.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/examples/20-published/plot_schenkel_morrison_casing.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_content_examples_20-published_plot_schenkel_morrison_casing.py: 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/simpeg/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! .. GENERATED FROM PYTHON SOURCE LINES 47-289 .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_001.png :alt: Simulation Mesh :srcset: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_001.png :class: sphx-glr-multi-img * .. image-sg:: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_002.png :alt: Log_10 (Sigma) :srcset: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_002.png :class: sphx-glr-multi-img * .. image-sg:: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_003.png :alt: Magnitude of Vertical Current in Casing, Magnitude of Vertical Current in Casing :srcset: /content/examples/20-published/images/sphx_glr_plot_schenkel_morrison_casing_003.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out .. code-block:: none Skin Depth: [5000000.0] Mesh Extent xmax: 30844847.844048,: zmin: -10732642.852316, zmax: 10732343.352316 Number of cells 878878 Time to solve 2 sources 7.282617807388306 | .. code-block:: Python import matplotlib.pylab as plt import numpy as np import discretize from SimPEG import maps, utils from SimPEG.electromagnetics import frequency_domain as FDEM import time try: from pymatsolver import Pardiso as Solver except Exception: from SimPEG import SolverLU as Solver def run(plotIt=True): # ------------------ MODEL ------------------ sigmaair = 1e-8 # air sigmaback = 1e-2 # background sigmacasing = 1e6 # casing casing_t = 0.006 # 1cm thickness casing_l = 300 # length of the casing casing_r = 0.1 casing_a = casing_r - casing_t / 2.0 # inner radius casing_b = casing_r + casing_t / 2.0 # outer radius casing_z = np.r_[-casing_l, 0.0] # ------------------ SURVEY PARAMETERS ------------------ freqs = np.r_[1e-6] # [1e-1, 1, 5] # frequencies dsz = -300 # down-hole z source location inf_loc = np.r_[0.0, 0.0, 1e4] print("Skin Depth: ", [(500.0 / np.sqrt(sigmaback * _)) for _ in freqs]) # ------------------ MESH ------------------ # fine cells near well bore csx1, csx2 = 2e-3, 60.0 pfx1, pfx2 = 1.3, 1.3 ncx1 = np.ceil(casing_b / csx1 + 2) # pad nicely to second cell size npadx1 = np.floor(np.log(csx2 / csx1) / np.log(pfx1)) hx1a = utils.unpack_widths([(csx1, ncx1)]) hx1b = utils.unpack_widths([(csx1, npadx1, pfx1)]) dx1 = sum(hx1a) + sum(hx1b) dx1 = np.floor(dx1 / csx2) hx1b *= (dx1 * csx2 - sum(hx1a)) / sum(hx1b) # second chunk of mesh dx2 = 300.0 # uniform mesh out to here ncx2 = np.ceil((dx2 - dx1) / csx2) npadx2 = 45 hx2a = utils.unpack_widths([(csx2, ncx2)]) hx2b = utils.unpack_widths([(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 = int(np.ceil(np.diff(casing_z)[0] / csz)) + 10, 68, 68 # vector of cell widths in the z-direction hz = utils.unpack_widths([(csz, npadzd, -1.3), (csz, ncz), (csz, npadzu, 1.3)]) # Mesh mesh = discretize.CylindricalMesh( [hx, 1.0, hz], [0.0, 0.0, -np.sum(hz[: npadzu + ncz - nza])] ) print( "Mesh Extent xmax: {0:f},: zmin: {1:f}, zmax: {2:f}".format( mesh.cell_centers_x.max(), mesh.cell_centers_z.min(), mesh.cell_centers_z.max(), ) ) print("Number of cells", mesh.nC) if plotIt is True: _, ax = plt.subplots(1, 1, figsize=(6, 4)) ax.set_title("Simulation Mesh") mesh.plot_grid(ax=ax) # Put the model on the mesh sigWholespace = sigmaback * np.ones((mesh.nC)) sigBack = sigWholespace.copy() sigBack[mesh.gridCC[:, 2] > 0.0] = sigmaair sigCasing = sigBack.copy() iCasingZ = (mesh.gridCC[:, 2] <= casing_z[1]) & (mesh.gridCC[:, 2] >= casing_z[0]) iCasingX = (mesh.gridCC[:, 0] >= casing_a) & (mesh.gridCC[:, 0] <= casing_b) iCasing = iCasingX & iCasingZ sigCasing[iCasing] = sigmacasing if plotIt is True: # plotting parameters xlim = np.r_[0.0, 0.2] zlim = np.r_[-350.0, 10.0] clim_sig = np.r_[-8, 6] # plot models fig, ax = plt.subplots(1, 1, figsize=(4, 4)) im = mesh.plot_image(np.log10(sigCasing), ax=ax)[0] im.set_clim(clim_sig) plt.colorbar(im, ax=ax) ax.grid(which="both") ax.set_title("Log_10 (Sigma)") ax.set_xlim(xlim) ax.set_ylim(zlim) # -------------- Sources -------------------- # Define Custom Current Sources # surface source sg_x = np.zeros(mesh.vnF[0], dtype=complex) sg_y = np.zeros(mesh.vnF[1], dtype=complex) sg_z = np.zeros(mesh.vnF[2], dtype=complex) nza = 2 # put the wire two cells above the surface # vertically directed wire # hook it up to casing at the surface sgv_indx = (mesh.gridFz[:, 0] > casing_a) & (mesh.gridFz[:, 0] < casing_a + csx1) sgv_indz = (mesh.gridFz[:, 2] <= +csz * nza) & (mesh.gridFz[:, 2] >= -csz * 2) sgv_ind = sgv_indx & sgv_indz sg_z[sgv_ind] = -1.0 # horizontally directed wire sgh_indx = (mesh.gridFx[:, 0] > casing_a) & (mesh.gridFx[:, 0] <= inf_loc[2]) sgh_indz = (mesh.gridFx[:, 2] > csz * (nza - 0.5)) & ( mesh.gridFx[:, 2] < csz * (nza + 0.5) ) sgh_ind = sgh_indx & sgh_indz sg_x[sgh_ind] = -1.0 # hook it up to casing at the surface sgv2_indx = (mesh.gridFz[:, 0] >= mesh.gridFx[sgh_ind, 0].max()) & ( mesh.gridFz[:, 0] <= inf_loc[2] * 1.2 ) sgv2_indz = (mesh.gridFz[:, 2] <= +csz * nza) & (mesh.gridFz[:, 2] >= -csz * 2) sgv2_ind = sgv2_indx & sgv2_indz sg_z[sgv2_ind] = 1.0 # assemble the source sg = np.hstack([sg_x, sg_y, sg_z]) sg_p = [FDEM.Src.RawVec_e([], _, sg / mesh.face_areas) for _ in freqs] # downhole source dg_x = np.zeros(mesh.vnF[0], dtype=complex) dg_y = np.zeros(mesh.vnF[1], dtype=complex) dg_z = np.zeros(mesh.vnF[2], dtype=complex) # vertically directed wire dgv_indx = mesh.gridFz[:, 0] < csx1 # go through the center of the well dgv_indz = (mesh.gridFz[:, 2] <= +csz * nza) & (mesh.gridFz[:, 2] > dsz + csz / 2.0) dgv_ind = dgv_indx & dgv_indz dg_z[dgv_ind] = -1.0 # couple to the casing downhole dgh_indx = mesh.gridFx[:, 0] < casing_a + csx1 dgh_indz = (mesh.gridFx[:, 2] < dsz + csz) & (mesh.gridFx[:, 2] >= dsz) dgh_ind = dgh_indx & dgh_indz dg_x[dgh_ind] = 1.0 # horizontal part at surface dgh2_indx = mesh.gridFx[:, 0] <= inf_loc[2] * 1.2 dgh2_indz = sgh_indz.copy() dgh2_ind = dgh2_indx & dgh2_indz dg_x[dgh2_ind] = -1.0 # vertical part at surface dgv2_ind = sgv2_ind.copy() dg_z[dgv2_ind] = 1.0 # assemble the source dg = np.hstack([dg_x, dg_y, dg_z]) dg_p = [FDEM.Src.RawVec_e([], _, dg / mesh.face_areas) for _ in freqs] # ------------ Problem and Survey --------------- survey = FDEM.Survey(sg_p + dg_p) problem = FDEM.Simulation3DMagneticField( mesh, survey=survey, sigmaMap=maps.IdentityMap(mesh), solver=Solver ) # ------------- Solve --------------------------- t0 = time.time() fieldsCasing = problem.fields(sigCasing) print("Time to solve 2 sources", time.time() - t0) # Plot current # current density jn0 = fieldsCasing[dg_p, "j"] jn1 = fieldsCasing[sg_p, "j"] # current in0 = [mesh.face_areas * jn0[:, i] for i in range(len(freqs))] in1 = [mesh.face_areas * jn1[:, i] for i in range(len(freqs))] in0 = np.vstack(in0).T in1 = np.vstack(in1).T # integrate to get z-current inside casing inds_inx = (mesh.gridFz[:, 0] >= casing_a) & (mesh.gridFz[:, 0] <= casing_b) inds_inz = (mesh.gridFz[:, 2] >= dsz) & (mesh.gridFz[:, 2] <= 0) inds_fz = inds_inx & inds_inz indsx = [False] * mesh.nFx inds = list(indsx) + list(inds_fz) in0_in = in0[np.r_[inds]] in1_in = in1[np.r_[inds]] z_in = mesh.gridFz[inds_fz, 2] in0_in = in0_in.reshape([in0_in.shape[0] // 3, 3]) in1_in = in1_in.reshape([in1_in.shape[0] // 3, 3]) z_in = z_in.reshape([z_in.shape[0] // 3, 3]) I0 = in0_in.sum(1).real I1 = in1_in.sum(1).real z_in = z_in[:, 0] if plotIt is True: fig, ax = plt.subplots(1, 2, figsize=(12, 4)) ax[0].plot(z_in, np.absolute(I0), z_in, np.absolute(I1)) ax[0].legend(["top casing", "bottom casing"], loc="best") ax[0].set_title("Magnitude of Vertical Current in Casing") ax[1].semilogy(z_in, np.absolute(I0), z_in, np.absolute(I1)) ax[1].legend(["top casing", "bottom casing"], loc="best") ax[1].set_title("Magnitude of Vertical Current in Casing") ax[1].set_ylim([1e-2, 1.0]) if __name__ == "__main__": run() plt.show() .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 9.653 seconds) **Estimated memory usage:** 2111 MB .. _sphx_glr_download_content_examples_20-published_plot_schenkel_morrison_casing.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_schenkel_morrison_casing.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_schenkel_morrison_casing.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_