Modeling Boundary-Dominated Flow in Hydraulically-Fractured Wells
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AbstractThis paper presents a simple method to model boundary-dominated flow in hydraulically fractured wells, including horizontal wells with multiple fractures. While these wells are almost always producedat more nearly constant BHP rather than constant rate, use of material-balance time transforms variable-rate production profiles to constant-rate profiles, allowing us to use the pseudo-steady-state (PSS) flow equation for modeling. However, the PSS equation requires use of shape factors in applications, and shape factors available in the literature are available only for square-shaped bounded reservoirs with hydraulic fractures. In this work, we derived shape factors for wells centered in rectangular-shaped drainage areas with different length-to-width aspect ratios.The superposition principle can be used to transform transient radial flow and transient linear flow solutions into bounded reservoir solutions. At large times (when boundary-dominated flow is established), results from these solutions are similar to those obtained from the PSS equation. Therefore, for a pre-defined reservoir geometry, pressure drop values from superimposed transient flow equationscan be substituted back into the PSS equation to calculate shape factors for that reservoir geometry.We used shape factors previously presented by other authors for square drainage areas to validate themethod before applying it to calculate shape factors for more general drainage area configurations.We present shape factors for different fracture half-length to fracture-spacing ratios ranging from 0.2 to 10. Calculated shape factors, when plotted against the fracture half-length to fracture-spacing ratio, produced a smooth curve which can be used to interpolate shape factor values for other fracture configurations. We present applications of this methodology to example low-permeability wells.The use of the PSS equation for wells with vertical fracturescan be extended to multi-fractured horizontal wells (MFHWs) by incorporating the number of fractures in the equation; hence, shape factorsderived for wells with vertical fractures can also be used for MFHWs. Although our results are rigorously correct only for fluids with constant compressibility, use of pseudo-pressure and pseudo-time transformations extend application to compressible fluids, notably gases. Using the PSS equation in production data analysis allows us to calculate contributing reservoir volume and drainage area in a simple manner not requiring use of specialized software.
