap(self.meshs))]
sec_problem = FDEM.Problem3D_e(self.meshs, sigmaMap=mapping)
sec_problem.Solver = Solver
print('... done setting up secondary problem')
return sec_problem
def prob(self):
if getattr(self, '_prob', None) is None:
self._prob = getattr(FDEM, 'Problem3D_{}'.format(
self.formulation))(self.meshGenerator.mesh,
sigmaMap=self.physprops.wires.sigma,
muMap=self.physprops.wires.mu,
Solver=Solver,
verbose=self.verbose)
if getattr(self, 'srcList') is not None:
self._survey = FDEM.Survey(self.srcList.srcList)
elif getattr(self, 'src') is not None:
self._survey = FDEM.Survey(self.src.srcList)
else:
raise Exception("one of src, srcList must be set")
self._prob.pair(self._survey)
return self._prob
def setUp(self):
cs = 10.
ncx, ncy, ncz = 30., 30., 30.
npad = 10.
hx = [(cs, npad, -1.5), (cs, ncx), (cs, npad, 1.5)]
hy = [(cs, npad, -1.5), (cs, ncy), (cs, npad, 1.5)]
hz = [(cs, npad, -1.5), (cs, ncz), (cs, npad, 1.5)]
self.mesh = Mesh.TensorMesh([hx, hy, hz], 'CCC')
mapping = Maps.ExpMap(self.mesh)
self.freq = 1.
self.prob_e = FDEM.Problem3D_e(self.mesh, mapping=mapping)
self.prob_b = FDEM.Problem3D_b(self.mesh, mapping=mapping)
self.prob_h = FDEM.Problem3D_h(self.mesh, mapping=mapping)
self.prob_j = FDEM.Problem3D_j(self.mesh, mapping=mapping)
loc = np.r_[0., 0., 0.]
self.loc = Utils.mkvc(
self.mesh.gridCC[Utils.closestPoints(self.mesh, loc, 'CC'), :])
def setUpClass(self):
print('\n------- Testing Primary Secondary Source HJ -> EB --------\n')
# receivers
self.rxlist = []
for rxtype in ['b', 'e']:
rx = getattr(FDEM.Rx, 'Point_{}'.format(rxtype))
for orientation in ['x', 'y', 'z']:
for comp in ['real', 'imag']:
self.rxlist.append(rx(rx_locs, component=comp,
orientation=orientation))
# primary
self.primaryProblem = FDEM.Problem3D_j(meshp, sigmaMap=primaryMapping)
self.primaryProblem.solver = Solver
s_e = np.zeros(meshp.nF)
inds = meshp.nFx + Utils.closestPoints(meshp, src_loc, gridLoc='Fz')
s_e[inds] = 1./csz
primarySrc = FDEM.Src.RawVec_e(
self.rxlist, freq=freq, s_e=s_e/meshp.area
)
self.primarySurvey = FDEM.Survey([primarySrc])
# Secondary Problem
self.secondaryProblem = FDEM.Problem3D_e(meshs, sigmaMap=mapping)
self.secondaryProblem.Solver = Solver
self.secondarySrc = FDEM.Src.PrimSecMappedSigma(
self.rxlist, freq, self.primaryProblem,
self.primarySurvey, primaryMap2Meshs
)
self.secondarySurvey = FDEM.Survey([self.secondarySrc])
self.secondaryProblem.pair(self.secondarySurvey)
# Full 3D problem to compare with
self.problem3D = FDEM.Problem3D_e(meshs, sigmaMap=mapping)
self.problem3D.Solver = Solver
s_e3D = np.zeros(meshs.nE)
inds = (meshs.nEx + meshs.nEy +
Utils.closestPoints(meshs, src_loc, gridLoc='Ez'))
s_e3D[inds] = [1./(len(inds))] * len(inds)
self.problem3D.model = model
src3D = FDEM.Src.RawVec_e(self.rxlist, freq=freq, s_e=s_e3D)
self.survey3D = FDEM.Survey([src3D])
self.problem3D.pair(self.survey3D)
# solve and store fields
print(' solving primary - secondary')
self.fields_primsec = self.secondaryProblem.fields(model)
print(' ... done')
self.fields_primsec = self.secondaryProblem.fields(model)
print(' solving 3D')
self.fields_3D = self.problem3D.fields(model)
print(' ... done')
return None
def __init__(self, **kwargs):
super(SimulationFDEM, self).__init__(**kwargs)
self._prob = getattr(
FDEM, 'Problem3D_{}'.format(self.formulation)
)(
self.meshGenerator.mesh,
sigmaMap=self.physprops.wires.sigma,
muMap=self.physprops.wires.mu,
Solver=Pardiso
)
if getattr(self.src, "physics", None) is None:
self.src.physics = "FDEM"
self._survey = FDEM.Survey(self.src.srcList)
self._prob.pair(self._survey)
def primaryProblem(self):
if getattr(self, '_primaryProblem', None) is None:
# define a custom prop map to include variable mu that we are not
# inverting for - This will change when we improve the propmap!
print('Getting Primary Problem')
# class CasingEMPropMap(Maps.PropMap):
# sigma = Maps.Property(
# "Electrical Conductivity", defaultInvProp=True,
# propertyLink=('rho', Maps.ReciprocalMap)
# )
# mu = Maps.Property(
# "Inverse Magnetic Permeability",
# defaultVal=self.muModel,
# propertyLink=('mui', Maps.ReciprocalMap)
# )
# rho = Maps.Property(
# "Electrical Resistivity",
# propertyLink=('sigma', Maps.ReciprocalMap)
# )
# mui = Maps.Property(
# "Inverse Magnetic Permeability",
# defaultVal=1./self.muModel,
# propertyLink=('mu', Maps.ReciprocalMap)
# )
# # set the problem's propmap
# FDEM.Problem3D_h.PropMap = CasingEMPropMap
# use H-J formulation for source with vertical current density and
# cylindrical symmetry (h faster on cyl --> less edges than faces)
primaryProblem = FDEM.Problem3D_h(
self.meshp, sigmaMap=self.primaryMapping
)
primaryProblem.mu = self.muModel
primaryProblem.Solver = Solver
self._primaryProblem = primaryProblem
print('... done building primary problem')
return self._primaryProblem
def setUpClass(self):
print('\n------- Testing Primary Secondary Source EB -> EB --------\n')
# receivers
self.rxlist = []
for rxtype in ['b', 'e']:
rx = getattr(FDEM.Rx, 'Point_{}'.format(rxtype))
for orientation in ['x', 'y', 'z']:
for comp in ['real', 'imag']:
self.rxlist.append(
rx(rx_locs, component=comp, orientation=orientation))
# primary
self.primaryProblem = FDEM.Problem3D_b(meshp, sigmaMap=primaryMapping)
self.primaryProblem.solver = Solver
primarySrc = FDEM.Src.MagDipole(self.rxlist, freq=freq, loc=src_loc)
self.primarySurvey = FDEM.Survey([primarySrc])
# Secondary Problem
self.secondaryProblem = FDEM.Problem3D_b(meshs, sigmaMap=mapping)
self.secondaryProblem.Solver = Solver
self.secondarySrc = FDEM.Src.PrimSecMappedSigma(
self.rxlist, freq, self.primaryProblem, self.primarySurvey,
primaryMap2Meshs)
self.secondarySurvey = FDEM.Survey([self.secondarySrc])
self.secondaryProblem.pair(self.secondarySurvey)
# Full 3D problem to compare with
self.problem3D = FDEM.Problem3D_b(meshs, sigmaMap=mapping)
self.problem3D.Solver = Solver
self.survey3D = FDEM.Survey([primarySrc])
self.problem3D.pair(self.survey3D)
# solve and store fields
print(' solving primary - secondary')
self.fields_primsec = self.secondaryProblem.fields(model)
print(' ... done')
self.fields_primsec = self.secondaryProblem.fields(model)
print(' solving 3D')
self.fields_3D = self.problem3D.fields(model)
print(' ... done')
return None
def setupProblem(
mesh, muMod, sigmaMod, prbtype='e', invertMui=False,
sigmaInInversion=False, freq=1.
