def setUpClass(self): print('\n------- Testing Primary Secondary Source HJ -> EB --------\n') # receivers self.rxlist = [] for rxtype in ['b', 'e']: rx = getattr(FDEM.Rx, 'Point_{}'.format(rxtype)) for orientation in ['x', 'y', 'z']: for comp in ['real', 'imag']: self.rxlist.append(rx(rx_locs, component=comp, orientation=orientation)) # primary self.primaryProblem = FDEM.Problem3D_j(meshp, sigmaMap=primaryMapping) self.primaryProblem.solver = Solver s_e = np.zeros(meshp.nF) inds = meshp.nFx + Utils.closestPoints(meshp, src_loc, gridLoc='Fz') s_e[inds] = 1./csz primarySrc = FDEM.Src.RawVec_e( self.rxlist, freq=freq, s_e=s_e/meshp.area ) self.primarySurvey = FDEM.Survey([primarySrc]) # Secondary Problem self.secondaryProblem = FDEM.Problem3D_e(meshs, sigmaMap=mapping) self.secondaryProblem.Solver = Solver self.secondarySrc = FDEM.Src.PrimSecMappedSigma( self.rxlist, freq, self.primaryProblem, self.primarySurvey, primaryMap2Meshs ) self.secondarySurvey = FDEM.Survey([self.secondarySrc]) self.secondaryProblem.pair(self.secondarySurvey) # Full 3D problem to compare with self.problem3D = FDEM.Problem3D_e(meshs, sigmaMap=mapping) self.problem3D.Solver = Solver s_e3D = np.zeros(meshs.nE) inds = (meshs.nEx + meshs.nEy + Utils.closestPoints(meshs, src_loc, gridLoc='Ez')) s_e3D[inds] = [1./(len(inds))] * len(inds) self.problem3D.model = model src3D = FDEM.Src.RawVec_e(self.rxlist, freq=freq, s_e=s_e3D) self.survey3D = FDEM.Survey([src3D]) self.problem3D.pair(self.survey3D) # solve and store fields print(' solving primary - secondary') self.fields_primsec = self.secondaryProblem.fields(model) print(' ... done') self.fields_primsec = self.secondaryProblem.fields(model) print(' solving 3D') self.fields_3D = self.problem3D.fields(model) print(' ... done') return None
def setupSecondarySurvey( self, primaryProblem, primarySurvey, map2meshSecondary ): print('Setting up Secondary Survey') nx = 41 ny = nx rx_x, rx_y = 2*[np.linspace(-2050, 2050, nx)] self.rxlocs = Utils.ndgrid([rx_x, rx_y, np.r_[-1]]) self.rx_x = self.rxlocs[:, 0].reshape(nx, ny, order='F') self.rx_y = self.rxlocs[:, 1].reshape(nx, ny, order='F') rx_ex = FDEM.Rx.Point_e(self.rxlocs, orientation='x', component='real') rx_ey = FDEM.Rx.Point_e(self.rxlocs, orientation='y', component='real') RxList = [rx_ex, rx_ey] sec_src = [ FDEM.Src.PrimSecMappedSigma( RxList, freq, primaryProblem, primarySurvey, map2meshSecondary=map2meshSecondary ) for freq in self.freqs ] print('... done secondary survey') return FDEM.Survey(sec_src)
def prob(self): if getattr(self, '_prob', None) is None: self._prob = getattr(FDEM, 'Problem3D_{}'.format( self.formulation))(self.meshGenerator.mesh, sigmaMap=self.physprops.wires.sigma, muMap=self.physprops.wires.mu, Solver=Solver, verbose=self.verbose) if getattr(self, 'srcList') is not None: self._survey = FDEM.Survey(self.srcList.srcList) elif getattr(self, 'src') is not None: self._survey = FDEM.Survey(self.src.srcList) else: raise Exception("one of src, srcList must be set") self._prob.pair(self._survey) return self._prob
