Abstract
In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D–3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law. The line source induces a logarithmic-type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D–3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower-regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D–3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth (i.e. non-singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.
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Acknowledgements
The authors thank B. Wohlmuth for her help in formulating the original splitting approach. The authors also thank P. Zunino for the interesting discussions on coupled 1D–3D flow problems. Finally, the authors would like to thank M. Kuchta, K. E. Holter and C. Daversin-Catty for developing and sharing with us code for the implementation of mixed-dimensional models in FEniCS.
Funding
This work was partially supported by the Research Council of Norway, project number 250223. The participation of the first author at ECMOR XVI 2018 was funded by the Academia Agreement between the University of Bergen and Equinor.
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Gjerde, I.G., Kumar, K. & Nordbotten, J.M. A singularity removal method for coupled 1D–3D flow models. Comput Geosci 24, 443–457 (2020). https://doi.org/10.1007/s10596-019-09899-4
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DOI: https://doi.org/10.1007/s10596-019-09899-4