Module 04 is the point in the WELLPROD™ Programme where the Darcy linear PI from earlier modules is replaced by something more realistic. Most producing wells spend the majority of their lives with flowing bottomhole pressure below the bubble point, in the two-phase, non-linear inflow regime that Vogel first characterised in 1968. Module 04 takes that empirical equation and builds outward from it: composite models for reservoirs that straddle the bubble point, skin-corrected deliverability for damaged wells, pseudo-pressure analysis for gas producers, the Fetkovich power-law fit for multi-rate tests, and finally the life-of-field IPR family that frames every artificial-lift and infill-drilling decision.
Why the Karama Field is used throughout
KA-07 and KA-G2 appear in every topic for the same reason GK-22 anchors Module 01: cumulative familiarity lets the learner concentrate on the new concept rather than decoding new numbers each time. More importantly, the Karama scenario is genuinely consequential, skin S′ = +8 is destroying $52,800 per day of potential revenue; the Fetkovich AOFP is 59% below the Vogel ideal; KA-07 alone cannot sustain the 1,800 stb/d plateau beyond Stage 1 of depletion. These are not illustrative discrepancies. They are the size of errors that mis-specify an ESP, delay an infill well, or fail a facilities gate.
KA-07 sits at p̅ = 5,100 psia above its bubble point of 4,500 psia — placing it precisely in the composite IPR regime where a Darcy segment above pb joins a Vogel curve below it. KA-G2 is a gas-cap producer with a four-period modified isochronal test, requiring LIT analysis in pseudo-pressure m(p) rather than the simpler p² form. Together they exercise every equation in the module in a single internally consistent dataset.
Learning integrationHow the six topics build to the Karama PBL
Each topic maps 1:1 to one sub-problem in the Karama problem set. Complete the topic before attempting its sub-problem. The sub-problems are sequenced — SP-6 (FDP report) requires confirmed outputs from SP-1 through SP-5.
4.1 + 4.2 Vogel & Composite → SP-1: Current IPR
4.3 Standing’s FE → SP-2: Acid-Job Economics
4.4 Gas Deliverability → SP-3: KA-G2 AOFP
4.5 Fetkovich → SP-4: ESP Sizing Basis
4.6 Future IPR → SP-5: Life-of-Field Forecast
All topics → SP-6: Karama FDP Report
Your learning path
1
Topic 4.1 · ~60 min · → SP-1
Two-Phase Inflow & Vogel’s IPR
When BHFP drops below the bubble point, how does the inflow relationship change — and what replaces the linear PI?
Below the bubble point, gas liberates from solution near the wellbore, introducing a second phase that reduces effective permeability to oil. The result is a non-linear inflow curve: the same pressure drawdown produces progressively less oil as BHFP decreases. Topic 4.1 derives Vogel’s (1968) empirical IPR equation, shows its origin in solution-gas-drive simulation studies, and establishes AOFP — the absolute open-flow potential at zero flowing BHP — as the primary deliverability metric for two-phase oil wells.
Karama context: KA-07 stabilised test T1 gives q = 820 stb/d at pwf = 1,800 psia — well below pb = 4,500 psia. Topic 4.1 shows how this single test point, via Vogel’s normalisation, determines the shape of the entire two-phase IPR curve and yields an AOFP estimate. The Vogel-based AOFP feeds directly into SP-1 and becomes the reference against which SP-4’s Fetkovich result is compared.
Vogel (1968): q/qmax = 1 − 0.2(pwf/p̅) − 0.8(pwf/p̅)²
qmax = AOFP [at pwf = 0]
KA-07 (T1): qmax = 820 / [1 − 0.2(0.400) − 0.8(0.400)²] = 820 / 0.792 ≈ 1,035 stb/d (Vogel-only baseline)
Two-phase flow near wellboreVogel’s equationAOFP definitionSingle-point calibrationIPR curve shapeLinear vs non-linear PI
2
Topic 4.2 · ~60 min · → SP-1
Composite IPR — Darcy Above, Vogel Below
When p̅ is above the bubble point but the well draws BHFP below it, which equation governs — and how are the two segments joined without a discontinuity?
Vogel’s equation was derived for a reservoir already below its bubble point. When average reservoir pressure sits above pb, there is a single-phase (Darcy) region between p̅ and pb, and a two-phase (Vogel) region below pb. Topic 4.2 constructs the composite IPR that joins these two segments continuously at the bubble-point anchor rate qb, establishes the maximum Vogel increment qv,max, and shows how the AOFP is assembled from both contributions.
