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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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
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
where tijk is the propagation arrival time of the cell with index (i,j,k). t∗x, t∗y, and t∗z are the upwind values along x, y, and z axes, respectively. For an anisotropic propagation speed tensor, Eq. 13 is discretised as [73]
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.
Label all boundary cells as frozen. These cells have t = 0.
- 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.
Find the cell with smallest t in the narrow band, mark it frozen, and remove it from narrow band.
- 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.
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
Inserting Eq. 33 into Eq. 13 leads to
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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DOI: https://doi.org/10.1007/s10596-019-09851-6