### Numerical Solution of the Multiphase Flow of Oil, Water and Gas in Horizontal Wells in Natural Petroleum Reservoirs

Arthur B. Soprano, António F. C. da Silva, Clovis R. Maliska

#### Abstract

The estimation of oil production in natural petroleum reservoirs requires the solution of the coupled flow between the reservoir and the wellbore. The oil, water and gas flow in the well is the result of the lateral flow coming from the reservoir, obeying different completions, aiming to have an oil production with almost constant rate along the horizontal well. The multiphase flow is governed by the Navier-Stokes equations and mass conservation for each phase. Due to the large difference in spatial scales between the wellbore and the petroleum reservoir domains, the flow in the wellbore can be considered as one dimensional, taking into account, through friction models, the pressure loss caused by the lateral mass income of oil, gas and water.
In this paper, the homogeneous flow model is adopted, and a slip between phases is specified according to some algebraic correlation available in the literature. The conservation equations are approximated using a finite volume method employing a staggered grid for the variable arrangement, what renders to the method robustness and guarantees a strong coupling between pressure and velocity in the wellbore.
Pressures and void fractions are located at the center of the mass conservation control volume, while velocities are at the surfaces. The block-matrix is solved using a Newton-like method, with the Jacobian matrix constructed using precise numerical derivatives. Comparisons with available numerical solutions for two-phase flow are performed, showing that the model performs well and allows the use of larger time steps than the ones reported in the literature. Examples demonstrating the ability of the method for solving three-phase flows are also presented. As an overall outcome, it can be said that the developed method is suitable for taking part in a more general application for estimating oil production in petroleum reservoirs.

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