Well modelling plays an important role in reservoir simulation. Commonly used well models are based on the 2D analytical pressure solution (radial flow) and on an "adequate" numerical productivity index calculation. These models applied to 3D simulations may give erroneous results, especially while modelling undulating wells (slanted wells), multi-lateral wells, finite-length horizontal wells or partially penetrating vertical wells. The main reasons are the lack of 3D pressure solution and the inaccurate flow calculation near the well.
In this paper, the layer potential is introduced for representing the 3D pressure distribution of the steady-state flow in the vicinity of wells. The flow approximation is improved by modifying the near well transmissibilities, taking into account this 3D pressure solution. This new well model is valid for any well configuration and allows to describe accurately 3D well behaviour.
The well model plays an important role in reservoir simulation since the precision of well flow rate or bottom hole pressure calculation is directly related to this model. The main well modelling difficulty is the problem of singularity due to the difference in scale between the small wellbore diameter and the large wellblock grid dimensions used in reservoir simulation. In practice, a numerical productivity index (PI) is introduced to relate the wellblock pressure, the wellbore pressure and the well flow rate1–4. The first theoretical study of the numerical productivity index (PI) was made by Peaceman in 1978 for vertical well modelling.
Peaceman3 demonstrated that the numerical PI depends on grid geometry through an equivalent wellblock radius r0, which equals to 0.2?x for square gridblocks. Later, he provided a general formula of the equivalent wellblock radius for uniform rectangular grid blocks and anisotropic media5 and extended the results to more general cases such as nonisolated wells6 or horizontal wells7. As a complement of Peaceman's work, many other contributions are found in the literature8–13, for example, Abou-Kassen and Aziz11 proposed analytical well model for rectangular grid and off-centre well; Shiralkar12 presented the productivity index formula for the nine-point scheme; Palagi and Aziz13 discussed well models for Voronoi grids, etc. However, all these works use a 2D pressure solution to calculate the numerical PI.
For several years, some authors have tried to improve the well modelling by using a 3D pressure solution4. Lee14 first introduced a boundary integral equation for representing the pressure distribution, but this method is complex. The unknown values are extended to the external geometry boundaries, and no straight formula can be obtained even for several simple cases such as an isolated horizontal well. Later, Lee15 introduced the slender body theory. But the formula is still complex and cannot represent general well configurations such as undulating wells or multi-lateral wells. In the meantime, Babu et al.4 developed a formula for horizontal wells in a box-shape reservoir, Sharpe and Ramesh16 used an analytical solution for partially penetrating vertical well modelling. However, these methods are restricted by the application range. Recently, Chen et al.17,18 introduced some transient analytical solutions, which are used in well testing. Their method is based on the point source and Green's function, and the reservoir geometry is limited to a rectangular box. Besides, all these well models are limited to the calculation of the numerical PI value or the pseudo-skin factor. They cannot improve the fluid flow calculation in the near well region. Therefore, these models cannot accurately describe the multi-phase flow, as hereafter shown in example 3.