def setUpClass(self): print('\n------- Testing Primary Secondary Source EB -> EB --------\n') # receivers self.rxlist = [] for rxtype in ['b', 'e']: rx = getattr(FDEM.Rx, 'Point_{}'.format(rxtype)) for orientation in ['x', 'y', 'z']: for comp in ['real', 'imag']: self.rxlist.append( rx(rx_locs, component=comp, orientation=orientation)) # primary self.primaryProblem = FDEM.Problem3D_b(meshp, sigmaMap=primaryMapping) self.primaryProblem.solver = Solver primarySrc = FDEM.Src.MagDipole(self.rxlist, freq=freq, loc=src_loc) self.primarySurvey = FDEM.Survey([primarySrc]) # Secondary Problem self.secondaryProblem = FDEM.Problem3D_b(meshs, sigmaMap=mapping) self.secondaryProblem.Solver = Solver self.secondarySrc = FDEM.Src.PrimSecMappedSigma( self.rxlist, freq, self.primaryProblem, self.primarySurvey, primaryMap2Meshs) self.secondarySurvey = FDEM.Survey([self.secondarySrc]) self.secondaryProblem.pair(self.secondarySurvey) # Full 3D problem to compare with self.problem3D = FDEM.Problem3D_b(meshs, sigmaMap=mapping) self.problem3D.Solver = Solver self.survey3D = FDEM.Survey([primarySrc]) self.problem3D.pair(self.survey3D) # solve and store fields print(' solving primary - secondary') self.fields_primsec = self.secondaryProblem.fields(model) print(' ... done') self.fields_primsec = self.secondaryProblem.fields(model) print(' solving 3D') self.fields_3D = self.problem3D.fields(model) print(' ... done') return None
def __init__(self, **kwargs): super(SimulationFDEM, self).__init__(**kwargs) self._prob = getattr( FDEM, 'Problem3D_{}'.format(self.formulation) )( self.meshGenerator.mesh, sigmaMap=self.physprops.wires.sigma, muMap=self.physprops.wires.mu, Solver=Pardiso ) if getattr(self.src, "physics", None) is None: self.src.physics = "FDEM" self._survey = FDEM.Survey(self.src.srcList) self._prob.pair(self._survey)
def 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 primarySurvey(self): if getattr(self, '_primarySurvey', None) is None: print('Setting up primary survey') def setupPrimarySource(plotIt=False): # Construct a downhole source that is coupled to the casing meshp = self.meshp src_a = self.src_a src_b = self.src_b casing_a = self.casing_a # downhole source dg_x = np.zeros(meshp.vnF[0], dtype=complex) dg_y = np.zeros(meshp.vnF[1], dtype=complex) dg_z = np.zeros(meshp.vnF[2], dtype=complex) # vertically directed wire in borehole # go through the center of the well dgv_indx = (meshp.gridFz[:, 0] < meshp.hx.min()) dgv_indz = ( (meshp.gridFz[:, 2] >= src_a[2]) & (meshp.gridFz[:, 2] <= src_b[2]) ) dgv_ind = dgv_indx & dgv_indz dg_z[dgv_ind] = -1. # couple to the casing downhole - top part dgh_indx = meshp.gridFx[:, 0] <= casing_a + meshp.hx.min()*2 # couple to the casing downhole - bottom part dgh_indz2 = ( (meshp.gridFx[:, 2] <= src_a[2]) & (meshp.gridFx[:, 2] > src_a[2] - meshp.hz.min()) ) dgh_ind2 = dgh_indx & dgh_indz2 dg_x[dgh_ind2] = 1. # horizontally directed wire sgh_indx = (meshp.gridFx[:, 0] <= src_b[0]) sgh_indz = ( (meshp.gridFx[:, 2] > meshp.hz.min()) & (meshp.gridFx[:, 2] < 2*meshp.hz.min()) ) sgh_ind = sgh_indx & sgh_indz dg_x[sgh_ind] = -1. # return electrode sgv_indx = ( (meshp.gridFz[:, 0] > src_b[0]*0.9) & (meshp.gridFz[:, 0] < src_b[0]*1.1) ) sgv_indz = ( (meshp.gridFz[:, 2] >= -meshp.hz.min()) & (meshp.gridFz[:, 2] < 2*meshp.hz.min()) ) sgv_ind = sgv_indx & sgv_indz dg_z[sgv_ind] = 1. # assemble the source (downhole grounded primary) dg = np.hstack([dg_x, dg_y, dg_z]) dg_p = [ FDEM.Src.RawVec_e([], _, dg/meshp.area) for _ in self.freqs ] # if plotIt: # # Plot the source to make sure the