def meshs(self):
if getattr(self, '_meshs', None) is None:
csx, ncx, npadx = 50, 21, 12
csy, ncy, npady = 50, 21, 12
csz, ncz, npadz = 25, 40, 14
pf = 1.5
hx = Utils.meshTensor(
[(csx, npadx, -pf), (csx, ncx), (csx, npadx, pf)]
)
hy = Utils.meshTensor(
[(csy, npady, -pf), (csy, ncy), (csy, npady, pf)]
)
hz = Utils.meshTensor(
[(csz, npadz, -pf), (csz, ncz), (csz, npadz, pf)]
)
x0 = np.r_[-hx.sum()/2., -hy.sum()/2., -hz[:npadz+ncz].sum()]
self._meshs = Mesh.TensorMesh([hx, hy, hz], x0=x0)
print('Secondary Mesh ... ')
print(
' xmin, xmax, zmin, zmax: ', self._meshs.vectorCCx.min(),
self._meshs.vectorCCx.max(), self._meshs.vectorCCy.min(),
self._meshs.vectorCCy.max(), self._meshs.vectorCCz.min(),
self._meshs.vectorCCz.max()
)
print(' nC, vnC', self._meshs.nC, self._meshs.vnC)
return self._meshs
def hx(self):
"""
cell spacings in the x-direction
"""
if getattr(self, '_hx', None) is None:
# finest uniform region
hx1a = Utils.meshTensor([(self.csx1, self.ncx1)])
# pad to second uniform region
hx1b = Utils.meshTensor([(self.csx1, self.npadx1, self.pfx1)])
# scale padding so it matches cell size properly
dx1 = sum(hx1a) + sum(hx1b)
dx1 = np.floor(dx1 / self.csx2)
hx1b *= (dx1 * self.csx2 - sum(hx1a)) / sum(hx1b)
# second uniform chunk of mesh
ncx2 = np.ceil((self.domain_x2 - dx1) / self.csx2)
hx2a = Utils.meshTensor([(self.csx2, ncx2)])
# pad to infinity
hx2b = Utils.meshTensor([(self.csx2, self.npadx, self.pfx2)])
self._hx = np.hstack([hx1a, hx1b, hx2a, hx2b])
return self._hx
def meshp(self):
if getattr(self, '_meshp', None) is None:
# -------------- Mesh Parameters ------------------ #
# x-direction
csx1, csx2 = 2.5e-3, 25. # fine cells near well bore
pfx1, pfx2 = 1.3, 1.4 # padding factors: fine -> uniform
ncx1 = np.ceil(self.casing_b/csx1+2) # number of fine cells
# (past casing wall)
dx2 = 1000. # uniform mesh out to here
npadx2 = 21 # padding out to infinity
# z-direction
csz = 0.05 # finest z-cells
nza = 10 # number of fine cells above air-earth interface
pfz = pfx2 # padding factor in z-direction
# ------------- Assemble the Cyl Mesh ------------- #
# 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
ncx2 = np.ceil((dx2 - dx1)/csx2)
hx2a = Utils.meshTensor([(csx2, ncx2)])
hx2b = Utils.meshTensor([(csx2, npadx2, pfx2)])
hx = np.hstack([hx1a, hx1b, hx2a, hx2b])
# cell size, number of core cells, number of padding cells in the
# x-direction
ncz = np.int(np.ceil(np.diff(self.casing_z)[0]/csz))+10
npadzu, npadzd = 43, 43
# vector of cell widths in the z-direction
hz = Utils.meshTensor(
[(csz, npadzd, -pfz), (csz, ncz), (csz, npadzu, pfz)]
)
# primary mesh
self._meshp = Mesh.CylMesh(
[hx, 1., hz], [0., 0., -np.sum(hz[:npadzu+ncz-nza])]
