Given. A damaged well ($S'=8$) with a single pre-frac stabilized test, then a single post-frac test at higher rate/water cut; both under the same current reservoir pressure.
Bubble-point / average reservoir pressure
4600 / 4500 psig
Wellbore radius, $r_w$ / drainage radius, $r_e$
0.5 ft / 3000 ft
Pre-frac skin, $S'$
8
Pre-frac test
$P_{wf}=4000$ psig, $q_o=500$ STB/D ($f_w=0$)
Post-frac test
$P_{wf}=3662$ psig, $Q_L=2000$ STBL/D, $f_w=0.25$
Find. (a) $q_o$ at $P_{wf}=2000$ psig under current (pre-frac) conditions; (b) anticipated $q_o$ at $P_{wf}=2000$ psig after the frac job, and whether the job succeeded.
Check: because $P_b(4600)>\bar P_R(4500)$, the reservoir is already at/below its bubble point — free gas exists throughout, so Vogel's equation (not a straight line) governs from the current reservoir pressure downward. Skin is converted to flow efficiency via the Hawkins/Golan&Whitson relation $X=\ln(0.472\,r_e/r_w)$, $FE=X/(X+S)$, then combined with Standing's FE-corrected Vogel equation to place both the damaged pre-frac test and the (much less damaged) post-frac test onto ONE common ideal ($FE=1$) reference curve.
Approach. (a) Fit Vogel's equation directly to the single pre-frac test point to get the well's as-is $q_{o,max}$, then read $q_o$ at $P_{wf}=2000$ psig. (b) Convert the given skin to $FE_{before}$, use it with the pre-frac test to find the ideal ($FE=1$) $q_{o,max}$ benchmark, then bisect for the $FE_{after}$ that reproduces the post-frac test against that SAME ideal benchmark; use $FE_{after}$ to project the post-frac rate at $P_{wf}=2000$ psig and compare against part (a).
Part (a) — as-is Vogel fit. $R=P_{wf}/\bar P_R=4000/4500=0.8889$. $q_{o,max}=\dfrac{q_{test}}{1-0.2R-0.8R^2}=\dfrac{500}{0.1901}$. $\boxed{q_{o,max}=2629.9\ \text{STB/day}}$.
Part (a) — rate at $P_{wf}=2000$ psig. $R=2000/4500=0.4444$: $q_o=2629.9\times(1-0.2(0.4444)-0.8(0.4444)^2)=2629.9\times0.7531$. $\boxed{q_o=1980.5\ \text{STB/day (pre-frac, current conditions)}}$.
Part (b) — skin to flow efficiency. $X=\ln(0.472\,r_e/r_w)=\ln(0.472\times3000/0.5)=\ln(2832)=7.949$. $FE_{before}=\dfrac{X}{X+S}=\dfrac{7.949}{15.949}$. $\boxed{FE_{before}=0.498}$ — the damaged well flows at roughly half its ideal rate.
Part (b) — ideal ($FE=1$) benchmark. Standing's FE-corrected Vogel, $\dfrac{q_o}{q_{o,max,FE=1}}=1.8y-0.8y^2$ with $y=FE(1-R)$: at the pre-frac test ($R=0.8889$, $y=0.498\times0.1111=0.0554$), $500=q_{o,max,FE=1}\times(1.8(0.0554)-0.8(0.0554)^2)=q_{o,max,FE=1}\times0.0972$. $\boxed{q_{o,max,FE=1}=5142.7\ \text{STB/day}}$.
Part (b) — post-frac flow efficiency. Post-frac oil rate $q_o=Q_L(1-f_w)=2000\times0.75=1500$ STB/day at $R=3662/4500=0.8138$. Bisecting Standing's equation for the $FE$ that reproduces $q_o/q_{o,max,FE=1}=1500/5142.7=0.2916$ against the SAME ideal benchmark gives $\boxed{FE_{after}=0.944}$.
Part (b) — flow-efficiency ratio. $FOI=FE_{after}/FE_{before}=0.944/0.498$. $\boxed{FOI=1.894}$ — the frac job nearly doubled the well's flow efficiency (skin damage largely removed).
Part (b) — anticipated post-frac rate at $P_{wf}=2000$ psig. $R=0.4444$, $y=0.944\times0.5556=0.5242$: $q_o=5142.7\times(1.8(0.5242)-0.8(0.5242)^2)=5142.7\times0.7237$. $\boxed{q_o=3722.8\ \text{STB/day (post-frac)}}$.
Comparing the two curves at the same $P_{wf}=2000$ psig: the post-frac rate (3722.8 STB/day) is nearly double the pre-frac rate (1980.5 STB/day), and $FOI=1.894\gt1$ confirms the improvement is real damage removal rather than measurement noise. $\boxed{\text{Yes, the hydraulic fracturing job was successful}}$ — despite the well now producing 25% water, oil deliverability at any common drawdown increased substantially.
Fig. 2 — Vogel IPR curves before (FE=0.50) and after (FE=0.94) the hydraulic fracturing job, both referenced to the same ideal (FE=1) benchmark; the post-frac curve delivers nearly double the oil rate at the same 2000 psig drawdown.
