Find. (a) Flow efficiency FE; (b) AOF at FE=1; (c) FE and Folds of Increase (FOI) after the frac; (d) before/after IPR curves.
Check: no permeability, net pay, viscosity or $B_o$ are given for this well, so FE cannot be built from Darcy's equation and a directly-measured skin. Instead this uses Couto's two-point method (Golan & Whitson §2.5), which solves for the single FE that makes Standing's flow-efficiency-corrected Vogel equation return the same $(q_o)_{max,FE=1}$ from both stabilized tests — it needs only $\bar P_R$ and the two ($P_{wf}$, $q_o$) pairs. $r_w$ and $r_e$ are then used separately in part (c), where the post-frac skin is converted to FE via the Dietz shape-factor form.
(a) Flow efficiency before the frac
Approach. Standing's FE-corrected Vogel equation gives, for any assumed FE, the ratio $q_o/(q_o)_{max,FE=1}=\text{FE}(1-R)\left[1.8-0.8\,\text{FE}(1-R)\right]$ where $R=P_{wf}/\bar P_R$. The correct FE is the one value for which both tests back out the same $(q_o)_{max,FE=1}$.
Set up the two ratios. $R_1=3400/4000=0.850$, $R_2=2500/4000=0.625$. Requiring $q_{o1}/\text{ratio}(FE,R_1)=q_{o2}/\text{ratio}(FE,R_2)$, i.e. $q_{o1}\cdot\text{ratio}(FE,R_2)=q_{o2}\cdot\text{ratio}(FE,R_1)$, is a single equation in FE.
Solve numerically (bisection). $\boxed{\text{FE}=0.651}$. Checking: at FE=0.651, ratio$(FE,R_1)=0.1680$ giving $(q_o)_{max,FE=1}=1000/0.1680=5952$ STB/day; ratio$(FE,R_2)=0.3915$ giving $(q_o)_{max,FE=1}=2330/0.3915=5952$ STB/day — both tests agree, confirming the solution.
Implied skin (context for part c). Using $X=\ln(0.472\,r_e/r_w)=\ln(0.472\times1500/0.3)=7.766$ and $\text{FE}=X/(X+S)$: $S_{\text{before}}=X(1-\text{FE})/\text{FE}=7.766\times(1-0.651)/0.651=\boxed{S_{\text{before}}\approx4.17}$ (positive $\Rightarrow$ a genuinely damaged well, consistent with FE $\lt 1$).
(b) AOF at FE = 1
Approach. $(q_o)_{max,FE=1}$ was already recovered identically from both tests in step 2 above.
Undamaged AOF. $\boxed{(q_o)_{max,FE=1}=5952\ \text{STB/day}}$ — this is the rate the well would produce at $P_{wf}=0$ if the near-wellbore damage were completely removed (FE raised to 1.0). The actual AOF today, still carrying FE=0.651, is lower: evaluating the ratio at $R=0$ gives $q_o/(q_o)_{max,FE=1}=0.832$, so actual current AOF $=0.832\times5952=4954$ STB/day.
(c) Flow efficiency, FOI, and success of the frac job
Approach. Convert the transient-test skin $S'=-1.3$ to FE via the same Dietz-shape-factor relation used in step 3 of part (a), then compare productivity index ratios (FOI) before and after.
Post-frac flow efficiency. $\text{FE}_{\text{after}}=\dfrac{X}{X+S'}=\dfrac{7.766}{7.766+(-1.3)}=\dfrac{7.766}{6.466}=\boxed{\text{FE}_{\text{after}}=1.201}$ — FE $>1$ confirms the completion is now better than an ideal, undamaged open hole.
Folds of Increase. FOI compares productivity index at the same drawdown, before vs. after: $\text{FOI}=J_{\text{after}}/J_{\text{before}}=(X+S_{\text{before}})/(X+S_{\text{after}})=\text{FE}_{\text{after}}/\text{FE}_{\text{before}}=1.201/0.651=\boxed{\text{FOI}=1.85}$.
Assessment. $\text{FOI}=1.85>1$: at the same drawdown the well now delivers 85% more oil than before the frac, and FE moved from 0.651 (damaged) to 1.201 (better than an ideal open hole) — $\boxed{\text{yes, this is a clearly successful fracturing job}}$, both removing the pre-existing damage and adding a stimulated, higher-conductivity flow path near the wellbore.
(d) IPR curves before and after the frac
Approach. Apply Standing's FE-corrected Vogel equation, $q_o=(q_o)_{max,FE=1}\cdot\text{FE}(1-R)[1.8-0.8\,\text{FE}(1-R)]$, across $P_{wf}=0$ to $\bar P_R$ using the same $(q_o)_{max,FE=1}=5952$ STB/day reference for both curves, with $\text{FE}_{\text{before}}=0.651$ and $\text{FE}_{\text{after}}=1.201$.
Fig. 2 — IPR before (FE=0.651, damaged) and after (FE=1.201, stimulated) the hydraulic fracturing job, both referenced to the same undamaged AOF of 5952 STB/day.
