Correction to: Probabilistic estimation of hydraulic fracture half-lengths: validating the Gaussian pressure-transient method with the traditional rate transient analysis-method (Wolfcamp case study)

Correction to: Journal of Petroleum Exploration and Production Technology https://doi.org/10.1007/s13202-023-01680-9

Full Reference for the Original Paper:

Alvayed, D., Khalid, M. S. A., Dafaalla, M., Ali, A., Ibrahim, A. F., & Weijermars, R. (2023). Probabilistic estimation of hydraulic fracture half-lengths: validating the Gaussian pressure-transient method with the traditional rate transient analysis-method (Wolfcamp case study). Journal of Petroleum Exploration and Production Technology, 1–16. https://doi.org/10.1007/S13202-023-01680-9

The original version of this article unfortunately contained serious typographical errors due to mishaps in the proof-printing process. This correction article aims to rectify these errors in order to do justice to our original work, which was executed with the greatest care for detail and accuracy.

Point 1: Due to two distracting errors in the Abstract, the correct Abstract is given here in full below:

Abstract—Despite significant advancements in geomodelling technologies, accurately estimating hydraulic fracture half-length remains a challenging task. This paper introduces a detailed estimation approach using the Gaussian pressure transient (GPT) method, which is relatively new. The GPT method is iterative, ensuring fast convergence and providing reliable estimations of hydraulic fracture half-length based on a predetermined hydraulic diffusivity value obtained from Gaussian decline curve analysis (DCA). To validate the GPT results, production data from two case study wells in the Wolfcamp Shale Formation, located in the Midland Basin of West Texas, are utilized alongside the traditional rate transient analysis (RTA) method. Moreover, the GPT method offers the capability to probabilistically estimate hydraulic fracture half-lengths, presenting two innovative approaches to evaluate the robustness of this newly developed method for both deterministic and probabilistic estimations. The simulation results demonstrate a close correlation between the Gaussian method and micro-seismic fracture half-lengths, with separate confirmation from the classic RTA method. Through the case studies presented in this paper, the GPT method showcases its utility in estimating hydraulic fracture half-lengths for two Wolfcamp case study wells, effectively demonstrating the validity and practical applicability of this novel method.

Point 2: Some italicizations and typographical errors in the List of symbols were missing, so the correct List of symbols is given here in full below:

List of symbols

$A$:

Area around the fracture, ft²

${B}_{0}$:

Oil formation volume factor at reservoir condition, bbl/STB

$C1$:

Conversion factor for field units, 0.178108 bbls/ft³

$C2$:

Conversion factor for field units, 1.06235E-14 ft²/mD

$C3$:

Conversion factor for field units, 1.67868E-12 psi.day/cPoise

${D}_{h}$:

Dimensionless reservoir diffusivity

$h$:

Reservoir thickness, ft

$k$:

Matrix permeability, Darcy

$n$:

Total number of fractures

${P}_{0}$:

Original reservoir pressure, psi

${q}_{i}$:

Initial production rate, bbl/day

${q}_{r}\left(t\right)$:

Production rate at time t, bbl/day

$t$:

Dimensionless time

$x$:

Fixed distance to the fracture plane (set at 1 ft if using field units), ft

${Y}_{f}$:

Fracture half-length, ft

$\mu$:

Fluid viscosity, cP

${P}_{BH}$:

Bottom hole pressure, psi

$\mathrm{DCA}$:

Decline curve analysis

$\mathrm{GPT}$:

Gaussian pressure transient

$\mathrm{RTA}$:

Rate Transient Analysis

Point 3: Equation (1) on page 2 was published incorrectly, which should be centered and italicized. The correct Eq. (1) is given below.

$$q\left(t\right)={q}_{i}\frac{1}{t}{e}^{\left(\frac{1}{4Dh}\right)\left(1-\frac{1}{t}\right)}$$

(1)

Point 4: The text “Formation volume factor (STB/ reservoir bbls)” in Table 1 on page 4 was published incorrectly, which should be aligned to the left. The correct Table 1 is given below.

Table 1 PVT mean values for Well 4H and 31H

Point 5: Equation (3) on page 6 was published incorrectly which should be centered and italicized, and use ${P}_{BH}$ instead of BHP. The correct Eq. (3) is given below.

$${q}_{r}\left(t\right)=\frac{2Ak\left({P}_{0}-{P}_{BH}\right)}{\mu {D}_{h}t}x{e}^{\left(\frac{-{x}^{2}}{4{D}_{h}t}\right)}$$

(3)

Point 6: In the sentence beginning with “BHP represents bottom…” on page 6, the symbol “BHP” was published incorrectly, which should be replaced with ${P}_{BH}$. The corrected part of the sentence is given below.

