Transmissivity Averaging in Fracture Flow on Self-affine Linear Profiles: Arithmetic, Harmonic, and Beyond

Abstract

Numerical modeling of unidirectional flow in self-affine fractures using the lubrication approximation requires averaging of the transmissivity between the nodes. Seven averaging techniques are reviewed: arithmetic averaging of transmissivity; harmonic averaging of transmissivity; two averaging techniques derived by cell-based collocation method; global reconstruction of profile by means of multiquadrics; arithmetic averaging of aperture; harmonic averaging of aperture. In order to evaluate the performance of the seven techniques in terms of pressure errors and hydraulic aperture errors, self-affine profiles of 1024 nodes with different Hurst exponents (0.4 to 0.8) are generated. Every second node is then removed, resulting in 512-node profiles. Apertures at removed nodes are used in reference flow simulations on the 512-node profiles. Then, simulations with the seven averaging techniques are performed on 512-node profiles. Errors are computed with regard to the results obtained in the reference simulations. Reconstruction with multiquadrics is found to provide superior accuracy on self-affine profiles, followed by harmonic averaging of transmissivity or harmonic averaging of the aperture (Some of the errors analyzed in this study are minimized with the two last mentioned schemes.). Multiquadrics reconstruction is found to provide the best accuracy also on a smooth periodic profile.