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 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 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 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
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 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.])