## Section: New Results

### Numerical simulation of petroleum reservoir

Participant : Daniela Capatina.

Figure 7. Computed temperature profile and sketch of velocity field (from [13] ).

A coupled 2.5D reservoir-1.5D wellbore model with heat transfer is implemented and analyzed in [13] . The flow equations are Darcy-Forchheimer in the porous media and compressible Navier-Stokes equations in the fluid; the result of a typical computation is shown in Figure 7 . The thermomechanical coupling of a petroleum reservoir with a vertical wellbore, both written in 2D axisymmetric form, has also been considered. The motivation is to interpret recorded temperatures in the wellbore as well as a flowrate history at the surface and thus to better characterize the reservoir. The reservoir is 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. Concerning the wellbore, 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. The coupling is then achieved by imposing transmission conditions at the perforations and yields, at each time step, a mixed formulation whose operator is mathematically non standard. A global solving of the coupled problem is implemented. The spatial discretization employs lowest-order Raviart-Thomas elements for the heat and mass fluxes, piecewise constant elements for the pressure and the temperature and Q1 continuous elements for the fluid's velocity; finally, 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. The well-posedness of the time-discretized coupled problem is proven, at both the continuous and the discrete level. Numerical tests including real cases are carried out, for the separate reservoir and wellbore codes and for the coupled one. The numerical modeling of multi-component multi-phase flows in petroleum reservoirs with heat transfer has been studied in [14] This work is supported by TOTAL.

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