Poroelasticity simulation in CLAWPACK for geophysical and biological applications

Transcription

1 Poroelasticity simulation in CLAWPACK for geophysical and biological applications October 27, 2012 Joint work with Randall J. LeVeque (University of Washington) and Yvonne Ou (University of Delaware) Supported in part by grants from the NIH and NSF

2 Outline 1 Poroelasticity 2 Numerical solution 3 Results Convergence studies Qualitative checks Mapped grid results 4 Summary and future work

3 Outline 1 Poroelasticity 2 Numerical solution 3 Results Convergence studies Qualitative checks Mapped grid results 4 Summary and future work

4 Poroelasticity theory Poroelasticity: study of mechanics of fluid-filled porous solids Originally developed by Maurice Biot in 1930s-1960s for soil and rock Major early interest from oil industry Recent interest for monitoring underground fluid injection (e.g. carbon sequestration) Recently applied to bone as well Understanding wave propagation in bone is original motivator for this work Pumice stone. Transverse section of cortical bone. Images courtesy Wikimedia Commons.

5 Equations of poroelasticity Glossing over a lot of details, can model poroelasticity as a first-order linear system of PDEs: t Q + A x Q + B z Q = DQ, Q = [ p σ xx σ zz σ xz v x v z q x q z ] T p is fluid pressure; σ is solid stress tensor; v is solid velocity; q is fluid flow rate WRT solid Left side is classic hyperbolic system Right side introduces dissipation, through viscous drag as fluid flows through pores Many important poroelastic materials are anisotropic A and B depend on axes orientation Show system matrices

6 Poroelastic waves Poroelasticity equations look like elastodynamics + acoustics. Three families of propagating waves: 1 Fast P-waves, where fluid and solid move (roughly) parallel to propagation direction and in phase 2 S-waves, where fluid and solid move transverse to propagation direction 3 Slow P-waves, where fluid and solid move (roughly) parallel to propation direction but 180 degrees out of phase Slow P-waves involve large motions of fluid relative to solid heavily damped by viscosity System also supports non-wave-like diffusive slow mode where fluid seeps through pores due to pressure gradient

7 Wave structure of viscous orthotropic poroelasticity Source term DQ causes dissipation and dispersion Anisotropy also causes wave speeds to differ depending on propagation direction Plots below: phase velocity vs. frequency and direction for orthotropic layered sandstone Phase velocity and decay rate for waves in the X direction Dashed lines are speeds without dissipation Frequency (Hz) Phase velocity (m/s) Decay rate (1/m) Phase velocity (m/s) versus direction, all wave families (no dissipation) Fast P-wave S-wave315 Slow P-wave 0

8 Stiffness of relaxation term t Q + A x Q + B z Q = DQ Source term DQ has its own intrinsic time scales May be stiff depending on problem solved Can expect difficulties with solution Source term is of relaxation type, so expect solution to be close to that of reduced system, t u + A r x u + B r z u = 0 obtained by restricting to null space N (D) A r = ΠAG, where G maps reduced variables to full variables and Π maps full to reduced Reduced system satisfies subcharacteristic condition wave speeds not faster than full system Can at least expect qualitatively correct solution

9 Outline 1 Poroelasticity 2 Numerical solution 3 Results Convergence studies Qualitative checks Mapped grid results 4 Summary and future work

10 Finite volume method We use high-resolution wave propagation finite volume methods (FVM) to solve the poroelasticity equations Overview of approach for a hyperbolic problem: Discretize domain into grid of cells Assume we know average value of solution on each cell At each time step: 1 Reconstruct approximate solution from cell averages (usually discontinuous between cells) 2 Solve Riemann problem (Cauchy problem with piecewise constant initial data) at each cell interface 3 Evolve Riemann solution for time step t 4 Average new solution onto grid

11 Advantages of high-resolution FVM Riemann solution explicitly reproduces full gamut of wave behavior Can greatly reduce numerical dispersion through wave limiters Limiters also improve accuracy (in typical computations) at low cost Recent advanced limiters can even increase order of accuracy Easy to incorporate non-cartesian grids Handles low-quality meshes gracefully

12 Fluid-poroelastic interface Riemann problem approach allows fairly natural handling of material discontinuities, special interface conditions Example of interest: interface between fluid and poroelastic medium Have set of interface conditions: Fluid continuity equation: Newton s third law: q fluid n = (v poro + q poro ) n p fluid n = τ poro n Hydraulic permeability condition: p poro p fluid = 1 K q poro n Can incorporate these as constraints on Riemann solution