)
print(
'Cyl Mesh Extent xmax: {},: zmin: {}, zmax: {}'.format(
self._meshp.vectorCCx.max(),
self._meshp.vectorCCz.min(),
self._meshp.vectorCCz.max()
)
)
return self._meshp
def run(plotIt=True):
M = Mesh.TensorMesh([100, 100])
h1 = Utils.meshTensor([(6, 7, -1.5), (6, 10), (6, 7, 1.5)])
h1 = h1 / h1.sum()
M2 = Mesh.TensorMesh([h1, h1])
V = Utils.ModelBuilder.randomModel(M.vnC, seed=79, its=50)
v = Utils.mkvc(V)
modh = Maps.Mesh2Mesh([M, M2])
modH = Maps.Mesh2Mesh([M2, M])
H = modH * v
h = modh * H
if not plotIt:
return
ax = plt.subplot(131)
M.plotImage(v, ax=ax)
ax.set_title('Fine Mesh (Original)')
ax = plt.subplot(132)
M2.plotImage(H, clim=[0, 1], ax=ax)
ax.set_title('Course Mesh')
ax = plt.subplot(133)
M.plotImage(h, clim=[0, 1], ax=ax)
ax.set_title('Fine Mesh (Interpolated)')
def run(plotIt=True):
M = Mesh.TensorMesh([100, 100])
h1 = Utils.meshTensor([(6, 7, -1.5), (6, 10), (6, 7, 1.5)])
h1 = h1/h1.sum()
M2 = Mesh.TensorMesh([h1, h1])
V = Utils.ModelBuilder.randomModel(M.vnC, seed=79, its=50)
v = Utils.mkvc(V)
modh = Maps.Mesh2Mesh([M, M2])
modH = Maps.Mesh2Mesh([M2, M])
H = modH * v
h = modh * H
if not plotIt:
return
ax = plt.subplot(131)
M.plotImage(v, ax=ax)
ax.set_title('Fine Mesh (Original)')
ax = plt.subplot(132)
M2.plotImage(H, clim=[0, 1], ax=ax)
ax.set_title('Course Mesh')
ax = plt.subplot(133)
M.plotImage(h, clim=[0, 1], ax=ax)
ax.set_title('Fine Mesh (Interpolated)')
def __init__(self, h_in, x0_in=None):
assert type(h_in) in [list, tuple], 'h_in must be a list'
assert len(h_in) in [1,2,3], 'h_in must be of dimension 1, 2, or 3'
h = range(len(h_in))
for i, h_i in enumerate(h_in):
if Utils.isScalar(h_i) and type(h_i) is not np.ndarray:
# This gives you something over the unit cube.
h_i = self._unitDimensions[i] * np.ones(int(h_i))/int(h_i)
elif type(h_i) is list:
h_i = Utils.meshTensor(h_i)
assert isinstance(h_i, np.ndarray), ("h[%i] is not a numpy array." % i)
assert len(h_i.shape) == 1, ("h[%i] must be a 1D numpy array." % i)
h[i] = h_i[:] # make a copy.
x0 = np.zeros(len(h))
if x0_in is not None:
assert len(h) == len(x0_in), "Dimension mismatch. x0 != len(h)"
for i in range(len(h)):
x_i, h_i = x0_in[i], h[i]
if Utils.isScalar(x_i):
x0[i] = x_i
elif x_i == '0':
x0[i] = 0.0
elif x_i == 'C':
x0[i] = -h_i.sum()*0.5
elif x_i == 'N':
x0[i] = -h_i.sum()
else:
raise Exception("x0[%i] must be a scalar or '0' to be zero, 'C' to center, or 'N' to be negative." % i)
BaseRectangularMesh.__init__(self, np.array([x.size for x in h]), x0)
# Ensure h contains 1D vectors
self._h = [Utils.mkvc(x.astype(float)) for x in h]
def run(plotIt=True):
"""
Maps: Mesh2Mesh
===============
This mapping allows you to go from one mesh to another.
