This paper presents an equation to calculate the productivity of horizontal wells and a derivation of that equation using potential-fluid theory. This equation may also be used to account for reservoir anisotrop and well eccentricity (i.e., horizontal well location other than midheight of a reservoir). The theoretical predictions were used to calculate the effective wellbore radius and the effective skin factors of horizontal wells. Laboratory experiments with an electrical analog were also conducted. These laboratory experimental data and also the laboratory data available in the literature show good agreement with the theoretical equation, indicating its accuracy. The paper also compares vertical-, slant-, and horizontal-well productivity indices, assuming an equal drainage area. In addition, the comparison also assumes an equal reservoir contact area for slant and horizontal wells. The results show that in a 100-ft [30.48-m] -thick reservoir, horizontal-well productivities are two to five times greater than unstimulated vertical- or slant-well productivities, depending on reservoir anisotropy. Conversely, in a 400-ft [122-m] -thick reservoir, slant wells perform better than horizontal wells if vertical permeability is less than horizontal permeability. Horizontal wells perform significantly better than vertical wells in reservoirs with gas cap and/or bottomwater. This study reports an equation to compare horizontal- and vertical-well gas-coning tendencies. The results indicate that horizontal wells are suitable for reservoirs that are thin, show high vertical permeability, or exhibit gas-and water-coning problems. The equations reported should be useful in initial evaluation of a horizontal-well drilling proposal.

Currently, about 30 horizontal wells are producing oil successfully worldwide. The wells have been drilled in Prudhoe Bay in Alaska, Empire Abo Unit in New Mexico, France, and offshore Italy. Because of a large reservoir contact area, horizontal-well oil-production rates are two to five times greater than unstimulated vertical-well rates. In addition, horizontal wells may intersect several fractures and help drain them effectively. Horizontal wells have also been known to reduce water- and gas-coning tendencies. The disadvantages of horizontal wells are that

they are ineffective in thick ( - 500 to 600 ft [ - 150 to 180 m]), low-vertical-permeability reservoirs;

reservoirs with several oil zones, separated by impermeable shale barriers, may require drilling of a horizontal hole in each reservoir layer to be drained;

some limitations currently exist in well-completion and stimulation technology; and

drilling costs are 1.4 to 2 times more than those for vertical wells.

The main goal of this work is to develop necessary mathematical equations for an initial evaluation of horizontal-well drilling prospects. This included the following objectives. 1. To develop a mathematical equation to calculate steady-state oil production with horizontal wells. 2. To determine the influence of reservoir anisotropy, height, well drainage area, and eccentricity(well location other than the reservoir midheight) on horizontal-well productivity. 3. To devise laboratory electrical analog experiments to measure horizontal-well productivities and to compare them with the theoretical equation. 4. To compare vertical-, slant-, and horizontal-well productivities. 5. To determine gas- and water-coning tendencies of horizontal wells and to compare them with those of vertical wells.

Borisov reported a theoretical equation to calculate steady-state oil production from a horizontal well; however, the report does not include the derivation of the equation. Later, using Borisov's equation, Giger et al. and Giger reported reservoir engineering aspects of horizontal drilling. Giger developed a concept of replacement ratio, FR, which indicates the number of vertical wells required to produce at the same rate as that of a single horizontal well. The replacement-ratio calculation assumes an equal drawdown for the horizontal and vertical wells. In addition, Giger studied fracturing of a horizontal well and provided a graphical solution to calculate reduction of water coning using horizontal wells. Giger et al. reported that horizontal wells are suitable for thin reservoirs, fractured reservoirs, and reservoirs with gas- and water-coning problems. Recently, Reiss reported a productivity-index equation for horizontal wells, but his equation is little different from that reported by Borisov and others. To clarify these differences, it was decided to derive the basic steady-state equation from fundamental fluid-flow theory. Such a derivation is reported here. Daviau et al. and others recently reported time-dependent theoretical analyses for horizontal wells. Their results, as well as our time-dependent horizontal-well theoretical results (not included here), indicate that if the length of a horizontal well is significantly larger than the reservoir height (L/h greater than 1), then in the lost time, horizontal-well production is the same as that obtained from a fully penetrating, infinite-conductivity vertical fracture. This is also shown by the steady-state equation derived in this paper. It is important to note that implications of these results are restricted to a single-phase flow.

Fig. 1 shows that a horizontal well of length L drains an ellipsoid, while a conventional vertical well drains a right circular cylindrical volume. Both of these wells drain a reservoir of height h, but their drainage volumes are different. To calculate oil production from a horizontal well mathematically, the three-dimensional (3D) equation ( p=0) needs to be solved first. If constant pressure at the drainage boundary and at the wellbore is assumed, the solution would give a pressure distribution within a reservoir. Once the pressure distribution is known, oil production rates can be calculated by Darcy's law. To simplify the mathematical solution, the 3D problem is subdivided into two two-dimensional (2D) problems. Fig. 2 shows the following subdivision of the ellipsoid drainage problem:

oil flow into a horizontal well in a horizontal plane and

oil flow into a horizontal well in a vertical plane.

Appendices A and B describe mathematical solutions to these two problems with potential-fluid-flow theory.

JPT

P. 729^