Sub-Problem 4 · Context
Your Task: Fetkovich Fit & ESP Sizing Check
FROM: Facilities Engineering · TO: Production Engineering
RE: KA-07 ESP Specification — Vogel AOF Sufficient Basis?
The ESP supplier has been given KA-07's Vogel AOFP of 2,232 stb/d (from SP-1) to size the pump. Before signing the equipment order, our reservoir engineer has asked: "Did you check with Fetkovich? The exponent n may not equal Vogel's implicit 0.77 — fit n directly from the tests, then compare the Fetkovich AOFP with Vogel at the same reservoir pressure to see whether Vogel is optimistic." Apply the Fetkovich two-point method to tests T1 and T2 to obtain n and C. Compute the Fetkovich AOFP and compare with Vogel. Report the ESP sizing implication.
SP-4 Data Slice
| Parameter | Value | Units | Source |
|---|---|---|---|
| Vogel AOFP (from SP-1) | 2,232 | stb/d | SP-1 composite IPR |
| p̄ (T1/T2 test state) | 3,600 | psia | Depleted bubble-point snapshot (p̄ = pb) |
| Test T1: q₁, pwf1 | 820 stb/d / 1,800 psia | stb/d / psia | 72-hr stabilised test |
| Test T2: q₂, pwf2 | 640 stb/d / 2,400 psia | stb/d / psia | 72-hr stabilised test |
Which reservoir pressure?
The deliverability tests T1 and T2 were acquired at the depleted bubble-point state, p̄ = pb = 3,600 psia — not the current composite p̄ = 5,100 psia used in SP-1. The Fetkovich two-point fit must be run at the state where the points were measured (3,600 psia); a single multi-rate test cannot sit at two reservoir pressures.Why NOT use Test T3?
Test T3 (q = 432 stb/d, pwf = 4,650 psia) is above the bubble point. Fetkovich's pressure-squared model applies to two-phase flow below pb. T3 must not be used — including it mixes single-phase and two-phase data, which invalidates the log-log slope calculation for n.Theory & Equations
Fetkovich Two-Parameter Deliverability
Fetkovich deliverability equation (Guo et al. Eq. 3.44)q_o = C × (p̄² − p_wf²)ⁿ
where:
C = deliverability coefficient [stb/d / psia²ⁿ]
n = flow exponent [dimensionless, 0.5 ≤ n ≤ 1.0]
n = 1.0 → Darcy (laminar) flow only
n = 0.5 → fully turbulent (non-Darcy) flow
Vogel assumes n ≈ 0.77 (average for two-phase systems)
Two-point method for n (log–log slope)n = log(q₁/q₂) / log[(p̄² − p_wf1²) / (p̄² − p_wf2²)]
Then: C = q₁ / (p̄² − p_wf1²)ⁿ (using T1 to calibrate C)
AOFP (at p_wf = 0):
q_AOFP = C × (p̄²)ⁿ = C × p̄^(2n)
Guided Tasks
Work Through in Sequence
- Compute (p̄² − pwf²) for tests T1 and T2. p̄² = 3,600² = 12,960,000 psia². T1: pwf1² = 1,800² = 3,240,000 → Δp²₁ = 9,720,000. T2: pwf2² = 2,400² = 5,760,000 → Δp²₂ = 7,200,000.
- Compute exponent n using the two-point log–log formula. n = log(q₁/q₂) / log(Δp²₁/Δp²₂). Use q₁ = 820, q₂ = 640, Δp²₁ and Δp²₂ from Task 1.
- Compute C using T1. C = q₁ / (Δp²₁)ⁿ.
- Compute Fetkovich AOFP. qAOF = C × (p̄²)ⁿ = C × 12,960,000ⁿ.
- Compare with Vogel AOFP from SP-1. Fetkovich AOFP vs. 2,232 stb/d. Express the discrepancy as a percentage. Explain the physical reason for the difference.
- Assess the ESP sizing implication. If the ESP was sized based on Vogel's 2,232 stb/d AOF, is the pump over-specified, under-specified, or correctly specified?
Worked Solution
Step-by-step solution — KA-07 Fetkovich Two-Point FitStep 1 — Pressure-squared differences (at the test state p̄ = p_b = 3,600 psia):
p̄² = 3,600² = 12,960,000 psia²
T1: Δp²₁ = 12,960,000 − 1,800² = 12,960,000 − 3,240,000 = 9,720,000 psia²
T2: Δp²₂ = 12,960,000 − 2,400² = 12,960,000 − 5,760,000 = 7,200,000 psia²
Step 2 — Exponent n:
n = log(q₁/q₂) / log(Δp²₁/Δp²₂)
= log(820/640) / log(9,720,000/7,200,000)
= log(1.28125) / log(1.35000)
= 0.10777 / 0.13033
= 0.826 (≈ 0.83)
n = 0.826 sits cleanly in the physical Fetkovich range
0.5 ≤ n ≤ 1.0 — slightly below Vogel's nominal 0.77 average,
indicating mild non-Darcy (turbulence) curvature at KA-07.