path is infact # # connected # fig, ax = plt.subplots(1, 1, figsize=(6, 4)) # meshp.plotGrid(ax=ax) # ax.plot(meshp.gridFz[dgv_ind, 0], meshp.gridFz[dgv_ind, 2], 'rd') # ax.plot(meshp.gridFx[dgh_ind2, 0], meshp.gridFx[dgh_ind2, 2], 'rd') # ax.plot(meshp.gridFz[sgv_ind, 0], meshp.gridFz[sgv_ind, 2], 'rd') # ax.plot(meshp.gridFx[sgh_ind, 0], meshp.gridFx[sgh_ind, 2], 'rd') # ax.set_title('downhole casing source on mesh') # ax.set_xlim([0, 1.1e4]) # ax.set_ylim([-1100., 0.5]) return dg_p srcList = setupPrimarySource() # create primary source self._primarySurvey = FDEM.Survey(srcList) # primary survey print('... done building primary survey') return self._primarySurvey
def run(plotIt=True, saveFig=False): # Set up cylindrically symmeric mesh cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh hx = [(cs, ncx), (cs, npad, 1.3)] hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)] mesh = Mesh.CylMesh([hx, 1, hz], '00C') # Conductivity model layerz = np.r_[-200., -100.] layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1]) active = mesh.vectorCCz < 0. sig_half = 1e-2 # Half-space conductivity sig_air = 1e-8 # Air conductivity sig_layer = 5e-2 # Layer conductivity sigma = np.ones(mesh.nCz) * sig_air sigma[active] = sig_half sigma[layer] = sig_layer # Mapping actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz) mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap mtrue = np.log(sigma[active]) # ----- FDEM problem & survey ----- # rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]]) bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real') bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag') freqs = np.logspace(2, 3, 5) srcLoc = np.array([0., 0., 0.]) print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half), 'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half)) print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(), 'max z ', mesh.vectorCCz.max()) srcList = [ FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z') for freq in freqs ] surveyFD = FDEM.Survey(srcList) prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver) prbFD.pair(surveyFD) std = 0.03 surveyFD.makeSyntheticData(mtrue, std) surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-5 # FDEM inversion np.random.seed(1) dmisfit = DataMisfit.l2_DataMisfit(surveyFD) regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]]) reg = Regularization.Simple(regMesh) opt = Optimization.InexactGaussNewton(maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt) # Inversion Directives beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) target = Directives.TargetMisfit() directiveList = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=directiveList) m0 = np.log(np.ones(mtrue.size) * sig_half) reg.alpha_s = 5e-1 reg.alpha_x = 1. prbFD.counter = opt.counter = Utils.Counter() opt.remember('xc') moptFD = inv.run(m0) # TDEM problem times = np.logspace(-4, np.log10(2e-3), 10) print('min diffusion distance ', 1.28 * np.sqrt(times.min() / (sig_half * mu_0)), 'max diffusion distance ', 1.28 * np.sqrt(times.max() / (sig_half * mu_0))) rx = TDEM.Rx.Point_b(rxlocs, times, 'z') src = TDEM.Src.MagDipole( [rx], waveform=TDEM.Src.StepOffWaveform(), loc=srcLoc # same src location as FDEM problem ) surveyTD = TDEM.Survey([src]) prbTD = TDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver) prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)] prbTD.pair(surveyTD) std = 0.03 surveyTD.makeSyntheticData(mtrue, std) surveyTD.std = std surveyTD.eps = np.linalg.norm(surveyTD.dtrue) * 1e-5 # TDEM inversion dmisfit = DataMisfit.l2_DataMisfit(surveyTD) regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]]) reg = Regularization.Simple(regMesh) opt = Optimization.InexactGaussNewton(maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt) # directives beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) target = Directives.TargetMisfit() directiveList = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=directiveList) m0 = np.log(np.ones(mtrue.size) * sig_half) reg.alpha_s = 5e-1 reg.alpha_x = 1. prbTD.counter = opt.counter = Utils.Counter() opt.remember('xc') moptTD = inv.run(m0) # Plot the results if plotIt: plt.figure(figsize=(10, 8)) ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2) ax1 = plt.subplot2grid((2, 2), (0, 1)) ax2 = plt.subplot2grid((2, 2), (1, 1)) fs = 13 # fontsize matplotlib.rcParams['font.size'] = fs # Plot the model ax0.semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2, label="True") ax0.semilogx(np.exp(moptFD), mesh.vectorCCz[active], 'bo', ms=6, markeredgecolor='k', markeredgewidth=0.5, label="FDEM") ax0.semilogx(np.exp(moptTD), mesh.vectorCCz[active], 'r*', ms=10, markeredgecolor='k', markeredgewidth=0.5, label="TDEM") ax0.set_ylim(-700, 0) ax0.set_xlim(5e-3, 1e-1) ax0.set_xlabel('Conductivity (S/m)', fontsize=fs) ax0.set_ylabel('Depth (m)', fontsize=fs) ax0.grid(which='both', color='k', alpha=0.5, linestyle='-', linewidth=0.2) ax0.legend(fontsize=fs, loc=4) # plot the data misfits - negative b/c we choose positive to be in the # direction of primary ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2, label="Obs (real)") ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2, label="Obs (imag)") dpredFD = surveyFD.dpred(moptTD) ax1.loglog(freqs, -dpredFD[::2], 'bo', ms=6, markeredgecolor='k', markeredgewidth=0.5, label="Pred (real)") ax1.loglog(freqs, -dpredFD[1::2], 'b+', ms=10, markeredgewidth=2., label="Pred (imag)") ax2.loglog(times, surveyTD.dobs, 'k-', lw=2, label='Obs') ax2.loglog(times, surveyTD.dpred(moptTD), 'r*', ms=10, markeredgecolor='k', markeredgewidth=0.5, label='Pred') ax2.set_xlim(times.min() - 1e-5, times.max() + 1e-4) # Labels, gridlines, etc ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax1.set_xlabel('Frequency (Hz)', fontsize=fs) ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs) ax2.set_xlabel('Time (s)', fontsize=fs) ax2.set_ylabel('Vertical magnetic field (T)', fontsize=fs) ax2.legend(fontsize=fs, loc=3) ax1.legend(fontsize=fs, loc=3) ax1.set_xlim(freqs.max() + 1e2, freqs.min() - 1e1) ax0.set_title("(a) Recovered Models", fontsize=fs) ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs) ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs) plt.tight_layout(pad=1.5) if saveFig is True: plt.savefig('example1.png', dpi=600)