“${P}_{BH}$ represents bottom hole pressure”

Point 7: Equation (4) on page 6 was published incorrectly, which should be centered and italicized. The correct Eq. (4) is given below.

$${q}_{w}\left(t\right)=\frac{{q}_{r}\left(t\right)}{B} \& {Y}_{f}=\frac{A}{2nh}$$

(4)

Point 8: The two symbols “${\mathrm{Y}}_{\mathrm{f}}$” in Table 3 on page 7 were published incorrectly, which should be in italics. The correct Table 3 is given below.

Table 3 Average fracture half-length ${Y}_{f}$ from micro-seismic for Wells 4H, 44H, 45H, and 46H

Point 9: Equation (5) on page 7 was published incorrectly which should be centered and italicized, and replace BHP by ${P}_{BH}$. The correct Eq. (5) is given below.

$${q}_{w}\left(t\right)=\frac{C1C2}{C3}\cdot \frac{2nhk{Y}_{f}\left({P}_{0}-{P}_{BH}\right)}{\mu {B}_{0}{D}_{h}t}x{e}^{\left(\frac{-{x}^{2}}{4{D}_{h}t}\right)}$$

(5)

Point 10: In the sentence beginning with “The conversion factors $\mathrm{C}1,\mathrm{ C}2,\mathrm{ and C}3$ are needed…” on page 7, the three symbols “$\mathrm{C}1,\mathrm{ C}2,\mathrm{ and C}3$” were published incorrectly, which should be in italicized. The correct symbols are given below.

“$C1, C2,\mathrm{ and }C3$”

Point 11: In the sentence beginning with “Rearranging Eq. (5) yields Eq. (6). Which was…” on page 7, the text. “Which” was published incorrectly. The dot should be removed, and the text “Which” should not be capitalized. The correct sentence is given below.

“Rearranging Eq. (5) yields Eq. (6), which was used in this study to stochastically model the fracture half-length:”

Point 12: The symbol “BHP” on page 7 was published incorrectly, which should be replaced with ${P}_{BH}$. The correct symbol is given below.

“${P}_{BH}$”

Point 13: Equation (6) on page 7 was published incorrectly which should be in the center and italics and replace BHP by ${P}_{BH}$. The correct Eq. (6) is given below.

$${Y}_{f}=\frac{C3}{C1C2}\cdot \frac{{q}_{w}\left(t\right).\mu {B}_{0}{D}_{h}t}{2nhk\left({P}_{0}-{P}_{BH}\right)}x{e}^{\left(\frac{{x}^{2}}{4{D}_{h}t}\right)}$$

(6)

Point 14: The text “Formation volume factor (STB/reservoir bbls)” in Table 4 on page 7 was published incorrectly which should be aligned to the left. The correct Table 4 is given below.

Table 4 Property values applied in history matching in approach A for both wells

Point 15: In the sentence beginning with “Two Approaches (A,B) were evaluated to…” on page 7, the text “Sections” was published incorrectly, which should not be capitalized. The correct sentence is given below.

“Two approaches (A,B) were evaluated to estimate the hydraulic fracture half-length, based on the GPT model explained in the previous sections.”

Point 16: The two texts “Formation volume factor (STB/ reservoir bbls)” in Table 5 on page 8 were published incorrectly, which should be aligned to the left. The correct Table 5 is given below.

Table 5 Properties of PDF used in approach B for both wells

Point 17: In the sentence beginning with “Fracture half-lengths were estimated with…” on page 8, the text “(Sect. 0)” should not be capitalized. The should be replaced by:.

“Fracture half-lengths were estimated with the GPT method using the proposed GPT approaches.”

Point 18: The text “bottom-hole” in Table 6 on page 8 was published incorrectly, which should be replaced by ${P}_{BH}$. The correct Table 6 is given below.

Table 6 FVF and viscosity data obtained from PVT tests for Well 4H

Point 19: In the sentence beginning with “Firstly, PVT data were obtained from…” on page 8, the symbol “FVF“ was published incorrectly, which should be replace by “formation volume factor (FVF)”. The correct sentence is:

“Firstly, PVT data were obtained from available reports to address suitable formation volume factor (FVF) and viscosity values”

Point 20: In the sentence beginning with “Accordingly, two values of…” on page 8, The text “condition’s FVF” was published incorrectly, which should be replaced by:

“Accordingly, two values of ${Y}_{f}$ were estimated using the reservoir condition’s FVF coupled with the viscosity at the reservoir and bottom hole conditions.”

Point 21: The symbol “${\mathrm{Y}}_{\mathrm{f}}$” in Table 7 on page 9 was published incorrectly, which should be in Italics. The correct Table 7 is given below.

