def run(plotIt=True):
M = discretize.TensorMesh([100, 100])
h1 = utils.meshTensor([(6, 7, -1.5), (6, 10), (6, 7, 1.5)])
h1 = h1 / h1.sum()
M2 = discretize.TensorMesh([h1, h1])
V = utils.model_builder.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):
# ------------------ 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 / 2.0 # inner radius
casing_b = casing_r + casing_t / 2.0 # outer radius
casing_z = np.r_[-casing_l, 0.0]
# ------------------ 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, 0.0, dsz]
inf_loc = np.r_[0.0, 0.0, 1e4]
print("Skin Depth: ", [(500.0 / np.sqrt(sigmaback * _)) for _ in freqs])
# ------------------ MESH ------------------
# fine cells near well bore
csx1, csx2 = 2e-3, 60.0
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 = 300.0 # 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 = discretize.CylMesh(
[hx, 1.0, hz], [0.0, 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] > 0.0] = 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, 0.2]
zlim = np.r_[-350.0, 10.0]
clim_sig = np.r_[-8, 6]
# plot models
fig, ax = plt.subplots(1, 1, figsize=(4, 4))
im = mesh.plotImage(np.log10(sigCasing), ax=ax)[0]
im.set_clim(clim_sig)
plt.colorbar(im, ax=ax)
ax.grid(which="both")
ax.set_title("Log_10 (Sigma)")
ax.set_xlim(xlim)
ax.set_ylim(zlim)
# -------------- 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] = -1.0
# 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] = -1.0
# 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] = 1.0
# 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 / 2.0)
dgv_ind = dgv_indx & dgv_indz
dg_z[dgv_ind] = -1.0
# 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] = 1.0
# 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] = -1.0
# vertical part at surface
dgv2_ind = sgv2_ind.copy()
dg_z[dgv2_ind] = 1.0
# 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.Simulation3DMagneticField(
mesh, survey=survey, sigmaMap=maps.IdentityMap(mesh), solver=Solver
)
# ------------- 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, 1.0])
def test_CylMeshEBDipoles(self, plotIt=plotIt):
print("Testing CylMesh Electric and Magnetic Dipoles in a wholespace-"
" Analytic: J-formulation")
sigmaback = 1.0
mur = 2.0
freq = 1.0
skdpth = 500.0 / 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)])
# define the cylindrical mesh
mesh = discretize.CylMesh([hx, 1, hz], [0.0, 0.0, -hz.sum() / 2])
if plotIt:
mesh.plotGrid()
# make sure mesh is big enough
self.assertTrue(mesh.hz.sum() > skdpth * 2.0)
self.assertTrue(mesh.hx.sum() > skdpth * 2.0)
# set up source
# test electric dipole
src_loc = np.r_[0.0, 0.0, 0.0]
s_ind = utils.closestPoints(mesh, src_loc, "Fz") + mesh.nFx
de = np.zeros(mesh.nF, dtype=complex)
de[s_ind] = 1.0 / csz
de_p = [fdem.Src.RawVec_e([], freq, de / mesh.area)]
dm_p = [fdem.Src.MagDipole([], freq, src_loc)]
# Pair the problem and survey
surveye = fdem.Survey(de_p)
surveym = fdem.Survey(dm_p)
prbe = fdem.Simulation3DMagneticField(mesh,
survey=surveye,
sigma=sigmaback,
mu=mur * mu_0)
prbm = fdem.Simulation3DElectricField(mesh,
survey=surveym,
sigma=sigmaback,
mu=mur * mu_0)
# solve
fieldsBackE = prbe.fields()
fieldsBackM = prbm.fields()
rlim = [20.0, 500.0]
# lookAtTx = de_p
r = mesh.vectorCCx[np.argmin(np.abs(mesh.vectorCCx - rlim[0])):np.
argmin(np.abs(mesh.vectorCCx - rlim[1]))]
z = 100.0
# where we choose to measure
XYZ = utils.ndgrid(r, np.r_[0.0], 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"]
SigmaBack = sigmaback * np.ones((mesh.nC))
Rho = utils.sdiag(1.0 / 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 = analytics.FDEM.ElectricDipoleWholeSpace(XYZ,
src_loc,
sigmaback,
freq,
"Z",
mu_r=mur)
exa = utils.mkvc(exa, 2)
eya = utils.mkvc(eya, 2)
eza = utils.mkvc(eza, 2)
bxa, bya, bza = analytics.FDEM.MagneticDipoleWholeSpace(XYZ,
src_loc,
sigmaback,
freq,
"Z",
mu_r=mur)
bxa = utils.mkvc(bxa, 2)
bya = utils.mkvc(bya, 2)
bza = 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.electromagnetics import time_domain
from scipy.constants import mu_0
import numpy as np
from SimPEG import maps
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 = 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 = time_domain.receivers.PointMagneticFluxTimeDerivative(
np.r_[10.0, 0.0, 30.0], times, orientation="z")
if src_type == "VMD":
src = time_domain.sources.CircularLoop(
[rxList],
loc=np.r_[0.0, 0.0, 30.0],
orientation="Z",
waveform=time_domain.sources.StepOffWaveform(),
radius=13.0,
)
elif src_type == "HMD":
src = time_domain.sources.MagDipole(
[rxList],
loc=np.r_[0.0, 0.0, 30.0],
orientation="X",
waveform=time_domain.sources.StepOffWaveform(),
)
SrcList = [src]
survey = time_domain.Survey(SrcList)
sig = 1e-2
sigma = np.ones(mesh.nC) * sig
sigma[mesh.gridCC[:, 2] > 0] = 1e-8
prb = time_domain.Simulation3DMagneticFluxDensity(
mesh,
sigmaMap=maps.IdentityMap(mesh),
verbose=True,
survey=survey,
solver=Pardiso,
)
prb.time_steps = [
(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)