Mixed finite element approximation for a coupled petroleum reservoir model

Abstract

Abstract.In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code.Mathematics Subject Classification. 35Q35, 65N15, 65N30, 76S05.

Stabilized mixed approximation of axisymmetric Brinkman flows

“…Recent contributions focusing on the design of numerical methods for axisymmetric formulations of coupled flow and transport problems can be found in e.g. [3,20,21,33]. On the other hand, time-dependent and static Maxwell equations in axisymmetric singular domains were studied in [8,9] by introducing a method based on a splitting of the space of solutions into a regular subspace and a singular one.…”

Numerical coupling of 2.5D reservoir and 1.5D wellbore models in order to interpret thermometrics

“…Problem (1) as well as the complete problem with convection were shown to be well-posed in [1], where a finite element discretization based on Raviart-Thomas elements for the fluxes and piecewise constant elements for the scalar unknowns was also proposed. We refer to [1] for the numerical analysis and the validation of this scheme.…”

“…Other codes exist dedicated to the wellbore but they do not take into account the evolution of the temperature in the reservoir. An axisymmetric reservoir model, notably including viscous dissipation and compressibility effect, and a quasi-1D vertical wellbore model were previously introduced and analysed in [1], respectively [2]. They were separately validated from theoretical and numerical points of view.…”

Coupling of Darcy–Forchheimer and Compressible Navier–Stokes Equations with Heat Transfer

“…A reservoir model, written in cylindrical coordinates and consisting of the Darcy-Forchheimer equation coupled with an exhaustive energy balance, has already been studied in [1]. The energy equation notably includes the temperature effects due to the decompression of the fluid (Joule-Thomson effect) and the frictional heating that occurs in the formation.…”

“…α = 1) was established by means of Fredholm's alternative, for Δt sufficiently small. We refer to [1] for the detailed proofs.…”