# Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells

Original Paper
Part of the following topical collections:
1. Numerical methods for processes in fractured porous media

## Abstract

We develop a mathematical model for the interaction of a three-dimensional reservoir with the flow through wells, namely narrow cylindrical channels cutting across the reservoir. Leak off or sink effects are taken into account. To enable the simulation of complex configurations featuring multiple wells, we apply a model reduction technique that represents the wells as one-dimensional channels. The challenge in this case is to account for the interaction of the reservoir with the embedded one-dimensional wells. The resulting problem consists of coupled partial differential equations defined on manifolds with heterogeneous dimensionality. The existence and regularity of weak solutions of such problem is thoroughly addressed. Afterwards, we focus on the numerical discretization of the problem in the framework of the finite element method. We notice that the numerical scheme does not require conformity between the computational mesh of the reservoir and the one of the wells. From the standpoint of the solvers, we discuss the application of multilevel algorithms, such as the algebraic multigrid method. Finally, the reduced mathematical model and the discretization method is applied to a few configurations of reservoir with wells, with the purpose of verifying the theoretical properties and to assess the generality of the approach.

## Keywords

Perforated reservoirs Dimensional model reduction Finite elements Multigrid solvers

## Mathematics Subject Classification

35J47 35Q35 35Q86 58J05 76S05 65N30

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Daniele Cerroni
• 1
• Federica Laurino
• 1
• 2
• Paolo Zunino
• 1
1. 1.MOX, Department of MathematicsPolitecnico di MilanoMilanItaly
2. 2.Istituto Italiano di TecnologiaGenoaItaly