Layal Lizaik
Co-authors : M. Amara, D. Capatina
Title :
Finite element approximation of a coupled reservoir-wellbore problem in order to
interpret thermometrics
Abstract :
In order to interpret recorded temperatures in the wellbore as well as a flowrate history at
the surface and thus to better characterize the reservoir, we are interested in the
thermomechanical coupling of a petroleum reservoir with a vertical wellbore, both written
in 2D axisymmetric form.
A reservoir model, assumed to be a monophasic multi-layered porous medium, described
by the Darcy-Forchheimer equation together with a non standard energy balance which
includes viscous dissipation and compressibility effects, has been studied in [1]. A wellbore
model, which is a compressible fluid medium based on the Navier-Stokes equations, a 1.5D
model is derived as a conforming approximation of the 2D axisymmetric one, in order to
take into account the privileged flow direction and also to reduce the computational cost.
Each model was previously time and space discretized and validated from theoretical,
numerical and physical points of view.
In order to achieve the coupling between the Darcy-Forchheimer and the Navier-Stokes
equations, adequate transmission conditions at the perforations are imposed and next
dualized by means of Lagrange multipliers. We finally obtain, at each time step, a mixed
formulation binding together the two formulations of the separated models.
Mathematically, its operator is non standard and the uniqueness of the solution is
established by means of a generalization of the Babuška-Brezzi theorem.
A global solving of the coupled problem is then envisaged. Concerning the spatial
discretization, we approximate the heat and mass fluxes by the lowest-order RaviartThomas mixed elements, the pressure and the temperature by piecewise constant
elements, the fluid’s velocity by Q1 continuous elements while the Lagrange multipliers on
the interface are taken piecewise constant. The density is updated by means of a
thermodynamic module and the convective terms are treated by appropriated upwind
schemes. A technical analysis of the discrete mixed formulation is carried on and the wellposedness of the problem is proved. This result is used to show the existence of a solution
for the continuous coupled problem by a Galerkin method.
Numerical tests including real cases as well as a comparison with the results obtained
separately by the two models will be presented. This work is supported by TOTAL.