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Pré-Publication, Document De Travail Année : 2007

Petroleum Wellbore Models

Résumé

In this paper, we introduce the well model which is based on the compressible Navier-Stokes equations coupled with an energy equation, both written in axisymmetric form. In order to take into account the privileged direction of the flow in the well and to reduce the cost of the calculation, we propose an approach to derive two 1.5D models, by constructing an explicit solution in terms of the radial coordinate. The nonlinear time-discretized problem is solved by means of a fixed point method with respect to the density $\rho$. More precisely, for a given $\rho$, we compute at any time step the specific flux from the mass equation and then we recover the velocity and the pressure, as well as the heat flux and the temperature by using mixed variational formulations. Finally, we update the density by means of a thermodynamic module and we loop until the convergence is achieved. The difference between our two 1.5D models consists in the computation of the radial velocity $u_{r}$ : in the first one, we neglect the momentum equation corresponding to $u_{r}$ and we compute it directly from the specific flux by a projection method, while in the second one, the whole velocity $(u_{r},u_{z})$ is computed by using the Navier-Stokes equation and by neglecting, this time, a boundary condition on the perforations. The first model thus obtained is a little bit simpler, but the second one is well adapted to the coupling with the reservoir. Once the time discretization achieved, we propose a well-posed finite element approximation, based on lowest-order Raviart-Thomas elements for the fluxes, $Q_{0}$ elements for the pressure and the temperature and $Q_{1}$-continuous finite elements for the velocity. Numerical tests are also presented, validating the code from both a numerical and a physical point of view.
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Dates et versions

hal-00192391 , version 1 (27-11-2007)

Identifiants

  • HAL Id : hal-00192391 , version 1

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Mohamed Amara, Daniela Capatina, Bertrand Denel. Petroleum Wellbore Models. 2007. ⟨hal-00192391⟩
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