def 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 run(plotIt=True): """ 1D FDEM Mu Inversion ==================== 1D inversion of Magnetic Susceptibility from FDEM data assuming a fixed electrical conductivity """ # Set up cylindrically symmeric mesh cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh hx = [(cs, ncx), (cs, npad, 1.3)] hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)] mesh = Mesh.CylMesh([hx, 1, hz], '00C') # Geologic Parameters model layerz = np.r_[-100., -50.] layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1]) active = mesh.vectorCCz < 0. # Electrical Conductivity sig_half = 1e-2 # Half-space conductivity sig_air = 1e-8 # Air conductivity sig_layer = 1e-2 # Layer conductivity sigma = np.ones(mesh.nCz) * sig_air sigma[active] = sig_half sigma[layer] = sig_layer # mur - relative magnetic permeability mur_half = 1. mur_air = 1. mur_layer = 2. mur = np.ones(mesh.nCz) * mur_air mur[active] = mur_half mur[layer] = mur_layer mtrue = mur[active] # Maps actMap = Maps.InjectActiveCells(mesh, active, mur_air, nC=mesh.nCz) surj1Dmap = Maps.SurjectVertical1D(mesh) murMap = Maps.MuRelative(mesh) # Mapping muMap = murMap * surj1Dmap * actMap # ----- FDEM problem & survey ----- rxlocs = Utils.ndgrid([np.r_[10.], np.r_[0], np.r_[30.]]) bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real') # bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag') freqs = np.linspace(2000, 10000, 10) #np.logspace(3, 4, 10) srcLoc = np.array([0., 0., 30.]) print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half), 'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half)) print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(), 'max z ', mesh.vectorCCz.max()) srcList = [ FDEM.Src.MagDipole([bzr], freq, srcLoc, orientation='Z') for freq in freqs ] surveyFD = FDEM.Survey(srcList) prbFD = FDEM.Problem3D_b(mesh, sigma=surj1Dmap * sigma, muMap=muMap, Solver=Solver) prbFD.pair(surveyFD) std = 0.03 surveyFD.makeSyntheticData(mtrue, std) surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-6 # FDEM inversion np.random.seed(13472) dmisfit = DataMisfit.l2_DataMisfit(surveyFD) regMesh = Mesh.TensorMesh([mesh.hz[muMap.maps[-1].indActive]]) reg = Regularization.Simple(regMesh) opt = Optimization.InexactGaussNewton(maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt) # Inversion Directives betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) target = Directives.TargetMisfit() directiveList = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=directiveList) m0 = mur_half * np.ones(mtrue.size) reg.alpha_s = 2e-2 reg.alpha_x = 1. prbFD.counter = opt.counter = Utils.Counter() opt.remember('xc') moptFD = inv.run(m0) dpredFD = surveyFD.dpred(moptFD) if plotIt: fig, ax = plt.subplots(1, 3, figsize=(10, 6)) fs = 13 # fontsize matplotlib.rcParams['font.size'] = fs # Plot the conductivity model ax[0].semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2) ax[0].set_ylim(-500, 0) ax[0].set_xlim(5e-3, 1e-1) ax[0].set_xlabel('Conductivity (S/m)', fontsize=fs) ax[0].set_ylabel('Depth (m)', fontsize=fs) ax[0].grid(which='both', color='k', alpha=0.5, linestyle='-', linewidth=0.2) ax[0].legend(['Conductivity Model'], fontsize=fs, loc=4) # Plot the permeability model ax[1].plot(mur[active], mesh.vectorCCz[active], 'k-', lw=2) ax[1].plot(moptFD, mesh.vectorCCz[active], 'b-', lw=2) ax[1].set_ylim(-500, 0) ax[1].set_xlim(0.5, 2.1) ax[1].set_xlabel('Relative Permeability', fontsize=fs) ax[1].set_ylabel('Depth (m)', fontsize=fs) ax[1].grid(which='both', color='k', alpha=0.5, linestyle='-', linewidth=0.2) ax[1].legend(['True', 