Interference – pressure at Well A from Well B's two-rate production: 24-Pet-B5 Reservoir Mechanics, December 2014, Q7

Question 7: Interference – pressure at Well A from Well B's two-rate production (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
Total compressibility	$c_t$	$1\times10^{-6}\ \text{psi}^{-1}$
Oil viscosity	$\mu$	1.1 cp
Oil formation volume factor	$B_o$	1.2 bbl/STB
Reservoir permeability	$k$	100 mD
Formation thickness	$h$	25 ft
Initial reservoir pressure	$p_i$	2000 psia
Formation porosity	$\phi$	0.25
Well A–B spacing	$r$	1000 ft
Well B rate, days 0–2	$q_1$	300 STB/D
Well B rate, days 2–3	$q_2$	500 STB/D

Fig. 6 – Well layout and Well B's two-rate production history.

Find. Pressure at the bottom of (observation) Well A, $r=1000$ ft from Well B, at $t=3$ days after Well B started producing.

Approach. Superpose the interference response of the two rate increments in time, each evaluated with the exact line-source ($Ei$) solution – since a quick check of $t_D$ at $r=1000$ ft shows it stays below 100 for both increments, so the log-approximation on the formula sheet does not apply and the $p_D$ chart/$Ei$ formula must be used instead.

Check which $p_D$ form applies. $\eta=0.0002637k/(\phi\mu c_t)=0.0002637(100)/[(0.25)(1.1)(1\times10^{-6})]\approx95{,}891\ \text{ft}^2/\text{hr}$. At $r=1000$ ft and $t=72$ hr (3 days), $t_{D1}=\eta t/r^2\approx6.9$; at $t=24$ hr (1 day since the rate change), $t_{D2}\approx2.3$. Both are $<100$, so the log approximation $p_D=0.5(\ln t_D+0.809)$ is NOT valid here – the exact $p_D=0.5[-Ei(-1/4t_D)]$ must be read from the formula sheet's $p_D$ chart (equivalently evaluated from the $Ei$-function directly).
Superposition in time (two rate increments). Treat Well B's history as a rate of $q_1=300$ STB/D starting at $t=0$ (running for the full 3 days = 72 hr by the time of interest) PLUS an incremental rate of $q_2-q_1=200$ STB/D starting at the 2-day mark (running for 1 day = 24 hr by the time of interest): $$\Delta p=\frac{q_1\mu B_o}{0.00708kh}p_D(t_{D1})+\frac{(q_2-q_1)\mu B_o}{0.00708kh}p_D(t_{D2})$$
Evaluate the two $p_D$ terms. $p_D(t_{D1}=6.9)\approx1.389$ and $p_D(t_{D2}=2.3)\approx0.874$ (read from the formula sheet's $p_D$-vs-$t_D$ chart / the exact $Ei$ formula). The shared coefficient is $$\frac{\mu B_o}{0.00708kh}=\frac{(1.1)(1.2)}{0.00708(100)(25)}\approx0.0746\ \text{psi per (STB/D)}$$
Combine and solve for $p_A$. $$\Delta p=(300)(0.0746)(1.389)+(200)(0.0746)(0.874)\approx31.1+13.0$$ $$\boxed{\Delta p\approx 44.1\ \text{psi}}$$ $$p_A=p_i-\Delta p=2000-44.1$$ $$\boxed{p_A(3\ \text{days})\approx 1956\ \text{psia}}$$

Result	Value
$t_{D1}$ (72 hr), $t_{D2}$ (24 hr)	6.90, 2.30
$p_D(t_{D1})$, $p_D(t_{D2})$	1.389, 0.874
Total interference pressure drop, $\Delta p$	≈ 44.1 psi
Pressure at Well A, $p_A$(3 days)	≈ 1956 psia

Topic. Interference testing and superposition in time.

Key relations. $\eta=0.0002637k/(\phi\mu c_t)$; $t_D=\eta t/r^2$; $p_D=0.5[-Ei(-1/4t_D)]$ (exact) or $0.5(\ln t_D+0.809)$ only if $t_D>100$; superposition sums each rate increment's own $p_D$ evaluated at its own elapsed time since that increment began.

Why this works. The diffusivity equation is linear, so the pressure response to an arbitrary rate history is exactly the sum of the responses to a sequence of constant-rate step changes, each one turned "on" at its own start time and left running from then on – this is the same superposition-in-time principle already applied to a rate-change well test in this discipline, just applied here between two separate wells (an interference test, Q1-d) rather than at a single well.

Common pitfall. Using the log approximation $p_D=0.5(\ln t_D+0.809)$ without first checking $t_D>100$ – at this well spacing (1000 ft) and these times, $t_D$ is only 2–7, so the log form would be badly wrong (it is only asymptotically valid at large $t_D$); the exact $Ei$-based $p_D$ must be used instead, exactly as this paper's own formula sheet notes.

Source. Earlougher, Advances in Well Test Analysis, ch.1 (superposition), ch.5 (interference tests); Lee, Well Testing, ch.2 (line-source/$Ei$ solution).

Question 7 of 7: Interference – pressure at Well A from Well B's two-rate production

Question 7: Interference – pressure at Well A from Well B's two-rate production (20 marks)