"""
M = Mesh.TensorMesh([100, 100])
h1 = Utils.meshTensor([(6, 7, -1.5), (6, 10), (6, 7, 1.5)])
h1 = h1 / h1.sum()
M2 = Mesh.TensorMesh([h1, h1])
V = Utils.ModelBuilder.randomModel(M.vnC, seed=79, its=50)
v = Utils.mkvc(V)
modh = Maps.Mesh2Mesh([M, M2])
modH = Maps.Mesh2Mesh([M2, M])
H = modH * v
h = modh * H
if not plotIt: return
import matplotlib.pyplot as plt
ax = plt.subplot(131)
M.plotImage(v, ax=ax)
ax.set_title('Fine Mesh (Original)')
ax = plt.subplot(132)
M2.plotImage(H, clim=[0, 1], ax=ax)
ax.set_title('Course Mesh')
ax = plt.subplot(133)
M.plotImage(h, clim=[0, 1], ax=ax)
ax.set_title('Fine Mesh (Interpolated)')
plt.show()
def hz(self):
if getattr(self, '_hz', None) is None:
self._hz = Utils.meshTensor([(self.csz, self.npadz, -self.pfz),
(self.csz, self.ncz),
(self.csz, self.npadz, self.pfz)])
return self._hz
def hz(self):
"""
cell spacings in the z-direction
:rtype: numpy.array
"""
if getattr(self, '_hz', None) is None:
self._hz = Utils.meshTensor([(self.csz, self.npadz, -self.pfz),
(self.csz, self.ncz),
(self.csz, self.npadz, self.pfz)])
return self._hz
def run(plotIt=True):
"""
Plot Mirrored Cylindrically Symmetric Model
===========================================
Here, we demonstrate plotting a model on a cylindrically
symmetric mesh with the plotting symmetric about x=0.
"""
sig_halfspace = 1e-6
sig_sphere = 1e0
sig_air = 1e-8
sphere_z = -50.
sphere_radius = 30.
# x-direction
cs = 1
nc = np.ceil(2.5 * (-(sphere_z - sphere_radius)) / cs)
# cell spacings in the x and z directions
h = Utils.meshTensor([(cs, nc)])
# define a mesh
mesh = Mesh.CylMesh([h, 1, h], x0='00C')
# Put the model on the mesh
sigma = sig_air * np.ones(mesh.nC) # start with air cells
sigma[mesh.gridCC[:, 2] < 0.] = sig_halfspace # cells below the earth
# indices of the sphere
sphere_ind = ((mesh.gridCC[:, 0]**2 +
(mesh.gridCC[:, 2] - sphere_z)**2) <= sphere_radius**2)
sigma[sphere_ind] = sig_sphere # sphere
# Plot a cross section through the mesh
fig, ax = plt.subplots(2, 1)
plt.colorbar(mesh.plotImage(np.log10(sigma), ax=ax[0])[0], ax=ax[0])
ax[0].set_title('mirror = False')
ax[0].axis('equal')
ax[0].set_xlim([-200., 200.])
plt.colorbar(mesh.plotImage(np.log10(sigma), ax=ax[1], mirror=True)[0],
ax=ax[1])
ax[1].set_title('mirror = True')
ax[1].axis('equal')
ax[1].set_xlim([-200., 200.])
plt.tight_layout()
def __init__(self, h_in, x0_in=None):
assert type(h_in) in [list, tuple], 'h_in must be a list'
assert len(h_in) in [1, 2, 3], 'h_in must be of dimension 1, 2, or 3'
h = list(range(len(h_in)))
for i, h_i in enumerate(h_in):
if Utils.isScalar(h_i) and type(h_i) is not np.ndarray:
# This gives you something over the unit cube.
h_i = self._unitDimensions[i] * np.ones(int(h_i)) / int(h_i)
elif type(h_i) is list:
h_i = Utils.meshTensor(h_i)
assert isinstance(
h_i, np.ndarray), ("h[{0:d}] is not a numpy array.".format(i))
assert len(h_i.shape) == 1, (
"h[{0:d}] must be a 1D numpy array.".format(i))
h[i] = h_i[:] # make a copy.
