**4. Implications of fracture behavior from** *G* **function superposition derivative**

In Figure 8, Barree, et al. (reference [5]) suggested that the normal leakoff (NL) from hydraulic fracture after shut-in leads to a straight line originated from the *G* function time in the form of the *G* function superposition derivative (i.e. G dP/dG). Fracture closure time is identified when the *G* function superposition derivative (GFSD) deviates from the straight line in a critical curve slope change.

**•** Case (d): pressure-dependent leakoff Case 1 (PDL-1) **•** Case (e): pressure-dependent leakoff Case 2 (PDL-2) **•** Case (f): pressure-dependent leakoff Case 3 (PDL-3)

**•** Case (g): fracture tip extension Case 1 (FTE-1)

fracture during the fracture closing.

straight line indicates normal leakoff.

**•** Case (h): fracture height recession Case 1 (FHR-1) **•** Case (i): fracture height recession Case 2 (FHR-2)

It is interesting to note that GFSD follows a straight line for NL from the beginning to the time when the GFSD changes slope, indicating a linear flow is maintained within the hydraulic

**Figure 8.** Determination of closure pressure from the GFSD curve to identify the critical curve slope change. The initial

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For PDL, GFSD is above the straight line and becomes convex from the beginning; then the GFSD follows a straight line before the GFSD changes slope. In the convex portion of GFSD,

Any initial response of GFSD that is different from a straight line as shown in Figure 8 can be interpreted as an abnormal leakoff. In addition, the non-straight-line behavior can be caused by events other than leakoff, such as fracture tip growth or height reduction, etc. In Figure 9, nine cases of the pressure to *G* time relationship are shown as follows:


**Figure 8.** Determination of closure pressure from the GFSD curve to identify the critical curve slope change. The initial straight line indicates normal leakoff.


**Figure 7.** Responses of BHP (in red) and corresponding injection rate (step rate, in blue) from a MiniFrac test: a) pres‐ sure increased sharply until generating a small fracture in LOP; b) formation breakdown at BP and small fracture prop‐

In Figure 8, Barree, et al. (reference [5]) suggested that the normal leakoff (NL) from hydraulic fracture after shut-in leads to a straight line originated from the *G* function time in the form of the *G* function superposition derivative (i.e. G dP/dG). Fracture closure time is identified when the *G* function superposition derivative (GFSD) deviates from the straight line in a critical

Any initial response of GFSD that is different from a straight line as shown in Figure 8 can be interpreted as an abnormal leakoff. In addition, the non-straight-line behavior can be caused by events other than leakoff, such as fracture tip growth or height reduction, etc. In Figure 9,

nine cases of the pressure to *G* time relationship are shown as follows:

**4. Implications of fracture behavior from** *G* **function superposition**

agation; c) shut-in, and d) pressure leakoff and fracture closing.

800 Effective and Sustainable Hydraulic Fracturing

**derivative**

curve slope change.

**•** Case (a): normal leakoff Case 1 (NL-1) **•** Case (b): normal leakoff Case 2 (NL-2) **•** Case (c): normal leakoff Case 3 (NL-3)


It is interesting to note that GFSD follows a straight line for NL from the beginning to the time when the GFSD changes slope, indicating a linear flow is maintained within the hydraulic fracture during the fracture closing.

For PDL, GFSD is above the straight line and becomes convex from the beginning; then the GFSD follows a straight line before the GFSD changes slope. In the convex portion of GFSD,

**•** Pressure-dependent leakoff cases (PDL-1 and PDL-3) result in the concave shapes of the P

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**•** Fracture tip extension cases (FTE-1) and fracture height recession cases (FHR-1) result in a concave shape of the P – G relationship in the beginning and then change to convex.

Cases c (NL-3) and e (PDL-2) have different pressure magnitudes than the other cases. Figure 11 indicates that PDL-2 maintains the concave shape of the P – G relationship, while NL-3 shows an extended linear line immediately after shut-in before changing to another linear line with a different slope. The initial linear lines after shut-in for Cases a (NL-1) and b (NL-2) are

**Figure 10.** Comparison of six cases of NL, PDL, FTE, and FHR: a) linear lines of the P – G relationship from normal leak‐ off (NL-1 and NL-2); b) concave curves of the P – G relationship from pressure-dependent leakoff (PDL-1 and PDL-3), and c) initial concave and later convex curves of the P – G relationship from fracture tip extension and fracture height


very short (see Figure 10).

recession (FTE-1 and FHR-1).

**Figure 9.** After shut-in fracture responses from the *G* function superposition derivative: normal leakoff Cases a, b, and c (NL-1, NL-2, and NL-3); pressure dependent leakoff Cases d, e, and f (PDL-1, PDL-2, and PDL-3); fracture tip extension Case g (FTE-1); and fracture height recession Cases h and i (FHR-1 and FHR-2).

the flow within the hydraulic fracture is affected by 'back stress' in the far field, a poro-elastic coupled response that demonstrates the interactive behavior between rock deformation and fluid flow. Another interpretation of pressure-dependent leakoff is that the existence of natural fractures deviates the path of GFSD from the straight line to a convex curve.

