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Fast flow computation methods on unstructured tetrahedral meshes for rapid reservoir modelling

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Abstract

Subsurface reservoir models have a high degree of uncertainty regarding reservoir geometry and structure. A range of conceptual models should therefore be generated to explore how fluids-in-place, reservoir dynamics, and development decisions are affected by such uncertainty. The rapid reservoir modelling (RRM) workflow has been developed to prototype reservoir models across scales and test their dynamic behaviour. RRM complements existing workflows in that conceptual models can be prototyped, explored, compared, and ranked rapidly prior to detailed reservoir modelling. Reservoir geology is sketched in 2D with geological operators and translated in real-time into geologically correct 3D models. Flow diagnostics provide quantitative information for these reservoir model prototypes about their static and dynamic behaviours. A tracing algorithm is reviewed and implemented to compute time-of-flight and tracer concentrations efficiently on unstructured grids. Numerical well testing (NWT) is adopted in RRM to further interrogate the reservoir model. A new edge-based fast marching method is developed and implemented to solve the diffusive time-of-flight for approximating pressure transients efficiently on unstructured tetrahedral meshes. We demonstrate that an implementation of the workflow consisting of integrated sketch-based interface modelling, unstructured mesh generation, flow diagnostics, and numerical well testing is possible.

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References

  1. Abushaikha, A.S., Blunt, M.J., Gosselin, O.R., Pain, C.C., Jackson, M.D.: Interface control volume finite element method for modelling multi-phase fluid flow in highly heterogeneous and fractured reservoirs. J. Comput. Phys. 298, 41–61 (2015)

    Google Scholar 

  2. Arnold, D., Demyanov, V., Christie, M., Bakay, A., Gopa, K.: Optimisation of decision making under uncertainty throughout field lifetime: A fractured reservoir example. Comput. Geosci. 95, 123–139 (2016)

    Google Scholar 

  3. Baliga, B., Patankar, S.: A new finite-element formulation for convection-diffusion problems. Numer. Heat Transfer 3(4), 393–409 (1980)

    Google Scholar 

  4. Bank, R., Falgout, R., Jones, T., Manteuffel, T.A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid domain and range decomposition (amg-dd/amg-rd). SIAM J. Sci. Comput. 37(5), S113–S136 (2015)

    Google Scholar 

  5. Batycky, R.P., Thiele, M.R., Baker, R.O., Chung, S.: Revisiting reservoir flood-surveillance methods using streamlines. SPE Paper 95402 (2005)

  6. Bentley, M.: Modelling for comfort? Pet. Geosci. 22(1), 3–10 (2016)

    Google Scholar 

  7. Bond, C., Gibbs, A., Shipton, Z., Jones, S.: What do you think this is? “Conceptual uncertainty” in geoscience interpretation. GSA Today 17(11), 4 (2007)

    Google Scholar 

  8. Bourdet, D.: Well Test Analysis: the Use of Advanced Interpretation Models, vol. 3. Elsevier, Amsterdam (2002)

    Google Scholar 

  9. Caumon, G., Collon-Drouaillet, P., De Veslud, C.L.C., Viseur, S., Sausse, J.: Surface-based 3d modeling of geological structures. Math. Geosci. 41(8), 927–945 (2009)

    Google Scholar 

  10. Cavero, J., Orellana, N.H., Yemez, I., Singh, V., Izaguirre, E.: Importance of conceptual geological models in 3d reservoir modelling. First Break 34(7), 39–49 (2016)

    Google Scholar 

  11. Chandra, V.S., Corbett, P.W., Geiger, S., Hamdi, H., et al.: Improving reservoir characterization and simulation with near-wellbore modeling. SPE Reserv. Eval. Eng. 16(02), 183–193 (2013)

    Google Scholar 

  12. Choudhuri, B., Thakuria, C., Belushi, A.A., Nurzaman, Z., Al Hashmi, K., Batycky, R.: Optimization of a large polymer flood with full-field streamline simulation. SPE Reserv. Eval. Eng. 18(2), 318–328 (2015)

    Google Scholar 

  13. Corbett, P., Geiger, S., Borges, L., Garayev, M., Valdez, C.: The third porosity system understanding the role of hidden pore systems in well-test interpretation in carbonates. Pet. Geosci. 18(1), 73–81 (2012)