x0 = np.zeros(len(h))
if x0_in is not None:
assert len(h) == len(x0_in), "Dimension mismatch. x0 != len(h)"
for i in range(len(h)):
x_i, h_i = x0_in[i], h[i]
if Utils.isScalar(x_i):
x0[i] = x_i
elif x_i == '0':
x0[i] = 0.0
elif x_i == 'C':
x0[i] = -h_i.sum() * 0.5
elif x_i == 'N':
x0[i] = -h_i.sum()
else:
raise Exception(
"x0[{0:d}] must be a scalar or '0' to be zero, "
"'C' to center, or 'N' to be negative.".format(i))
if isinstance(self, BaseRectangularMesh):
BaseRectangularMesh.__init__(self, np.array([x.size for x in h]),
x0)
else:
BaseMesh.__init__(self, np.array([x.size for x in h]), x0)
# Ensure h contains 1D vectors
self._h = [Utils.mkvc(x.astype(float)) for x in h]
def get_3d_mesh(
self, dx=None, dy=None, dz=None,
npad_x=0, npad_y=0, npad_z=0,
core_z_length=None,
nx=100,
ny=100,
):
xmin, xmax = self.topography[:, 0].min(), self.topography[:, 0].max()
ymin, ymax = self.topography[:, 1].min(), self.topography[:, 1].max()
zmin, zmax = self.topography[:, 2].min(), self.topography[:, 2].max()
zmin -= self.mesh_1d.vectorNx.max()
lx = xmax-xmin
ly = ymax-ymin
lz = zmax-zmin
if dx is None:
dx = lx/nx
print ((">> dx:%.1e")%(dx))
if dy is None:
dy = ly/ny
print ((">> dx:%.1e")%(dy))
if dz is None:
dz = np.median(self.mesh_1d.hx)
nx = int(np.floor(lx/dx))
ny = int(np.floor(ly/dy))
nz = int(np.floor(lz/dz))
if nx*ny*nz > 1e6:
warnings.warn(
("Size of the mesh (%i) will greater than 1e6")%(nx*ny*nz)
)
hx = [(dx, npad_x, -1.2), (dx, nx), (dx, npad_x, -1.2)]
hy = [(dy, npad_y, -1.2), (dy, ny), (dy, npad_y, -1.2)]
hz = [(dz, npad_z, -1.2), (dz, nz)]
zmin = self.topography[:, 2].max() - Utils.meshTensor(hz).sum()
self._mesh_3d = Mesh.TensorMesh([hx, hy, hz], x0=[xmin, ymin, zmin])
return self.mesh_3d
def run(plotIt=True):
"""
Maps: Mesh2Mesh
===============
This mapping allows you to go from one mesh to another.
"""
M = Mesh.TensorMesh([100, 100])
h1 = Utils.meshTensor([(6, 7, -1.5), (6, 10), (6, 7, 1.5)])
h1 = h1 / h1.sum()
M2 = Mesh.TensorMesh([h1, h1])
V = Utils.ModelBuilder.randomModel(M.vnC, seed=79, its=50)
v = Utils.mkvc(V)
modh = Maps.Mesh2Mesh([M, M2])
modH = Maps.Mesh2Mesh([M2, M])
H = modH * v
h = modh * H
if not plotIt:
return
import matplotlib.pyplot as plt
ax = plt.subplot(131)
M.plotImage(v, ax=ax)
ax.set_title("Fine Mesh (Original)")
ax = plt.subplot(132)
M2.plotImage(H, clim=[0, 1], ax=ax)
ax.set_title("Course Mesh")
ax = plt.subplot(133)
M.plotImage(h, clim=[0, 1], ax=ax)
ax.set_title("Fine Mesh (Interpolated)")
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()
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):
"""
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):
# ------------------ 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):
# ------------------ 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):
# ## 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
# sphinx_gallery_thumbnail_number = 3
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()
def test_CylMeshEBDipoles(self):
print(
"Testing CylMesh Electric and Magnetic Dipoles in a wholespace-"
" Analytic: J-formulation")
sigmaback = 1.
mur = 2.
freq = 1.
skdpth = 500. / np.sqrt(sigmaback * freq)
csx, ncx, npadx = 5, 50, 25
csz, ncz, npadz = 5, 50, 25
hx = Utils.meshTensor([(csx, ncx), (csx, npadx, 1.3)])
hz = Utils.meshTensor([(csz, npadz, -1.3), (csz, ncz),
(csz, npadz, 1.3)])
mesh = Mesh.CylMesh(
[hx, 1, hz],
[0., 0., -hz.sum() / 2]) # define the cylindrical mesh
if plotIt:
mesh.plotGrid()
# make sure mesh is big enough
self.assertTrue(mesh.hz.sum() > skdpth * 2.)
self.assertTrue(mesh.hx.sum() > skdpth * 2.)
