Question 2 of 5: Vogel IPR and Standing's Flow-Efficiency Method — Hydraulic-Fracturing Evaluation

Question 2: Vogel IPR and Standing's Flow-Efficiency Method — Hydraulic-Fracturing Evaluation (25 points)

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.

(a) Current-condition Vogel IPR

Given. The reservoir is defined by its geometry, skin, and a single stabilized test.

Find. The current Vogel IPR (rate vs. $P_{wf}$), including its AOF.

Approach. Because the whole system is below $P_b$, use Vogel's dimensionless IPR directly with the single test point to solve for $(q_o)_{max}$.

1. Vogel ratio at the test point. $\dfrac{P_{wf}}{\bar P_R}=\dfrac{3000}{4000}=0.75$. $1-0.2(0.75)-0.8(0.75)^2=1-0.15-0.45=0.400$.
2. Solve for the current AOF. $(q_o)_{max}=\dfrac{q_o}{0.400}=\dfrac{1615}{0.400}$. $\boxed{(q_o)_{max}=4038\ \text{STB/day (current)}}$.

(b) Post-fracturing evaluation — Standing's flow-efficiency method

Given. Two post-frac stabilized tests, plus the pre-frac skin $S'=4.18$ already used in part (a).

Find. Whether the fracturing job improved the well's flow efficiency (FE), and hence whether it was successful.

Approach. Standing's method compares a well's actual IPR to the IPR it would have with zero skin ($FE=1$) via $q_o/(q_o)_{max,FE=1}=FE(1-R)\left[1.8-0.8\,FE(1-R)\right]$, $R=P_{wf}/\bar P_R$. First get the pre-frac $FE$ from the known skin; then, since two post-frac tests are available, solve simultaneously for the post-frac $FE$ and its own ideal AOF (Couto's two-point method) — no skin measurement is needed for that half.

1. Pre-frac flow efficiency from skin. $FE_{before}=\dfrac{\ln(r_e/r_w)-0.75}{\ln(r_e/r_w)-0.75+S'}$, with $\ln(2000/0.4)=\ln(5000)=8.517$: $FE_{before}=\dfrac{8.517-0.75}{8.517-0.75+4.18}=\dfrac{7.767}{11.947}$. $\boxed{FE_{before}=0.650}$.
2. Post-frac $FE$ (Couto's method). Find the single $FE_{after}$ for which both post-frac tests project the same ideal ($FE=1$) AOF through $q_o/(q_o)_{max,FE=1}=FE(1-R)[1.8-0.8FE(1-R)]$. With $R_1=3430/4000=0.8575$ and $R_2=2500/4000=0.625$, bisecting on $FE$ until $\dfrac{q_{o1}}{\text{ratio}(FE,R_1)}=\dfrac{q_{o2}}{\text{ratio}(FE,R_2)}$ gives $\boxed{FE_{after}=1.303}$, with both tests projecting the same ideal AOF of 3588 STB/day (test 1: 3587.8; test 2: 3587.8 — agreement to 4 significant figures confirms the fit).
3. Compare. $FE_{after}/FE_{before}=1.303/0.650=2.00$ — the fracturing job roughly doubled the well's flow efficiency, and $FE_{after}\gt1$ means the completion is now flowing better than a theoretical zero-skin (undamaged) well would. $\boxed{\text{Job successful: } FE\ 0.650\rightarrow1.303}$.

The two independently-derived $FE=1$ reference AOFs (5950 STB/day implied by the pre-frac test combined with its skin-derived $FE$, vs. 3588 STB/day implied by the two post-frac tests alone) do not have to agree exactly — each is a separate application of the same empirical correlation to different data, and some scatter between them is normal. The comparison that actually answers "was the job successful" is the flow-efficiency ratio itself, which is unambiguous: near-zero skin damage before the job (FE well below 1) became a mild negative-skin, stimulated completion afterward (FE above 1).