    Google Scholar 

  14. Corbett, P.W., Geiger-Boschung, S., Borges, L.P., Garayev, M., Gonzalez, J.G., Valdez, C., et al.: Limitations in numerical well test modelling of fractured carbonate rocks. In: SPE EUROPEC/EAGE Annual Conference and Exhibition. Society of Petroleum Engineers (2010)

  15. Corbett, P.W., Mesmari, A., Stewart, G., et al.: A method for using the naturally-occurring negative geoskin in the description of fluvial reservoirs. In: European Petroleum Conference. Society of Petroleum Engineers (1996)

  16. Dahiya, D., Baskar, S., Coulouvrat, F.: Characteristic fast marching method for monotonically propagating fronts in a moving medium. SIAM J. Sci. Comput. 35(4), A1880–A1902 (2013)

    Google Scholar 

  17. Datta-Gupta, A., King, M.J.: Streamline Simulation: Theory and practice. Vol. 11. Society of Petroleum Engineers Richardson (2007)

  18. Datta-Gupta, A., Xie, J., Gupta, N., King, M.J., Lee, W.J., et al.: Radius of investigation and its generalization to unconventional reservoirs. J. Petrol. Tech. 63(07), 52–55 (2011)

    Google Scholar 

  19. Edwards, M.G.: Higher-resolution hyperbolic-coupled-elliptic flux-continuous CVD schemes on structured and unstructured grids in 2-d. Int. J. Numer. Methods Fluids 51(9-10), 1059–1077 (2006)

    Google Scholar 

  20. Egya, D., Geiger, S., Corbett, P., March, R., Bisdom, K., Bertotti, G., Bezerra, F.: Analysing the limitations of the dual-porosity response during well tests in naturally fractured reservoirs. Pet. Geosci. 25, 30–49 (2019)

    Google Scholar 

  21. Elias, R.N., Martins, M.A., Coutinho, A.L.: Simple finite element-based computation of distance functions in unstructured grids. Int. J. Numer. Methods Eng. 72(9), 1095–1110 (2007)

    Google Scholar 

  22. Forsyth, P.A., et al.: A control-volume, finite-element method for local mesh refinement in thermal reservoir simulation. SPE Reserv. Eng. 5(04), 561–566 (1990)

    Google Scholar 

  23. Gomes, J., Pavlidis, D., Salinas, P., Xie, Z., Percival, J.R., Melnikova, Y., Pain, C.C., Jackson, M.D.: A force-balanced control volume finite element method for multiphase porous media flow modelling. International Journal for Numerical Methods Fluids, pp. 1097–0363 (2016)

  24. Graham, G.H., Jackson, M.D., Hampson, G.J.: Three-dimensional modeling of clinoforms in shallow-marine reservoirs: part 1. Concepts and application clinoform modeling in shallow-marine reservoirs: Part 1. Concepts and application. AAPG Bull. 99(6), 1013–1047 (2015)

    Google Scholar 

  25. Graham, G.H., Jackson, M.D., Hampson, G.J.: Three-dimensional modeling of clinoforms in shallow-marine reservoirs: part 2. Impact on fluid flow and hydrocarbon recovery in fluvial-dominated deltaic reservoirs. AAPG Bull. 99(6), 1049–1080 (2015)

    Google Scholar 

  26. Gries, S.: On the convergence of System-AMG in reservoir simulation. SPE Paper 182630 (2017)

  27. Hægland, H.: Streamline methods with application to flow and transport in fractured media. Ph.D. thesis, University of Bergen (2009)

  28. Hægland, H., Kaufmann, R., Aavatsmark, I.: Comparison of vertex- and cell-centered methods for flow and transport simulation in 3d. SPE paper 163593 (2012)

  29. Hassanpour, M.M., Pyrcz, M.J., Deutsch, C.V.: Improved geostatistical models of inclined heterolithic strata for McMurray Formation, Alberta, Canada. AAPG Bull. 97(7), 1209–1224 (2013)

    Google Scholar 

  30. Hoffman, K.S., Neave, J.W.: Horizon modeling using a three-dimensional fault restoration technique. SPE paper 56445 (1999)

  31. Jackson, M., Hampson, G., Rood, D., Geiger, S., Zhang, Z., Sousa, M., Amorim, R., Brazil, E., Samavati, F., Guimaraes, L.: Rapid reservoir modeling: psrototyping of reservoir models, well trajectories and development options using an intuitive, sketch-based interface. SPE Paper 173237 (2015)