):
rxcomp = ['real', 'imag']
loc = Utils.ndgrid(
[mesh.vectorCCx, np.r_[0.], mesh.vectorCCz]
)
if prbtype in ['e', 'b']:
rxfields_y = ['e', 'j']
rxfields_xz = ['b', 'h']
elif prbtype in ['h', 'j']:
rxfields_y = ['b', 'h']
rxfields_xz = ['e', 'j']
rxList_edge = [
getattr(FDEM.Rx, 'Point_{f}'.format(f=f))(
loc, component=comp, orientation=orient
)
for f in rxfields_y
for comp in rxcomp
for orient in ['y']
]
rxList_face = [
getattr(FDEM.Rx, 'Point_{f}'.format(f=f))(
loc, component=comp, orientation=orient
)
for f in rxfields_xz
for comp in rxcomp
for orient in ['x', 'z']
]
rxList = rxList_edge + rxList_face
src_loc = np.r_[0., 0., 0.]
if prbtype in ['e', 'b']:
src = FDEM.Src.MagDipole(
rxList=rxList, loc=src_loc, freq=freq
)
elif prbtype in ['h', 'j']:
ind = Utils.closestPoints(mesh, src_loc, 'Fz') + mesh.vnF[0]
vec = np.zeros(mesh.nF)
vec[ind] = 1.
src = FDEM.Src.RawVec_e(rxList=rxList, freq=freq, s_e=vec)
survey = FDEM.Survey([src])
if sigmaInInversion:
wires = Maps.Wires(
('mu', mesh.nC),
('sigma', mesh.nC)
)
muMap = Maps.MuRelative(mesh) * wires.mu
sigmaMap = Maps.ExpMap(mesh) * wires.sigma
if invertMui:
muiMap = Maps.ReciprocalMap(mesh)*muMap
prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))(
mesh, muiMap=muiMap, sigmaMap=sigmaMap
)
# m0 = np.hstack([1./muMod, sigmaMod])
else:
prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))(
mesh, muMap=muMap, sigmaMap=sigmaMap
)
m0 = np.hstack([muMod, sigmaMod])
else:
muMap = Maps.MuRelative(mesh)
if invertMui:
muiMap = Maps.ReciprocalMap(mesh) * muMap
prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))(
mesh, sigma=sigmaMod, muiMap=muiMap
)
# m0 = 1./muMod
else:
prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))(
mesh, sigma=sigmaMod, muMap=muMap
)
m0 = muMod
prob.pair(survey)
return m0, prob, survey
def setUpClass(self):
"""
Set up a cyl symmetric EM problem on 2D and 3D meshes.
"""
sigma_back = 1e-1 # wholespace
modelParameters = casingSimulations.model.Wholespace(
src_a=np.r_[0., 0., -9.],
src_b=np.r_[0., 0., -1.],
freqs=np.r_[0.1, 1., 2.],
sigma_back=sigma_back, # wholespace
)
# Set up the meshes
npadx, npadz = 11, 26
mesh2D = casingSimulations.CylMeshGenerator(
modelParameters=modelParameters, npadx=npadx, npadz=npadz, csz=2.)
mesh3D = casingSimulations.CylMeshGenerator(
modelParameters=modelParameters,
hy=np.ones(4) * 2 * np.pi / 4.,
csz=2.,
npadx=npadx,
npadz=npadz)
# get wirepath on mesh
wire2D = getSrcWire(mesh2D.mesh, modelParameters)
wire3D = getSrcWire(mesh3D.mesh, modelParameters)
# create sources
srcList2D = [
FDEM.Src.RawVec_e(s_e=wire2D, freq=freq, rxList=[])
for freq in modelParameters.freqs
]
srcList3D = [
FDEM.Src.RawVec_e(s_e=wire3D, freq=freq, rxList=[])
for freq in modelParameters.freqs
]
# get phys prop models
physprops2D = casingSimulations.model.PhysicalProperties(
mesh2D, modelParameters)
physprops3D = casingSimulations.model.PhysicalProperties(
mesh3D, modelParameters)
# create the problems and surveys
prb2D = FDEM.Problem3D_h(mesh2D.mesh,
sigmaMap=physprops2D.wires.sigma,
muMap=physprops2D.wires.mu,
Solver=Pardiso)
prb3D = FDEM.Problem3D_h(mesh3D.mesh,
sigmaMap=physprops3D.wires.sigma,
muMap=physprops3D.wires.mu,
Solver=Pardiso)
survey2D = FDEM.Survey(srcList2D)
survey3D = FDEM.Survey(srcList3D)
prb2D.pair(survey2D)
prb3D.pair(survey3D)
print('starting 2D solve ... ')
fields2D = prb2D.fields(physprops2D.model)
print(' ... done \n')
print('starting 3D solve ...')
fields3D = prb3D.fields(physprops3D.model)
print(' ... done \n')
# assign the properties that will be helpful
self.mesh2D = mesh2D
self.mesh3D = mesh3D
self.srcList2D = srcList2D
self.srcList3D = srcList3D
self.prb2D = prb2D
self.prb3D = prb3D
self.survey2D = survey2D
self.survey3D = survey3D
self.fields2D = fields2D
self.fields3D = fields3D
# Get cells inside the sphere
sph_ind = PF.MagAnalytics.spheremodel(mesh, 0., 0., 0., rad)
# Adjust susceptibility for volume difference
Vratio = (4. / 3. * np.pi * rad**3.) / (np.sum(sph_ind) * cs**3.)
model = np.ones(mesh.nC) * 1e-8
model[sph_ind] = 0.01
rxLoc = np.asarray([np.r_[0, 0, 4.]])
bzi = FDEM.Rx.Point_bSecondary(rxLoc, 'z', 'real')
bzr = FDEM.Rx.Point_bSecondary(rxLoc, 'z', 'imag')
freqs = [400] #np.logspace(2, 3, 5)
srcLoc = np.r_[0, 0, 4.]
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z')
for freq in freqs
]
mapping = Maps.IdentityMap(mesh)
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=PardisoSolver)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(model, std)
#Mesh.TensorMesh.writeUBC(mesh,'MeshGrav.msh')
#Mesh.TensorMesh.writeModelUBC(mesh,'MeshGrav.den',model)
#PF.Gravity.writeUBCobs("Obs.grv",survey,d)
def setUpClass(self):
"""
Set up a cyl symmetric EM problem on 2D and 3D meshes.