SigmaBack = sigmaback * np.ones((mesh.nC))
MuBack = mur * mu_0 * np.ones((mesh.nC))
# set up source
# test electric dipole
src_loc = np.r_[0., 0., 0.]
s_ind = Utils.closestPoints(mesh, src_loc, 'Fz') + mesh.nFx
de = np.zeros(mesh.nF, dtype=complex)
de[s_ind] = 1. / csz
de_p = [EM.FDEM.Src.RawVec_e([], freq, de / mesh.area)]
dm_p = [EM.FDEM.Src.MagDipole([], freq, src_loc)]
# Pair the problem and survey
surveye = EM.FDEM.Survey(de_p)
surveym = EM.FDEM.Survey(dm_p)
mapping = [('sigma', Maps.IdentityMap(mesh)),
('mu', Maps.IdentityMap(mesh))]
prbe = EM.FDEM.Problem3D_h(mesh, mapping=mapping)
prbm = EM.FDEM.Problem3D_e(mesh, mapping=mapping)
prbe.pair(surveye) # pair problem and survey
prbm.pair(surveym)
# solve
fieldsBackE = prbe.fields(np.r_[SigmaBack, MuBack]) # Done
fieldsBackM = prbm.fields(np.r_[SigmaBack, MuBack]) # Done
rlim = [20., 500.]
lookAtTx = de_p
r = mesh.vectorCCx[np.argmin(np.abs(mesh.vectorCCx - rlim[0])):np.
argmin(np.abs(mesh.vectorCCx - rlim[1]))]
z = 100.
# where we choose to measure
XYZ = Utils.ndgrid(r, np.r_[0.], np.r_[z])
Pf = mesh.getInterpolationMat(XYZ, 'CC')
Zero = sp.csr_matrix(Pf.shape)
Pfx, Pfz = sp.hstack([Pf, Zero]), sp.hstack([Zero, Pf])
jn = fieldsBackE[de_p, 'j']
bn = fieldsBackM[dm_p, 'b']
Rho = Utils.sdiag(1. / SigmaBack)
Rho = sp.block_diag([Rho, Rho])
en = Rho * mesh.aveF2CCV * jn
bn = mesh.aveF2CCV * bn
ex, ez = Pfx * en, Pfz * en
bx, bz = Pfx * bn, Pfz * bn
# get analytic solution
exa, eya, eza = EM.Analytics.FDEM.ElectricDipoleWholeSpace(
XYZ, src_loc, sigmaback, freq, orientation='Z', mu=mur * mu_0)
exa, eya, eza = Utils.mkvc(exa, 2), Utils.mkvc(eya,
2), Utils.mkvc(eza, 2)
bxa, bya, bza = EM.Analytics.FDEM.MagneticDipoleWholeSpace(
XYZ, src_loc, sigmaback, freq, orientation='Z', mu=mur * mu_0)
bxa, bya, bza = Utils.mkvc(bxa, 2), Utils.mkvc(bya,
2), Utils.mkvc(bza, 2)
print ' comp, anayltic, numeric, num - ana, (num - ana)/ana'
print ' ex:', np.linalg.norm(exa), np.linalg.norm(ex), np.linalg.norm(
exa - ex), np.linalg.norm(exa - ex) / np.linalg.norm(exa)
print ' ez:', np.linalg.norm(eza), np.linalg.norm(ez), np.linalg.norm(
eza - ez), np.linalg.norm(eza - ez) / np.linalg.norm(eza)
print ' bx:', np.linalg.norm(bxa), np.linalg.norm(bx), np.linalg.norm(
bxa - bx), np.linalg.norm(bxa - bx) / np.linalg.norm(bxa)
print ' bz:', np.linalg.norm(bza), np.linalg.norm(bz), np.linalg.norm(
bza - bz), np.linalg.norm(bza - bz) / np.linalg.norm(bza)
if plotIt is True:
# Edipole
plt.subplot(221)
plt.plot(r, ex.real, 'o', r, exa.real, linewidth=2)
plt.grid(which='both')
plt.title('Ex Real')
plt.xlabel('r (m)')
plt.subplot(222)
plt.plot(r, ex.imag, 'o', r, exa.imag, linewidth=2)
plt.grid(which='both')
plt.title('Ex Imag')
plt.legend(['Num', 'Ana'], bbox_to_anchor=(1.5, 0.5))
plt.xlabel('r (m)')
plt.subplot(223)