'Predicted'], fontsize=fs, loc=4) # plot the data misfits - negative b/c we choose positive to be in the # direction of primary ax[2].plot(freqs, -surveyFD.dobs, 'k-', lw=2) # ax[2].plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2) ax[2].loglog(freqs, -dpredFD, 'bo', ms=6) # ax[2].loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10) # Labels, gridlines, etc ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax[2].set_xlabel('Frequency (Hz)', fontsize=fs) ax[2].set_ylabel('Vertical magnetic field (-T)', fontsize=fs) # ax[2].legend(("Obs", "Pred"), fontsize=fs) ax[2].legend(("z-Obs (real)", "z-Pred (real)"), fontsize=fs) ax[2].set_xlim(freqs.max(), freqs.min()) ax[0].set_title("(a) Conductivity Model", fontsize=fs) ax[1].set_title("(b) $\mu_r$ Model", fontsize=fs) ax[2].set_title("(c) FDEM observed vs. predicted", fontsize=fs) # ax[2].set_title("(c) TDEM observed vs. predicted", fontsize=fs) plt.tight_layout(pad=1.5)
# Get cells inside the sphere sph_ind = PF.MagAnalytics.spheremodel(mesh, 0., 0., 0., rad) # Adjust susceptibility for volume difference Vratio = (4. / 3. * np.pi * rad**3.) / (np.sum(sph_ind) * cs**3.) model = np.ones(mesh.nC) * 1e-8 model[sph_ind] = 0.01 rxLoc = np.asarray([np.r_[0, 0, 4.]]) bzi = FDEM.Rx.Point_bSecondary(rxLoc, 'z', 'real') bzr = FDEM.Rx.Point_bSecondary(rxLoc, 'z', 'imag') freqs = [400] #np.logspace(2, 3, 5) srcLoc = np.r_[0, 0, 4.] srcList = [ FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z') for freq in freqs ] mapping = Maps.IdentityMap(mesh) surveyFD = FDEM.Survey(srcList) prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=PardisoSolver) prbFD.pair(surveyFD) std = 0.03 surveyFD.makeSyntheticData(model, std) #Mesh.TensorMesh.writeUBC(mesh,'MeshGrav.msh') #Mesh.TensorMesh.writeModelUBC(mesh,'MeshGrav.den',model) #PF.Gravity.writeUBCobs("Obs.grv",survey,d)
def setUpClass(self): """ Set up a cyl symmetric EM problem on 2D and 3D meshes. """ sigma_back = 1e-1 # wholespace cp = casingSimulations.CasingParameters( casing_l=10., src_a=np.r_[0., 0., -9.], src_b=np.r_[0., 0., -1.], freqs=np.r_[0.1, 1., 2.], sigma_back=sigma_back, # wholespace sigma_layer=sigma_back, sigma_air=sigma_back, ) # Set up the meshes npadx, npadz = 11, 26 dx2 = 500. mesh2D = casingSimulations.CylMeshGenerator(cp=cp, npadx=npadx, npadz=npadz, domain_x2=dx2).mesh mesh3D = casingSimulations.CylMeshGenerator(cp=cp, ncy=4, npadx=npadx, npadz=npadz, domain_x2=dx2).mesh # get wirepath on mesh wire2D = getSrcWire(mesh2D, cp) wire3D = getSrcWire(mesh3D, cp) # create sources srcList2D = [ FDEM.Src.RawVec_e(s_e=wire2D, freq=freq, rxList=[]) for freq in cp.freqs ] srcList3D = [ FDEM.Src.RawVec_e(s_e=wire3D, freq=freq, rxList=[]) for freq in cp.freqs ] # get phys prop models physprops2D = casingSimulations.PhysicalProperties(mesh2D, cp) physprops3D = casingSimulations.PhysicalProperties(mesh3D, cp) # plot the phys prop models fig, ax = plt.subplots(1, 1) plt.colorbar(mesh2D.plotImage(np.log10(physprops2D.sigma), ax=ax, mirror=True)[0], ax=ax) ax.set_xlim([-1., 1.]) ax.set_ylim([-20., 10.]) if plotIt: plt.show() # create the problems and surveys prb2D = FDEM.Problem3D_h(mesh2D, sigmaMap=physprops2D.wires.sigma, muMap=physprops2D.wires.mu, Solver=Pardiso) prb3D = FDEM.Problem3D_h(mesh3D, sigmaMap=physprops3D.wires.sigma, muMap=physprops3D.wires.mu, Solver=Pardiso) survey2D = FDEM.Survey(srcList2D) survey3D = FDEM.Survey(srcList3D) prb2D.pair(survey2D) prb3D.pair(survey3D) print('starting 2D solve ... ') fields2D = prb2D.fields(physprops2D.model) print(' ... done \n') print('starting 3D solve ...') fields3D = prb3D.fields(physprops3D.model) print(' ... done \n') # assign the properties that will be helpful self.mesh2D = mesh2D self.mesh3D = mesh3D self.srcList2D = srcList2D self.srcList3D = srcList3D self.prb2D = prb2D self.prb3D = prb3D self.survey2D = survey2D self.survey3D = survey3D self.fields2D = fields2D self.fields3D = fields3D
def setupProblem( mesh, muMod, sigmaMod, prbtype='e', invertMui=False, sigmaInInversion=False, freq=1. ): rxcomp = ['real', 'imag'] loc = Utils.ndgrid( [mesh.vectorCCx, np.r_[0.], mesh.vectorCCz] ) if prbtype in ['e', 'b']: rxfields_y = ['e', 'j'] rxfields_xz = ['b', 'h'] elif prbtype in ['h', 'j']: rxfields_y = ['b', 'h'] rxfields_xz = ['e', 'j'] rxList_edge = [ getattr(FDEM.Rx, 'Point_{f}'.format(f=f))( loc, component=comp, orientation=orient ) for f in rxfields_y for comp in rxcomp for orient in ['y'] ] rxList_face = [ getattr(FDEM.Rx, 'Point_{f}'.format(f=f))( loc, component=comp, orientation=orient ) for f in rxfields_xz for comp in rxcomp for orient in ['x', 'z'] ] rxList = rxList_edge + rxList_face src_loc = np.r_[0., 0., 0.] if prbtype in ['e', 'b']: src = FDEM.Src.MagDipole( rxList=rxList, loc=src_loc, freq=freq ) elif prbtype in ['h', 'j']: ind = Utils.closestPoints(mesh, src_loc, 'Fz') + mesh.vnF[0] vec = np.zeros(mesh.nF) vec[ind] = 1. src = FDEM.Src.RawVec_e(rxList=rxList, freq=freq, s_e=vec) survey = FDEM.Survey([src]) if sigmaInInversion: wires = Maps.Wires( ('mu', mesh.nC), ('sigma', mesh.nC) ) muMap = Maps.MuRelative(mesh) * wires.mu sigmaMap = Maps.ExpMap(mesh) * wires.sigma if invertMui: muiMap = Maps.ReciprocalMap(mesh)*muMap prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, muiMap=muiMap, sigmaMap=sigmaMap ) # m0 = np.hstack([1./muMod, sigmaMod]) else: prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, muMap=muMap, sigmaMap=sigmaMap ) m0 = np.hstack([muMod, sigmaMod]) else: muMap = Maps.MuRelative(mesh) if invertMui: muiMap = Maps.ReciprocalMap(mesh) * muMap prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, sigma=sigmaMod, muiMap=muiMap ) # m0 = 1./muMod else: prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, sigma=sigmaMod, muMap=muMap ) m0 = muMod prob.pair(survey) return m0, prob, survey
def 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.])
rx_real = FDEM.Rx.Point_bSecondary(locs=rx_locs, orientation=orientation, component='real') rx_imag = FDEM.Rx.Point_bSecondary(locs=rx_locs, orientation=orientation, component='imag') src = FDEM.Src.MagDipole(rxList=[rx_real, rx_imag], loc=src_loc, orientation=orientation, freq=freq) srcList.append(src) # create the survey and problem objects for running the forward simulation survey = FDEM.Survey(srcList) prob = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver) prob.pair(survey) ############################################################################### # Data # ---- # # Generate clean, synthetic data t = time.time() dclean = survey.dpred(m_true) print("Done forward simulation. Elapsed time = {:1.2f} s".format(time.time() - t))