13 Fluid-poroelastic interface Some algorithmic complexity introduced at fluid-poroelastic interface High-resolution FVM originates with conservation laws, e.g. q t + F(q) x + G(q) z = 0 Some ideas most easily written in terms of fluxes F and G Can t define single flux between materials with different physics! Higher-order correction fluxes harder to formulate, some accuracy lost Not a serious problem in practice nonsmooth solution at interface anyway

14 Source term Source term handled via operator splitting Q t = DQ solved exactly with matrix exponential Best accuracy available for this part of system No stability restriction Used Strang splitting scheme: 1 Take 1 2 t step of source term 2 Take full t step of wave propagation terms 3 Take another 1 2 t step of source term Formally second-order accurate as t 0, but will see what we actually get for realistic step size

15 CLAWPACK Solved poroelasticity equations using CLAWPACK CLAWPACK: Conservation LAWs PACKage Can really handle any hyperbolic system, not just conservation laws High-resolution finite volume package for wave propagation Low memory overhead, parallel (multicore, PETSc) Operator splitting built in for source terms Adaptive mesh refinement available too (Berger-Colella-Oliger approach, AMRCLAW) Handles code that is common across all high-resolution FVM solvers User only needs to write routines for Riemann solve, source terms, grid mapping (if used)

16 Outline 1 Poroelasticity 2 Numerical solution 3 Results Convergence studies Qualitative checks Mapped grid results 4 Summary and future work

17 Overview of results Two general classes of test problems run so far: 1 Convergence studies comparing against known analytic solutions Advantage: Can quantitatively measure accuracy, convergence rate Disadvantage: Limited library of solutions to compare against 2 Qualitative sanity-check problems and eyeball norm comparisons to published solutions Advantage: Useful when analytic solution not available, good for ruling out some types of bug Disadvantage: Not very precise Most results are for Cartesian grids, but also have mapped grid results with more interesting geometry

18 Convergence studies Conducted convergence studies against analytic solutions Solutions used: plane waves of the form Q(x, z, t) = V exp(i(k x x + k z z ωt)) with some real ω specified. Important to test with waves propagating in variety of directions θ wave Also need to test variety of material principal directions θ mat For each (θ wave, θ mat ) pair, need to sweep over grid size to check convergence z (meters) v z (m/s) at time t = z θ mat θ wave 1 x Wave direction x (meters) Snapshot of plane wave solution showing θ wave, principal axes, and θ mat

19 Convergence results: inviscid First convergence study: ignore viscous dissipation, validate hyperbolic solver by itself Results are generally good Slow P wave was underresolved on coarse grids worse apparent performance Error measured using energy max-norm, relative to amplitude in energy norm of true solution Convergence rate Error on finest grid Wave family Best Worst Best Worst Fast P S Slow P

20 Convergence results: viscous high-frequency Ran more detailed convergence studies at lower and upper frequency ranges of Biot system High frequency chosen: 10 khz Ran slow P waves on different grid from others because of extremely rapid damping Results are good, though not as good as inviscid for the fast P and S waves Convergence rate Error on finest grid Wave family Best Worst Best Worst Fast P S Slow P

21 Convergence results: viscous low-frequency Also examined results at low frequency: 10 Hz Again ran slow P waves on different grid because of rapid damping Results are not so good Convergence rate Error on finest grid Wave family Best Worst Best Worst Fast P S Slow P

22 Convergence results: discussion What s going on here? Trouble only for timesteps comparable to relaxation time or longer Asymptotic error estimates only good as t 0 not inconsistent with bad results for Strang at large t. Literature suggests hyperbolic systems with stiff relaxation terms are hard to model accurately Dissipation causes change in Riemann solution structure at longer times Solution structure approaches that of reduced system Reduced system waves are blurred into erf-shapes, so reduced wave structure is not quite right either Possible ways to improve accuracy: Modeling full Riemann solution structure explicitly Using method of lines approach with appropriate integrator

23 Poroelastic code validation: two-material test case Test case: wave reflections from a material interface, from de la Puente et al. (2008) Excitation is a point source with a Ricker wavelet profile in time Forcing acts on σ z and fluid pressure terms with equal magnitude and opposite sign Source is located in shale overlying sandstone bed Material properties isotropic for this case, viscosity ignored AMR used to capture fine details