  32. Jackson, M., Percival, J., Mostaghimi, P., Tollit, B., Pavlidis, D., Pain, C., Gomes, J., Elsheikh, A.H., Salinas, P., Muggeridge, A., Blunt, M.: Reservoir modeling for flow simulation by use of surfaces, adaptive unstructured meshes, and an overlapping-control-volume finite-element method. SPE Reserv. Eval. Eng. 18(02), 115–132 (2015)

    Google Scholar 

  33. Jacquemyn, C., Jackson, M., Hampson, G.: Surface-based geological reservoir modelling using grid-free nurbs curves and surfaces. Mathematical Geosciences. https://doi.org/10.1007/s11004-018-9764-8 (2018)

  34. Jacquemyn, C., Melnikova, Y., Jackson, M., Hampson, G., John, C.: Geologic modelling using parametric nurbs surfaces. In: ECMOR XV-15th European Conference on the Mathematics of Oil Recovery (2016)

  35. Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95(15), 8431–8435 (1998)

    Google Scholar 

  36. Konukoglu, E., Sermesant, M., Clatz, O., Peyrat, J.-M., Delingette, H., Ayache, N.: A recursive anisotropic fast marching approach to reaction diffusion equation: application to tumor growth modeling. In: Information Processing in Medical Imaging, pp. 687–699. Springer (2007)

  37. Krogstad, S., Lie, K.-A., Nilsen, H.M., Berg, C.F., Kippe, V.: Efficient flow diagnostics proxies for polymer flooding. Comput. Geosci. 21(5-6), 1203–1218 (2017)

    Google Scholar 

  38. Mallet, J.-L.: Geomodelling. Oxford University Press, Oxford (2002)

    Google Scholar 

  39. Massart, B.Y., Jackson, M.D., Hampson, G.J., Johnson, H.D., Legler, B., Jackson, C. A.-L.: Effective flow properties of heterolithic, cross-bedded tidal sandstones: part 1. Surface-based modeling. AAPG Bull. 100(5), 697–721 (2016)

    Google Scholar 

  40. Massonnat, G., Bandiziol, D., et al.: Interdependence between geology and well test interpretation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (1991)

  41. Matringe, S.F., Juanes, R., Tchelepi, H.A.: Tracing streamlines on unstructured grids from finite volume discretizations. SPE J. 13(4), 423–431 (2008)

    Google Scholar 

  42. Mello, U.T., Rodrigues, J.R.P., Rossa, A.L.: A control-volume finite-element method for three-dimensional multiphase basin modeling. Mar. Pet. Geol. 26(4), 504–518 (2009)

    Google Scholar 

  43. Milliotte, C., Matthäi, S.: From seismic interpretation to reservoir model: an integrated study accounting for the structural complexity of the Vienna Basin using an unstructured reservoir grid. First Break 32(5), 95–101 (2014)

    Google Scholar 

  44. Moog, G.: Advanced discretization methods for flow simulation using unstructured grids. Department of Energy Resources Engineering, Stanford University CA (2013)

  45. Møyner, O., Krogstad, S., Lie, K.-A.: The application of flow diagnostics for reservoir management. SPE J. 20(02), 306–323 (2015)

    Google Scholar 

  46. Natvig, J.R., Lie, K.-A., Eikemo, B., Berre, I.: An efficient discontinuous Galerkin method for advective transport in porous media. Adv. Water Resour. 30(12), 2424–2438 (2007)

    Google Scholar 

  47. Olsen, L., Samavati, F.F., Sousa, M.C., Jorge, J.A.: Sketch-based modeling: a survey. Comput. Graph. 33(1), 85–103 (2009)

    Google Scholar 

  48. Pyrcz, M.J., Boisvert, J.B., Deutsch, C.V.: Alluvsim: a program for event-based stochastic modeling of fluvial depositional systems. Comput. Geosci. 35(8), 1671–1685 (2009)

    Google Scholar 

  49. Rankey, E.C., Mitchell, J.C.: That’s why it’s called interpretation: impact of horizon uncertainty on seismic attribute analysis. Lead. Edge 22(9), 820–828 (2003)

    Google Scholar 

  50. Rasmussen, A.F.: Streamline tracing on irregular geometries. In: EC- MOR XII-12th European Conference on the Mathematics of Oil Recovery (2010)

  51. Rood, M., Jackson, M., Hampson, G., Brazil, E., de Carvalho, F., Coda, C., Sousa, M., Zhang, Z., Geiger, S.: Sketch-based geologic modeling. In: AGU Fall Meeting Abstracts (2015)

