Modelling Multi-lateral Wells
Authors: Su H.J. , Fong W.S. 
Published: March 2010

Abstract
A computation scheme was implemented in a general-purpose, finite-difference simulator to model multi-lateral wells. Frictional pressure drop along the wellbore and proper fluid mixing at lateral connection points were included in the calculation of the wellbore pressure profile for each lateral. The formulation is general and a lateral can be branched out from any point along the main well path.
Since the production profile of each lateral is influenced by the wellbore pressure distribution, the prediction accuracy depends on a good wellbore flow model. The Beggs and Brill(1) correlation and the homogeneous flow model were implemented, along with a new model proposed by Ouyang et al.(2) which included an acceleration term and accounted for the lubrication effect due to radial influx. Well performance prediction results using the three models were compared.
The impacts of different tubing sizes on the well performance and the production contribution from each lateral were studied.

Introduction
Recently multi-lateral wells have been employed in off-shore platforms or remote and hard to access terrain areas to save costs comparing with drilling multiple vertical or horizontal wells. This work is to study the pressure and production profiles along each lateral using different pipe flow models.
The vertical standoffs of laterals to GOC or WOC, the optimal length of laterals, the angles between laterals, the completion PI values used in simulation, and the optimal number of laterals are other important factors in designing multi-lateral wells. However, these issues will not be addressed here.

Numerical Method
The completion production rate and wellbore pressure profiles along each lateral are closely related. There is also interaction between laterals. The fluid dynamics calculation inside the wellbore can be described as follows:
Pipeflow Calculation
There are many steady-state correlations available to calculate frictional pressure losses. In this work, we implemented three models, namely, the Beggs and Brill model, the homogeneous model, and a modified homogeneous model (including radial inflow effect) in a general-purpose reservoir simulator. The homogeneous model gave a better pressure match for the Stanford/Marathon two-phase flow tests(3,4). It was a very simplified model. For two-phase flow, it used a mixing rule to calculate the density and viscosity values of the gas-liquid mixture. Then it used the mixture properties in the single-phase fluid-wall frictional loss formula.
We did not use a transient pipeflow model(5) because it would require much longer computing time.
Radial Inflow Effect
Rigorously speaking, the regular pipe flow calculation is not suitable for open hole or perforation intervals due to the presence of radial inflow from the reservoir. For turbulent flow, the radial inflow provides a so called "lubrication effect" and reduces the frictional loss(6,7). For laminar flow (rarely occurring in oil field practice), it increases the frictional loss(8). Here we used a formula proposed by Ouyang et al.(3) to model radial inflow effect,
Equation (1) (Available In Full Paper)
where ƒo is the corrected frictional coefficient due to radial inflow and ƒ is the wall friction coefficient. For the open hole, Rew can be expressed as
Equation (2) (Available In Full Paper)