← GK-22 PBL Hub  ·  Sub-Problem 4 of 5

SP-4: Compute the Full Geometric Factor with Non-Darcy Turbulence Correction

Calculate the pseudo-steady-state geometric factor G = ln(re/rw) − ¾ + S for GK-22's 640-acre drainage area with provisional skin S = +2.5. Then apply the non-Darcy (Forchheimer) turbulence correction D·q, which is uniquely significant for high-rate gas wells and omitting which overestimates AOF by ~30%. Evaluate the relative PI benefit of stimulation versus rate management.

Topic 1.4 · Drainage Geometry
~25 min
Bloom: Apply & Evaluate
5 MCQ + Carry-Forward
Sub-Problem 4 · The Question

What Is the Full Denominator of the GK-22 Deliverability Equation?

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Engineering Context
SP-4 delivers the last piece before SP-5 integration, the complete geometric denominator

The pseudo-steady-state gas deliverability equation assembled in SP-5 has the form:

qg [Mscfd] = keff · h · (p̅² − pwf²) / [1,424 · μg · Z · T · Gtotal]

From SP-3 you have keff = 35.7 mD and h = 94 ft. From SP-2 you have μg = 0.0312 cp, Z = 1.40, T = 745°R. What remains is the complete geometric denominator Gtotal. For a vertical oil well, G = ln(re/rw) − ¾ + S. For GK-22, a high-rate gas well, there is an additional rate-dependent term: the non-Darcy (Forchheimer) turbulence skin S′ = D·qg.

Your tasks in SP-4 are:

  1. Compute ln(re/rw) for the 640-acre drainage and 9⅝" wellbore.
  2. Assemble Gbasic = ln(re/rw) − ¾ + S with provisional skin S = +2.5.
  3. Compute the non-Darcy skin S′ = D·q at the DST rate q = 12,400 Mscfd with D = 0.00035 d/Mscf.
  4. Assemble Gtotal = Gbasic + D·q and break it into its four components.
  5. Evaluate two engineering options, stimulation (remove S) versus rate restriction (reduce D·q), and quantify which yields greater PI improvement.
  6. Assess the sensitivity of Gbasic to a 30% change in re to demonstrate the logarithm's damping effect.
Non-Darcy Flow: Why It Is Uniquely Important for Gas Wells
At the DST rate of 12.4 MMscfd, the near-wellbore gas velocity in GK-22 creates Reynolds numbers well above the Darcy (laminar) regime. The additional inertial (turbulent) pressure drop is proportional to q². Expressed as an apparent skin, D·q = 4.34 at the DST rate, 74% larger than the mechanical skin S = 2.5. Ignoring D·q therefore overestimates AOF by (15.13 − 10.79)/10.79 = 40%. This is not a minor correction for a high-rate gas well.
SP-4 Data Slice

Data Relevant to This Sub-Problem

ParameterSymbolValueUnitsRelevance to SP-4
Drainage radiusre2,980ftPrimary geometric input for ln(re/rw)
Wellbore radiusrw0.354ft9⅝" casing ID ÷ 24 in/ft → rw in ft
Mechanical skin (DST provisional)S+2.5—Positive skin = near-wellbore damage; adds to denominator
Non-Darcy coefficientD0.00035d/MscfTurbulence coefficient: compute D·q at each rate of interest
DST stabilised gas rateqg,DST12,400MscfdCalculate D·q at DST rate for initial Gtotal
Gas viscosityμg0.0312cpFrom SP-2, used in SP-5 deliverability equation
Background Knowledge

Key Concepts for SP-4: Drainage Geometry and Non-Darcy Flow

Full treatment in Topic 1.4: Drainage Geometry.

Pseudo-Steady-State Geometric Factor: Derivation

For a vertical well centred in a circular drainage area under pseudo-steady-state flow, Darcy's radial flow equation integrates to give a denominator:

PSS Geometric Factor: Component Breakdown Gbasic = ln(re/rw) − 3/4 + S Where: ln(re/rw) Dietz geometric term: set by drainage area and wellbore size Logarithm DAMPENS the effect of re uncertainty significantly −3/4 Lewis & Horner PSS correction — converts from boundary pressure pe to average reservoir pressure p̅ (Changes to −1/2 for steady-state flow at constant pe) +S Mechanical skin: damage or stimulation effect on G (+ damage → larger G → lower PI; − stimulation → smaller G → higher PI) Physical interpretation: A larger Gbasic means larger pressure drop per unit flow rate → lower PI Each unit increase in S reduces PI by approximately 1/G × 100%

Non-Darcy Flow: Forchheimer Extension to Darcy's Law

Darcy's law assumes linear (laminar, viscous-dominated) flow: dP/dr = μq/(kA). At high gas velocities near the wellbore of a high-rate gas well, inertial forces become significant — the Forchheimer equation adds a velocity-squared (quadratic) term:

