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A singularity removal method for coupled 1D-3D flow models
Univ Bergen, Dept Math, Bergen, Norway..
Univ Bergen, Dept Math, Bergen, Norway.;Univ Karlstad, Dept Math, Karlstad, Sweden..
Univ Bergen, Dept Math, Bergen, Norway..
(English)In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
SPRINGER.
Keywords [en]
Singularities, Green's functions, Finite elements, Improved well modelling
Identifiers
URN: urn:nbn:se:kau:diva-76247DOI: 10.1007/s10596-019-09899-4ISI: 000503673500002OAI: oai:DiVA.org:kau-76247DiVA, id: diva2:1384209
Available from: 2020-01-09 Created: 2020-01-09 Last updated: 2020-01-09

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