"""
sigma_back = 1e-1 # wholespace
cp = casingSimulations.CasingParameters(
casing_l=10.,
src_a=np.r_[0., 0., -9.],
src_b=np.r_[0., 0., -1.],
freqs=np.r_[0.1, 1., 2.],
sigma_back=sigma_back, # wholespace
sigma_layer=sigma_back,
sigma_air=sigma_back,
)
# Set up the meshes
npadx, npadz = 11, 26
dx2 = 500.
mesh2D = casingSimulations.CylMeshGenerator(cp=cp,
npadx=npadx,
npadz=npadz,
domain_x2=dx2).mesh
mesh3D = casingSimulations.CylMeshGenerator(cp=cp,
ncy=4,
npadx=npadx,
npadz=npadz,
domain_x2=dx2).mesh
# get wirepath on mesh
wire2D = getSrcWire(mesh2D, cp)
wire3D = getSrcWire(mesh3D, cp)
# create sources
srcList2D = [
FDEM.Src.RawVec_e(s_e=wire2D, freq=freq, rxList=[])
for freq in cp.freqs
]
srcList3D = [
FDEM.Src.RawVec_e(s_e=wire3D, freq=freq, rxList=[])
for freq in cp.freqs
]
# get phys prop models
physprops2D = casingSimulations.PhysicalProperties(mesh2D, cp)
physprops3D = casingSimulations.PhysicalProperties(mesh3D, cp)
# plot the phys prop models
fig, ax = plt.subplots(1, 1)
plt.colorbar(mesh2D.plotImage(np.log10(physprops2D.sigma),
ax=ax,
mirror=True)[0],
ax=ax)
ax.set_xlim([-1., 1.])
ax.set_ylim([-20., 10.])
if plotIt:
plt.show()
# create the problems and surveys
prb2D = FDEM.Problem3D_h(mesh2D,
sigmaMap=physprops2D.wires.sigma,
muMap=physprops2D.wires.mu,
Solver=Pardiso)
prb3D = FDEM.Problem3D_h(mesh3D,
sigmaMap=physprops3D.wires.sigma,
muMap=physprops3D.wires.mu,
Solver=Pardiso)
survey2D = FDEM.Survey(srcList2D)
survey3D = FDEM.Survey(srcList3D)
prb2D.pair(survey2D)
prb3D.pair(survey3D)
print('starting 2D solve ... ')
fields2D = prb2D.fields(physprops2D.model)
print(' ... done \n')
print('starting 3D solve ...')
fields3D = prb3D.fields(physprops3D.model)
print(' ... done \n')
# assign the properties that will be helpful
self.mesh2D = mesh2D
self.mesh3D = mesh3D
self.srcList2D = srcList2D
self.srcList3D = srcList3D
self.prb2D = prb2D
self.prb3D = prb3D
self.survey2D = survey2D
self.survey3D = survey3D
self.fields2D = fields2D
self.fields3D = fields3D
def primarySurvey(self):
if getattr(self, '_primarySurvey', None) is None:
print('Setting up primary survey')
def setupPrimarySource(plotIt=False):
# Construct a downhole source that is coupled to the casing
meshp = self.meshp
src_a = self.src_a
src_b = self.src_b
casing_a = self.casing_a
# downhole source
dg_x = np.zeros(meshp.vnF[0], dtype=complex)
dg_y = np.zeros(meshp.vnF[1], dtype=complex)
dg_z = np.zeros(meshp.vnF[2], dtype=complex)
# vertically directed wire in borehole
# go through the center of the well
dgv_indx = (meshp.gridFz[:, 0] < meshp.hx.min())
dgv_indz = (
(meshp.gridFz[:, 2] >= src_a[2]) &
(meshp.gridFz[:, 2] <= src_b[2])
)
dgv_ind = dgv_indx & dgv_indz
dg_z[dgv_ind] = -1.
# couple to the casing downhole - top part
dgh_indx = meshp.gridFx[:, 0] <= casing_a + meshp.hx.min()*2
# couple to the casing downhole - bottom part
dgh_indz2 = (
(meshp.gridFx[:, 2] <= src_a[2]) &
(meshp.gridFx[:, 2] > src_a[2] - meshp.hz.min())
)
dgh_ind2 = dgh_indx & dgh_indz2
dg_x[dgh_ind2] = 1.
# horizontally directed wire
sgh_indx = (meshp.gridFx[:, 0] <= src_b[0])
sgh_indz = (
(meshp.gridFx[:, 2] > meshp.hz.min()) &
(meshp.gridFx[:, 2] < 2*meshp.hz.min())
)
sgh_ind = sgh_indx & sgh_indz
dg_x[sgh_ind] = -1.
# return electrode
sgv_indx = (
(meshp.gridFz[:, 0] > src_b[0]*0.9) &
(meshp.gridFz[:, 0] < src_b[0]*1.1)
)
sgv_indz = (
(meshp.gridFz[:, 2] >= -meshp.hz.min()) &
(meshp.gridFz[:, 2] < 2*meshp.hz.min())
)
sgv_ind = sgv_indx & sgv_indz
dg_z[sgv_ind] = 1.
# assemble the source (downhole grounded primary)
dg = np.hstack([dg_x, dg_y, dg_z])
dg_p = [
FDEM.Src.RawVec_e([], _, dg/meshp.area) for _ in self.freqs
]
# if plotIt:
# # Plot the source to make sure the path is infact
# # connected
# fig, ax = plt.subplots(1, 1, figsize=(6, 4))
# meshp.plotGrid(ax=ax)
# ax.plot(meshp.gridFz[dgv_ind, 0], meshp.gridFz[dgv_ind, 2], 'rd')
# ax.plot(meshp.gridFx[dgh_ind2, 0], meshp.gridFx[dgh_ind2, 2], 'rd')
# ax.plot(meshp.gridFz[sgv_ind, 0], meshp.gridFz[sgv_ind, 2], 'rd')
# ax.plot(meshp.gridFx[sgh_ind, 0], meshp.gridFx[sgh_ind, 2], 'rd')
# ax.set_title('downhole casing source on mesh')
# ax.set_xlim([0, 1.1e4])
# ax.set_ylim([-1100., 0.5])
return dg_p
srcList = setupPrimarySource() # create primary source
self._primarySurvey = FDEM.Survey(srcList) # primary survey
print('... done building primary survey')
return self._primarySurvey
def run(plotIt=True):
"""
1D FDEM Mu Inversion
====================
1D inversion of Magnetic Susceptibility from FDEM data assuming a fixed
electrical conductivity
"""
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Geologic Parameters model
layerz = np.r_[-100., -50.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
# Electrical Conductivity
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 1e-2 # Layer conductivity
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# mur - relative magnetic permeability
mur_half = 1.
mur_air = 1.
mur_layer = 2.
mur = np.ones(mesh.nCz) * mur_air
mur[active] = mur_half
mur[layer] = mur_layer
mtrue = mur[active]
# Maps
actMap = Maps.InjectActiveCells(mesh, active, mur_air, nC=mesh.nCz)
surj1Dmap = Maps.SurjectVertical1D(mesh)
murMap = Maps.MuRelative(mesh)
# Mapping
muMap = murMap * surj1Dmap * actMap
# ----- FDEM problem & survey -----
rxlocs = Utils.ndgrid([np.r_[10.], np.r_[0], np.r_[30.]])