Step 3 — Coefficient C (calibrate on T1):
C = q₁ / (Δp²₁)ⁿ = 820 / (9,720,000)^0.826
(9,720,000)^0.826 ≈ 591,300
C = 820 / 591,300 = 1.39 × 10⁻³ stb/d/(psia²)^(2n)
Step 4 — Fetkovich AOFP (at p_wf = 0):
AOFP = C × (p̄²)ⁿ = 1.39×10⁻³ × (12,960,000)^0.826
(12,960,000)^0.826 ≈ 749,900
AOFP = 1.39×10⁻³ × 749,900 ≈ 1,040 stb/d
SUMMARY — Two-point Fetkovich results (p̄ = 3,600):
n = 0.826 (≈ 0.83, physically valid)
C = 1.39 × 10⁻³ stb/d/(psia²)^(2n)
AOFP = 1,040 stb/d
LIKE-FOR-LIKE COMPARISON (same depleted state, p̄ = 3,600):
Vogel qmax (from 820 @ 1,800, p̄ = 3,600) = 1,171 stb/d
r = pwf/p̄ = 1,800/3,600 = 0.5
qmax = 820 / (1 − 0.2(0.5) − 0.8(0.5)²) = 820 / 0.70 = 1,171
Fetkovich AOFP = 1,040 stb/d
Fetkovich (1,040) is ~11% BELOW Vogel (1,171) at the same
reservoir pressure. Vogel's fixed curvature (its implicit
n ≈ 0.77) is slightly more optimistic than the fitted n = 0.83;
Fetkovich is therefore the more conservative method here.
ROOT CAUSE OF THE GAP: pure model curvature, NOT skin.
Both numbers describe the SAME well at the SAME depleted state
(p̄ = 3,600). The ~11% difference is the intrinsic Vogel-vs-
Fetkovich curvature difference — Vogel assumes one fixed
exponent, Fetkovich fits the actual exponent from the tests.
(NOTE — different state: SP-1's 2,232 stb/d is the CURRENT-state
composite AOF at p̄ = 5,100 psia. That is a different reservoir
pressure, so it is NOT a like-for-like method comparison with
the Fetkovich fit above.)
ESP SIZING IMPLICATION:
Sizing the ESP on the Vogel qmax (1,171) rather than the
Fetkovich AOFP (1,040) modestly OVER-specifies the pump
(by ~11%) at this depleted state. Vogel is the optimistic
bound; Fetkovich is the conservative one. The prudent basis
is the lower (Fetkovich) deliverability — and, for the
stimulated well, the post-acid composite AOFP from SP-2/SP-5.
Key Results
Fetkovich n
0.83
two-point fit, valid 0.5–1.0
Fetkovich AOFP
~1,040
stb/d — p̄ = 3,600
Vogel qmax (same state)
1,171
stb/d — p̄ = 3,600
Method gap
−11%
Fetkovich below Vogel (curvature)
ESP Sizing Note
Compared like-for-like at the depleted state (p̄ = 3,600 psia), the Vogel qmax of 1,171 stb/d is ~11% higher than the Fetkovich AOFP of 1,040 stb/d — so a pump sized on the Vogel number is modestly over-specified. The gap is intrinsic model curvature (Vogel's fixed exponent vs the fitted n = 0.83), not skin. SP-1's current-state composite AOF of 2,232 stb/d is at a different reservoir pressure (p̄ = 5,100) and is not a like-for-like method comparison.Comparison Chart
Fetkovich vs. Vogel IPR — KA-07
Both curves are drawn at the same depleted state (p̄ = 3,600 psia). Blue: Vogel (qmax = 1,171). Orange: Fetkovich n = 0.83 (AOFP = 1,040). Green dots: tests T1 and T2. The modest gap between curves is intrinsic model curvature (~11%), not skin — Vogel is the optimistic bound, Fetkovich the conservative one.
Just-in-Time Resources
Need a Refresher? Pull These at Point of Need
Each links straight to the Module-04 asset that builds the method behind this sub-problem — open one only if you are stuck.
Watch
Lecture 4.5C — Fetkovich for Oil Wells: the Two-Point Method
Produced lecture
Self-check
../../courses/c01/production/m04/topic-4-5-fetkovich/file-pack/fetkovich.py
verified calculator — reproduce your numbers
Knowledge Check
SP-4 · 5 Questions
Fetkovich Deliverability — Knowledge Check
1. In the Fetkovich equation q = C(p̄² − pwf²)ⁿ, what does the exponent n represent and what are its theoretical limits?
Correct — C. n characterises the flow regime in the reservoir: n = 1.0 when all flow is laminar (Darcy), applicable at low rates or high-permeability wells; n = 0.5 when turbulence completely dominates near the wellbore. Most oil wells at production rates fall between these limits; Vogel's empirical simulation work found n ≈ 0.77 on average for solution-gas drive systems. A two-point Fetkovich fit using test data at two different rates determines the actual n for a specific well, which may differ from 0.77.