The response of GFSD from fracture tip extension (FTE) is similar to PDL except that there is no linear leakoff period after the convex portion of the GFSD curve.

For fracture height recession (FHR), the GFSD curve is concave from the beginning until the GFSD changes slope. The greater pressure drop within the hydraulic fracture is the result of fracture geometric reduction or fracture height recession.

Comparing the selected cases using the relationship between BHP and the linear *G* function in Figure 10, the following interesting observations can be made:

**•** Normal leakoff cases (NL-1 and NL-2) lead to straight lines between P and G.


Cases c (NL-3) and e (PDL-2) have different pressure magnitudes than the other cases. Figure 11 indicates that PDL-2 maintains the concave shape of the P – G relationship, while NL-3 shows an extended linear line immediately after shut-in before changing to another linear line with a different slope. The initial linear lines after shut-in for Cases a (NL-1) and b (NL-2) are very short (see Figure 10).

the flow within the hydraulic fracture is affected by 'back stress' in the far field, a poro-elastic coupled response that demonstrates the interactive behavior between rock deformation and fluid flow. Another interpretation of pressure-dependent leakoff is that the existence of natural

**Figure 9.** After shut-in fracture responses from the *G* function superposition derivative: normal leakoff Cases a, b, and c (NL-1, NL-2, and NL-3); pressure dependent leakoff Cases d, e, and f (PDL-1, PDL-2, and PDL-3); fracture tip extension

The response of GFSD from fracture tip extension (FTE) is similar to PDL except that there is

For fracture height recession (FHR), the GFSD curve is concave from the beginning until the GFSD changes slope. The greater pressure drop within the hydraulic fracture is the result of

Comparing the selected cases using the relationship between BHP and the linear *G* function

**•** Normal leakoff cases (NL-1 and NL-2) lead to straight lines between P and G.

fractures deviates the path of GFSD from the straight line to a convex curve.

no linear leakoff period after the convex portion of the GFSD curve.

Case g (FTE-1); and fracture height recession Cases h and i (FHR-1 and FHR-2).

802 Effective and Sustainable Hydraulic Fracturing

in Figure 10, the following interesting observations can be made:

fracture geometric reduction or fracture height recession.

**Figure 10.** Comparison of six cases of NL, PDL, FTE, and FHR: a) linear lines of the P – G relationship from normal leak‐ off (NL-1 and NL-2); b) concave curves of the P – G relationship from pressure-dependent leakoff (PDL-1 and PDL-3), and c) initial concave and later convex curves of the P – G relationship from fracture tip extension and fracture height recession (FTE-1 and FHR-1).

The quality of fracture closure analysis also affects the quality of cuttings injection manage‐ ment. Among various methods, using GFSD in interpreting BHP responses to the cuttings injection operation appears to be an efficient way to identify the occurrence of fracture closure. Interpreting the shapes of GFSD over the P – G relationship may help identify the different responses from various formations such as injection in naturally fractured reservoirs, reflect various behaviors involved in the injection process such as the pressure/stress dependency in a poro-elastic formation, or reveal the outcome from the complex, coupled process of rock

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and Juan Reyna1

[1] Nagel, N. and McLennan, J.D., editors, Solids Injection Monograph, Society of Petro‐

[2] Carter, R. D. Derivation of the general equation for estimating the extent of the frac‐ tured area, Appendix I of Optimum Fluid Characteristics for Fracture Extension, Drilling and Production Practice, G.C. Howard and C.R. Fast, New York, New York,

[3] Horner, D. R. Pressure buildup in wells, Proceeding, Third World Petroleum Con‐

[4] Nolte, K. G. Determination of fracture parameters from fracturing pressure decline, paper SPE 8341, presented at the SPE Annual Technical Conference and Exhibition,

[5] Barree, R. D, Barree, V. L, & Craig, D. P. Holistic fracture diagnostics: consistent in‐ terpretation of prefrac injection tests using multiple analysis methods, SPE Produc‐

[6] Economides, M. J, & Nolte, K. J. Reservoir Stimulation, 3rd Edition, Wiley, (2000).

deformation and fluid flow under various reservoir boundary conditions.

, John McLennan2

USA, American Petroleum Institute, (1957). , 261-269.

Las Vegas, Nevada, USA, Sept., (1979).

tion and Operations, August, (2009). , 396-406.

gress, The Hague, Netherlands (SPE), Sec. II, (1951). , 503-523.

\*Address all correspondence to: mao.bai@halliburton.com

**Author details**

, Arturo Diaz1

1 Halliburton, Houston, USA

2 University of Utah, Salt Lake City, USA

leum Engineers (SPE), (2010).

Mao Bai1

**References**

**Figure 11.** Comparison of two cases of (NL-3 and PDL-2): a) initial linear line and late linear line with a different slope for the P - G relationship with normal leakoff; and b) concave curves for the P – G relationship with pressure-depend‐ ent leakoff.
