Double-porosity buildup – storativity, fracture permeability, skin, average pressure: 24-Pet-B5 Reservoir Mechanics, December 2014, Q4

Question 4: Double-porosity buildup – storativity, fracture permeability, skin, average pressure (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	$q$	500 STB/D
Formation thickness	$h$	5 ft
Porosity	$\phi$	0.1
Wellbore radius	$r_w$	0.3 ft
Oil formation volume factor	$B_o$	1.2 bbl/STB
Total compressibility	$c_t$	$5\times10^{-6}\ \text{psi}^{-1}$
Oil viscosity	$\mu$	3 cp
Flowing pressure at shut-in, $\Delta t=0$	$p_{wf}(\Delta t=0)$	700 psia
Production time before shut-in	$t_p$	10,000 hr

Find. Storativity ratio $\omega$, average fracture permeability $\hat{k}_f$, skin factor $S$, and average reservoir pressure.

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 as the early line. Fit both lines (forcing a shared slope), then read $\hat{k}_fh$ from that slope, $\omega$ from the vertical separation between the two lines, $S$ from $p_{1hr}$ on the late line, and $p^*$ by extrapolating the late line to a Horner ratio of 1.

Fig. 3 – Horner plot of the double-porosity buildup (digitized markers, with the fitted early/fracture and late/total-system straight lines). The characteristic S-shape – two parallel-slope lines separated by a flatter transition – is the diagnostic signature of a naturally fractured reservoir.

Fit the two parallel semilog lines. A shared-slope least-squares fit through the digitized early-time points (fracture line) and late-time points (total-system line) gives $$m\approx 493\ \text{psi/cycle}\quad(\text{check: digitized})$$ with early-line intercept $b_e\approx3322$ psia and late-line intercept $b_l\approx2761$ psia (both at Horner ratio $=1$ on their own respective straight-line trends).
Average fracture permeability. Both lines have the same slope in the ideal Warren–Root model, so either one gives the same $(kh)_f$ via the formula sheet's double-porosity slope relation: $$(kh)_f=\frac{162.6\,q\mu B_o}{m}=\frac{162.6(500)(3)(1.2)}{493}\approx 594\ \text{mD-ft}$$ $$\boxed{\hat{k}_f=(kh)_f/h\approx 118.8\ \text{mD}}$$
Storativity ratio $\omega$. The vertical gap between the two parallel lines (at any common Horner ratio) is $\Delta p=b_l-b_e\approx2761-3322=-561$ psi (i.e. the fracture-only line sits above the eventual total-system line by 561 psi at the same Horner ratio, since the fracture-only system depletes faster before matrix support kicks in), which converts to the storativity ratio via $$\omega=10^{-|\Delta p|/m}=10^{-561/493}$$ $$\boxed{\omega\approx 0.073}$$ – a physically reasonable value (real naturally fractured reservoirs typically show $\omega\sim0.001$–0.1).
Skin factor. Extrapolating the LATE (total-system) line back to $\Delta t=1$ hr, i.e. Horner ratio $(t_p+1)/1=10{,}001$, gives $p_{1hr}\approx790$ psia. With the given $p_{wf}(\Delta t=0)=700$ psia, the buildup-test skin formula from the formula sheet gives $$S=1.151\left[\frac{p_{1hr}-p_{wf}(\Delta t=0)}{|m|}-\log\!\left(\frac{\hat{k}_f}{\phi\mu c_t r_w^2}\right)+3.23\right]$$ $$S=1.151\left[\frac{790-700}{493}-\log\!\left(\frac{118.8}{(0.1)(3)(5\times10^{-6})(0.3)^2}\right)+3.23\right]$$ $$\boxed{S\approx -6.4}$$
Average reservoir pressure. Extrapolating the late (total-system) line all the way to a Horner ratio of 1 ($\Delta t\to\infty$, i.e. an infinite shut-in) gives the classic $p^*$ extrapolated pressure, taken as the average reservoir pressure estimate (no Dietz shape-factor correction is available from the given data, so $p^*$ is used directly – check): $$\boxed{p^*\approx 2761\ \text{psia}}$$

Result	Value
Semilog slope (both lines), $m$	≈ 493 psi/cycle
Average fracture permeability, $\hat{k}_f$	≈ 118.8 mD
Storativity ratio, $\omega$	≈ 0.073
Skin factor, $S$	≈ −6.4
*Average reservoir pressure, $p^$**	≈ 2761 psia

Topic. Warren–Root double-porosity buildup analysis.

Key relations. $(kh)_f=162.6q\mu B_o/m$; $\omega=10^{-\Delta p/m}$; buildup skin $S=1.151[(p_{1hr}-p_{wf,0})/|m|-\log(k/\phi\mu c_tr_w^2)+3.23]$; $p^*$ from Horner-ratio-1 extrapolation.

Why this works. The fracture network has very high permeability but very little storage capacity, so it responds – and depletes toward its own local semilog trend – almost immediately; the matrix has the opposite properties (huge storage, negligible flow capacity) and only begins delivering fluid into the fractures once a large enough pressure differential has built up between the two systems, which is what produces the transition dip and, eventually, the parallel late-time line once fracture and matrix pressures re-equilibrate.

Common pitfall. Fitting the early and late lines independently (different slopes) instead of forcing a common slope – the Warren–Root model requires the two lines to be exactly parallel, and treating them as independent throws away that constraint, degrading the permeability and storativity estimates (an unconstrained fit here would put the two slopes anywhere from roughly parity to nearly double one another purely from chart-reading noise).

Source. Warren & Root (1963), SPE Journal; Bourdet, Well Test Analysis, ch.1 (double-porosity type curves and storativity ratio).

Question 4 of 7: Double-porosity buildup – storativity, fracture permeability, skin, average pressure

Question 4: Double-porosity buildup – storativity, fracture permeability, skin, average pressure (20 marks)