A singularity removal method for coupled 1D–3D flow models

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 -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.

3D-1D coupling on non conforming meshes via three-field optimization based domain decomposition

“…In the next section an original formulation of the coupled 3D-1D problem is derived, starting from a variational formulation of the fully dimensional 3D-3D problem (1)-( 2) and ( 5)- (6), and defining the proper functional spaces and operators required to reduce this formulation to a well posed 3D-1D coupling.…”

“…Clearly this topological reduction can be a viable approach only if one-dimensional modeling assumptions can be applied to the problem at hand. Examples of application are: capillary networks exchanging flux with the surrounding tissue [1,2], the interaction of tree roots with the soil [3,4], a system of wells for fluid in geological applications [5,6,7], or the modeling of fiberreinforced materials [8,9].…”

On the Diffuse Interface Models for High Codimension Dispersed Inclusions

“…The stationary solution (Φ, φ) of ( 12), ( 13) is a pair of functions minimizing the functional (10) for φ satisfying BCs (16). It is natural to solve the problem for φ ∈ W 1 2 (Ω), where W q p (Ω), Ω ⊂ R n , is the standard Sobolev space (see, e.g., [33,34]).…”

“…The model [12] is considered as a specific example of a phase field model of "codimension two". Nevertheless, we believe that the presented considerations are of the general interest and importance and potentially can be applied to the construction of phase field models for a such effectively mixed-dimensional problems as elastic multi-structures [13], dislocation lines dynamics [14], coupled 1D/3D tissue perfusion models [15] and well models in geophysics [16], and numerous electrostatic problems which requires analysis of thin charged or conducting lines, which is a classical problem of electrostatics with numerous application, see, e.g., [17].…”

A gradient based resolution strategy for a PDE-constrained optimization approach for 3D-1D coupled problems

“…This allows to avoid the complexity related to the building of a three-dimensional grid within the inclusions. Examples of applications range from the description of biological tissues [2,3], roots-soil interaction [4,5], geological reservoir simulations [6,7,8], to fiber-reinforced materials [9,10].…”