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
# bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.linspace(2000, 10000, 10) #np.logspace(3, 4, 10)
srcLoc = np.array([0., 0., 30.])
print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = [
FDEM.Src.MagDipole([bzr], freq, srcLoc, orientation='Z')
for freq in freqs
]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh,
sigma=surj1Dmap * sigma,
muMap=muMap,
Solver=Solver)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-6
# FDEM inversion
np.random.seed(13472)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[muMap.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
directiveList = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = mur_half * np.ones(mtrue.size)
reg.alpha_s = 2e-2
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
dpredFD = surveyFD.dpred(moptFD)
if plotIt:
fig, ax = plt.subplots(1, 3, figsize=(10, 6))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the conductivity model
ax[0].semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2)
ax[0].set_ylim(-500, 0)
ax[0].set_xlim(5e-3, 1e-1)
ax[0].set_xlabel('Conductivity (S/m)', fontsize=fs)
ax[0].set_ylabel('Depth (m)', fontsize=fs)
ax[0].grid(which='both',
color='k',
alpha=0.5,
linestyle='-',
linewidth=0.2)
ax[0].legend(['Conductivity Model'], fontsize=fs, loc=4)
# Plot the permeability model
ax[1].plot(mur[active], mesh.vectorCCz[active], 'k-', lw=2)
ax[1].plot(moptFD, mesh.vectorCCz[active], 'b-', lw=2)
ax[1].set_ylim(-500, 0)
ax[1].set_xlim(0.5, 2.1)
ax[1].set_xlabel('Relative Permeability', fontsize=fs)
ax[1].set_ylabel('Depth (m)', fontsize=fs)
ax[1].grid(which='both',
color='k',
alpha=0.5,
linestyle='-',
linewidth=0.2)
ax[1].legend(['True', 'Predicted'], fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax[2].plot(freqs, -surveyFD.dobs, 'k-', lw=2)
# ax[2].plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2)
ax[2].loglog(freqs, -dpredFD, 'bo', ms=6)
# ax[2].loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10)
# Labels, gridlines, etc
ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax[2].set_xlabel('Frequency (Hz)', fontsize=fs)
ax[2].set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
# ax[2].legend(("Obs", "Pred"), fontsize=fs)
ax[2].legend(("z-Obs (real)", "z-Pred (real)"), fontsize=fs)
ax[2].set_xlim(freqs.max(), freqs.min())
ax[0].set_title("(a) Conductivity Model", fontsize=fs)
ax[1].set_title("(b) $\mu_r$ Model", fontsize=fs)
ax[2].set_title("(c) FDEM observed vs. predicted", fontsize=fs)
# ax[2].set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
def run(plotIt=True, saveFig=False):
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Conductivity model
layerz = np.r_[-200., -100.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 5e-2 # Layer conductivity
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# Mapping
actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
mtrue = np.log(sigma[active])
# ----- FDEM problem & survey ----- #
rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]])
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0., 0., 0.])
print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z')
for freq in freqs
]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
directiveList = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
# TDEM problem
times = np.logspace(-4, np.log10(2e-3), 10)
print('min diffusion distance ',
1.28 * np.sqrt(times.min() / (sig_half * mu_0)),
'max diffusion distance ',
1.28 * np.sqrt(times.max() / (sig_half * mu_0)))
rx = TDEM.Rx.Point_b(rxlocs, times, 'z')
src = TDEM.Src.MagDipole(
[rx],
waveform=TDEM.Src.StepOffWaveform(),
loc=srcLoc # same src location as FDEM problem
)
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
prbTD.pair(surveyTD)
std = 0.03
surveyTD.makeSyntheticData(mtrue, std)
surveyTD.std = std
surveyTD.eps = np.linalg.norm(surveyTD.dtrue) * 1e-5
# TDEM inversion
dmisfit = DataMisfit.l2_DataMisfit(surveyTD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
directiveList = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbTD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptTD = inv.run(m0)
# Plot the results
if plotIt:
plt.figure(figsize=(10, 8))
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the model
ax0.semilogx(sigma[active],
mesh.vectorCCz[active],
'k-',
lw=2,
label="True")
ax0.semilogx(np.exp(moptFD),
mesh.vectorCCz[active],
'bo',
ms=6,
markeredgecolor='k',
markeredgewidth=0.5,
label="FDEM")
ax0.semilogx(np.exp(moptTD),
mesh.vectorCCz[active],
'r*',
ms=10,
markeredgecolor='k',
markeredgewidth=0.5,
label="TDEM")
ax0.set_ylim(-700, 0)
ax0.set_xlim(5e-3, 1e-1)
ax0.set_xlabel('Conductivity (S/m)', fontsize=fs)
ax0.set_ylabel('Depth (m)', fontsize=fs)
ax0.grid(which='both',
color='k',
alpha=0.5,
linestyle='-',
linewidth=0.2)
ax0.legend(fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2, label="Obs (real)")
ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2, label="Obs (imag)")
dpredFD = surveyFD.dpred(moptTD)
ax1.loglog(freqs,
-dpredFD[::2],
'bo',
ms=6,
markeredgecolor='k',
markeredgewidth=0.5,
label="Pred (real)")
ax1.loglog(freqs,
-dpredFD[1::2],
'b+',
ms=10,
markeredgewidth=2.,
label="Pred (imag)")
ax2.loglog(times, surveyTD.dobs, 'k-', lw=2, label='Obs')
ax2.loglog(times,
surveyTD.dpred(moptTD),
'r*',
ms=10,
markeredgecolor='k',
markeredgewidth=0.5,
label='Pred')
ax2.set_xlim(times.min() - 1e-5, times.max() + 1e-4)
# Labels, gridlines, etc
ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.set_xlabel('Frequency (Hz)', fontsize=fs)
ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.set_xlabel('Time (s)', fontsize=fs)
ax2.set_ylabel('Vertical magnetic field (T)', fontsize=fs)
ax2.legend(fontsize=fs, loc=3)
ax1.legend(fontsize=fs, loc=3)
ax1.set_xlim(freqs.max() + 1e2, freqs.min() - 1e1)
ax0.set_title("(a) Recovered Models", fontsize=fs)
ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs)
ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
if saveFig is True:
plt.savefig('example1.png', dpi=600)
def run(plotIt=True):
# ------------------ MODEL ------------------
sigmaair = 1e-8 # air
sigmaback = 1e-2 # background
sigmacasing = 1e6 # casing
sigmainside = sigmaback # inside the casing
casing_t = 0.006 # 1cm thickness
casing_l = 300 # length of the casing
casing_r = 0.1
casing_a = casing_r - casing_t / 2. # inner radius
casing_b = casing_r + casing_t / 2. # outer radius
casing_z = np.r_[-casing_l, 0.]
