This paper presents an "equivalent drawdown time" for hydraulically fractured wells. This new equivalent time is derived from a general elliptical flow model. This new equivalent time is helpful in post-fracture pressure buildup test analysis for wells with finite-conductivity fractures including wellbore storage and fracture-face skin. Examples are provided.
In 1980, Agarwal derived an "equivalent drawdown time" to account for producing time effects when drawdown type curves are used to analyze pressure buildup and other test data. Agarwal's equivalent drawdown time was derived for radial flow by assuming that the logarithmic approximation to the original solution is appropriate. In the rest of this paper, we call Agarwal's equivalent drawdown time radial equivalent time.
Since the introduction of Agarwal's radial equivalent drawdown time, its importance has been widely recognized in well test analysis. Similarly, an equivalent drawdown time for linear flow problems, which we call linear equivalent time in the remainder of this paper, has been introduced.
For more than a decade, engineers have routinely used radial equivalent time or linear equivalent time to analyze well test data for hydraulically fractured wells due to the lack of a suitable equivalent drawdown time for such wells. Engineers find that radial and linear equivalent times work under some specific conditions and do not work under other conditions.
Fluid flow toward a well with a hydraulic fracture is an example of elliptical flow problems. In this paper, we present a method to determine "equivalent drawdown time" for pressure buildup test analysis for hydraulically fractured wells. We derived this method from a general elliptical flow model. We call this new equivalent drawdown time elliptical equivalent time. At "early" flow times, this elliptical equivalent drawdown time approaches the equivalent drawdown time derived for linear flow problems. At "large" flow times, this new equivalent drawdown time approaches Agarwal's equivalent drawdown time derived for radial flow problems.
We also provide examples to show how to use our elliptical equivalent time for hydraulically fractured wells.
We consider a hydraulic fracture in a single-layer, isotropic reservoir. The reservoir can be either finite or infinite. The fracture shape is elliptical. The fracture has a maximum width, bmax, at the wellbore and half-length, Lf. Fig. 1 is a schematic of the physical model.
The half focal-length of the elliptical fracture is L, which we treat as equal to the fracture half-length, Lf, for practical purposes. The fracture conductivity is generally finite; an infinite conductivity fracture is a limiting case of finite conductivity fractures.
If the reservoir is finite, we assume that the outer reservoir boundary is also elliptical and is "confocal" with the elliptical fracture.
The reservoir can be either homogeneous or elliptically composite. For an elliptically composite reservoir, each interface between two adjacent zones is assumed to be confocal with the elliptical fracture.