Karama context: KA-07 has p̅ = 5,100 psia and pb = 4,500 psia. The Darcy segment yields qb = J* × (p̅ − pb) = 0.72 × 600 = 432 stb/d. The Vogel increment qv,max = J* × pb/1.8 = 1,800 stb/d. Total ideal AOFP = 2,232 stb/d. This confirms KA-07 can theoretically reach the 1,800 stb/d plateau target — but only at BHFP ≈ 1,820 psia with an ESP.
qb = J* × (p̅ − pb) = 0.72 × 600 = 432 stb/d [Darcy anchor]
qv,max = J* × pb / 1.8 = 0.72 × 4,500 / 1.8 = 1,800 stb/d [Vogel max]
AOFP = qb + qv,max = 432 + 1,800 = 2,232 stb/d [composite ideal, S = 0]
Composite IPR derivationBubble-point anchor qbMaximum Vogel incrementIdeal vs actual PIAOFP from composite modelInflow model selection
3
Topic 4.3 · ~60 min · → SP-2
Standing’s Flow Efficiency & Skin-Corrected IPR
How much production is being lost to skin damage — and what is the acid job actually worth in dollars per day?
Flow Efficiency (FE) is the ratio of actual to ideal deliverability for the same pressure drawdown. For a well with skin S, Standing showed that FE = 7/(7+S) and that a modified Vogel equation, with an effective flowing pressure p′wf corrected for the skin pressure drop, gives the actual IPR from any skin state. Topic 4.3 develops this correction, shows how to build pre-acid and post-acid IPR curves from skin data alone, and provides the economic framework for stimulation decision-making.
Karama context: KA-07 skin S′ = +8 gives FEpre = 0.467 — the well delivers less than half its undamaged potential. After a matrix acid job to S = +1, FEpost = 0.875 and production at BHFP = 2,000 psia rises from 548 to 1,459 stb/d (Δq = +911 stb/d). At $58/stb netback the daily revenue uplift is $52,838 — the $450K acid job pays back in 8.5 days.
FE = 7 / (7 + S) [Standing approximation]
FEpre = 7 / (7 + 8) = 0.467 ⇒ Jactual,pre = 0.72 × 0.467 = 0.336 stb/d/psi
FEpost = 7 / (7 + 1) = 0.875 ⇒ Jactual,post = 0.72 × 0.875 = 0.630 stb/d/psi
Payback = $450,000 / (911 stb/d × $58/stb) = 8.5 days
Flow Efficiency definitionStanding’s FE approximationSkin-corrected IPRPre- and post-acid curvesStimulation economicsAcid-before-ESP sequencing
4
Topic 4.4 · ~75 min · → SP-3
Gas Well Deliverability & LIT Analysis
How is an AOFP derived from a modified isochronal test — and why does pseudo-pressure m(p) matter at high reservoir pressures?
Gas well deliverability requires a different framework from oil: viscosity μg and compressibility factor Z both vary with pressure, so the simple Darcy PI breaks down. Topic 4.4 develops the LIT (Laminar-Inertial-Turbulent) deliverability equation in pseudo-pressure form, shows how to extract the Darcy coefficient A and non-Darcy coefficient B from a multi-rate test, applies the stabilised extended-flow point to anchor the long-term deliverability curve, and solves the AOFP quadratic at a specified back-pressure.
Karama context: KA-G2 has been tested via a four-period modified isochronal test (MIF) at rates of 4,000 to 16,000 Mscf/d, plus a stabilised extended-flow point at 10,000 Mscf/d and pwf = 2,200 psia. LIT analysis in m(p) gives Astab = 66,690 and B = −0.709. AOFP at separator back-pressure 800 psia is ≈ 15,779 Mscf/d per well — 2.6× the 6,000 Mscf/d contract allocation, confirming the three-well gas cap can sustain the 18,000 Mscf/d contract at first gas without compression.
LIT: Δm(p) / q = A + B·q [plot Y vs q → slope = B, intercept = A]
Yext = [m(p̅) − m(pwf,ext)] / qext = (928.00 − 332.00)×10^6 / 10,000 = 59,600
AOFP: B·q² + Astab·q − [m(p̅) − m(pbp)] = 0 → q ≈ 15,779 Mscfd
Pseudo-pressure m(p)LIT deliverability equationIsochronal vs stabilised testsA & B coefficient fittingAOFP quadratic solutionGas contract assessment
5
Topic 4.5 · ~60 min · → SP-4
Fetkovich’s Deliverability Equation
When you have two stabilised test points at different rates, how do you fit the deliverability curve — and does the result agree with Vogel?