# ------------------ SURVEY PARAMETERS ------------------
freqs = np.r_[1e-6] # [1e-1, 1, 5] # frequencies
dsz = -300 # down-hole z source location
src_loc = np.r_[0., 0., dsz]
inf_loc = np.r_[0., 0., 1e4]
print('Skin Depth: ', [(500. / np.sqrt(sigmaback * _)) for _ in freqs])
# ------------------ MESH ------------------
# fine cells near well bore
csx1, csx2 = 2e-3, 60.
pfx1, pfx2 = 1.3, 1.3
ncx1 = np.ceil(casing_b / csx1 + 2)
# pad nicely to second cell size
npadx1 = np.floor(np.log(csx2 / csx1) / np.log(pfx1))
hx1a = Utils.meshTensor([(csx1, ncx1)])
hx1b = Utils.meshTensor([(csx1, npadx1, pfx1)])
dx1 = sum(hx1a) + sum(hx1b)
dx1 = np.floor(dx1 / csx2)
hx1b *= (dx1 * csx2 - sum(hx1a)) / sum(hx1b)
# second chunk of mesh
dx2 = 300. # uniform mesh out to here
ncx2 = np.ceil((dx2 - dx1) / csx2)
npadx2 = 45
hx2a = Utils.meshTensor([(csx2, ncx2)])
hx2b = Utils.meshTensor([(csx2, npadx2, pfx2)])
hx = np.hstack([hx1a, hx1b, hx2a, hx2b])
# z-direction
csz = 0.05
nza = 10
# cell size, number of core cells, number of padding cells in the
# x-direction
ncz, npadzu, npadzd = np.int(np.ceil(
np.diff(casing_z)[0] / csz)) + 10, 68, 68
# vector of cell widths in the z-direction
hz = Utils.meshTensor([(csz, npadzd, -1.3), (csz, ncz),
(csz, npadzu, 1.3)])
# Mesh
mesh = Mesh.CylMesh([hx, 1., hz],
[0., 0., -np.sum(hz[:npadzu + ncz - nza])])
print('Mesh Extent xmax: {0:f},: zmin: {1:f}, zmax: {2:f}'.format(
mesh.vectorCCx.max(), mesh.vectorCCz.min(), mesh.vectorCCz.max()))
print('Number of cells', mesh.nC)
if plotIt is True:
fig, ax = plt.subplots(1, 1, figsize=(6, 4))
ax.set_title('Simulation Mesh')
mesh.plotGrid(ax=ax)
# Put the model on the mesh
sigWholespace = sigmaback * np.ones((mesh.nC))
sigBack = sigWholespace.copy()
sigBack[mesh.gridCC[:, 2] > 0.] = sigmaair
sigCasing = sigBack.copy()
iCasingZ = ((mesh.gridCC[:, 2] <= casing_z[1]) &
(mesh.gridCC[:, 2] >= casing_z[0]))
iCasingX = ((mesh.gridCC[:, 0] >= casing_a) &
(mesh.gridCC[:, 0] <= casing_b))
iCasing = iCasingX & iCasingZ
sigCasing[iCasing] = sigmacasing
if plotIt is True:
# plotting parameters
xlim = np.r_[0., 0.2]
zlim = np.r_[-350., 10.]
clim_sig = np.r_[-8, 6]
# plot models
fig, ax = plt.subplots(1, 1, figsize=(4, 4))
f = plt.colorbar(mesh.plotImage(np.log10(sigCasing), ax=ax)[0], ax=ax)
ax.grid(which='both')
ax.set_title('Log_10 (Sigma)')
ax.set_xlim(xlim)
ax.set_ylim(zlim)
f.set_clim(clim_sig)
# -------------- Sources --------------------
# Define Custom Current Sources
# surface source
sg_x = np.zeros(mesh.vnF[0], dtype=complex)
sg_y = np.zeros(mesh.vnF[1], dtype=complex)
sg_z = np.zeros(mesh.vnF[2], dtype=complex)
nza = 2 # put the wire two cells above the surface
# vertically directed wire
# hook it up to casing at the surface
sgv_indx = ((mesh.gridFz[:, 0] > casing_a) &
(mesh.gridFz[:, 0] < casing_a + csx1))
sgv_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] >= -csz * 2))
sgv_ind = sgv_indx & sgv_indz
sg_z[sgv_ind] = -1.
# horizontally directed wire
sgh_indx = ((mesh.gridFx[:, 0] > casing_a) &
(mesh.gridFx[:, 0] <= inf_loc[2]))
sgh_indz = ((mesh.gridFx[:, 2] > csz * (nza - 0.5)) &
(mesh.gridFx[:, 2] < csz * (nza + 0.5)))
sgh_ind = sgh_indx & sgh_indz
sg_x[sgh_ind] = -1.
# hook it up to casing at the surface
sgv2_indx = ((mesh.gridFz[:, 0] >= mesh.gridFx[sgh_ind, 0].max()) &
(mesh.gridFz[:, 0] <= inf_loc[2] * 1.2))
sgv2_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] >= -csz * 2))
sgv2_ind = sgv2_indx & sgv2_indz
sg_z[sgv2_ind] = 1.
# assemble the source
sg = np.hstack([sg_x, sg_y, sg_z])
sg_p = [FDEM.Src.RawVec_e([], _, sg / mesh.area) for _ in freqs]
# downhole source
dg_x = np.zeros(mesh.vnF[0], dtype=complex)
dg_y = np.zeros(mesh.vnF[1], dtype=complex)
dg_z = np.zeros(mesh.vnF[2], dtype=complex)
# vertically directed wire
dgv_indx = (mesh.gridFz[:, 0] < csx1) # go through the center of the well
dgv_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] > dsz + csz / 2.))
dgv_ind = dgv_indx & dgv_indz
dg_z[dgv_ind] = -1.
# couple to the casing downhole
dgh_indx = mesh.gridFx[:, 0] < casing_a + csx1
dgh_indz = (mesh.gridFx[:, 2] < dsz + csz) & (mesh.gridFx[:, 2] >= dsz)
dgh_ind = dgh_indx & dgh_indz
dg_x[dgh_ind] = 1.
# horizontal part at surface
dgh2_indx = mesh.gridFx[:, 0] <= inf_loc[2] * 1.2
dgh2_indz = sgh_indz.copy()
dgh2_ind = dgh2_indx & dgh2_indz
dg_x[dgh2_ind] = -1.
# vertical part at surface
dgv2_ind = sgv2_ind.copy()
dg_z[dgv2_ind] = 1.
