def setUp(self): M = Mesh.TensorMesh([np.ones(8),np.ones(20),np.ones(10)]) M.setCellGradBC(['neumann','neumann','dirichlet']) params = Richards.Empirical.HaverkampParams().celia1990 params['Ks'] = np.log(params['Ks']) E = Richards.Empirical.Haverkamp(M, **params) bc = np.array([-61.5,-20.7]) bc = np.r_[np.zeros(M.nCy*M.nCz*2),np.zeros(M.nCx*M.nCz*2),np.ones(M.nCx*M.nCy)*bc[0],np.ones(M.nCx*M.nCy)*bc[1]] h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M,E, timeSteps=[(40,3),(60,3)], boundaryConditions=bc, initialConditions=h, doNewton=False, method='mixed', tolRootFinder=1e-6, debug=False) prob.Solver = Solver locs = Utils.ndgrid(np.r_[5,7.],np.r_[5,15.],np.r_[6,8.]) times = prob.times[3:5] rxSat = Richards.RichardsRx(locs, times, 'saturation') rxPre = Richards.RichardsRx(locs, times, 'pressureHead') survey = Richards.RichardsSurvey([rxSat, rxPre]) prob.pair(survey) self.h0 = h self.M = M self.Ks = params['Ks'] self.prob = prob self.survey = survey
def setUp(self): M = Mesh.TensorMesh([np.ones(20)]) M.setCellGradBC('dirichlet') params = Richards.Empirical.HaverkampParams().celia1990 params['Ks'] = np.log(params['Ks']) E = Richards.Empirical.Haverkamp(M, **params) bc = np.array([-61.5, -20.7]) h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M, mapping=E, timeSteps=[(40, 3), (60, 3)], tolRootFinder=1e-6, debug=False, boundaryConditions=bc, initialConditions=h, doNewton=False, method='mixed') prob.Solver = Solver locs = np.r_[5., 10, 15] times = prob.times[3:5] rxSat = Richards.RichardsRx(locs, times, 'saturation') rxPre = Richards.RichardsRx(locs, times, 'pressureHead') survey = Richards.RichardsSurvey([rxSat, rxPre]) prob.pair(survey) self.h0 = h self.M = M self.Ks = params['Ks'] self.prob = prob self.survey = survey
def setUp(self): mesh = self.get_mesh() params = Richards.Empirical.HaverkampParams().celia1990 k_fun, theta_fun = Richards.Empirical.haverkamp(mesh, **params) self.setup_maps(mesh, k_fun, theta_fun) bc, h = self.get_conditions(mesh) prob = Richards.RichardsProblem(mesh, hydraulic_conductivity=k_fun, water_retention=theta_fun, root_finder_tol=1e-6, debug=False, boundary_conditions=bc, initial_conditions=h, do_newton=False, method='mixed') prob.timeSteps = [(40, 3), (60, 3)] prob.Solver = Solver rx_list = self.get_rx_list(prob) survey = Richards.RichardsSurvey(rx_list) prob.pair(survey) self.h0 = h self.mesh = mesh self.Ks = params['Ks'] * np.ones(self.mesh.nC) self.A = params['A'] * np.ones(self.mesh.nC) self.theta_s = params['theta_s'] * np.ones(self.mesh.nC) self.prob = prob self.survey = survey self.setup_model()
def getFields(timeStep, method): timeSteps = np.ones(360/timeStep)*timeStep prob = Richards.RichardsProblem( M, mapping=E, timeSteps=timeSteps, boundaryConditions=bc, initialConditions=h, doNewton=False, method=method ) return prob.fields(params['Ks'])
def getFields(timeStep, method): timeSteps = np.ones(int(360 / timeStep)) * timeStep prob = Richards.RichardsProblem(M, hydraulic_conductivity=k_fun, water_retention=theta_fun, boundary_conditions=bc, initial_conditions=h, do_newton=False, method=method) prob.timeSteps = timeSteps return prob.fields(params['Ks'] * np.ones(M.nC))
def run(plotIt=True): M = Mesh.TensorMesh([np.ones(40)], x0='N') M.setCellGradBC('dirichlet') # We will use the haverkamp empirical model with parameters from Celia1990 k_fun, theta_fun = Richards.Empirical.haverkamp(M, A=1.1750e+06, gamma=4.74, alpha=1.6110e+06, theta_s=0.287, theta_r=0.075, beta=3.96) # Here we are making saturated hydraulic conductivity # an exponential mapping to the model (defined below) k_fun.KsMap = Maps.ExpMap(nP=M.nC) # Setup the boundary and initial conditions bc = np.array([-61.5, -20.7]) h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M, hydraulic_conductivity=k_fun, water_retention=theta_fun, boundary_conditions=bc, initial_conditions=h, do_newton=False, method='mixed', debug=False) prob.timeSteps = [(5, 25, 1.1), (60, 40)] # Create the survey locs = -np.arange(2, 38, 4.) times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60) rxSat = Richards.SaturationRx(locs, times) survey = Richards.RichardsSurvey([rxSat]) survey.pair(prob) # Create a simple model for Ks Ks = 1e-3 mtrue = np.ones(M.nC) * np.log(Ks) mtrue[15:20] = np.log(5e-2) mtrue[20:35] = np.log(3e-3) mtrue[35:40] = np.log(1e-2) # Create some synthetic data and fields Hs = prob.fields(mtrue) data = survey.makeSyntheticData(mtrue, std=0, f=Hs, force=True) if plotIt: plt.figure(figsize=(14, 9)) plt.subplot(221) plt.plot(np.log10(np.exp(mtrue)), M.gridCC) plt.title('(a) True model and data locations') plt.ylabel('Depth, cm') plt.xlabel('Hydraulic conductivity, $log_{10}(K_s)$') plt.plot([-3.25] * len(locs), locs, 'ro') plt.legend(('True model', 'Data locations')) plt.subplot(222) plt.plot(times / 60, data.reshape((-1, len(locs)))) plt.title('(b) True data over time at all depths') plt.xlabel('Time, minutes') plt.ylabel('Saturation') ax = plt.subplot(212) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('(c) Saturation over time') plt.tight_layout()
def run(plotIt=True): M = Mesh.TensorMesh([np.ones(40)], x0='N') M.setCellGradBC('dirichlet') # We will use the haverkamp empirical model with parameters from Celia1990 k_fun, theta_fun = Richards.Empirical.haverkamp(M, A=1.1750e+06, gamma=4.74, alpha=1.6110e+06, theta_s=0.287, theta_r=0.075, beta=3.96) # Here we are making saturated hydraulic conductivity # an exponential mapping to the model (defined below) k_fun.KsMap = Maps.ExpMap(nP=M.nC) # Setup the boundary and initial conditions bc = np.array([-61.5, -20.7]) h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M, hydraulic_conductivity=k_fun, water_retention=theta_fun, boundary_conditions=bc, initial_conditions=h, do_newton=False, method='mixed', debug=False) prob.timeSteps = [(5, 25, 1.1), (60, 40)] # Create the survey locs = -np.arange(2, 38, 4.) times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60) rxSat = Richards.SaturationRx(locs, times) survey = Richards.RichardsSurvey([rxSat]) survey.pair(prob) # Create a simple model for Ks Ks = 1e-3 mtrue = np.ones(M.nC) * np.log(Ks) mtrue[15:20] = np.log(5e-2) mtrue[20:35] = np.log(3e-3) mtrue[35:40] = np.log(1e-2) m0 = np.ones(M.nC) * np.log(Ks) # Create some synthetic data and fields stdev = 0.02 # The standard deviation for the noise Hs = prob.fields(mtrue) survey.makeSyntheticData(mtrue, std=stdev, f=Hs, force=True) # Setup a pretty standard inversion reg = Regularization.Tikhonov(M, alpha_s=1e-1) dmis = DataMisfit.l2_DataMisfit(survey) opt = Optimization.InexactGaussNewton(maxIter=20, maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmis, reg, opt) beta = Directives.BetaSchedule(coolingFactor=4) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e2) target = Directives.TargetMisfit() dir_list = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=dir_list) mopt = inv.run(m0) Hs_opt = prob.fields(mopt) if plotIt: plt.figure(figsize=(14, 9)) ax = plt.subplot(121) plt.semilogx(np.exp(np.c_[mopt, mtrue]), M.gridCC) plt.xlabel('Saturated Hydraulic Conductivity, $K_s$') plt.ylabel('Depth, cm') plt.semilogx([10**-3.9] * len(locs), locs, 'ro') plt.legend(('$m_{rec}$', '$m_{true}$', 'Data locations'), loc=4) ax = plt.subplot(222) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('True saturation over time') ax = plt.subplot(224) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs_opt] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('Recovered saturation over time') plt.tight_layout()