Fetkovich (1973) generalised the back-pressure deliverability equation to q = C(p̅² − pwf²)n, where n is a flow exponent (1.0 for pure Darcy, 0.5 for fully turbulent flow) and C is a deliverability coefficient. Topic 4.5 derives the two-point log–log method for fitting n and C from any two stabilised test points, explains why n physically cannot exceed 1.0, and frames the critical comparison with Vogel’s result: when the two methods disagree, the root cause is almost always skin.
Karama context: The two-point Fetkovich fit on KA-07 tests T1 and T2 yields n → 1.0 (Darcy-limited, confirming the composite well) and AOFP ≈ 905 stb/d — 59% below the Vogel ideal of 2,232 stb/d. The discrepancy is not a method error: Vogel was calibrated on the ideal S = 0 PI, while Fetkovich was fitted to the actual damaged well (S′ = +8). An ESP sized on the Vogel ideal would be 2.5× over-specified, causing cavitation. The correct ESP design basis is the post-acid Fetkovich AOFP of ≈ 1,700 stb/d.
Fetkovich: q = C × (p̅² − pwf²)n
n = log(q1/q2) / log[(Δp²1) / (Δp²2)]
KA-07 (T1, T2): n ≈ 1.0 (Darcy-dominated) ⇒ AOFP ≈ 905 stb/d [actual, S′ = +8]
vs Vogel ideal AOFP = 2,232 stb/d ⇒ −59% gap = skin S′ = +8
Fetkovich power-law equationFlow exponent nTwo-point log–log methodn physical limits (0.5–1.0)Fetkovich vs Vogel comparisonESP sizing basis
6
Topic 4.6 · ~75 min · → SP-5
Predicting Future IPRs
As the reservoir depletes and PVT properties degrade, how does the IPR evolve — and when does the artificial-lift system need to be upgraded?
The IPR is not fixed: as reservoir pressure falls, kro declines (rising gas saturation), μo increases (oil becomes heavier), and Bo decreases. Standing’s J* scaling method uses the mobility ratio [kro/(μoBo)] to track J* through depletion using only PVT data. Fetkovich’s depletion method provides a simpler pressure-ratio scaling for a P10 downside check. Together they generate the life-of-field IPR family that underlies every artificial-lift decision.
Karama context: applying Standing’s mobility scaling to the PVT table, KA-07’s post-acid J* declines from 0.630 stb/d/psi at p̅ = 5,100 psia to 0.464 at Stage 2 (p̅ = 3,200 psia). At Stage 2, even with the ESP at maximum drawdown (BHFP = 800 psia), the achievable rate is only 743 stb/d — less than half the 1,800 stb/d plateau target. KA-07 alone cannot sustain plateau beyond current conditions; the FDP must incorporate either pressure maintenance or an infill well before Stage 2 onset.
J*future = J*present × [kro/(μoBo)]future / [kro/(μoBo)]present
Stage 2 (p̅ = 3,200): J* = 0.630 × 0.737 = 0.464 stb/d/psi → AOFP = 825 stb/d
Max q at BHFP = 800 psia: 743 stb/d ⇒ −1,057 stb/d below 1,800 target
⇒ Infill well required before p̅ reaches 3,200 psia
Standing’s J* mobility scalingPVT mobility ratioLife-of-field IPR familyFetkovich depletion methodLift-escalation scheduleInfill-well trigger design
★
Capstone · ~5.5 hrs · → SP-6
Integration & the Karama FDP recommendation
Once all six topics are complete, launch the Karama PBL hub. Six sequenced sub-problems take you from composite IPR construction through to a graded Field Development Plan report, assembling every result from Topics 4.1–4.6 into an integrated well performance study. SP-6 produces six specific FDP recommendations covering stimulation sequencing, ESP specification, gas compression timing, infill-well trigger, pressure maintenance evaluation, and facilities sizing — submitted as a leadership forum deliverable.
Module learning outcomes
By the end of Module 04 you will be able to…
- Construct Vogel’s two-phase IPR from a single stabilised test point; read AOFP and rate at any flowing BHP; and justify when Vogel applies versus when a composite or Darcy-only model is required.
- Build a composite (Darcy + Vogel) IPR for a well where p̅ > pb, correctly computing the bubble-point anchor rate qb and maximum Vogel increment qv,max, and confirm continuity at the join point.
- Calculate Flow Efficiency from skin using Standing’s approximation; apply the FE-corrected modified Vogel to generate pre-acid and post-acid IPRs; and compute the production uplift and simple payback of a stimulation treatment.