# assemble the source
dg = np.hstack([dg_x, dg_y, dg_z])
dg_p = [FDEM.Src.RawVec_e([], _, dg / mesh.area) for _ in freqs]
# ------------ Problem and Survey ---------------
survey = FDEM.Survey(sg_p + dg_p)
problem = FDEM.Problem3D_h(mesh,
sigmaMap=Maps.IdentityMap(mesh),
Solver=Solver)
problem.pair(survey)
# ------------- Solve ---------------------------
t0 = time.time()
fieldsCasing = problem.fields(sigCasing)
print('Time to solve 2 sources', time.time() - t0)
# Plot current
# current density
jn0 = fieldsCasing[dg_p, 'j']
jn1 = fieldsCasing[sg_p, 'j']
# current
in0 = [
mesh.area * fieldsCasing[dg_p, 'j'][:, i] for i in range(len(freqs))
]
in1 = [
mesh.area * fieldsCasing[sg_p, 'j'][:, i] for i in range(len(freqs))
]
in0 = np.vstack(in0).T
in1 = np.vstack(in1).T
# integrate to get z-current inside casing
inds_inx = ((mesh.gridFz[:, 0] >= casing_a) &
(mesh.gridFz[:, 0] <= casing_b))
inds_inz = (mesh.gridFz[:, 2] >= dsz) & (mesh.gridFz[:, 2] <= 0)
inds_fz = inds_inx & inds_inz
indsx = [False] * mesh.nFx
inds = list(indsx) + list(inds_fz)
in0_in = in0[np.r_[inds]]
in1_in = in1[np.r_[inds]]
z_in = mesh.gridFz[inds_fz, 2]
in0_in = in0_in.reshape([in0_in.shape[0] // 3, 3])
in1_in = in1_in.reshape([in1_in.shape[0] // 3, 3])
z_in = z_in.reshape([z_in.shape[0] // 3, 3])
I0 = in0_in.sum(1).real
I1 = in1_in.sum(1).real
z_in = z_in[:, 0]
if plotIt is True:
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
ax[0].plot(z_in, np.absolute(I0), z_in, np.absolute(I1))
ax[0].legend(['top casing', 'bottom casing'], loc='best')
ax[0].set_title('Magnitude of Vertical Current in Casing')
ax[1].semilogy(z_in, np.absolute(I0), z_in, np.absolute(I1))
ax[1].legend(['top casing', 'bottom casing'], loc='best')
ax[1].set_title('Magnitude of Vertical Current in Casing')
ax[1].set_ylim([1e-2, 1.])
def run(plotIt=True):
"""
EM: Schenkel and Morrison Casing Model
======================================
Here we create and run a FDEM forward simulation to calculate the
vertical current inside a steel-cased. The model is based on the
Schenkel and Morrison Casing Model, and the results are used in a 2016
SEG abstract by Yang et al.
.. code-block:: text
Schenkel, C.J., and H.F. Morrison, 1990, Effects of well casing on
potential field measurements using downhole current sources:
Geophysical prospecting, 38, 663-686.
The model consists of:
- Air: Conductivity 1e-8 S/m, above z = 0
- Background: conductivity 1e-2 S/m, below z = 0
- Casing: conductivity 1e6 S/m
- 300m long
- radius of 0.1m
- thickness of 6e-3m
Inside the casing, we take the same conductivity as the background.
We are using an EM code to simulate DC, so we use frequency low enough
that the skin depth inside the casing is longer than the casing length
(f = 1e-6 Hz). The plot produced is of the current inside the casing.
These results are shown in the SEG abstract by Yang et al., 2016: 3D DC
resistivity modeling of steel casing for reservoir monitoring using
equivalent resistor network. The solver used to produce these results
and achieve the CPU time of ~30s is Mumps, which was installed using
pymatsolver_
.. _pymatsolver: https://github.com/rowanc1/pymatsolver
This example is on figshare:
https://dx.doi.org/10.6084/m9.figshare.3126961.v1
If you would use this example for a code comparison, or build upon it,
a citation would be much appreciated!
"""
if plotIt:
import matplotlib.pylab as plt
# ------------------ MODEL ------------------
sigmaair = 1e-8 # air
sigmaback = 1e-2 # background
sigmacasing = 1e6 # casing
sigmainside = sigmaback # inside the casing
casing_t = 0.006 # 1cm thickness
casing_l = 300 # length of the casing
casing_r = 0.1
casing_a = casing_r - casing_t / 2. # inner radius
casing_b = casing_r + casing_t / 2. # outer radius
casing_z = np.r_[-casing_l, 0.]
# ------------------ SURVEY PARAMETERS ------------------
freqs = np.r_[1e-6] # [1e-1, 1, 5] # frequencies
dsz = -300 # down-hole z source location
src_loc = np.r_[0., 0., dsz]
inf_loc = np.r_[0., 0., 1e4]
print('Skin Depth: ', [(500. / np.sqrt(sigmaback * _)) for _ in freqs])
# ------------------ MESH ------------------
# fine cells near well bore
csx1, csx2 = 2e-3, 60.
pfx1, pfx2 = 1.3, 1.3
ncx1 = np.ceil(casing_b / csx1 + 2)
# pad nicely to second cell size
npadx1 = np.floor(np.log(csx2 / csx1) / np.log(pfx1))
hx1a = Utils.meshTensor([(csx1, ncx1)])
hx1b = Utils.meshTensor([(csx1, npadx1, pfx1)])
dx1 = sum(hx1a) + sum(hx1b)
dx1 = np.floor(dx1 / csx2)
hx1b *= (dx1 * csx2 - sum(hx1a)) / sum(hx1b)
# second chunk of mesh
dx2 = 300. # uniform mesh out to here
ncx2 = np.ceil((dx2 - dx1) / csx2)
npadx2 = 45
hx2a = Utils.meshTensor([(csx2, ncx2)])
hx2b = Utils.meshTensor([(csx2, npadx2, pfx2)])
hx = np.hstack([hx1a, hx1b, hx2a, hx2b])
# z-direction
csz = 0.05
nza = 10
# cell size, number of core cells, number of padding cells in the
# x-direction
ncz, npadzu, npadzd = np.int(np.ceil(
np.diff(casing_z)[0] / csz)) + 10, 68, 68
# vector of cell widths in the z-direction
hz = Utils.meshTensor([(csz, npadzd, -1.3), (csz, ncz),
(csz, npadzu, 1.3)])
# Mesh
mesh = Mesh.CylMesh([hx, 1., hz],
[0., 0., -np.sum(hz[:npadzu + ncz - nza])])
print('Mesh Extent xmax: {0:f},: zmin: {1:f}, zmax: {2:f}'.format(
mesh.vectorCCx.max(), mesh.vectorCCz.min(), mesh.vectorCCz.max()))
print('Number of cells', mesh.nC)
if plotIt is True:
fig, ax = plt.subplots(1, 1, figsize=(6, 4))
ax.set_title('Simulation Mesh')
mesh.plotGrid(ax=ax)
plt.show()
# Put the model on the mesh
sigWholespace = sigmaback * np.ones((mesh.nC))
sigBack = sigWholespace.copy()
sigBack[mesh.gridCC[:, 2] > 0.] = sigmaair
sigCasing = sigBack.copy()
iCasingZ = ((mesh.gridCC[:, 2] <= casing_z[1]) &
(mesh.gridCC[:, 2] >= casing_z[0]))
iCasingX = ((mesh.gridCC[:, 0] >= casing_a) &
(mesh.gridCC[:, 0] <= casing_b))
iCasing = iCasingX & iCasingZ
sigCasing[iCasing] = sigmacasing
if plotIt is True:
# plotting parameters
xlim = np.r_[0., 0.2]
zlim = np.r_[-350., 10.]
clim_sig = np.r_[-8, 6]
# plot models
fig, ax = plt.subplots(1, 1, figsize=(4, 4))
f = plt.colorbar(mesh.plotImage(np.log10(sigCasing), ax=ax)[0], ax=ax)
ax.grid(which='both')
ax.set_title('Log_10 (Sigma)')
ax.set_xlim(xlim)
ax.set_ylim(zlim)
f.set_clim(clim_sig)
plt.show()
# -------------- Sources --------------------
# Define Custom Current Sources
# surface source
sg_x = np.zeros(mesh.vnF[0], dtype=complex)
sg_y = np.zeros(mesh.vnF[1], dtype=complex)
sg_z = np.zeros(mesh.vnF[2], dtype=complex)
nza = 2 # put the wire two cells above the surface
ncin = 2
# vertically directed wire
# hook it up to casing at the surface
sgv_indx = ((mesh.gridFz[:, 0] > casing_a) &
(mesh.gridFz[:, 0] < casing_a + csx1))
sgv_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] >= -csz * 2))
sgv_ind = sgv_indx & sgv_indz
sg_z[sgv_ind] = -1.