  52. Salinas, P., Pavlidis, D., Xie, Z., Pain, C.C., Jackson, M.: A double control volume finite element method with dynamic unstructured mesh optimization. SPE paper 182647 (2017)

  53. Sech, R.P., Jackson, M.D., Hampson, G.J.: Three-dimensional modeling of a shoreface-shelf parasequence reservoir analog: part 1. Surface-based modeling to capture high-resolution facies architecture. AAPG Bulletin 93 (9), 1155–1181 (2009)

    Google Scholar 

  54. Sermesant, M., Konukoglu, E., Delingette, H., Coudière, Y., Chinchapatnam, P., Rhode, K., Razavi, R., Ayache, N.: An anisotropic multi-front fast marching method for real-time simulation of cardiac electrophysiology. Functional Imaging and Modeling of the Heart, 160–169 (2007)

  55. Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93 (4), 1591–1595 (1996)

    Google Scholar 

  56. Sethian, J.A., Vladimirsky, A.: Fast methods for the Eikonal and related Hamilton–Jacobi equations on unstructured meshes. Proc. Natl. Acad. Sci. 97(11), 5699–5703 (2000)

    Google Scholar 

  57. Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton–Jacobi equations: Theory and algorithms. SIAM J. Numer. Anal. 41(1), 325–363 (2003)

    Google Scholar 

  58. Shahvali, M., Mallison, B., Wei, K., Gross, H.: An alternative to streamlines for flow diagnostics on structured and unstructured grids. SPE J. 17(03), 768–778 (2012)

    Google Scholar 

  59. Sharifi, M., Kelkar, M., Bahar, A., Slettebo, T., et al.: Dynamic ranking of multiple realizations by use of the fast-marching method. SPE J. 19(06), 1–069 (2014)

    Google Scholar 

  60. Shook, G.M., Mitchell, K.M.: A robust measure of heterogeneity for ranking earth models: The F PHI curve and dynamic Lorenz coefficient. SPE Paper 124625 (2009)

  61. Si, H.: Tetgen, a Delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. (TOMS) 41 (2), 11 (2015)

    Google Scholar 

  62. Spooner, V., Geiger, S., Dan, A.: Flow diagnostics for naturally fractured reservoirs. SPE Paper 190877 (2018)

  63. Stüben, K., Clees, T., Klie, H., Lu, B., Wheeler, M.: Algebraic multigrid methods (AMG) for the efficient solution of fully implicit formulations in reservoir simulation. SPE Paper 105832 (2007)

  64. Thiele, M.R., Batycky, R.: Water injection optimization using a streamline-based workflow. SPE Paper 84080 (2003)

  65. Vasco, D., Keers, H., Karasaki, K.: Estimation of reservoir properties using transient pressure data: An asymptotic approach. Water Resour. Res. 36(12), 3447–3465 (2000)

    Google Scholar 

  66. Velho, L., Zorin, D.: 4–8 subdivision. Comput. Aided Geom. Des. 18(5), 397–427 (2001)

    Google Scholar 

  67. Xie, J., Yang, C., Gupta, N., King, M.J., Datta-Gupta, A., et al.: Depth of investigation and depletion in unconventional reservoirs with fast-marching methods. SPE J. 20(04), 831–841 (2015)

    Google Scholar 

  68. Xie, J., Yang, C., Gupta, N., King, M.J., Datta-Gupta, A., et al.: Integration of shale-gas-production data and microseismic for fracture and reservoir properties with the fast marching method. SPE J. 20(02), 347–359 (2015)

    Google Scholar 

  69. Yang, C., Vyas, A., Data-Gupta, A., Ley, B., Biswas, P.: Rapid multistage hydraulic fracture design and optimization in unconventional reservoirs using a novel fast marching method. J. Pet. Sci. Eng. 156, 91–101 (2017)

    Google Scholar 

  70. Yang, J., Stern, F.: A highly scalable massively parallel fast marching method for the Eikonal equation. J. Comput. Phys. 332, 333–362 (2017)

    Google Scholar 

  71. Zhang, X., Pyrcz, M.J., Deutsch, C.V.: Stochastic surface modeling of deepwater depositional systems for improved reservoir models. J. Pet. Sci. Eng. 68(1-2), 118–134 (2009)

