# Step 1: Magma inflation ```{include} step01_inflation-synopsis.md ``` ## Simulation parameters This example uses poroelasticity to model flow of magma up through a conduit and into a magma reservoir. The magma reservoir and conduit have a higher permeability than the surrounding crust. We generate flow by imposing a pressure on the external boundary of the conduit that is higher than the uniform initial pressure in the domain. {numref}`fig:example:magma:2d:step01:diagram` shows the boundary conditions on the domain. The parameters specific to this example are in `step01_inflation.cfg`. :::{figure-md} fig:example:magma:2d:step01:diagram Boundary and initial conditions for magma inflation. We apply roller boundary conditions on the +x, -x, and -y boundaries. We impose zero pressure (undrained conditions) on the +y boundary and a pressure on the external boundary of the conduit to generate fluid flow. ::: ```{code-block} cfg --- caption: Time stepping parameters for Step 1. --- [pylithapp.timedependent] start_time = -0.2*year initial_dt = 0.2*year end_time = 10.0*year ``` ```{code-block} cfg --- caption: Initial condition parameters for Step 1. We impose an initial fluid pressure over the entire domain that increases from 0 at the top boundary to 5 MPa at the bottom boundary. --- [pylithapp.problem] ic = [domain] ic.domain = pylith.problems.InitialConditionDomain [pylithapp.problem.ic.domain] subfields = [pressure] db = spatialdata.spatialdb.SimpleDB db.description = Initial conditions for domain db.iohandler.filename = initial_pressure.spatialdb ``` We create an array of 5 Dirichlet boundary conditions: 3 for displacement and 2 for fluid pressure. We have zero displacement perpendicular to the -x, +x, and -y boundaries, zero pressure on the +y boundary, and 10 MPa of fluid pressure on the external boundary of the conduit. ```{code-block} cfg --- caption: Dirichlet boundary conditions for Step 1. --- [pylithapp.problem] bc = [bc_xneg, bc_xpos, bc_yneg, bc_ypos, bc_flow] bc.bc_xneg = pylith.bc.DirichletTimeDependent bc.bc_xpos = pylith.bc.DirichletTimeDependent bc.bc_yneg = pylith.bc.DirichletTimeDependent bc.bc_ypos = pylith.bc.DirichletTimeDependent bc.bc_flow = pylith.bc.DirichletTimeDependent [pylithapp.problem.bc.bc_xneg] constrained_dof = [0] label = boundary_xneg field = displacement db_auxiliary_field = pylith.bc.ZeroDB db_auxiliary_field.description = Dirichlet BC -x [pylithapp.problem.bc.bc_xpos] constrained_dof = [0] label = boundary_xpos field = displacement db_auxiliary_field = pylith.bc.ZeroDB db_auxiliary_field.description = Dirichlet BC +x [pylithapp.problem.bc.bc_yneg] constrained_dof = [1] label = boundary_yneg field = displacement db_auxiliary_field = pylith.bc.ZeroDB db_auxiliary_field.description = Dirichlet BC -y [pylithapp.problem.bc.bc_ypos] constrained_dof = [0] label = boundary_ypos field = pressure db_auxiliary_field = pylith.bc.ZeroDB db_auxiliary_field.description = Dirichlet BC +z [pylithapp.problem.bc.bc_flow] constrained_dof = [0] label = boundary_flow field = pressure db_auxiliary_field = spatialdata.spatialdb.UniformDB db_auxiliary_field.description = Flow into external boundary of conduit db_auxiliary_field.values = [initial_amplitude] db_auxiliary_field.data = [10.0*MPa] ``` ## Running the simulation ```{code-block} console --- caption: Run Step 1 simulation --- $ pylith step01_inflation.cfg # The output should look something like the following. >> software/pylith-debug/lib/python3.12/site-packages/pylith/apps/PyLithApp.py:80:main -- info (application-flow) -- Running on 1 process(es). >> src/cig/pylith/libsrc/pylith/utils/PetscOptions.cc:251:static void pylith::utils::_PetscOptions::write(pythia::journal::info_t &, const char *, const PetscOptions &) -- info (application-flow) -- Setting PETSc options: fieldsplit_displacement_pc_type = lu fieldsplit_pressure_pc_type = bjacobi