# horizontally directed wire
sgh_indx = ((mesh.gridFx[:, 0] > casing_a) &
(mesh.gridFx[:, 0] <= inf_loc[2]))
sgh_indz = ((mesh.gridFx[:, 2] > csz * (nza - 0.5)) &
(mesh.gridFx[:, 2] < csz * (nza + 0.5)))
sgh_ind = sgh_indx & sgh_indz
sg_x[sgh_ind] = -1.
# hook it up to casing at the surface
sgv2_indx = ((mesh.gridFz[:, 0] >= mesh.gridFx[sgh_ind, 0].max()) &
(mesh.gridFz[:, 0] <= inf_loc[2] * 1.2))
sgv2_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] >= -csz * 2))
sgv2_ind = sgv2_indx & sgv2_indz
sg_z[sgv2_ind] = 1.
# assemble the source
sg = np.hstack([sg_x, sg_y, sg_z])
sg_p = [FDEM.Src.RawVec_e([], _, sg / mesh.area) for _ in freqs]
# downhole source
dg_x = np.zeros(mesh.vnF[0], dtype=complex)
dg_y = np.zeros(mesh.vnF[1], dtype=complex)
dg_z = np.zeros(mesh.vnF[2], dtype=complex)
# vertically directed wire
dgv_indx = (mesh.gridFz[:, 0] < csx1) # go through the center of the well
dgv_indz = ((mesh.gridFz[:, 2] <= +csz * nza) &
(mesh.gridFz[:, 2] > dsz + csz / 2.))
dgv_ind = dgv_indx & dgv_indz
dg_z[dgv_ind] = -1.
# couple to the casing downhole
dgh_indx = mesh.gridFx[:, 0] < casing_a + csx1
dgh_indz = (mesh.gridFx[:, 2] < dsz + csz) & (mesh.gridFx[:, 2] >= dsz)
dgh_ind = dgh_indx & dgh_indz
dg_x[dgh_ind] = 1.
# horizontal part at surface
dgh2_indx = mesh.gridFx[:, 0] <= inf_loc[2] * 1.2
dgh2_indz = sgh_indz.copy()
dgh2_ind = dgh2_indx & dgh2_indz
dg_x[dgh2_ind] = -1.
# vertical part at surface
dgv2_ind = sgv2_ind.copy()
dg_z[dgv2_ind] = 1.
# assemble the source
dg = np.hstack([dg_x, dg_y, dg_z])
dg_p = [FDEM.Src.RawVec_e([], _, dg / mesh.area) for _ in freqs]
# ------------ Problem and Survey ---------------
survey = FDEM.Survey(sg_p + dg_p)
mapping = [('sigma', Maps.IdentityMap(mesh))]
problem = FDEM.Problem3D_h(mesh, mapping=mapping, Solver=solver)
problem.pair(survey)
# ------------- Solve ---------------------------
t0 = time.time()
fieldsCasing = problem.fields(sigCasing)
print('Time to solve 2 sources', time.time() - t0)
# Plot current
# current density
jn0 = fieldsCasing[dg_p, 'j']
jn1 = fieldsCasing[sg_p, 'j']
# current
in0 = [
mesh.area * fieldsCasing[dg_p, 'j'][:, i] for i in range(len(freqs))
]
in1 = [
mesh.area * fieldsCasing[sg_p, 'j'][:, i] for i in range(len(freqs))
]
in0 = np.vstack(in0).T
in1 = np.vstack(in1).T
# integrate to get z-current inside casing
inds_inx = ((mesh.gridFz[:, 0] >= casing_a) &
(mesh.gridFz[:, 0] <= casing_b))
inds_inz = (mesh.gridFz[:, 2] >= dsz) & (mesh.gridFz[:, 2] <= 0)
inds_fz = inds_inx & inds_inz
indsx = [False] * mesh.nFx
inds = list(indsx) + list(inds_fz)
in0_in = in0[np.r_[inds]]
in1_in = in1[np.r_[inds]]
z_in = mesh.gridFz[inds_fz, 2]
in0_in = in0_in.reshape([in0_in.shape[0] // 3, 3])
in1_in = in1_in.reshape([in1_in.shape[0] // 3, 3])
z_in = z_in.reshape([z_in.shape[0] // 3, 3])
I0 = in0_in.sum(1).real
I1 = in1_in.sum(1).real
z_in = z_in[:, 0]
if plotIt is True:
fig, ax = plt.subplots(1, 2, figsize=(12, 4))
ax[0].plot(z_in, np.absolute(I0), z_in, np.absolute(I1))
ax[0].legend(['top casing', 'bottom casing'], loc='best')
ax[0].set_title('Magnitude of Vertical Current in Casing')
ax[1].semilogy(z_in, np.absolute(I0), z_in, np.absolute(I1))
ax[1].legend(['top casing', 'bottom casing'], loc='best')
ax[1].set_title('Magnitude of Vertical Current in Casing')
ax[1].set_ylim([1e-2, 1.])
plt.show()
def run(plotIt=True):
"""
Mesh: Plotting with defining range
==================================
When using a large Mesh with the cylindrical code, it is advantageous
to define a :code:`range_x` and :code:`range_y` when plotting with
vectors. In this case, only the region inside of the range is
interpolated. In particular, you often want to ignore padding cells.
"""
# ## Model Parameters
#
# We define a
# - resistive halfspace and
# - conductive sphere
# - radius of 30m
# - center is 50m below the surface
# electrical conductivities in S/m
sig_halfspace = 1e-6
sig_sphere = 1e0
sig_air = 1e-8
# depth to center, radius in m
sphere_z = -50.
sphere_radius = 30.
