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The Radial Basis Functions Method for Improved Numerical Approximations of Geological Processes in Heterogeneous Systems

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Abstract

A robust, high order modeling approach is introduced, based on the finite difference-based radial basis functions method, for solving the groundwater flow equation in the presence of an active well, in the case of a confined aquifer in a complex geological environment. The two important novelties of this work are the analytical handling of the wells’ singularities and the ability to do this accurately and efficiently in a heterogeneous medium. It is argued that the most commonly used methods for this type of problem have severe weaknesses in both the treatment of the singularities associated with the well, and in representing heterogeneities which commonly occur in geological processes. The method presented here is first applied to the groundwater flow problem in a homogeneous medium for which the analytical solution is known, to show its high order algebraic convergence. The method is then compared against the United States geological survey’s MODFLOW software on a quasi-realistic benchmark test case in a heterogeneous medium. It is shown that much fewer nodes are needed by the proposed method to yield a similar accuracy.

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References

  1. Anderson MP, Woessner WW, Hunt RJ (2015) Applied groundwater modeling: simulation of flow and advective transport. Academic Press, Cambridge. ISBN 0080916384

  2. Bayona V, Kindelan M (2013) Propagation of premixed laminar flames in 3D narrow open ducts using RBF-generated finite differences. Combust Theory Model 17(5):789–803 (ISSN 1364-7830)

  3. Bayona V, Flyer N, Fornberg B, Barnett GA (2017) On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J Comput Phys 332:257–273 (ISSN 0021-9991)

  4. Bayona V, Flyer N, Fornberg B (2019) On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries. J Comput Phys 380:378–399 (ISSN 0021-9991)

  5. Bianchi M, Kearsey T, Kingdon A (2015) Integrating deterministic lithostratigraphic models in stochastic realizations of subsurface heterogeneity. Impact on predictions of lithology, hydraulic heads and groundwater fluxes. J Hydrol 531:557–573 (ISSN 0022-1694)

  6. Bochner S (1933) Monotone funktionen, stieltjessche integrale und harmonische analyse. Math Ann 108(1):378–410 (ISSN 0025-5831)

  7. Bollig EF, Flyer N, Erlebacher G (2012) Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs. J Comput Phys 231(21):7133–7151 (ISSN 0021-9991)

  8. Brunner P, Simmons CT (2012) Hydrogeosphere: a fully integrated, physically based hydrological model. Groundwater 50(2):170–176 (ISSN 0017-467X)

  9. Chan Y, Mullineux N, Reed J (1976) Analytic solutions for drawdowns in rectangular artesian aquifers. J Hydrol 31(1–2):151–160 (ISSN 0022-1694)

  10. Chow R, Frind ME, Frind EO, Jones JP, Sousa MR, Rudolph DL, Molson JW, Nowak W (2016) Delineating baseflow contribution areas for streams—a model and methods comparison. J Contam Hydrol 195:11–22

  11. De Marsily G, Delay F, Goncalves J, Renard P, Teles V, Violette S (2005) Dealing with spatial heterogeneity. Hydrogeol J 13(1):161–183 (ISSN 1431-2174)

  12. Dehghan M, Abbaszadeh M (2018) A combination of proper orthogonal decomposition-discrete empirical interpolation method (POD-DEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination. Comput Math Appl 75(4):1390–1412 (ISSN 0898-1221)

  13. Dickinson JE, James SC, Mehl S, Hill MC, Leake S, Zyvoloski GA, Faunt CC, Eddebbarh AA (2007) A new ghost-node method for linking different models and initial investigations of heterogeneity and nonmatching grids. Adv Water Resour 30(8):1722–1736 (ISSN 0309-1708)

  14. Driscoll TA, Fornberg B (2002) Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl 43(3–5):413–422 (ISSN 0898-1221)

  15. Famiglietti JS, Murdoch L, Lakshmi V, Arrigo J, Hooper R (2011) Establishing a framework for community modeling in hydrologic science. In: Report from the 3rd workshop on a community hydrologic modeling platform (CHyMP): a strategic and implementation plan, vol 10. Technical report, pp 15–17

  16. Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, Singapore. ISBN 981270633X

  17. Fitts C (2012) Groundwater science, 2nd edn. Elsevier Science Publishing Co Inc, San Diego. ISBN 9780123847058 0123847052

  18. Flyer N, Fornberg B (2011) Radial basis functions: developments and applications to planetary scale flows. Comput Fluids 46(1):23–32 (ISSN 0045-7930)

  19. Flyer N, Lehto E, Blaise S, Wright GB, St-Cyr A (2012) A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere. J Comput Phys 231(11):4078–4095 (ISSN 0021-9991)

