Technical note: Finite element formulations to map discrete fracture elements in three-dimensional groundwater models

de Rooij, Rob

doi:https://doi.org/10.5194/hess-2024-289

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https://doi.org/10.5194/hess-2024-289

Preprints

14 Oct 2024

| 14 Oct 2024

Status: this preprint is currently under review for the journal HESS.

Technical note: Finite element formulations to map discrete fracture elements in three-dimensional groundwater models

Rob de Rooij

Abstract. Typically, in finite element groundwater models, fractures are represented by two-dimensional triangular or quadrilateral elements. When embedded in a three-dimensional space, the Jacobian matrix governing the transformation from the global three-dimensional space to the local two-dimensional space is rectangular and thus not invertible. There exist different approaches to obtain a unique mapping from local to global space even though the Jacobian matrix is not invertible. These approaches are discussed in this study. It is illustrated that all approaches yield the same result and may be applied to curved elements. The mapping of anisotropic hydraulic conductivity tensors for possibly curved fracture elements is also discussed.

Received: 16 Sep 2024 – Discussion started: 14 Oct 2024

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Rob de Rooij

Status: final response (author comments only)

RC1:
'Comment on hess-2024-289', Anonymous Referee #1, 24 Dec 2024

This manuscript presents a methodology for including two-dimensional elements, such as fractures, in a three-dimensional porous media matrix. The authors claim that the proposed methodology is easier to implement than the commonly adopted method composed of a two-dimensional mapping on an orthonormal space, followed by a three-dimensional rotation. In addition, the proposed methodology works for curved two-dimensional surfaces with the integration performed using the Gaussian quadrature method, a case that cannot be addressed with the existing methodology. The proposed new approach cannot be applied when the integration is performed analytically (in cases where this is possible).
The technical note is "dry" in the sense that it discusses only the new projection method showing how it works only for one of the integrals (Eq. 10) that should be computed in the FEM solution of the flow equation. The discussion on the hydraulic conductivity projection of the 2-D elements (the fractures) presented in Section 6 is appreciable, but the manuscript lacks an illustrative example, which would be certainly appreciated by the HESS readership and myself too.
The proposed methodology is sound. However, as mentioned above, the presentation is somewhat "dry," and including an application example would enhance the reader's understanding of the benefits offered by the proposed approach. The example does not need to be geometrically complex; for instance, a cubic homogeneous formation with a single curvilinear fracture, an imposed head gradient between two opposing faces, and the remaining faces set as impervious would be sufficient to demonstrate the methodology's capabilities. In addition, I suggest that the authors compare the solution obtained using their proposed approach with that obtained through the existing and widely adopted projection and rotation methodology.

Citation: https://doi.org/10.5194/hess-2024-289-RC1
- AC1: 'Reply on RC1', Rob de Rooij, 02 Jan 2025
  
  I agree with the assessment that the manuscript is somewhat 'dry'. However, this being a technical note this is by design. Nonetheless, I agree that an illustrative example could be a useful addition. The reason why I did not include one was to keep the manuscript concise.
  If an illustrative example should be added, I think that an even simpler example would suffice (simpler than the one proposed by the reviewer). Namely, if we take a single flat triangular element in 3D space with a given hydraulic conductivity tensor, then the example could work out the element conductance matrix (Equation 10), using the different mapping techniques.
  
  Citation: https://doi.org/10.5194/hess-2024-289-AC1
RC2:
'Comment on hess-2024-289', Anonymous Referee #2, 16 Jan 2025

The authors propose several approaches to map local spaces of fractures embedded in higher-dimensional domain. My main concern is related to the fact that the proposed approaches should be numerically validate, showing their performances for complex fracture networks. This makes the reader difficult to evaluate which algorithm should be preferred. I have also the following minor comments:

- eq (3) use a different symbol since it resembles a partial derivative

- the part on the Penrose-Moore inverse can be moved to an appendix

Citation: https://doi.org/10.5194/hess-2024-289-RC2
- AC2: 'Reply on RC2', Rob de Rooij, 30 Jan 2025
  
  Thanks for noting the error in denoting the Kronecker delta symbol (equation 3).
  I am not sure if the latter part on the Penrose-Moore inverse should be moved to an appendix as that part is integral to the overall explanation of this approach. Moreover, it would create an imbalance w.r.t. to how the first approach is presented.
  Regarding the suggestion of a numerical validation on a complex fracture network, I would like to refer to my comments to Referee #1. Namely, I think it is preferable to present a rather simple example on a single fracture element and work out either the gradient matrix or the conductance matrix using different approaches. This would illustrate that the formulation based on the first two approaches are simpler to work out even though the underlying math is more complicated (which is the key message here). The problem with a model application is that all approaches are correct and once implemented there is little reason to prefer one over the other. This paper is merely concerned with picking a particular approach from the ones that are available. The choice is not deemed important for accuracy or efficiency, but the choice matters in terms of how difficult they are to implement. It is interesting that the approach which is easiest to understand mathematically is the most complicated to implement. Moreover, in existing literature this approach is implemented such that it only applies to non-curved elements, while the other approaches are not only easier to implement but work straight away on curved elements.
  
  Citation: https://doi.org/10.5194/hess-2024-289-AC2

Rob de Rooij

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Short summary

Coordinate transformations are often used in the finite element method. Here, three existing approaches are discussed that can be applied to the same mapping problem. From a mathematical viewpoint, the approaches are quite different. It is shown that all three approaches provide the same result. Interestingly, the more mathematically challenging approaches provide an expression that is the most practical to implement into a finite element code.


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