plt.plot(r, ez.real, 'o', r, eza.real, linewidth=2)
plt.grid(which='both')
plt.title('Ez Real')
plt.xlabel('r (m)')
plt.subplot(224)
plt.plot(r, ez.imag, 'o', r, eza.imag, linewidth=2)
plt.grid(which='both')
plt.title('Ez Imag')
plt.xlabel('r (m)')
plt.tight_layout()
# Bdipole
plt.subplot(221)
plt.plot(r, bx.real, 'o', r, bxa.real, linewidth=2)
plt.grid(which='both')
plt.title('Bx Real')
plt.xlabel('r (m)')
plt.subplot(222)
plt.plot(r, bx.imag, 'o', r, bxa.imag, linewidth=2)
plt.grid(which='both')
plt.title('Bx Imag')
plt.legend(['Num', 'Ana'], bbox_to_anchor=(1.5, 0.5))
plt.xlabel('r (m)')
plt.subplot(223)
plt.plot(r, bz.real, 'o', r, bza.real, linewidth=2)
plt.grid(which='both')
plt.title('Bz Real')
plt.xlabel('r (m)')
plt.subplot(224)
plt.plot(r, bz.imag, 'o', r, bza.imag, linewidth=2)
plt.grid(which='both')
plt.title('Bz Imag')
plt.xlabel('r (m)')
plt.tight_layout()
self.assertTrue(
np.linalg.norm(exa - ex) / np.linalg.norm(exa) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(eza - ez) / np.linalg.norm(eza) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(bxa - bx) / np.linalg.norm(bxa) < tol_EBdipole)
self.assertTrue(
np.linalg.norm(bza - bz) / np.linalg.norm(bza) < tol_EBdipole)
def test_CylMeshEBDipoles(self):
print ("Testing CylMesh Electric and Magnetic Dipoles in a wholespace-"
" Analytic: J-formulation")
sigmaback = 1.
mur = 2.
freq = 1.
skdpth = 500./np.sqrt(sigmaback*freq)
csx, ncx, npadx = 5, 50, 25
csz, ncz, npadz = 5, 50, 25
hx = Utils.meshTensor([(csx,ncx), (csx,npadx,1.3)])
hz = Utils.meshTensor([(csz,npadz,-1.3), (csz,ncz), (csz,npadz,1.3)])
mesh = Mesh.CylMesh([hx, 1, hz], [0., 0., -hz.sum()/2]) # define the cylindrical mesh
if plotIt:
mesh.plotGrid()
# make sure mesh is big enough
self.assertTrue(mesh.hz.sum() > skdpth*2.)
self.assertTrue(mesh.hx.sum() > skdpth*2.)
SigmaBack = sigmaback*np.ones((mesh.nC))
MuBack = mur*mu_0*np.ones((mesh.nC))
# set up source
# test electric dipole
src_loc = np.r_[0., 0., 0.]
s_ind = Utils.closestPoints(mesh, src_loc, 'Fz') + mesh.nFx
de = np.zeros(mesh.nF, dtype=complex)
de[s_ind] = 1./csz
de_p = [EM.FDEM.Src.RawVec_e([], freq, de/mesh.area)]
dm_p = [EM.FDEM.Src.MagDipole([], freq, src_loc)]
# Pair the problem and survey
surveye = EM.FDEM.Survey(de_p)
surveym = EM.FDEM.Survey(dm_p)
mapping = [('sigma', Maps.IdentityMap(mesh)),
('mu', Maps.IdentityMap(mesh))]
prbe = EM.FDEM.Problem3D_h(mesh, mapping=mapping)
prbm = EM.FDEM.Problem3D_e(mesh, mapping=mapping)
prbe.pair(surveye) # pair problem and survey
prbm.pair(surveym)
# solve
fieldsBackE = prbe.fields(np.r_[SigmaBack, MuBack]) # Done
fieldsBackM = prbm.fields(np.r_[SigmaBack, MuBack]) # Done
rlim = [20., 500.]
lookAtTx = de_p
r = mesh.vectorCCx[np.argmin(np.abs(mesh.vectorCCx-rlim[0])) : np.argmin(np.abs(mesh.vectorCCx-rlim[1]))]
z = 100.