  20. Flyer N, Wright GB, Fornberg B (2014) Radial basis function-generated finite differences: a mesh-free method for computational geosciences. Handbook of geomathematics, pp 1–30. ISSN 3642277934

  21. Flyer N, Fornberg B, Bayona V, Barnett GA (2016) On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J Comput Phys 321:21–38 (ISSN 0021-9991)

  22. Fornberg B, Flyer N (2015a) A primer on radialbasis functions with applications to the geosciences. SIAM. ISBN 161197402X

  23. Fornberg B, Flyer N (2015b) Solving PDEs with radial basis functions. Acta Numer 24:215–258 (ISSN 0962-4929)

  24. Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230(6):2270–2285 (ISSN 0021-9991)

  25. Haitjema HM (1995) Analytic element modeling of groundwater flow. Elsevier, Amsterdam. ISBN 0080499104

  26. Harbaugh AW, Banta ER, Hill MC, McDonald MG (2000) MODFLOW-2000, the U.S. geological survey modular ground-water model—user guide to modularization concepts and the ground-water flow process. Technical report, U.S. Geological Survey

  27. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915 (ISSN 0148-0227)

  28. Jackson CR (2001) The development and validation of the object-oriented quasi three-dimensional regional groundwater model ZOOMQ3D. Technical report, British Geological Survey

  29. Kansa EJ (1990a) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput Math Appl 19(8–9):127–145 (ISSN 0898-1221)

  30. Kansa EJ (1990b) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8–9):147–161 (ISSN 0898-1221)

  31. Langevin C, Panday S, Niswonger R, Hughes J, Ibaraki M, Mehl S (2011) Local grid refinement with an unstructured grid version of modflow. In: Maxwell P, Hill Z (eds) MODFLOW and more 2011: integrated hydrologic modeling—conference proceedings, pp 47–51

  32. Larsson E, Lehto E, Heryudono A, Fornberg B (2013) Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J Sci Comput 35(4):A2096–A2119 (ISSN 1064-8275)

  33. Majumder P, Eldho TI (2015) Groundwater flow and transport simulation for a confined aquifer using analytic element method and random walk particle tracking. In: E-proceedings of the 36th IAHR world congress. IAHR, Madrid, Spain, pp 1–10

  34. Martin B, Fornberg B, St-Cyr A (2015) Seismic modeling with radial-basis-function-generated finite differences. Geophysics 80(4):T137–T146 (ISSN 0016-8033)

  35. Micchelli CA (1986) Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr Approx 2(1):11–22 (ISSN 1432-0940)

  36. Panday S, Langevin CD (2012) Improving sub-grid scale accuracy of boundary features in regional finite-difference models. Adv Water Resour 41:65–75 (ISSN 0309-1708)

  37. Patel S, Rastogi AK (2015) Application of Kansa’s multiquadric radial basis function based meshfree model for groundwater simulation. In: HYDRO 2015 INTERNATIONAL: 20th international conference on hydraulics, water resources and river engineering, pp 1–8

  38. Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 811–841 (ISSN 0003-486X)

  39. Shu C, Ding H, Yeo K (2003) Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 192(7–8):941–954 (ISSN 0045-7825)

  40. Singh A (2014) Groundwater resources management through the applications of simulation modeling: a review. Sci Total Environ 499:414–423 (ISSN 0048-9697)

  41. Tolstykh AI (2000) On using RBF-based differencing formulas for unstructured and mixed structured–unstructured grid calculations. In: Proceedings of the 16th IMACS world congress, vol 228, Lausanne, pp 4606–4624

  42. Tolstykh AI, Shirobokov DA (2003) On using radial basis functions in a “finite difference mode” with applications to elasticity problems. Comput Mech 33(1):68–79 (ISSN 1432-0924)

  43. Trefry MG, Muffels C (2007) FEFLOW: a finite-element ground water flow and transport modeling tool. Groundwater 45(5):525–528

  44. Watson C, Richardson J, Wood B, Jackson C, Hughes A (2015) Improving geological and process model integration through TIN to 3D grid conversion. Comput Geosci 82:45–54 (ISSN 0098-3004)

  45. Wright GB (2003) Radial basis function interpolation: numerical and analytical developments. Dissertation, University of Colorado at Boulder

  46. Zyvoloski G (2007) FEHM: a control volume finite element code for simulating subsurface multi-phase multi-fluid heat and mass transfer. Technical report LAUR-07-3359, Los Alamos National Laboratory

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Correspondence to Cécile Piret.

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Piret, C., Dissanayake, N., Gierke, J.S. et al. The Radial Basis Functions Method for Improved Numerical Approximations of Geological Processes in Heterogeneous Systems. Math Geosci (2019). https://doi.org/10.1007/s11004-019-09820-w

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Keywords

  • RBF-FD
  • Drawdown by pumping
  • Groundwater flow
  • Heterogeneous media