Given. Reservoir/wellbore geometry and two stabilized tests, with no independent $k,h,\mu$ or pre-frac skin measurement.
Average reservoir pressure, $\bar P_R$
4000 psig
Wellbore radius, $r_w$
0.3 ft
Drainage radius, $r_e$
1500 ft
Test 1
$P_{wf}=3400$ psig, $q_o=1000$ STB/day
Test 2
$P_{wf}=2500$ psig, $q_o=2330$ STB/day
Post-frac skin, $S'$
$-1.3$
Find. (a) $FE$; (b) the ideal ($FE=1$) reference AOF; (c) $FE'$ and FOI after the frac job; (d) the before/after IPR curves.
Approach. With no direct $k,h,\mu$ or skin available, use Couto's method (Golan & Whitson §2.5): find the single $FE$ for which Standing's FE-corrected Vogel equation projects the same ideal AOF, $(q_o)_{max,FE=1}$, from both test points (bisection). Then convert the post-frac skin to $FE'$ through $r_e/r_w$, and re-apply Standing's equation with the same reference AOF.
Standing's FE-corrected Vogel equation. $\dfrac{q_o}{(q_o)_{max,FE=1}}=FE(1-R)\left[1.8-0.8\,FE(1-R)\right]$, $R=P_{wf}/\bar P_R$. Each test point implies $(q_o)_{max,FE=1}=q_o/\text{ratio}(FE,R)$ for any trial $FE$; Couto's condition is that both tests give the same value.
Bisect on FE. $R_1=3400/4000=0.850$, $R_2=2500/4000=0.625$. Solving $q_1/\text{ratio}(FE,R_1)=q_2/\text{ratio}(FE,R_2)$ numerically gives $\boxed{FE=0.651}$ — a moderately damaged completion.
(b) Ideal ($FE=1$) reference AOF. Substituting $FE=0.651$ back into either test point: $(q_o)_{max,FE=1}=1000/\text{ratio}(0.651,0.850)$. $\boxed{(q_o)_{max,FE=1}=5952\ \text{STB/day}}$.
(c) Flow efficiency after the frac job. $X=\ln(0.472\,r_e/r_w)=\ln(0.472\times1500/0.3)=\ln(2360)=7.766$. $FE'=\dfrac{X}{X+S'}=\dfrac{7.766}{7.766-1.3}$. $\boxed{FE'=1.201}$ — greater than 1, i.e. the completion is now better than an undamaged (zero-skin) well.
Fold of increase. $\text{FOI}=FE'/FE=1.201/0.651$. $\boxed{\text{FOI}=1.85}$. Since $\text{FOI}>1$ (indeed $FE'>1$, a negative-skin, stimulated completion), the job is judged a successful fracturing treatment.
Fig. 2 — Standing FE-corrected IPR before ($FE=0.651$, red) and after ($FE'=1.201$, blue) the frac job, both referenced to the same ideal ($FE=1$) AOF of 5952 STB/day; the two stabilized test points are marked on the pre-frac curve.