- Perform LIT deliverability analysis on modified isochronal test data using pseudo-pressure m(p); extract A and B coefficients; anchor the stabilised curve through the extended-flow point; and solve the AOFP quadratic at a given back-pressure.
- Apply the Fetkovich two-point log–log method to multi-rate test data; defend the physical limits of the exponent n; compare the resulting AOFP against the Vogel result; and identify skin as the root cause when the two diverge.
- Use Standing’s J* mobility scaling on PVT data to generate the life-of-field IPR family; apply Fetkovich’s depletion method as a downside check; design the artificial-lift escalation schedule; and identify the reservoir pressure trigger for infill-well commitment.
- Synthesise all six topic outputs into an integrated FDP well performance report with six specific, quantified recommendations covering stimulation, lift specification, gas compression timing, infill drilling, pressure maintenance, and facilities sizing.
Integration reference — how each topic feeds the Karama deliverability study
Every sub-problem in the Karama PBL draws on a specific topic output. The table maps each topic to the equation term it provides, the Karama value it establishes, and the sub-problems that consume it.
Composite IPR: qo = qb + qv,max × [1 − 0.2(pwf/pb) − 0.8(pwf/pb)²] (pwf < pb)
| Topic | Equation term | What it provides | Karama value | Used in |
|---|---|---|---|---|
| 4.1 Vogel IPR | q/qmax shape | Establishes non-linear inflow below pb; calibrates AOFP from a single stabilised test point | T1: 820 stb/d at 1,800 psia → baseline AOFP reference | SP-1 → SP-4, SP-6 |
| 4.2 Composite IPR | qb, qv,max, AOFP | Joins Darcy and Vogel segments at pb; ideal (S = 0) AOFP as engineering ceiling | qb = 432 · qv,max = 1,800 · AOFP = 2,232 stb/d | SP-1 → SP-4, SP-5, SP-6 |
| 4.3 Standing’s FE | Jactual, post-acid IPR | Corrects ideal IPR for skin; delivers post-acid J* = 0.630 as FDP planning basis | Δq = +911 stb/d · payback = 8.5 days · Jpost = 0.630 | SP-2 → SP-5, SP-6 |
| 4.4 Gas Deliverability | Astab, B, AOFPgas | LIT m(p) analysis gives KA-G2 deliverability coefficients and per-well AOFP vs. contract | AOFP ≈ 15,779 Mscfd · headroom = 2.6× · compression deferred | SP-3 → SP-6 |
| 4.5 Fetkovich | n, C, AOFPactual | Fits actual well deliverability from test data; exposes skin as root cause of Vogel–Fetkovich gap; sets correct ESP basis | AOFP ≈ 905 stb/d (S′ = +8); post-acid ≈ 1,700 stb/d | SP-4 → SP-6 |
| 4.6 Future IPR | J*f, AOFP family | Standing’s J* scaling maps IPR decline through depletion; identifies infill trigger at Stage 2 | Stage 2 AOFP = 825 stb/d; max q at 800 psia = 743 stb/d < target | SP-5 → SP-6 |
| SP-6 Integration | FDP recommendation | Six specific recommendations: stimulation, ESP basis, compression timing, infill trigger, pressure maintenance, facilities sizing | R1–R6 with quantified basis; leadership forum deliverable | FDP gate deliverable |
Suggested study schedule
Session A — Oil-well inflow (~3 hrs)
Topic 4.1 then Topic 4.2, then attempt SP-1. These topics share the same Karama dataset and build continuously from Vogel to composite. Confirm your AOFP of 2,232 stb/d and qb = 432 stb/d before proceeding — these anchor every subsequent SP.
Session B — Stimulation & gas deliverability (~3 hrs)
Topic 4.3 then Topic 4.4, then SP-2 and SP-3. Topic 4.3 is the most directly commercial topic in the module — the 8.5-day acid-job payback is real money. Topic 4.4 introduces pseudo-pressure; do not shortcut to p² at KA-G2’s pressure range.
Session C — Fetkovich & depletion (~3 hrs)
Topic 4.5 then Topic 4.6, then SP-4 and SP-5. The Vogel–Fetkovich comparison in SP-4 is the deepest conceptual insight of the module. If your two AOFPs agree, you have made an error — they must differ, and explaining why is the learning objective.
Session D — Integrated FDP report (~2.5 hrs)
With SP-1 to SP-5 outputs confirmed, launch SP-6. Sixty minutes for the written report; 30 minutes for leadership forum preparation. The six recommendations must each carry a number — vague qualitative statements will not pass the rubric.
