Use of horizontal and multilateral wells for oil and gas recovery has become a common practice in the petroleum industry. It is generally recognized that most horizontal wells do not produce oil and gas at the expected production rate. The discrepancy is mainly due to the fact that it is unclear as what constitutes the "expected production rate." The expected production rate is usually the production rate predicted by the mathematical models that were derived on the basis of the assumption of infinite-conductivity drain hole. Although a number of mathematical models are available for predicting the productivity of horizontal wells, they are not easy to use due to their common nature of mathematical complication and numerical treatments. This paper presents a simple and rigorous mathematical model for estimating the productivity of horizontal wells.

By rigorously coupling reservoir inflow and drain hole hydraulics we derived an equation for pressure distribution in the horizontal drain hole in this study. A simple function for fluid flow rate distribution in the drain hole was then developed. An equation for deliverability of the horizontal drain hole was obtained by evaluating the flow rate function at heel. This new model was compared with three existing mathematical models and field data. The comparison shows that the new model has two advantages over other models:

  1. it is more accurate than other models, and

  2. it is very easy to use.

This paper provides petroleum engineers a simple tool for accurately predicting productivity of horizontal drain holes.


Accurate prediction of productivity of horizontal wells is vitally important for horizontal well planning in the oil and gas field development. Mathematical models used for predicting horizontal well productivity can be classified into three categories:

  • simple analytical solutions derived in late 1980's and early 1990's based on the assumption of infinite drain hole conductivity,

  • sophisticated analytical models developed after 1990's for drain holes of finite conductivity, and

  • Numerical models considering wellbore hydraulics.

The simple analytical models were presented by Giger (1985), Joshi (1988), Babu and Odeh (1989), Goode and Kuchuk (1991), Butler (1994), and Furui et al. (2005). The sophisticated analytical models are those approximate solutions presented by Dikken (1990), Landman (1994), Halvorsen (1994), Novy (1995), Penmatcha et al. (1997), Asheim and Oudeman (1997), and Kamkom and Zhu (2005), and more rigorous models given by Ozkan et al. (1993), Ihara and Shimizu (1993), Sarica et al. (1994), Suzuki (1997) and Yildiz and Ozkan (1998). The numerical models were described by Folefac et al. (1991), Seines et al. (1993), Su and Lee (1995), Siu and Subramanian (1995), Yuan et al. (1998), Ouyang et al. (1998), Ouyang and Huang (2005), and Guo et al. (2006).

The simple analytical solutions in the first category of the above models are widely adopted in the industry because they are easy to use. However, they give over-estimates of well productivity due to the fact that the frictional pressure drop in the drain hole is not considered. According to the study of Hill and Zhu (2006), the frictional pressure drop along a 5,000 ft long, 4-in. ID drain hole in a 100-ft thick reservoir of 100 to 200 md can be from 2% to 8% of the total pressure drawdown, and this value increases with drain hole length and becomes more significant in small drain holes in high-permeability reservoirs.

The sophisticated models in the second category were developed to account for the effects of wellbore hydraulics on drain hole productivity. Although these models are more rigorous in terms of coupling the reservoir inflow and drain hole outflow, they require sophisticated computational algorithms. This hinders their field applications.

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