24 Poroelastic code validation: two-material test case v z (m/s) at time t = Shale a) b) z (meters) Sandstone x (meters) Figure 9. a Snapshot of the y-direction of the solid particle velocity at t 0.25 s. ers by empty circles. b Zoomed region showing the fine mesh required to capture z component of matrix velocity field. Left: CLAWPACK. Right: de la Puente. Rectangular outlines indicate boundaries of AMR grids. Note: Different value-to-shade maps on each plot. Downloaded 02 Jan 2011 to Redistribution subject to SEG lice

25 Poroelastic code validation: two-material test case DG methods for poroelastic media T93 15 a) u (m/s) v z (m/s) at gauge 1 8 Receiver 1 (y = 750m) time Time (s) c) ADER DG O5 FD Residuals d) ADER DG O5 FD 6 Residuals Left: CLAWPACK, right: 4 de la Puente Time (s) Time-history of matrix z velocity at topmost gauge. b) v (m/s) ADER DG O5 FD Residuals Receiver 1 (y = 750m) ADER DG O5 FD Residuals 1 2 u (m/s) 0 v (m/s)

26 Cylindrical scatterer More challenging test case: poroelastic cylinder in fluid bath Time-harmonic behavior of cylinder scattering acoustic waves Problem is not in stiff regime Tests mapped grid, fluid-poro interface Analytic solution provided by Prof. Yvonne Ou, University of Delaware Results: Observed 2nd-order convergence in 1- and 2-norms, 1st-order in max-norm Accuracy degraded by fluid-poro interface, highly distorted grid map

27 Cylindrical scatterer results 0.06 grids at time t = e z (meters) x (meters) Results for cylindrical scatterer in biological fluid. Incoming wavelength is 1/4 cylinder diameter. Left: fluid pressure ( grid); right: grid illustrating mapping Logically rectangular circle map from Calhoun, Helzel, and LeVeque (2007)

28 Simple bone model Have some quick results with simplified model of human femur Model uses mapped grid for concentric circular cross-sections Could do more complex geometry as long as mapping subroutine is available Have multiple materials, fluid-poroelastic interface: Inner core of cancellous bone Outer annulus of cortical bone Surrounding domain is water Model is not in stiff regime relaxation time is much longer than time step

29 Simple bone model p (Pa) at time t = e-05 grids at time t = z (meters) z (meters) x (meters) x (meters) Results for simplified model of human femur in water, struck by acoustic wave with Gaussian profile. Left: fluid pressure ( grid); right: grid illustrating mapping

30 Outline 1 Poroelasticity 2 Numerical solution 3 Results Convergence studies Qualitative checks Mapped grid results 4 Summary and future work

31 Summary Poroelasticity is a rich and complex system, with a wide variety of behaviors Have developed poroelasticity solver, validated against known solutions Solver capabilities: Multiple materials Fluid-poroelastic interfaces Mapped grids for moderately complex geometries Parallel execution (thanks to CLAWPACK framework) Convergence and accuracy are good away from stiff regime Convergence degraded in stiff regime, but magnitude of error is generally not bad Good results obtained for mapped grid models

32 Future work Deal with convergence problems in stiff regime More advanced Riemann solver? Exponential integrator for method of lines scheme Implement property averaging across material boundaries for geometry too complex for mapped grids? Look at micro-scale modeling to obtain poroelastic properties Examine cylindrical scatterer in more detail resonance, surface waves Extend to 3D

33 Poroelasticity system matrices [ ] [ ] 0 Ăsv 0 0 Ă = n xa + n zb =, D = Ă vs 0 0 D v n xα 1 M n zα 3 M n xm n zm Ă sv = n xc u 11 n zc u 13 n xα 1 M n zα 1 M n xc u 13 n zc u 33 n xα 3 M n zα 3 M n zc u 55 n xc u ρ n f m x n 1 m x 0 n 1 1 z 1 ρ 1 n f m z 0 n 3 m z n 3 Ă vs = 3 x 3 3 ρ ρ n x n f ρ x 0 n f 1 z 1 1 ρ ρ n z 0 n f ρ z n f 3 x 3 3 ρ 0 0 f η 0 1 κ 1 ρ f η D v = 3 κ ρη 0 1 κ ρη 3 κ 3 Note: these are in principal axes of orthotropic material Back to main