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)