2. Compared like-for-like at the depleted state (p̄ = 3,600 psia), the Fetkovich AOFP for KA-07 is ~1,040 stb/d while the same-state Vogel qmax is ~1,171 stb/d. What is the primary reason Fetkovich sits ~11% below Vogel?
Correct — C. This is the key conceptual insight in SP-4. Both numbers describe KA-07 at the same depleted state (p̄ = pb = 3,600 psia), so the comparison is genuinely like-for-like. The ~11% gap is the intrinsic difference between Vogel's fixed-curvature model and the Fetkovich fit's actual exponent (n = 0.83). Vogel is the optimistic bound; Fetkovich the conservative one. (SP-1's 2,232 stb/d AOF is the CURRENT-state composite at p̄ = 5,100 — a different reservoir pressure, and therefore not a like-for-like method comparison.)
3. The two-point Fetkovich fit on T1 and T2 at the test state p̄ = 3,600 psia returns n = 0.83 and AOFP ≈ 1,040 stb/d. What do these values tell you about KA-07's flow regime?
Correct — B. Run at the correct test state (p̄ = pb = 3,600 psia), the points give Δp²₁ = 9.72 M and Δp²₂ = 7.20 M psia² (a ratio of 1.35) against q₁/q₂ = 1.28, so n = log(1.28)/log(1.35) = 0.826 — a clean, physical exponent. An earlier attempt that placed these points at the current composite p̄ = 5,100 produced n > 1 (impossible); that was an artefact of using the wrong reservoir pressure, not a real flow-regime result. With n = 0.83, C = 1.39×10⁻³ and AOFP = C × (3,600²)^0.826 ≈ 1,040 stb/d — no forcing of the exponent is required.
4. At the depleted state, Vogel qmax = 1,171 stb/d and Fetkovich AOFP = 1,040 stb/d. If the ESP is sized on the Vogel number rather than Fetkovich, what is the implication?
Correct — C. Compared like-for-like at p̄ = 3,600 psia, the Vogel qmax (1,171) is ~11% above the Fetkovich AOFP (1,040), so a Vogel-based ESP spec is modestly over-sized — not the gross 2.5× mismatch that would arise from comparing against a different reservoir state. The gap is pure model curvature (Vogel's fixed exponent vs the fitted n = 0.83), not skin. Best practice: size on the conservative Fetkovich deliverability, confirm against the pump's minimum-flow cavitation limit, and for the stimulated well use the post-acid composite AOFP from SP-2/SP-5.
5. After the acid job (SP-2, S = +1), Jactual,post = 0.630 stb/d/psi. What is the post-acid deliverability relative to the pre-acid Fetkovich AOFP (~1,040 stb/d)?
Correct — A: ~1,700 stb/d. Post-acid Jactual = 0.630 stb/d/psi. Recalculating with FE-corrected q values: at BHFP = 2,000 psia post-acid q ≈ 1,459 stb/d (from SP-2). Using this to anchor a post-acid deliverability estimate gives ≈ 1,700 stb/d — well above the pre-acid depleted-state Fetkovich AOFP of ~1,040 stb/d. The ESP should be designed for the post-acid well performance (see also the post-acid composite AOFP carried forward in SP-5), confirming the recommendation to acid before installing the ESP.
SP-4 Output
Record Before Moving to SP-5
SP-4 Key Outputs — KA-07 Fetkovich Analysis
Fetkovich exponent n (two-point, T1 & T2, at p̄ = pb = 3,600): n = 0.826 (≈ 0.83 — physical, 0.5–1.0)
Fetkovich C: 1.39 × 10⁻³ stb/d/(psia²)^(2n)
Fetkovich AOFP (depleted state, p̄ = 3,600): ~1,040 stb/d
Vogel qmax (same depleted state, from 820 @ 1,800): 1,171 stb/d
Method gap (like-for-like, p̄ = 3,600): Fetkovich ~11% below Vogel — intrinsic model curvature, not skin
Note — different state: SP-1's 2,232 stb/d is the current-state composite AOF (p̄ = 5,100), not a like-for-like comparison
ESP sizing recommendation: Size on conservative Fetkovich deliverability; use post-acid composite AOFP (SP-2/SP-5) for the stimulated well
→ Carry Fetkovich vs. Vogel comparison (depleted state) and ESP sizing basis to SP-6 integrated report.