Matrix stimulation methods that utilize acid injection can enhance formation permeability out to a few feet away from the wellbore, which limits their impact on well production rate. Hydraulic fractures can penetrate into the reservoir hundreds to thousands of feet, extending the wellbore reach and enabling significant well productivity improvement through increasing the surface area directly connected to the wellbore and creating a highly conductive pathway that allows fluids to flow more easily from the reservoir to the wellbore. After a fracture is created, reservoir fluids flow from the formation into the fracture and then along the fracture to the well. For the fracture to significantly improve production, it must offer much less flow resistance than the formation. The degree to which the fracture can enhance fluid transmission to the wellbore can be quantified using the dimensionless fracture conductivity, defined as1Cinco-Ley, H., & Samaniego-V., F. (1981). Transient pressure analysis for fractured wells. Journal of Petroleum Technology, 33(9), 1749–1766. https://doi.org/10.2118/7490-PA:
where:
- \(F_{CD}\): dimensionless fracture conductivity
- \(k\): reservoir permeability
- \(k_f\): fracture permeability which is dictated by the proppant
- \(w_f\): fracture width
- \(x_f\): fracture half-length
The term \(k_f w_f\) represents the flow capacity of the fracture, while \(k x_f\) represents the reservoir’s ability to supply fluid to the fracture. If \(F_{CD}\) is small, the fracture itself restricts flow, and the production improvement can be limited. If \(F_{CD}\) is large, the fracture provides an efficient pathway for fluid flow and behaves as a high-conductivity fracture. An \(F_{CD} > 70\) is considered equivalent to an infinite conductivity fracture2Valkó, P., & Economides, M. J. (1995). Hydraulic fracture mechanics. Wiley. (relative to the formation) such that flow resistance within the fracture can be neglected for the purpose of flow modeling.
Although \(F_{CD}\) is a measure of the effectiveness of a fracture in a reservoir, well productivity is more complicated, and depends independently on fracture half-length (\(x_f\)) and fracture conductivity (\(k_f w_f\)), and fracture effectiveness also varies with time.
\(J/J_o\) is strongly time dependent in vertical wells with vertical hydraulic fractures, such that \(J/J_o\) can be hundreds of times higher during the early production history of a well as compared to later3Morse, R. A., & Von Gonten, W. D. (1972). Productivity of vertically fractured wells prior to stabilized flow. Journal of Petroleum Technology, 24(7), 807–811. . For wells producing from a finite, bounded drainage area, early vs late time is defined in the following way:
- early time – Production behavior is defined as infinite acting because the depletion front moving out from the well has not yet contacted any flow boundaries (also known as transient behavior).
- late time – Production behavior after the depletion driven pressure front reaches a boundary, when flow transitions from the infinite-acting to finite reservoir behavior. This late time behavior is termed pseudo-steady state, and is characterized by a dramatic increase in the rate of decline of the production.
The time in days at which pseudo-steady state behavior begins for a well, \(t_{pss}\), can be calculated as4Earlougher, R. C. (1977). Advances in well test analysis. Henry L. Doherty Memorial Fund of AIME. https://doi.org/10.2118/3631-PA
where
- \(\phi\) is porosity (in decimal format),
- \(\mu\) is viscosity in cp,
- \(c_t\) is total compressibility in psi−1,
- \(A\) is drainage area in ft2, and
- \(k\) is permeability in md.
Assuming reasonable oilfield properties (\(\phi = 0.15\), \(\mu = 1\) cp, \(c_t = 1 \times 10^{-5}\) psi−1, \(A = 1{,}742{,}400\) ft2), pseudo-steady state for the 100 md conventional reservoir permeability of our previous example starts at
For the unconventional example permeability of \(k = 0.1\) md, pseudo-steady state starts 1,000 times later at 372 days. Taking permeability reduction one step further, to include a shale with \(k = 0.001\) md (1 microdarcy), the time to start pseudo-steady state would be 37,000 days or 100 years. Since the definition of the beginning of pseudo-steady state is the time at which the depletion pressure front reaches the reservoir boundary, for the 40 acre square drainage area of our example, in the 100 md conventional reservoir, the depletion pressure front will travel the 660 ft from the well to the boundary in ½ day, but in the unconventional case with \(k = 0.001\) md, that same journey will take 100 years!