# ## Survey Parameters
#
# - Transmitter and receiver 20m above the surface
# - Receiver offset from transmitter by 8m horizontally
# - 25 frequencies, logaritmically between $10$ Hz and $10^5$ Hz
boom_height = 20.
rx_offset = 8.
freqs = np.r_[1e1, 1e5]
# source and receiver location in 3D space
src_loc = np.r_[0., 0., boom_height]
rx_loc = np.atleast_2d(np.r_[rx_offset, 0., boom_height])
# print the min and max skin depths to make sure mesh is fine enough and
# extends far enough
def skin_depth(sigma, f):
return 500. / np.sqrt(sigma * f)
print('Minimum skin depth (in sphere): {:.2e} m '.format(
skin_depth(sig_sphere, freqs.max())))
print('Maximum skin depth (in background): {:.2e} m '.format(
skin_depth(sig_halfspace, freqs.min())))
# ## Mesh
#
# Here, we define a cylindrically symmetric tensor mesh.
#
# ### Mesh Parameters
#
# For the mesh, we will use a cylindrically symmetric tensor mesh. To
# construct a tensor mesh, all that is needed is a vector of cell widths in
# the x and z-directions. We will define a core mesh region of uniform cell
# widths and a padding region where the cell widths expand "to infinity".
# x-direction
csx = 2 # core mesh cell width in the x-direction
ncx = np.ceil(
1.2 * sphere_radius / csx
) # number of core x-cells (uniform mesh slightly beyond sphere radius)
npadx = 50 # number of x padding cells
# z-direction
csz = 1 # core mesh cell width in the z-direction
ncz = np.ceil(
1.2 * (boom_height - (sphere_z - sphere_radius)) / csz
) # number of core z-cells (uniform cells slightly below bottom of sphere)
npadz = 52 # number of z padding cells
# padding factor (expand cells to infinity)
pf = 1.3
# cell spacings in the x and z directions
hx = Utils.meshTensor([(csx, ncx), (csx, npadx, pf)])
hz = Utils.meshTensor([(csz, npadz, -pf), (csz, ncz), (csz, npadz, pf)])
# define a SimPEG mesh
mesh = Mesh.CylMesh([hx, 1, hz],
x0=np.r_[0., 0., -hz.sum() / 2. - boom_height])
# ### Plot the mesh
#
# Below, we plot the mesh. The cyl mesh is rotated around x=0. Ensure that
# each dimension extends beyond the maximum skin depth.
#
# Zoom in by changing the xlim and zlim.
# X and Z limits we want to plot to. Try
xlim = np.r_[0., 2.5e6]
zlim = np.r_[-2.5e6, 2.5e6]
fig, ax = plt.subplots(1, 1)
mesh.plotGrid(ax=ax)
ax.set_title('Simulation Mesh')
ax.set_xlim(xlim)
ax.set_ylim(zlim)
print('The maximum skin depth is (in background): {:.2e} m. '
'Does the mesh go sufficiently past that?'.format(
skin_depth(sig_halfspace, freqs.min())))
# ## Put Model on Mesh
#
# Now that the model parameters and mesh are defined, we can define
# electrical conductivity on the mesh.
#
# The electrical conductivity is defined at cell centers when using the
# finite volume method. So here, we define a vector that contains an
# electrical conductivity value for every cell center.
# create a vector that has one entry for every cell center
sigma = sig_air * np.ones(
mesh.nC) # start by defining the conductivity of the air everwhere
sigma[mesh.gridCC[:, 2] <
0.] = sig_halfspace # assign halfspace cells below the earth
# indices of the sphere (where (x-x0)**2 + (z-z0)**2 <= R**2)
sphere_ind = ((mesh.gridCC[:, 0]**2 +
(mesh.gridCC[:, 2] - sphere_z)**2) <= sphere_radius**2)
sigma[sphere_ind] = sig_sphere # assign the conductivity of the sphere
# Plot a cross section of the conductivity model
fig, ax = plt.subplots(1, 1)
cb = plt.colorbar(mesh.plotImage(np.log10(sigma), ax=ax, mirror=True)[0])
# plot formatting and titles
cb.set_label('$\log_{10}\sigma$', fontsize=13)
ax.axis('equal')
ax.set_xlim([-120., 120.])
ax.set_ylim([-100., 30.])
ax.set_title('Conductivity Model')
# ## Set up the Survey
#
# Here, we define sources and receivers. For this example, the receivers
# are magnetic flux recievers, and are only looking at the secondary field
# (eg. if a bucking coil were used to cancel the primary). The source is a
# vertical magnetic dipole with unit moment.
# Define the receivers, we will sample the real secondary magnetic flux
# density as well as the imaginary magnetic flux density
bz_r = FDEM.Rx.Point_bSecondary(
locs=rx_loc, orientation='z',
component='real') # vertical real b-secondary
bz_i = FDEM.Rx.Point_b(
locs=rx_loc, orientation='z',
component='imag') # vertical imag b (same as b-secondary)
rxList = [bz_r, bz_i] # list of receivers
# Define the list of sources - one source for each frequency. The source is
# a point dipole oriented in the z-direction
srcList = [
FDEM.Src.MagDipole(rxList, f, src_loc, orientation='z') for f in freqs
]
print(
'There are {nsrc} sources (same as the number of frequencies - {nfreq}). '
'Each source has {nrx} receivers sampling the resulting b-fields'.
format(nsrc=len(srcList), nfreq=len(freqs), nrx=len(rxList)))
# ## Set up Forward Simulation
#
# A forward simulation consists of a paired SimPEG problem and Survey.
# For this example, we use the E-formulation of Maxwell's equations,
# solving the second-order system for the electric field, which is defined
# on the cell edges of the mesh. This is the `prob` variable below. The
# `survey` takes the source list which is used to construct the RHS for the
# problem. The source list also contains the receiver information, so the
# `survey` knows how to sample fields and fluxes that are produced by
# solving the `prob`.
# define a problem - the statement of which discrete pde system we want to
# solve
prob = FDEM.Problem3D_e(mesh, sigmaMap=Maps.IdentityMap(mesh))
prob.solver = Solver
survey = FDEM.Survey(srcList)
# tell the problem and survey about each other - so the RHS can be
# constructed for the problem and the
# resulting fields and fluxes can be sampled by the receiver.
prob.pair(survey)
# ### Solve the forward simulation
#
# Here, we solve the problem for the fields everywhere on the mesh.
fields = prob.fields(sigma)
# ### Plot the fields
#
# Lets look at the physics!
# log-scale the colorbar
from matplotlib.colors import LogNorm
fig, ax = plt.subplots(1, 2, figsize=(12, 6))
def plotMe(field, ax):
plt.colorbar(mesh.plotImage(field,
vType='F',
view='vec',
range_x=[-100., 100.],
range_y=[-180., 60.],
pcolorOpts={
'norm': LogNorm(),
'cmap': plt.get_cmap('viridis')
},
streamOpts={'color': 'k'},
ax=ax,
mirror=True)[0],
ax=ax)
plotMe(fields[srcList[0], 'bSecondary'].real, ax[0])
ax[0].set_title('Real B-Secondary, {}Hz'.format(freqs[0]))
plotMe(fields[srcList[1], 'bSecondary'].real, ax[1])
ax[1].set_title('Real B-Secondary, {}Hz'.format(freqs[1]))
plt.tight_layout()
if plotIt:
plt.show()
rx_real = FDEM.Rx.Point_bSecondary(locs=rx_locs,
orientation=orientation,
component='real')
rx_imag = FDEM.Rx.Point_bSecondary(locs=rx_locs,
orientation=orientation,
component='imag')
src = FDEM.Src.MagDipole(rxList=[rx_real, rx_imag],
loc=src_loc,
orientation=orientation,
freq=freq)
srcList.append(src)
# create the survey and problem objects for running the forward simulation
survey = FDEM.Survey(srcList)
prob = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prob.pair(survey)
###############################################################################
# Data
# ----
#
# Generate clean, synthetic data
t = time.time()
dclean = survey.dpred(m_true)
print("Done forward simulation. Elapsed time = {:1.2f} s".format(time.time() -
t))