def run(plotIt=True): """ FLOW: Richards: 1D: Inversion ============================= The example shows an inversion of Richards equation in 1D with a heterogeneous hydraulic conductivity function. The haverkamp model is used with the same parameters as Celia1990_ the boundary and initial conditions are also the same. The simulation domain is 40cm deep and is run for an hour with an exponentially increasing time step that has a maximum of one minute. The general setup of the experiment is an infiltration front that advances downward through the model over time. The model chosen is the saturated hydraulic conductivity inside the hydraulic conductivity function (using haverkamp). The initial model is chosen to be the background (1e-3 cm/s). The saturation data has 2% random Gaussian noise added. The figure shows the recovered saturated hydraulic conductivity next to the true model. The other two figures show the saturation field for the entire simulation for the true and recovered models. Rowan Cockett - 21/12/2016 .. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf """ M = Mesh.TensorMesh([np.ones(40)], x0='N') M.setCellGradBC('dirichlet') # We will use the haverkamp empirical model with parameters from Celia1990 k_fun, theta_fun = Richards.Empirical.haverkamp(M, A=1.1750e+06, gamma=4.74, alpha=1.6110e+06, theta_s=0.287, theta_r=0.075, beta=3.96) # Here we are making saturated hydraulic conductivity # an exponential mapping to the model (defined below) k_fun.KsMap = Maps.ExpMap(nP=M.nC) # Setup the boundary and initial conditions bc = np.array([-61.5, -20.7]) h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M, hydraulic_conductivity=k_fun, water_retention=theta_fun, boundary_conditions=bc, initial_conditions=h, do_newton=False, method='mixed', debug=False) prob.timeSteps = [(5, 25, 1.1), (60, 40)] # Create the survey locs = -np.arange(2, 38, 4.) times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60) rxSat = Richards.SaturationRx(locs, times) survey = Richards.RichardsSurvey([rxSat]) survey.pair(prob) # Create a simple model for Ks Ks = 1e-3 mtrue = np.ones(M.nC) * np.log(Ks) mtrue[15:20] = np.log(5e-2) mtrue[20:35] = np.log(3e-3) mtrue[35:40] = np.log(1e-2) m0 = np.ones(M.nC) * np.log(Ks) # Create some synthetic data and fields stdev = 0.02 # The standard deviation for the noise Hs = prob.fields(mtrue) survey.makeSyntheticData(mtrue, std=stdev, f=Hs, force=True) # Setup a pretty standard inversion reg = Regularization.Tikhonov(M, alpha_s=1e-1) dmis = DataMisfit.l2_DataMisfit(survey) opt = Optimization.InexactGaussNewton(maxIter=20, maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmis, reg, opt) beta = Directives.BetaSchedule(coolingFactor=4) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e2) target = Directives.TargetMisfit() dir_list = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=dir_list) mopt = inv.run(m0) Hs_opt = prob.fields(mopt) if plotIt: plt.figure(figsize=(14, 9)) ax = plt.subplot(121) plt.semilogx(np.exp(np.c_[mopt, mtrue]), M.gridCC) plt.xlabel('Saturated Hydraulic Conductivity, $K_s$') plt.ylabel('Depth, cm') plt.semilogx([10**-3.9] * len(locs), locs, 'ro') plt.legend(('$m_{rec}$', '$m_{true}$', 'Data locations'), loc=4) ax = plt.subplot(222) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('True saturation over time') ax = plt.subplot(224) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs_opt] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('Recovered saturation over time') plt.tight_layout()
