Hydraulically fractured well – fracture half-length and skin: 24-Pet-B5 Reservoir Mechanics, May 2015, Q6

Question 6: Hydraulically fractured well – fracture half-length and skin (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
Oil rate	$q$	200 STBD
Reservoir permeability	$k$	1.95 mD
Formation thickness	$h$	12 ft
Oil formation volume factor	$B_o$	1.325 bbl/STB
Initial pressure	$p_i$	3343.40 psia
Porosity	$\phi$	11.8%
Total compressibility	$c_t$	$14.7\times10^{-6}\ \text{psi}^{-1}$
Wellbore radius	$r_w$	0.25 ft
Oil viscosity	$\mu_o$	0.49 cP

Early-time pressure data
$t$ (hr)	$p_{wf}$ (psia)
0.0010	3314
0.0040	3292
0.0090	3269
0.0128	3257
0.0239	3228
0.0320	3211
0.0426	3192
0.0564	3172

Find. Fracture half-length $x_f$ and skin factor $S$.

Approach. Early time in an infinite-conductivity vertical fracture is dominated by LINEAR flow into the fracture face, for which $\Delta p$ is linear in $\sqrt t$ (not $\log t$); the printed pressure-time table already covers this early window, so the slope of $\Delta p$ vs. $\sqrt t$ can be regressed directly. The fracture half-length converts to an equivalent (negative) skin via the standard infinite-conductivity-fracture relation, since the question specifies that assumption directly – no additional late-time (radial-flow) chart reading is needed.

Fig. 6 – Linear-flow plot, Δp = p_i−p_wf vs. √t, from the printed early-time table, with the least-squares regression line.

Regress $\Delta p$ against $\sqrt t$. With $\Delta p=p_i-p_{wf}$ computed at each tabulated time, a least-squares fit through the eight early-time points gives $$\Delta p\approx 8.0+692.6\sqrt{t}\qquad\Rightarrow\qquad m_{LF}\approx 692.6\ \text{psi}/\sqrt{\text{hr}}$$ (the small intercept, $\approx8$ psi against $\Delta p$ values of 30–170 psi, confirms the eight points lie close to a straight line through the origin, as linear-flow theory requires).
Fracture half-length. For linear flow into an infinite-conductivity vertical fracture, the formula-sheet relation is $$x_f=\frac{4.064\,qB_o}{h\,m_{LF}}\sqrt{\frac{\mu_o}{k\phi c_t}}=\frac{4.064(200)(1.325)}{(12)(692.6)}\sqrt{\frac{0.49}{(1.95)(0.118)(14.7\times10^{-6})}}$$ $$\boxed{x_f\approx 49\ \text{ft}}$$
Equivalent skin factor. An infinite-conductivity vertical fracture behaves, once flow becomes effectively radial far from the fracture, like an unfractured well with an enlarged effective wellbore radius $r_w'=x_f/2$; the resulting apparent skin (Cinco-Ley & Samaniego) is $$S=-\ln\!\left(\frac{x_f}{2r_w}\right)=-\ln\!\left(\frac{49}{2(0.25)}\right)$$ $$\boxed{S\approx -4.6}$$

Result	Value
Linear-flow slope, $m_{LF}$	≈ 692.6 psi/√hr
Fracture half-length, $x_f$	≈ 49 ft
Equivalent skin factor, $S$	≈ −4.6

Topic. Linear-flow analysis of an infinite-conductivity hydraulic fracture.

Key relations. $\Delta p=\dfrac{4.064qB_o}{hx_f}\sqrt{\dfrac{\mu t}{k\phi c_t}}$; equivalent skin $S=-\ln(x_f/2r_w)$.

Why this works. With an infinite-conductivity fracture, pressure drop near the well is dominated (at early time) by fluid flowing essentially perpendicular into the two fracture faces – a 1-D linear-flow problem – rather than by radial convergence toward a cylindrical wellbore, which is exactly why $\Delta p$ tracks $\sqrt t$, not $\log t$, in this window; the large NEGATIVE apparent skin simply expresses, in the language of an equivalent unfractured well, how much larger the effective drainage entry surface has become.

Common pitfall. Applying the usual $\log t$ semilog (radial-flow) skin formula to this early-time data – a fractured well's early data is never in radial flow, and forcing it through the wrong model produces a meaningless "permeability" and skin. Linear flow must be diagnosed (and analysed) on its own $\sqrt t$ axis first.

Source. Cinco-Ley & Samaniego, “Transient Pressure Analysis for Fractured Wells,” JPT, 1981; Bourdet, Well Test Analysis, ch.6 (fractured wells); Lee, Well Testing, ch.6.

Question 6 of 7: Hydraulically fractured well – fracture half-length and skin

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