    Google Scholar 

  72. Zhang, Y., Bansal, N., Fujita, Y., Datta-Gupta, A., King, M.J., Sankaran, S., et al.: From streamlines to fast marching: rapid simulation and performance assessment of shale-gas reservoirs by use of diffusive time of flight as a spatial coordinate. SPE J. 21(05), 1883–1898 (2016)

    Google Scholar 

  73. Zhang, Y., Yang, C., King, M.J., Datta-Gupta, A., et al.: Fast-marching methods for complex grids and anisotropic permeabilities: application to unconventional reservoirs. SPE Paper 163637 (2013)

  74. Zhang, Z.: Unstructured mesh methods for stratified turbulent flows. Ph.D. thesis, Loughborough University (2015)

  75. Zhang, Z., Geiger, S., Rood, M., Jacquemyn, C., Jackson, M., Hampson, G., De Carvalho, F.M., Silva, C.C.M.M., Silva, J.D.M., Sousa, M.C., et al.: A tracing algorithm for flow diagnostics on fully unstructured grids with multipoint flux approximation. SPE J. 22(6), 1946–1962 (2017)

    Google Scholar 

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Acknowledgements

Sebastian Geiger thanks Energi Simulation for supporting his Chair in Carbonate Reservoir Simulation.

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This research is funded by members of the Rapid Reservoir Modelling consortium (Equinor, ExxonMobil, Shell, Petrobras, and IBM Research/CAS Alberta).

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Appendix: A review of fast marching methods for structured and unstructured grids

Appendix: A review of fast marching methods for structured and unstructured grids

The fast marching method (FMM) is an efficient non-iterative algorithm developed by [55] to solve the boundary value problems of the Eikonal equation for monotonically advancing fronts. Here, monotonicity means that the front or interface can only expand or shrink. A number of practical applications have been modelled based on the principles of FMM. For example, [16] presented a characteristic FMM for wave propagation in a moving medium. Elias et al. [21] developed a FMM based on the finite element method for computing the distance field in computer graphics. Sermesant et al. [54] presented an anisotropic multi-front FMM for real-time simulation of cardiac electrophysiology. Sharifi et al. [59] used FMM to approximate the propagation time of pressure fronts of flows in porous media for reservoir characterisation. Zhang et al. [73] applied FMM to estimate drainage volumes and pressure depletion for numerical well testing in shale gas reservoirs. Zhang et al. [70] presented a scalable massively parallel implementation of FMM for large industrial models.

The FMM as presented in [55] is based on solving the Eikonal equation using upwind finite difference approximation. The method is efficient, consistent, and monotone on Cartesian grids with a computational complexity of \(\mathcal {O}(N\log N)\) where N is the number of unknowns [35]. On unstructured meshes where the connectivity between elements are often not aligned with the main coordinate axes, the gradient in the Eikonal equation could be discretised based on directional derivatives along edges [56]. However, this approach is not monotone. Thus, a causality condition is introduced to ensure that only monotone results are obtained. The causality condition requires that the computed propagation time on an unknown node must be higher than the propagation time on known nodes. But, it is possible that no neighbouring element satisfies the causality condition for a node in an unstructured mesh containing skewed elements. Sethian and Vladimirsky [56] suggest to solve this problem by splitting the obtuse angles and constructing virtual supportive triangles. The extension of this approach to 3D unstructured tetrahedral meshes, where skewed elements are more common, is rather cumbersome. An alternative solution can be defined for the Eikonal equation using the finite element method [21], although similar difficulties arise when fulfilling the causality condition.

1.1 Fast marching methods on Cartesian grids

FMM solves Eq. 12 numerically subject to boundary condition

$$ t=g(\mathbf{x})~, \quad x\in {\Gamma} $$
(30)

where Γ is a subset of Ω. Both F and g are given, and the computation starts from Γ. Here we let g(x) = 0 similarly as in [55]. For isotropic propagation speeds, the finite difference discretisation for Eq. 12 on 3D Cartesian grids is

$$ \left( \frac{t_{ijk}-t^{*x}}{{\Delta} x}\right)^{2}+ \left( \frac{t_{ijk}-t^{*y}}{{\Delta} y}\right)^{2}+ \left( \frac{t_{ijk}-t^{*z}}{{\Delta} z}\right)^{2}=\frac{1}{f^{2}}~, $$
(31)

where tijk is the propagation arrival time of the cell with index (i,j,k). tx, ty, and tz are the upwind values along x, y, and z axes, respectively. For an anisotropic propagation speed tensor, Eq. 13 is discretised as [73]