fieldsplit_trace_strain_pc_type = bjacobi ksp_atol = 1.0e-7 ksp_converged_reason = true ksp_error_if_not_converged = true ksp_guess_pod_size = 8 ksp_guess_type = pod ksp_rtol = 1.0e-14 pc_fieldsplit_0_fields = 2 pc_fieldsplit_1_fields = 1 pc_fieldsplit_2_fields = 0 pc_fieldsplit_type = multiplicative pc_type = fieldsplit snes_atol = 5.0e-7 snes_converged_reason = true snes_error_if_not_converged = true snes_monitor = true snes_rtol = 1.0e-14 ts_error_if_step_fails = true ts_exact_final_time = matchstep ts_monitor = true ts_type = beuler viewer_hdf5_collective = true >> src/cig/pylith/libsrc/pylith/meshio/MeshIOPetsc.cc:205:virtual void pylith::meshio::MeshIOPetsc::_read() -- info (application-flow) -- Component 'meshiopetsc.reader': Reading finite-element mesh from 'mesh_tri.msh'. >> src/cig/pylith/libsrc/pylith/meshio/MeshIO.cc:76:void pylith::meshio::MeshIO::read(pylith::topology::Mesh *, const bool) -- info (application-flow) -- Component 'meshiopetsc.reader': Domain bounding box: (0, 20000) (-20000, 0) >> src/cig/pylith/libsrc/pylith/problems/TimeDependent.cc:316:virtual void pylith::problems::TimeDependent::verifyConfiguration() const -- info (application-flow) -- Component 'timedependent.problem': Verifying problem configuration. >> software/pylith-debug/lib/python3.12/site-packages/pylith/problems/Problem.py:238:_printInfo -- info (application-flow) -- Scales for nondimensionalization: Length scale: 5000*m Displacement scale: 10*m Time scale: 4.16667e+07*s Rigidity scale: 6e+09*m**-1*kg*s**-2 Temperature scale: 1*K >> src/cig/pylith/libsrc/pylith/problems/TimeDependent.cc:342:virtual void pylith::problems::TimeDependent::initialize() -- info (application-flow) -- Component 'timedependent.problem': Initializing problem. >> src/cig/pylith/libsrc/pylith/problems/TimeDependent.cc:473:void pylith::problems::TimeDependent::solve() -- info (application-flow) -- Component 'timedependent.problem': Solving equations. 0 TS dt 0.151476 time -0.151476 0 SNES Function norm 9.951368195027e-01 Linear solve converged due to CONVERGED_ATOL iterations 160 1 SNES Function norm 3.315089885638e-08 Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1 # -- many lines omitted -- 50 TS dt 0.151476 time 7.42235 0 SNES Function norm 1.755481622154e-03 Linear solve converged due to CONVERGED_ATOL iterations 44 1 SNES Function norm 3.243946411840e-08 Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1 51 TS dt 0.151476 time 7.57382 >> software/pylith-debug/lib/python3.12/site-packages/pylith/problems/Problem.py:222:finalize -- info (application-flow) -- Finalizing problem. ``` At the beginning of the output written to the terminal, we see that PyLith is reading the mesh using the `MeshIOPetsc` reader and that it found the domain to extend from 0 to 20 km in the x direction and from -20 km to 0 in the y direction. PyLith detects the use of poroelasticity without a fault and selects appropriate preconditioning options as discussed in {ref}`sec-user-run-pylith-petsc-options`. At the end of the output written to the terminal, we see that the solver advanced the solution 51 time steps. The linear solver converges in about 50 or less iteration at most time steps and the norm of the residual met the absolute convergence tolerance (`ksp_atol`) . The nonlinear solve converged in 1 iteration, which we expect because this is a linear problem, and the residual met the absolute convergence tolerance (`snes_atol`). ## Visualizing the results In {numref}`fig:example:magma:2d:step01:solution` we use the `pylith_viz` utility to visualize the pressure field. You can move the slider or use the `p` and `n` keys to change the increment or decrement time. The initial linear variation in fluid pressure adjusts to match the boundary conditions and spatial variations in constitutive behavior. ```{code-block} console --- caption: Visualize PyLith output