# A Singularity Removal Method for Coupled 1D-3D Flow Models

## TL;DR

This paper introduces a method to remove singularities in coupled 1D-3D reservoir flow models, enabling more accurate numerical simulations by reformulating the equations with smooth variables.

## Contribution

The authors develop a singularity removal technique for coupled flow models, allowing the use of standard numerical methods on smooth reformulated equations.

## Key findings

- Singularity removal leads to smooth reformulated equations.
- The method improves numerical stability and accuracy.
- Reformulation resembles a continuous Peaceman well correction.

## 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.