Two-rate flow test – initial reservoir pressure by superposition: 24-Pet-B5 Reservoir Mechanics, May 2015, Q7

Question 7: Two-rate flow test – initial reservoir pressure by superposition (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
First rate, duration	$q_1,\ t_1$	25 STBD, 24 hr
Second rate, duration	$q_2,\ \Delta t_2$	50 STBD, 72 hr (to $t=96$ hr total)
Wellbore pressure at $t=96$ hr	$p_{wf}$	1000 psia
Formation permeability	$k$	162.6 mD
Skin factor	$S$	2
Formation thickness	$h$	50 ft
Oil formation volume factor	$B_o$	1.143 bbl/STB
Porosity	$\phi$	0.082
Total compressibility	$c_t$	$10.5\times10^{-6}\ \text{psi}^{-1}$
Oil viscosity	$\mu_o$	1.278 cp
Wellbore radius	$r_w$	0.45 ft

Find. Initial reservoir pressure $p_i$.

Approach. Since $k$ and $S$ are already given, superposition in time directly gives $\Delta p$ from $p_i$ to the observed $p_{wf}$ at $t=96$ hr: split the rate history into two increments (0→$q_1$ at $t=0$, then $q_1\rightarrow q_2$ at $t=24$ hr) and sum each increment's own $p_D$ contribution, evaluated at its own elapsed time since that increment began. The supplied $p_D$-vs-$t_D$ chart is the exact-$Ei$ curve; checking $t_D$ first shows both increments are comfortably in the $t_D\gt 100$ log-approximation range, so the chart itself is not needed for a numeric reading here (Check).

Fig. 7 – Rate history: 25 STBD for the first 24 hr, then 50 STBD for a further 72 hr, with p_wf=1000 psia recorded at t=96 hr.

Dimensionless times. Using $t_D=0.0002637kt/(\phi\mu c_tr_w^2)$ with the given $k=162.6$ mD, evaluated at each increment's own elapsed time since it began (96 hr for the $q_1$ term, 72 hr for the $q_2-q_1$ term): $$t_{D}(96\text{ hr})\approx1.85\times10^{7},\qquad t_{D}(72\text{ hr})\approx1.39\times10^{7}$$ both far above 100, so the log approximation applies to both terms.
Superposition in time. With rate increments $\Delta q_1=q_1=25$ (from $t=0$) and $\Delta q_2=q_2-q_1=25$ (from $t=24$ hr), and skin $S=2$ carried in each term (the telescoping sum $\Delta q_1+\Delta q_2=q_2$ automatically applies the skin once, at the current total rate), $$\Delta p=\frac{141.2\mu B_o}{kh}\Big[\Delta q_1\big(0.5[\ln t_{D,1}+0.809]+S\big)+\Delta q_2\big(0.5[\ln t_{D,2}+0.809]+S\big)\Big]$$ $$\Delta p=\frac{141.2(1.278)(1.143)}{(162.6)(50)}\Big[25(0.5[\ln(1.85\times10^{7})+0.809]+2)+25(0.5[\ln(1.39\times10^{7})+0.809]+2)\Big]$$ $$\boxed{\Delta p\approx 13.6\ \text{psi}}$$
Initial reservoir pressure. $$p_i=p_{wf}+\Delta p=1000+13.6$$ $$\boxed{p_i\approx 1014\ \text{psia}}$$

Result	Value
$t_D$ (96 hr term)	≈ 1.85×10⁷
$t_D$ (72 hr term)	≈ 1.39×10⁷
Total pressure drop, $\Delta p$	≈ 13.6 psi
Initial reservoir pressure, $p_i$	≈ 1014 psia

Topic. Superposition in time for a multi-rate flow test.

Key relations. $t_D=0.0002637kt/(\phi\mu c_tr_w^2)$; $\Delta p=141.2\mu B_o/(kh)\sum\Delta q_j\,p_D(t_{D,elapsed\ since\ j})$, with $p_D=0.5[\ln t_D+0.809]+S$.

Why this works. The diffusivity equation is linear, so the pressure response to any rate HISTORY is the superposed sum of the responses to each individual rate CHANGE, each one evaluated at its own elapsed time since it happened – not at the total test time. Because the rate increments always telescope back to the current total rate ($\sum\Delta q_j=q_{now}$), a skin term carried additively inside each increment's own $p_D$ sums correctly to the physically correct single skin-driven pressure drop at the well's current rate.

Common pitfall. Using the TOTAL elapsed time (96 hr) for BOTH rate increments, rather than each increment's own elapsed time since it started – the second rate change has only been in effect for 72 hr, not 96, and using the wrong time systematically biases the extrapolated $p_i$.

Source. Earlougher, Advances in Well Test Analysis, ch.2 (superposition in time, multi-rate tests); Lee, Well Testing, ch.4.

Question 7 of 7: Two-rate flow test – initial reservoir pressure by superposition

Question 7: Two-rate flow test – initial reservoir pressure by superposition (20 marks)