Wellbore storage – end of storage and dimensionless coefficient: 24-Pet-B5 Reservoir Mechanics, December 2015, Q7

Question 7: Wellbore storage – end of storage and dimensionless coefficient (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. The 8-point buildup table above, with $q$, $p_{wf}(\Delta t=0)$, $h$, $\phi$, $r_w$, $B_o$, $c_t$ as listed.

Find. The approximate time at which wellbore-storage distortion ends, and the dimensionless wellbore storage coefficient $C_D$.

Approach. Plot $\Delta p=p_{ws}-p_{wf}(0)$ against $\Delta t$ on log-log axes; a $45^\circ$ unit-slope trend identifies the wellbore-storage-dominated early period, from which $C$ is read directly, and the point where the data departs from that unit slope marks the approximate end of storage.

Dashed red line: unit-slope ($45^\circ$) reference through the first two points. The data tracks it closely through $\Delta t\approx0.006$ hr, then progressively departs – wellbore storage is essentially over by $\Delta t\approx0.05$ hr.

Confirm the unit-slope region. The log-log slope between the first two points is $0.99$ (essentially unit slope); it drops to $0.97$ by the third point and $0.91$ by the fourth – the departure crosses roughly 10% by $\Delta t\approx0.0125$ hr, so $$\boxed{\text{end of wellbore storage}\approx0.01\text{-}0.05\text{ hr}}$$ (the derivative-style slope check is materially off unity by 0.05 hr, where it has fallen to 0.74).
Wellbore storage coefficient from the unit-slope segment. On the unit-slope line, $\Delta p/\Delta t$ is constant; averaging the two cleanest unit-slope points: $$C=\frac{qB_o}{24}\left(\frac{\Delta t}{\Delta p}\right)_{\text{USL}}=\frac{1000(1.2)}{24}\times\left(\frac{1}{51{,}976}\right)$$ $$\boxed{C\approx 9.6\times10^{-4}\text{ bbl/psi}}$$
Dimensionless wellbore storage coefficient. Using this exam's own formula-sheet constant: $$C_D=\frac{0.8939\,C}{\phi c_th r_w^2}=\frac{0.8939(9.6\times10^{-4})}{(0.20)(7\times10^{-6})(70)(0.25)^2}$$ $$\boxed{C_D\approx 140}$$

Quantity	Result
Approximate end of wellbore storage	0.01 – 0.05 hr
Wellbore storage coefficient, $C$	$9.6\times10^{-4}$ bbl/psi
Dimensionless storage coefficient, $C_D$	≈ 140

Topic. Log-log wellbore-storage diagnostic and the dimensionless storage coefficient.

Key relations. Unit-slope region: $\Delta p\propto\Delta t$; $C=qB_o\Delta t/(24\Delta p)|_{\text{USL}}$; $C_D=0.8939C/(\phi c_th r_w^2)$.

Why this works. During pure wellbore-storage-dominated flow, essentially all produced/injected fluid comes from (or goes into) the wellbore itself rather than the formation, so $\Delta p$ is directly proportional to the cumulative volume $q\Delta t$ – a straight $45^\circ$ line on log-log axes is the direct fingerprint of that constant-proportionality regime.

Common pitfall. Reading $C$ from a point that is already past the unit-slope region (e.g. the third or fourth data point here) – the ratio $\Delta t/\Delta p$ is no longer constant there, silently biasing $C$ (and therefore $C_D$) low.

Source. Bourdet, Well Test Analysis, ch.1–2 (log-log diagnostic, wellbore storage); Lee, Well Testing, ch.1 (wellbore storage coefficient).

Question 7 of 7: Wellbore storage – end of storage and dimensionless coefficient

Question 7: Wellbore storage – end of storage and dimensionless coefficient (20 marks)