In using hydraulically fractured wells for aggressive reservoir exploitation, the spacing of vertical wells and/or transverse fractures in long horizontal wells becomes quite an important consideration. The geometry and conductivity of single fractures in infinite-acting reservoirs have been related to productivity by the well known Cinco-Ley and Samaniego work. Later work by Valkó and Economides provided physical optimization under pseudosteady-state conditions in regularly shaped (circular or square reservoirs) using the novel concept of the Proppant Number as a normalizing and descriptive parameter.

However, the shape of the drainage is rarely square and, especially it is not so when a given drainage area is split by the drilling of in-field vertical wells or, in the case of a horizontal well, multiple transverse fractures are executed. The shape of the drainage, encompassing parallel, tightly spaced fractures, becomes a controlling variable for the calculation of the individual and total productivity indices. This is particularly important in our quest to maximize well performance, not just to fracture wells. Until now, such work was the realm of numerical simulation with all the associated benefits and problems.

In this work, the pseudosteady-state productivity index of a fractured well in reservoirs with different aspect ratios (ye/xe) was calculated using the Direct Boundary Method. For relatively small Proppant Numbers, there is a simple relationship that allows for the use of already developed correlations by Valkó and Economides for maximum productivity index and optimum fracture conductivity (and the indicated fracture geometry). For larger Proppant Numbers, this work provides factors labeled Fopt that account for the departure from the square optimum. These factors facilitate the calculation of the maximum JD in different reservoirs for any Proppant Number. The work allows for drainages of a large range of aspect ratios and any fracture penetration.


The predominant vision of a fracture, which is supported by rock and fracture mechanics theories and well established principles, is a linear, bi-wing structure. This geometry has lent itself to production and flow behavior mechanisms by past researchers such as McGuire and Sikora[1], Prats and co-workers [2,3], Cinco and Samaniego and co-workers[4–7], and Agarwal et al[8]. Among other findings and conclusions these works provided means with which to assess the performance of hydraulic fractures and account for the stimulation effects. Well known fracture variables are the fracture (half-) length and the dimensionless fracture conductivity. However, even more useful from a practical point of view is the coupling of these variables, which leads to an equivalent skin effect, long established in production engineering as an appropriate way to assess stimulation or damage.

Starting from these principles, Valkó and Economides[9,10] presented physical optimization methods of hydraulic fractures to maximize their performance. The technique known as Unified Fracture Design (UFD) provides correlations for both optimum fracture conductivity and maximum productivity index using a novel parameter, the Proppant Number (Np). In that work the drainage was approximated by a square with no-flow boundaries. The Proppant Number is a unique quantity in the sense that it is independent of ultimate fracture geometry and is based on just the ratio of the injected fracture volume to the reservoir drainage volume, adjusted by the two important permeabilities, those of the reservoir and the proppant pack.

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