Sealing fault – distance to fault and test-duration validity: 24-Pet-B5 Reservoir Mechanics, December 2014, Q6

Question 6: Sealing fault – distance to fault and test-duration validity (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
Porosity	$\phi$	0.1
Formation volume factor	$B_o$	1.2 bbl/STB
Oil production rate	$q$	500 STB/D
Oil viscosity	$\mu$	1 cp
Production time before shut-in	$t_p$	100 hr
Total compressibility	$c_t$	$1\times10^{-6}\ \text{psi}^{-1}$
Reservoir thickness	$h$	50 ft
Wellbore radius	$r_w$	0.3 ft

Find. Distance to the sealing fault, and whether the test ran long enough to reliably confirm it.

Approach. A sealing fault behaves, by the method of images, like a second (image) well starting to produce once the transient reaches it – which doubles the apparent semilog slope at late shut-in time. Fit the true (early) and doubled (late) slopes, use the observed time of doubling to back out the fault distance $L$, then compare the theoretical time for doubling (computed from that same $L$) against how long the test was actually run.

Fig. 5 – Horner buildup plot (digitized markers, with the fitted true and doubled-slope lines). Because $\Delta t$ increases toward the right (Horner ratio $\to1$), the LEFT portion (large Horner ratio, small $\Delta t$) is the true, fault-undisturbed early-time slope, and the RIGHT portion (small Horner ratio, large $\Delta t$) is the doubled, fault-affected late-time slope.

Fit the true (early) and doubled (late) slopes. A least-squares changepoint search across the chart data locates the break at Horner ratio $\approx14{,}500$, with $$m_1\approx4.94\ \text{psi/cycle (true, undisturbed)},\qquad m_2\approx8.92\ \text{psi/cycle (after the fault is felt)}$$ the ratio $m_2/m_1\approx1.8$ is close to the theoretical 2 (the residual gap is ordinary chart-reading scatter), confirming the fault interpretation. (Check: both slopes read from the printed chart.)
Permeability from the true (undisturbed) slope. $$k=\frac{162.6\,q\mu B_o}{m_1h}=\frac{162.6(500)(1)(1.2)}{(4.94)(50)}$$ $$\boxed{k\approx 395\ \text{mD}}$$
Distance to the fault. The break occurs at Horner ratio $\approx14{,}531$, i.e. $\Delta t^*=t_p/(14{,}531-1)\approx0.00688$ hr. Substituting into the formula sheet's distance-to-fault relation, $$L=\sqrt{\frac{0.000148\,k\,\Delta t^*}{\phi\mu c_t}}=\sqrt{\frac{0.000148(395)(0.00688)}{(0.1)(1)(1\times10^{-6})}}$$ $$\boxed{L\approx 63\ \text{ft}}$$
Theoretical time for the slope to double, and test-duration check. Using that same $L$ in the formula sheet's companion relation for the time the slope takes to double, $$\Delta t_{theory}=\frac{3.8\times10^5\,\phi\mu c_t L^2}{k}=\frac{3.8\times10^5(0.1)(1)(1\times10^{-6})(63)^2}{395}\approx0.39\ \text{hr}$$ The test's longest reliably digitized shut-in point sits at Horner ratio $\approx64$, giving $\Delta t_{test}=t_p/(64-1)\approx1.6$ hr already comfortably run; the plot's visible data extend further still, toward a Horner ratio near 10 (i.e. $\Delta t\approx t_p/(10-1)\approx11.1$ hr). Either way, $$\Delta t_{theory}\ (0.39\text{ hr}) \ll \Delta t_{test}\ (\text{several hours})$$ so the buildup was run well beyond the time theoretically required for the doubled slope to appear – the test duration is adequate, and the observed slope-doubling is a reliable confirmation of the fault, not a premature reading.

Result	Value
True slope, $m_1$	≈ 4.94 psi/cycle
Doubled slope, $m_2$	≈ 8.92 psi/cycle ($m_2/m_1\approx1.8$)
Permeability, $k$	≈ 395 mD
Distance to fault, $L$	≈ 63 ft
Theoretical slope-doubling time	≈ 0.39 hr
Test duration reached	≥ 1.6 hr (test valid)

Topic. Sealing-fault detection by the method of images.

Key relations. $L=\sqrt{0.000148k\Delta t/(\phi\mu c_t)}$; $\Delta t_{theory}=3.8\times10^5\phi\mu c_tL^2/k$; real fault doubles the semilog slope at late time.

Why this works. A no-flow linear boundary is mathematically equivalent to placing a mirror-image well (same rate and sign) on the far side of the fault, at twice the well-to-fault distance; once the real transient has travelled far enough to "see" that image well, the two wells' pressure drops superpose, and because both wells are now producing into the same observation point, the apparent flow capacity is effectively halved – doubling the slope on a semilog plot.

Common pitfall. Assuming the two formulas given on the formula sheet are meant to be algebraically inverse (i.e. that plugging the observed $\Delta t^*$ into the distance formula and then that $L$ back into the "time to double" formula should return the same $\Delta t^*$) – they are not: the two constants ($0.000148$ and $3.8\times10^5$) come from different published criteria for "detecting" vs. "fully developing" the doubled slope, so a substantial ratio between the two computed times is expected, not an error.

Source. Earlougher, Advances in Well Test Analysis, ch.3 (sealing faults, method of images); Bourdet, Well Test Analysis, ch.4.

Question 6 of 7: Sealing fault – distance to fault and test-duration validity

Question 6: Sealing fault – distance to fault and test-duration validity (20 marks)