# where we choose to measure
XYZ = Utils.ndgrid(r, np.r_[0.], np.r_[z])
Pf = mesh.getInterpolationMat(XYZ, 'CC')
Zero = sp.csr_matrix(Pf.shape)
Pfx,Pfz = sp.hstack([Pf,Zero]),sp.hstack([Zero,Pf])
jn = fieldsBackE[de_p,'j']
bn = fieldsBackM[dm_p,'b']
Rho = Utils.sdiag(1./SigmaBack)
Rho = sp.block_diag([Rho,Rho])
en = Rho*mesh.aveF2CCV*jn
bn = mesh.aveF2CCV*bn
ex, ez = Pfx*en, Pfz*en
bx, bz = Pfx*bn, Pfz*bn
# get analytic solution
exa, eya, eza = EM.Analytics.FDEM.ElectricDipoleWholeSpace(XYZ, src_loc, sigmaback, freq,orientation='Z',mu= mur*mu_0)
exa, eya, eza = Utils.mkvc(exa, 2), Utils.mkvc(eya, 2), Utils.mkvc(eza, 2)
bxa, bya, bza = EM.Analytics.FDEM.MagneticDipoleWholeSpace(XYZ, src_loc, sigmaback, freq,orientation='Z',mu= mur*mu_0)
bxa, bya, bza = Utils.mkvc(bxa, 2), Utils.mkvc(bya, 2), Utils.mkvc(bza, 2)
print(' comp, anayltic, numeric, num - ana, (num - ana)/ana')
print(' ex:', np.linalg.norm(exa), np.linalg.norm(ex), np.linalg.norm(exa-ex), np.linalg.norm(exa-ex)/np.linalg.norm(exa))
print(' ez:', np.linalg.norm(eza), np.linalg.norm(ez), np.linalg.norm(eza-ez), np.linalg.norm(eza-ez)/np.linalg.norm(eza))
print(' bx:', np.linalg.norm(bxa), np.linalg.norm(bx), np.linalg.norm(bxa-bx), np.linalg.norm(bxa-bx)/np.linalg.norm(bxa))
print(' bz:', np.linalg.norm(bza), np.linalg.norm(bz), np.linalg.norm(bza-bz), np.linalg.norm(bza-bz)/np.linalg.norm(bza))
if plotIt is True:
# Edipole
plt.subplot(221)
plt.plot(r,ex.real,'o',r,exa.real,linewidth=2)
plt.grid(which='both')
plt.title('Ex Real')
plt.xlabel('r (m)')
plt.subplot(222)
plt.plot(r,ex.imag,'o',r,exa.imag,linewidth=2)
plt.grid(which='both')
plt.title('Ex Imag')
plt.legend(['Num','Ana'],bbox_to_anchor=(1.5,0.5))
plt.xlabel('r (m)')
plt.subplot(223)
plt.plot(r,ez.real,'o',r,eza.real,linewidth=2)
plt.grid(which='both')
plt.title('Ez Real')
plt.xlabel('r (m)')
plt.subplot(224)
plt.plot(r,ez.imag,'o',r,eza.imag,linewidth=2)
plt.grid(which='both')
plt.title('Ez Imag')
plt.xlabel('r (m)')
plt.tight_layout()
# Bdipole
plt.subplot(221)
plt.plot(r,bx.real,'o',r,bxa.real,linewidth=2)
plt.grid(which='both')
plt.title('Bx Real')
plt.xlabel('r (m)')
plt.subplot(222)
plt.plot(r,bx.imag,'o',r,bxa.imag,linewidth=2)
plt.grid(which='both')
plt.title('Bx Imag')
plt.legend(['Num','Ana'],bbox_to_anchor=(1.5,0.5))
plt.xlabel('r (m)')
plt.subplot(223)
plt.plot(r,bz.real,'o',r,bza.real,linewidth=2)
plt.grid(which='both')
plt.title('Bz Real')
plt.xlabel('r (m)')
plt.subplot(224)
plt.plot(r,bz.imag,'o',r,bza.imag,linewidth=2)
plt.grid(which='both')
plt.title('Bz Imag')
plt.xlabel('r (m)')
plt.tight_layout()
self.assertTrue(np.linalg.norm(exa-ex)/np.linalg.norm(exa) < tol_EBdipole)