Table 7 Deterministic fracture half-length ${Y}_{f}$-values estimated using Approach A for Well 4H

Point 22: The symbol “BHP” in Table 8 on page 9 was published incorrectly, which should be replaced with ${P}_{BH}$. The correct Table 8 is given below.

Table 8 FVF and viscosity data obtained from PVT tests for Well 31H

Point 23: The symbol “${\mathrm{Y}}_{\mathrm{f}}$” in Table 9 on page 9 was published incorrectly, which should be in italics. The correct Table 9 is given below.

Table 9 Facture half-length values using deterministic approach for Well 31H

Point 24: In the sentence beginning with “This section will summarize..” on page 9, the text “data shown in Sect. 0” on page 9 should be removed. The correct sentence is given below.

“This section will summarize the micro-seismic data to validate GPT approaches.”

Point 25: The two symbols “${\mathrm{Y}}_{\mathrm{f}}$” in Table 11 on page 10 were published incorrectly, which should be in italics. The correct Table 11 is given below.

Table 11 Probabilistic ${Y}_{f}$ values from Approach B for Well 4H

Point 26: The term “$\mathrm{P}-{\mathrm{P}}_{\mathrm{wf}}$” on page 10 was published incorrectly, which should be in italics. The correct term is given below.

“$P-{P}_{wf}$”

Point 27: The term “function,$t\hspace{0.33em}[\Delta (p)/{q}_{o}]{\prime}$” on page 10 was published incorrectly. A space needs to be inserted after the comma. The correct term is given below.

“function, $t\hspace{0.33em}[\Delta (p)/{q}_{o}]{\prime}$”

Point 28: The text “Minimum Extreme Value” in Table 12 on page 10 was published incorrectly, which should be aligned to the left. The correct Table 12 is given below.

Table 12 Best fit distributions for input parameters of Eq. (6) for Well 31H

Point 29: The two symbols “${\mathrm{Y}}_{\mathrm{f}}$” in Table 13 on page 11 were published incorrectly, which should be in italics. The correct Table 13 is given below.

Table 13 Probabilistic ${Y}_{f}$ values from Approach B for Well 31H

Point 30: The two symbols “${\mathrm{Y}}_{\mathrm{f}}$” in Table 14 on page 11 were published incorrectly, which should be in italics. The correct Table 14 is given below.

Table 14 Average fracture half-lengths reported from micro-seismic measurements

Point 31: In the sentence beginning with “In the Cartesian plot, a straight…” on page 11 was published incorrectly. The correct sentence is given below.

“In the Cartesian plot, a straight line with a slope (m) can be estimated.”

Point 32: Eq. (7) on page 11 should be centered. The correct Eq. (7) is given below.

$$\sqrt{k}{A}_{c}=\frac{79.65B\mu }{\sqrt{\hspace{0.33em}(\varphi \mu {c}_{t}{)}_{i}}}\left(\frac{1}{m}\right)$$

(7)

Point 33: The symbol “$\mathrm{Nf}$” on page 11 was published incorrectly. The correct symbol is given below.

“${N}_{f}$”

Point 34: The symbol “$\mathrm{Hf}$” on page 11 was published incorrectly. The correct symbol is given below.

“${H}_{f}$”

Point 35: Eq. (8) on page 11 was published incorrectly and should be in the Center. The correct Eq. (8) is given below.

$${A}_{c}=4{N}_{f}{Y}_{f}{H}_{f}$$

(8)

Point 36: In the sentence beginning with “For Well4H, GPT-Approach B successfully simulated…” on page 11, the term “Well4H” was published incorrectly. A space needs to be inserted after the word “Well”. The correct term is given below.

“Well 4H”

Point 37: The three symbols “${\mathrm{Y}}_{\mathrm{f}}$” in Table 15 on page 12 were published incorrectly, which should be in italics. The correct Table 15 is given below.

Table 15 Comparison of ${Y}_{f}$-estimations from GPT, RTA, and MS methods for Well 4H

Point 38: The term “$\left[{\mathrm{p}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{wf}}\right]/{\mathrm{q}}_{\mathrm{o}}$” in the caption of Fig. 12 on page 12 was published incorrectly, without italics. The correct term is “$\left[{p}_{i}-{p}_{wf}\right]/{q}_{o}$”

The below mentioned Figs. 2, 3, 13 and 14 are published incorrectly and the correct figures are given below.

Fig. 2

Permeability probability density functions (a) 4H (118 sample size) and (b) 31H (95 sample size)

Fig. 3

History matching of production rates on Gaussian DCA model (a) Well 4H and (b) Well 31H

Fig. 13

Hydraulic fracture half-length with time for Well 4H

Fig. 14

Hydraulic fracture half-length with time for Well 31H