Hydraulically fractured well – fracture half-length and flow regimes: 24-Pet-B5 Reservoir Mechanics, May 2016, Q6

Question 6: Hydraulically fractured well – fracture half-length and flow regimes (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Symbol	Value
Permeability	$k$	4 mD
Flow rate	$q$	200 STBD
Formation thickness	$h$	12 ft
Formation volume factor	$B_o$	1.325 bbl/STB
Initial pressure	$p_i$	3343.40 psia
Porosity	$\phi$	0.118
Total compressibility	$c_t$	$14.7\times10^{-6}\ \text{psi}^{-1}$
Wellbore radius	$r_w$	0.25 ft
Oil viscosity	$\mu$	0.49 cP

Early-time flowing pressure (printed on source)
$t$ (hr)	$p_{wf}$ (psia)
0.0062	3280.78
0.0176	3243.17
0.0426	3192.27
0.0743	3148.39
0.1279	3095.23

Find. (a) Fracture half-length $x_f$; (b) The flow regimes visible on the log-log pressure-change/derivative plot.

Approach. For an infinite-conductivity vertical fracture, early time shows linear flow into the fracture face, where $\Delta p$ is proportional to $\sqrt t$ through the origin (no separate skin term); fit that proportionality constant to the printed early-time table and invert the linear-flow equation for $x_f$. Separately, walk the full printed log-log $\Delta p$/derivative plot from early to late time to name each flow regime by its characteristic log-log slope.

Linear-flow slope. $\Delta p=p_i-p_{wf}$ at each table point; an origin-constrained least-squares fit of $\Delta p$ vs. $\sqrt t$ gives $$\frac{\Delta p}{\sqrt t}\approx 712\ \text{psi/hr}^{0.5}$$
Fracture half-length. Rearranging the infinite-conductivity-fracture linear-flow equation $\Delta p=4.064\,(qB_o/h)\sqrt{\mu/(k\phi c_t)}\,\sqrt t\,/x_f$, $$x_f=\frac{4.064\,(qB_o/h)\sqrt{\mu/(k\phi c_t)}}{\Delta p/\sqrt t}=\frac{4.064\left(\dfrac{200(1.325)}{12}\right)\sqrt{\dfrac{0.49}{(4)(0.118)(14.7\times10^{-6})}}}{712}$$ $$\boxed{x_f\approx 33.5\ \text{ft}}$$
Flow-regime identification (log-log $\Delta p$ and derivative plot). Walking the digitized full-range plot from early to late time:
- Fracture linear flow ($t\lesssim0.5$ hr): both $\Delta p$ and its derivative climb on parallel trends with the derivative running at about half the $\Delta p$ value (half-slope signature) — this is the segment used for the $x_f$ fit above.
- Transition ($t\approx0.5$–40 hr): the two curves gradually converge as the pressure disturbance outgrows the fracture's own linear-flow zone and the flow lines bend toward the wellbore.
- Infinite-acting (pseudo-)radial flow ($t\approx40$–200 hr): the derivative flattens to a plateau ($\approx410$–460 psi) while $\Delta p$ keeps climbing — the classic derivative-plateau IARF signature, now centred on an effective (fracture-equivalent) wellbore.
- Late-time boundary effect ($t\gtrsim200$ hr): the derivative departs the plateau and climbs again (from ≈460 up to ≈1100 psi by the last recorded point), signalling the transient has begun to feel a reservoir boundary as the 725-hour test approaches its end.

Digitized log-log pressure-change (red triangles) and derivative (blue circles) data with early-time half-slope linear-flow reference lines (dashed).

Result	Value
Linear-flow slope, $\Delta p/\sqrt t$	≈ 712 psi/hr^0.5
Fracture half-length, $x_f$	≈ 33.5 ft
Flow regimes (early → late)	Fracture linear → transition → pseudo-radial (derivative plateau) → late boundary effect

Topic. Hydraulically fractured well diagnosis via log-log pressure/derivative type-curve behaviour.

Key relations. Linear flow $\Delta p=4.064(qB_o/h)\sqrt{\mu/(k\phi c_t)}\sqrt t/x_f$ (through the origin, infinite-conductivity fracture); derivative plateau $\Rightarrow$ infinite-acting radial flow.

Why this works. An infinite-conductivity fracture forces flow perpendicular to the fracture face at early time (linear flow, half-slope on log-log), which only transitions to ordinary radial flow once the pressure disturbance has moved a few fracture half-lengths into the reservoir; the derivative diagnostic makes each regime's characteristic slope directly readable without needing a type-curve match.

Common pitfall. Fitting the linear-flow slope through points already in the transition regime inflates $\Delta p/\sqrt t$ and understates $x_f$; only points where the derivative-to-$\Delta p$ ratio is close to the theoretical 0.5 should be used.

Source. Cinco-Ley & Samaniego, “Transient Pressure Analysis for Fractured Wells,” JPT 1981; Bourdet, Well Test Analysis, Ch. 6 (fractured-well type curves).

Question 6 of 6: Hydraulically fractured well – fracture half-length and flow regimes

Question 6: Hydraulically fractured well – fracture half-length and flow regimes (20 marks)