self.assertTrue(np.linalg.norm(eza-ez)/np.linalg.norm(eza) < tol_EBdipole)
self.assertTrue(np.linalg.norm(bxa-bx)/np.linalg.norm(bxa) < tol_EBdipole)
self.assertTrue(np.linalg.norm(bza-bz)/np.linalg.norm(bza) < tol_EBdipole)
def run_simulation(fname="tdem_vmd.h5", sigma_halfspace=0.01, src_type="VMD"):
from SimPEG.EM import TDEM, Analytics, mu_0
import numpy as np
from SimPEG import Mesh, Maps, Utils, EM, Survey
from pymatsolver import Pardiso
cs = 20.0
ncx, ncy, ncz = 5, 3, 4
npad = 10
npadz = 10
pad_rate = 1.3
hx = [(cs, npad, -pad_rate), (cs, ncx), (cs, npad, pad_rate)]
hy = [(cs, npad, -pad_rate), (cs, ncy), (cs, npad, pad_rate)]
hz = Utils.meshTensor([(cs, npadz, -1.3), (cs / 2.0, ncz), (cs, 5, 2)])
mesh = Mesh.TensorMesh([hx, hy, hz],
x0=["C", "C", -hz[:int(npadz + ncz / 2)].sum()])
sigma = np.ones(mesh.nC) * sigma_halfspace
sigma[mesh.gridCC[:, 2] > 0.0] = 1e-8
xmin, xmax = -600.0, 600.0
ymin, ymax = -600.0, 600.0
zmin, zmax = -600, 100.0
times = np.logspace(-5, -2, 21)
rxList = EM.TDEM.Rx.Point_dbdt(np.r_[10.0, 0.0, 30.0],
times,
orientation="z")
if src_type == "VMD":
src = EM.TDEM.Src.CircularLoop(
[rxList],
loc=np.r_[0.0, 0.0, 30.0],
orientation="Z",
waveform=EM.TDEM.Src.StepOffWaveform(),
radius=13.0,
)
elif src_type == "HMD":
src = EM.TDEM.Src.MagDipole(
[rxList],
loc=np.r_[0.0, 0.0, 30.0],
orientation="X",
waveform=EM.TDEM.Src.StepOffWaveform(),
)
SrcList = [src]
survey = EM.TDEM.Survey(SrcList)
sig = 1e-2
sigma = np.ones(mesh.nC) * sig
sigma[mesh.gridCC[:, 2] > 0] = 1e-8
prb = EM.TDEM.Problem3D_b(mesh,
sigmaMap=Maps.IdentityMap(mesh),
verbose=True)
prb.pair(survey)
prb.Solver = Pardiso
prb.timeSteps = [
(1e-06, 5),
(5e-06, 5),
(1e-05, 10),
(5e-05, 10),
(1e-4, 15),
(5e-4, 16),
]
f = prb.fields(sigma)
xyzlim = np.array([[xmin, xmax], [ymin, ymax], [zmin, zmax]])
actinds, meshCore = Utils.ExtractCoreMesh(xyzlim, mesh)
Pex = mesh.getInterpolationMat(meshCore.gridCC, locType="Ex")
Pey = mesh.getInterpolationMat(meshCore.gridCC, locType="Ey")
Pez = mesh.getInterpolationMat(meshCore.gridCC, locType="Ez")
Pfx = mesh.getInterpolationMat(meshCore.gridCC, locType="Fx")
Pfy = mesh.getInterpolationMat(meshCore.gridCC, locType="Fy")
Pfz = mesh.getInterpolationMat(meshCore.gridCC, locType="Fz")
sigma_core = sigma[actinds]
def getEBJcore(src0):
B0 = np.r_[Pfx * f[src0, "b"], Pfy * f[src0, "b"], Pfz * f[src0, "b"]]
E0 = np.r_[Pex * f[src0, "e"], Pey * f[src0, "e"], Pez * f[src0, "e"]]
J0 = Utils.sdiag(np.r_[sigma_core, sigma_core, sigma_core]) * E0
return E0, B0, J0
E, B, J = getEBJcore(src)
tdem_is = {
"E": E,
"B": B,
"J": J,
"sigma": sigma_core,
"mesh": meshCore.serialize(),
"time": prb.times,
}
dd.io.save(fname, tdem_is)
