Naturally fractured reservoir buildup – skin, initial pressure, storativity: 24-Pet-B5 Reservoir Mechanics, May 2015, Q5

Question 5: Naturally fractured reservoir buildup – skin, initial pressure, storativity (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 production rate	$q$	125 STB/D
Formation thickness	$h$	17 ft
Oil formation volume factor	$B_o$	1.054 rb/STB
Production time before shut-in	$t_p$	1200 hr
Formation porosity	$\phi$	13.0%
Total compressibility	$c_t$	$7.19\times10^{-6}\ \text{psi}^{-1}$
Flowing wellbore pressure before shut-in	$p_{wf}$	211.20 psia
Wellbore radius	$r_w$	0.30 ft
Oil viscosity	$\mu_o$	1.72 cp

Find. Skin factor $S$, initial reservoir pressure $p_i$, and storativity ratio $\omega$.

Approach. A double-porosity (Warren–Root) buildup traces an S-shaped Horner curve: an early straight line (fracture system alone), a transition dip (matrix starts feeding the fractures), and a late straight line (total system) of the same slope. Fitting both lines with a shared-slope least-squares regression gives $kh$ from that common slope, $\omega$ from the vertical separation between the two lines, $S$ from $p_{1hr}$ read off the late line, and $p^{*}$ by extrapolating the late line to a Horner ratio of 1.

Fig. 5 – Horner plot of the double-porosity buildup (digitized markers), with the shared-slope early (fracture) and late (total-system) straight lines fitted by least squares.

Shared-slope fit of the two semilog lines. Fitting the early-time (large Horner-ratio, fracture-only) cluster and the late-time (small Horner-ratio, total-system) cluster with a common slope by least squares gives $$m\approx 26.1\ \text{psi/cycle}\quad(\text{check: digitized})$$ with early-line intercept $b_e\approx548$ psia and late-line intercept $b_l\approx514$ psia (both at Horner ratio $=1$ on their own respective trends).
Permeability-thickness (total system). Using the formula sheet's double-porosity slope relation with the shared slope, $$(kh)_f=\frac{162.6\,q\mu B_o}{m}=\frac{162.6(125)(1.72)(1.054)}{26.1}\approx1410\ \text{mD-ft}\qquad(\hat{k}\approx83\ \text{mD})$$
Storativity ratio $\omega$. The vertical gap between the two parallel lines, at any common Horner ratio, is $|b_e-b_l|\approx548-514=34$ psi, which converts to the storativity ratio via $$\omega=10^{-|b_e-b_l|/m}=10^{-34/26.1}$$ $$\boxed{\omega\approx 0.050}$$ – a physically reasonable value (real naturally fractured reservoirs typically show $\omega\sim0.001$–0.1).
Skin factor. Extrapolating the LATE (total-system) line to $\Delta t=1$ hr, i.e. Horner ratio $(1200+1)/1=1201$, gives $p_{1hr}\approx434$ psia. With the given $p_{wf}(\Delta t=0)=211.20$ psia, $$S=1.151\left[\frac{p_{1hr}-p_{wf}}{m}-\log_{10}\!\left(\frac{\hat{k}}{\phi\mu_oc_tr_w^2}\right)+3.23\right]$$ $$S=1.151\left[\frac{434-211.2}{26.1}-\log_{10}\!\left(\frac{83}{(0.13)(1.72)(7.19\times10^{-6})(0.30)^2}\right)+3.23\right]$$ $$\boxed{S\approx 3.4}$$
Initial reservoir pressure. Extrapolating the late (total-system) line all the way to a Horner ratio of 1 ($\Delta t\to\infty$, an infinite shut-in) gives the classic $p^{*}$ extrapolated pressure, taken as the estimate of the initial reservoir pressure (no Dietz shape-factor correction is available from the given data – check): $$\boxed{p_i\approx p^{*}\approx 514\ \text{psia}}$$

Result	Value
Shared semilog slope, $m$	≈ 26.1 psi/cycle
Permeability-thickness, $(kh)_f$	≈ 1410 mD-ft ($\hat{k}\approx$83 mD)
Storativity ratio, $\omega$	≈ 0.050
Skin factor, $S$	≈ 3.4
Initial reservoir pressure, $p_i$	≈ 514 psia

Topic. Warren–Root double-porosity buildup analysis.

Key relations. $m=162.6q\mu B_o/(kh)_f$; $\omega=10^{-\Delta p/m}$ (vertical separation of the parallel lines); $S=1.151[(p_{1hr}-p_{wf})/m-\log(\hat{k}/\phi\mu c_tr_w^2)+3.23]$.

Why this works. Early in the buildup only the high-permeability, low-storage fracture network has time to respond, giving one semilog straight line set by the fracture storativity; once the low-permeability matrix blocks begin bleeding into the fractures (the transition dip), the system behaves as a single continuum with the TOTAL storativity, giving a second, parallel line offset from the first by an amount that depends only on the storativity ratio $\omega=(\phi c_t)_f/[(\phi c_t)_f+(\phi c_t)_m]$.

Common pitfall. Fitting the early and late clusters independently (two different slopes) rather than forcing a SHARED slope – the shared-slope constraint is itself the correct way to average out that noise. A second, easy slip in the skin formula itself: the log term uses the PERMEABILITY $\hat{k}$ alone, not the permeability-thickness $(kh)_f$ that the slope directly yields – dividing $(kh)_f$ back out by $h$ first is a required step, and skipping it silently understates $S$ by a wide margin.

Source. Warren & Root, “The Behavior of Naturally Fractured Reservoirs,” SPE Journal, 1963; Bourdet, Well Test Analysis, ch.1, 5; Lee, Well Testing, ch.4.

Question 5 of 7: Naturally fractured reservoir buildup – skin, initial pressure, storativity

Question 5: Naturally fractured reservoir buildup – skin, initial pressure, storativity (20 marks)