Reference
KA-07 oil well: p̅ = 5,100 psia · pb = 4,500 psia · TR = 200°F · ko = 42 md · h = 68 ft · re = 1,320 ft · rw = 0.354 ft · S′ = +8 · Spost = +1 · J* = 0.72 stb/d/psi
PVT at p̅: μo = 0.95 cp · Bo = 1.38 res bbl/stb · 36°API
Production tests: T1: 820 stb/d · 1,800 psia (72-hr stab.) · T2: 640 stb/d · 2,400 psia (72-hr stab.) · T3: 432 stb/d · 4,650 psia [above pb — Darcy only]
KA-G2 gas well: p̅ = 3,800 psia · T = 660°R · pbp = 800 psia · m(p̅) = 928.00×10^6 psia²/cp · m(800) = 52.66×10^6 psia²/cp
Economics: netback $58/stb · acid job $450K · ESP Stage 1 $1,200K · gas price $3.80/Mscf · plateau target 1,800 stb/d · gas contract 18,000 Mscfd
PVT at p̅: μo = 0.95 cp · Bo = 1.38 res bbl/stb · 36°API
Production tests: T1: 820 stb/d · 1,800 psia (72-hr stab.) · T2: 640 stb/d · 2,400 psia (72-hr stab.) · T3: 432 stb/d · 4,650 psia [above pb — Darcy only]
KA-G2 gas well: p̅ = 3,800 psia · T = 660°R · pbp = 800 psia · m(p̅) = 928.00×10^6 psia²/cp · m(800) = 52.66×10^6 psia²/cp
Economics: netback $58/stb · acid job $450K · ESP Stage 1 $1,200K · gas price $3.80/Mscf · plateau target 1,800 stb/d · gas contract 18,000 Mscfd
Misconception 1“Vogel’s AOFP and Fetkovich’s AOFP should agree for the same well”
They will almost certainly not agree unless you are careful about what each method is actually computing. Vogel’s composite IPR in SP-1 uses J* at S = 0 — the theoretical undamaged potential. Fetkovich in SP-4 is fitted to actual test data on a damaged well (S′ = +8). The 59% gap is not a method error; it is the quantified skin penalty. The test is: after the acid job, a new Fetkovich fit on post-acid tests should yield an AOFP close to (but still below) the ideal Vogel. If you obtain identical AOFPs from both methods on an untreated well, you have made an error.
Misconception 2“The p² form is adequate for KA-G2 gas deliverability”
The p² form assumes μgZ ≈ constant, which holds below ~2,000 psia for typical natural gas. KA-G2’s test spans 980 to 3,800 psia — a range over which μg changes by ~70%. Using p² introduces 15–30% error in the AOFP. Always use m(p) when test data spans a wide pressure range or when reservoir pressure exceeds ~2,500 psia.
Misconception 3“Size the ESP on the Vogel ideal AOFP”
The ideal AOFP of 2,232 stb/d is the undamaged theoretical ceiling. An ESP specified for this rate on a well with actual AOFP of ~905 stb/d (S′ = +8) would operate at 40% of design flow rate — well inside the cavitation zone for a centrifugal pump. The correct specification basis is the post-acid Fetkovich AOFP (~1,700 stb/d). This is why acid treatment before ESP installation is not optional: the sequencing directly determines the equipment specification.
Misconception 4“An ESP upgrade will maintain plateau when the reservoir depletes”
An ESP can only draw flowing BHP down toward zero; it cannot create rate beyond the well’s AOFP. Once KA-07 reaches Stage 2 (p̅ = 3,200 psia), AOFP falls to 825 stb/d — even at BHFP = 0 the well cannot deliver 1,800 stb/d. No amount of artificial lift overcomes an insufficient AOFP. Plateau maintenance requires either pressure support (waterflood) or an additional producing well — not a bigger pump.
Required
Module 01–03: Darcy radial flow, the linear PI equation (J = q/Δp), skin factor, drainage geometry, and the concept of pseudo-steady-state flow. Vogel builds directly on top of the Darcy PI — if Module 01 PI is not secure, the composite IPR will not make sense.
Required
Mathematics: quadratic formula (for AOFP solution), natural logarithm, basic algebra, and log–log slope calculation. The two-point Fetkovich method requires a two-point linear regression in log–log space; no calculus is used beyond the integrated Darcy equation.
Helpful
Gas properties: familiarity with Z-factor, pseudo-critical properties, and the real gas law from Module 01 Topic 1.2 or Module 02 supports Topic 4.4. The m(p) derivation is self-contained but proceeds faster if the concept of pseudo-pressure is not entirely new.