In numerical reservoir simulation practise the orientation of the spatial grid is often dictated by geological features of the reservoir model. As a consequence the trajectory of, in particular, a horizontal or slanted well or a well with a complex geometry, possibly with multilaterals, generally is not aligned to the numerical grid. This situation may lead to significant numerical artefacts, as an irregular inflow profile along the well. Although the total flow rate of the well may be reasonably accurate, a non-smooth profile is unwanted if the simulator is coupled to a dedicated wellbore flow model. Another artefact occurs if the ratio of the vertical to horizontal permeability is small (i. e. kv/kh< 0.001): standard models result in an overestimation of the total flow rate.

Well inflow modelling relates the so-called well index, the ratio between well rate times viscosity and the difference between the well pressure and the gridblock pressure, to quantities such as gridblock dimensions and permeability. The origin of the numerical issues is the limitation of the well index estimation as proposed by Peaceman, and generalised by Alvestad for wells not aligned to the grid.

We combined simulation results of a semi-analytic model and simulation results from a dedicated numerical simulation to calculate so-called semi-analytical well indices. We demonstrate that by using these semi-analytical well indices in a numerical reservoir simulator the aforementioned issues are resolved. The semi-analytic model that has been used solves the equation for steady state flow in 3D for line sink(s) with variable influx. The variable influx is then exploited to approximate a uniform pressure condition along the wellbore. This modelling approach allows for a complex geometry of the well such as a multilateral.

To arrive at a practical implementation of our method we propose to use the standard well index along with a corrective skin factor based on our approach. Correlations for such a corrective skin factor then need to be developed based on a great number of well scenarios, anisotropy ratios and grid aspect ratios.

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