using `pylith_viz`. --- pylith_viz --filenames=output/step01_inflation-domain.h5 warp_grid --field=pressure ``` :::{figure-md} fig:example:magma:2d:step01:solution Solution for Step 1 at t=100 yr. The colors indicate the fluid pressure, and the deformation is exaggerated by a factor of 1000. Solution for Step 1 at t=100 yr. The colors of the shaded surface indicate the fluid pressure, and the deformation is exaggerated by a factor of 1000. ::: ## Step 1b: Adaptive Time Stepping In Step 1b we demonstrate the use adaptive time stepping. We start with an initial time step of 0.2 years and let the adaptive time stepping algorithm increase the time step as the rate of deformation decreases. We dcrease the tolerances slightly (the default tolerances are 0.05) for better agreement with the solution from uniform time stepping. ```{code-block} cfg --- caption: Adaptive time stepping parameters for Step 1b. --- [pylithapp.timedependent] start_time = -0.2*year initial_dt = 0.2*year end_time = 10.0*year petsc_defaults.adaptive_time_stepping = True [pylithapp.petsc] ts_atol = 0.02 ts_rtol = 0.02 ``` ```{code-block} console --- caption: Run Step 1b simulation --- $ pylith step01_inflation.cfg step01b_inflation.cfg # The output should look something like the following. >> software/pylith-debug/lib/python3.12/site-packages/pylith/utils/PetscManager.py:55:initialize -- info (application-flow) -- Initialized PETSc with user options ts_atol = 0.02 ts_rtol = 0.02 >> software/pylith-debug/lib/python3.12/site-packages/pylith/apps/PyLithApp.py:80:main -- info (application-flow) -- Running on 1 process(es). >> src/cig/pylith/libsrc/pylith/utils/PetscOptions.cc:251:static void pylith::utils::_PetscOptions::write(pythia::journal::info_t &, const char *, const PetscOptions &) -- info (application-flow) -- Setting PETSc options: fieldsplit_displacement_pc_type = lu fieldsplit_pressure_pc_type = bjacobi fieldsplit_trace_strain_pc_type = bjacobi ksp_atol = 1.0e-7 ksp_converged_reason = true ksp_error_if_not_converged = true ksp_guess_pod_size = 8 ksp_guess_type = pod ksp_rtol = 1.0e-14 pc_fieldsplit_0_fields = 2 pc_fieldsplit_1_fields = 1 pc_fieldsplit_2_fields = 0 pc_fieldsplit_type = multiplicative pc_type = fieldsplit snes_atol = 5.0e-7 snes_converged_reason = true snes_error_if_not_converged = true snes_monitor = true snes_rtol = 1.0e-14 ts_adapt_monitor = true ts_adapt_reject_safety = 0.1 ts_adapt_safety = 0.2 ts_adapt_type = basic ts_error_if_step_fails = true ts_exact_final_time = matchstep ts_monitor = true ts_type = beuler viewer_hdf5_collective = true # -- many lines omitted -- 19 TS dt 1.2231 time 6.35073 0 SNES Function norm 1.893597646523e-03 Linear solve converged due to CONVERGED_ATOL iterations 77 1 SNES Function norm 2.064447881404e-08 Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1 TSAdapt basic beuler 0: step 19 accepted t=6.35073 + 1.223e+00 dt=1.412e+00 wlte= 0.03 wltea= -1 wlter= -1 20 TS dt 1.41194 time 7.57382 >> software/pylith-debug/lib/python3.12/site-packages/pylith/problems/Problem.py:222:finalize -- info (application-flow) -- Finalizing problem. ``` The adaptive time stepping algorithm shortens a few of the early time steps but then lengthens the time step as the rate of deformation decreases. We end up with 20 time steps with adaptive time stepping compared to 51 with uniform time steps. ```{code-block} console --- caption: Compare the results from Step 1 ad 1b using the `plot_compare.py` Python script. --- ./plot_compare.py ``` :::{figure-md} fig:example:magma:2d:step01b:solution Vertical displacement and fluid pressure time histories for Step 1 and 1b. Vertical displacement and fluid pressure time histories for Step 1 and 1b at locations with the largest changes. We find very little difference between the solutions from uniform time stepping (51 time steps) and adaptive time stepping (20 time steps). :::