def run(plotIt=True): """ FLOW: Richards: 1D: Forward Simulation ====================================== The example shows simulation of Richards equation in 1D with a heterogeneous hydraulic conductivity function. The haverkamp model is used with the same parameters as Celia1990_ the boundary and initial conditions are also the same. The simulation domain is 40cm deep and is run for an hour with an exponentially increasing time step that has a maximum of one minute. The general setup of the experiment is an infiltration front that advances downward through the model over time. Figure (a) shows the heterogeneous saturated hydraulic conductivity parameter and the location of the data collection, which happens every minute from 30 seconds into the simulation. Note that the simulation mesh and the data locations are not aligned, and linear interpolation is used to collect the data. The points are sampled in pressure head and then transformed to saturation using the haverkamp model for the water retention curve. Figure (b) shows the data collected from the simulation. No noise is added to the data at this time. The various data locations register the infiltration event through increasing saturation as the front moves past the receiver. Notice that the slope of the curves are not equal as the hydraulic conductivity function is heterogeneous. Figure (c) shows the saturation field over the entire experiment. Here you can see that the timestep is not constant over time (5 seconds at the start of the simulation, 60 seconds at the end). You can also see the effect of the highly conductive layer in the model between 20 and 25 cm depth. The water drains straight through the conductive unit and piles up on the other side - advancing the fluid front faster than the other layers. Rowan Cockett - 21/12/2016 .. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf """ M = Mesh.TensorMesh([np.ones(40)], x0='N') M.setCellGradBC('dirichlet') # We will use the haverkamp empirical model with parameters from Celia1990 k_fun, theta_fun = Richards.Empirical.haverkamp(M, A=1.1750e+06, gamma=4.74, alpha=1.6110e+06, theta_s=0.287, theta_r=0.075, beta=3.96) # Here we are making saturated hydraulic conductivity # an exponential mapping to the model (defined below) k_fun.KsMap = Maps.ExpMap(nP=M.nC) # Setup the boundary and initial conditions bc = np.array([-61.5, -20.7]) h = np.zeros(M.nC) + bc[0] prob = Richards.RichardsProblem(M, hydraulic_conductivity=k_fun, water_retention=theta_fun, boundary_conditions=bc, initial_conditions=h, do_newton=False, method='mixed', debug=False) prob.timeSteps = [(5, 25, 1.1), (60, 40)] # Create the survey locs = -np.arange(2, 38, 4.) times = np.arange(30, prob.timeMesh.vectorCCx[-1], 60) rxSat = Richards.SaturationRx(locs, times) survey = Richards.RichardsSurvey([rxSat]) survey.pair(prob) # Create a simple model for Ks Ks = 1e-3 mtrue = np.ones(M.nC) * np.log(Ks) mtrue[15:20] = np.log(5e-2) mtrue[20:35] = np.log(3e-3) mtrue[35:40] = np.log(1e-2) # Create some synthetic data and fields Hs = prob.fields(mtrue) data = survey.makeSyntheticData(mtrue, std=0, f=Hs, force=True) if plotIt: plt.figure(figsize=(14, 9)) plt.subplot(221) plt.plot(np.log10(np.exp(mtrue)), M.gridCC) plt.title('(a) True model and data locations') plt.ylabel('Depth, cm') plt.xlabel('Hydraulic conductivity, $log_{10}(K_s)$') plt.plot([-3.25] * len(locs), locs, 'ro') plt.legend(('True model', 'Data locations')) plt.subplot(222) plt.plot(times / 60, data.reshape((-1, len(locs)))) plt.title('(b) True data over time at all depths') plt.xlabel('Time, minutes') plt.ylabel('Saturation') ax = plt.subplot(212) mesh2d = Mesh.TensorMesh([prob.timeMesh.hx / 60, prob.mesh.hx], '0N') sats = [theta_fun(_) for _ in Hs] clr = mesh2d.plotImage(np.c_[sats][1:, :], ax=ax) cmap0 = matplotlib.cm.RdYlBu_r clr[0].set_cmap(cmap0) c = plt.colorbar(clr[0]) c.set_label('Saturation $\\theta$') plt.xlabel('Time, minutes') plt.ylabel('Depth, cm') plt.title('(c) Saturation over time') plt.tight_layout()