$$ \left( F_{x}\cdot\frac{t_{ijk}-t^{*x}}{{\Delta} x}\right)^{2}+ \left( F_{y}\cdot\frac{t_{ijk}-t^{*y}}{{\Delta} y}\right)^{2}+ \left( F_{z}\cdot\frac{t_{ijk}-t^{*z}}{{\Delta} z}\right)^{2}=1~, $$
(32)

where Fx, Fy, and Fz are the propagation speeds at cell (i,j,k) along x, y, and z directions, respectively. If the speed is isotropic (Fx = Fy = Fz = f), Eq. 32 is equal to Eq. 31. The steps of FMM for computing the propagation time on a Cartesian grid are summarised as follows [55]:

  1. 1.

    Label all boundary cells as frozen. These cells have t = 0.

  2. 2.

    Compute t for all cells that have at least one frozen neighbour and label them as candidate. All candidate cells form the narrow band [55]. Because of monotonicity, a cell can only have at most one frozen neighbour along a direction which is also the upwind cell. If there is no neighbour along a direction, then the corresponding term in Eq. 2 or 3 is simply neglected.

  3. 3.

    Find the cell with smallest t in the narrow band, mark it frozen, and remove it from narrow band.

  4. 4.

    Solve t for all neighbours of the recently frozen cell and move them to the narrow band. If a neighbour is already in the narrow band, it is recomputed taking the recently frozen cell into account.

  5. 5.

    If the narrow band is not empty, go to step 3. The loop continues until all cells become frozen.

1.2 Difficulty of applying fast marching methods on unstructured grids

In unstructured finite element meshes, the connectivity between adjacent elements is not necessarily aligned with the coordinate axes. Elemental edges are addressed as edges for conciseness. We could use a linear combination of directional derivatives along edges to compute the gradient in Eq. 12 in an element. The propagation time and speed are defined on nodes and elements, respectively [56]. Let P denote the matrix storing in rows the vectors along edges and td denote the vector of directional derivatives of propagation time along edges in an element. We have

$$ P\nabla t=\mathbf{t}_{d}~. $$
(33)

Inserting Eq. 33 into Eq. 13 leads to

$$ \mathbf{t}^{T}_{d}(P^{-1})^{T}FP^{-1}\mathbf{t}_{d}=1~. $$
(34)

The dimension of td is two for 2D triangular meshes and three for 3D tetrahedral meshes. The directional derivatives are set to be piecewise constant on the edges. Consequently, Eq. 34 will always result in a quadratic equation for t regardless of the number of dimensions.

The problem with this method is the lack of monotonicity. Suppose we have a triangle ijm with the propagation time known at nodes i and j and unknown at node m. We could compute tm using Eq. 34 and adopt the larger propagation time for tm. However, tm might be smaller than either ti or tj, which is against the principle of monotonicity. Therefore, [56] introduced a causality condition requiring that tm can only be solved in adjacent triangles where both ti and tj are known and smaller than the computed tm. However, the causality condition might not hold in any neighbouring triangles of a node in a mesh containing obtuse angles. As a result, FMM may terminate much earlier before sweeping the entire domain. Sethian and Vladimirsky [56] proposed a possible solution by splitting obtuse angles and building extra supportive triangles. However, the extension to 3D tetrahedral meshes is difficult.

An alternative finite element discretisation of the Eikonal equation on unstructured meshes is presented in [21]. Here, ∇t is discretised as ∇Niti where Ni is a linear basis function. However, the same problem of fulfilling the causality condition arises. Thus, they modified the FMM algorithm and frozen a node as soon as it is computed as a candidate without building a narrow band, but this is against the principle of FMM and will produce inaccurate results in heterogeneous models.

Further, the causality condition may not be satisfied even on a Cartesian grid when the propagation speed is anisotropic with main axes not aligned with the connectivity between cells. Sethian and Vladimirsky [57] presented an enlarged neighbourhood method to include more cells for computing t when the causality condition fails. Konukoglu et al. [36] includes a recursive correction step in the FMM main loop where the neighbours of an updated node are recomputed such that the causality condition could be satisfied. However, it is not clear whether these methods can solve the monotonicity problem in unstructured tetrahedral meshes.

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Zhang, Z., Geiger, S., Rood, M. et al. Fast flow computation methods on unstructured tetrahedral meshes for rapid reservoir modelling. Comput Geosci 24, 641–661 (2020). https://doi.org/10.1007/s10596-019-09851-6

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