Forchheimer Equation and Rate-Dependent Skin Darcy (laminar): dP/dr = μq / (kA) Forchheimer (inertial): dP/dr = μq/(kA) + βρq²/A² The inertial term βρq²/A² causes additional near-wellbore pressure drop. In practical deliverability calculations this is expressed as an apparent rate-dependent skin S′: S′ = D · qg where D has units of d/Mscf Total apparent skin: Sapp = S + D·qg Full geometric factor: Gtotal = ln(re/rw) − 3/4 + S + D·qg Note: D·q makes the deliverability equation NONLINEAR — the denominator changes with flow rate. A rigorous treatment requires iteration; this PBL uses the simplified approach of evaluating Gtotal at the design rate. Jones correlation for estimating D when not measured directly: D ≈ (2.222 × 10⁻¹⁵ βγg) / (μg,w rw h²) where β = Firoozabadi-Katz turbulence factor ≈ f(k) (For GK-22: D = 0.00035 d/Mscf is provided from Jones estimate)
The Jones correlation gives D as an estimate. A rigorous D is obtained from a multi-rate gas well test (isochronal or modified isochronal test) by fitting the back-pressure equation. The DST at GK-22 was a single rate, so D is estimated. Include this as a sensitivity item in the SP-5 recommendation.

Component Decomposition of Gtotal: What Drives the Denominator?

ComponentGK-22 ValueFraction of GtotalSensitivityEngineering lever
ln(re/rw)9.038~60%LOW: logarithm dampens re uncertaintyLarger wellbore (rw) reduces G slightly; diminishing returns
−¾ (PSS correction)−0.750—Fixed for PSS circular drainageNone, fixed geometry term
Mechanical skin S+2.500~16%HIGH: each unit S changes G by ~1/G × 100%Acid stimulation, perforation optimisation
Non-Darcy D·q+4.340 (at DST rate)~29%HIGH and rate-dependent: D·q increases with qRate management, larger perforated interval, horizontal well
Step-by-Step Tasks

SP-4 Tasks: Work Through in Order

  1. Task 4.1: Calculate ln(re/rw)
    Compute re/rw = 2,980/0.354. Then calculate the natural logarithm (ln, base e, not log10). Express to three decimal places. Confirm by checking: ln(8,000) ≈ 8.99, so your answer for a ratio of ~8,418 should be close to 9.04. State clearly: this is the natural log (ln), not the common log (log10), getting this wrong by a factor of 2.303 is the most common calculation error in radial flow equations.
  2. Task 4.2: Assemble Gbasic
    Add the three components: Gbasic = ln(re/rw) − 0.750 + S. With S = +2.5 and your ln result from Task 4.1: compute Gbasic to three decimal places. Calculate the percentage of Gbasic contributed by the skin term alone (S/Gbasic × 100%). This tells you how much PI improvement is available from stimulation that removes the skin.
  3. Task 4.3: Compute Non-Darcy Apparent Skin at DST Rate
    Using D = 0.00035 d/Mscf and qg = 12,400 Mscfd (= 12.4 MMscfd expressed in Mscfd): calculate S′ = D × qg. Show units explicitly (d/Mscf × Mscfd = Mscf/Mscf × Mscf/d × d/Mscf, simplify to dimensionless). Compare S′ to mechanical skin S = 2.5. Which dominates? By what factor? State the physical interpretation: what does S′ = 4.34 mean for the well's ability to produce at the DST rate?
  4. Task 4.4: Assemble Gtotal and Decompose
    Compute Gtotal = Gbasic + D·q. Compare to Gbasic. Calculate the percentage by which non-Darcy turbulence increases the geometric denominator (and therefore reduces AOF) at the DST rate. Present a four-row decomposition table: component name, value, and percentage of Gtotal.
  5. Task 4.5: Stimulation vs. Rate Management: Quantify the PI Benefit
    Evaluate two options, each starting from Gtotal = 15.13 at 12.4 MMscfd:
    Option A — Acid stimulation removes mechanical skin (S: +2.5 → 0); D·q unchanged at 4.34. Calculate new Gtotal,A and PI improvement (PI ∝ 1/G).
    Option B: Rate restriction to qg = 8,000 Mscfd; S unchanged at +2.5. Recalculate D·q and new Gtotal,B. Calculate the PI improvement (note: rate restriction also directly reduces production revenue — consider this in your assessment).
    State which option provides superior PI improvement per unit engineering effort. Is this result surprising? What does it imply about the value of stimulation for this well?
  6. Task 4.6, re Sensitivity: How Much Does Drainage Area Uncertainty Matter?
    If the actual effective drainage radius is 2,100 ft (a 30% reduction from the assumed 2,980 ft, possible if adjacent wells drain overlapping areas), recalculate Gbasic with re = 2,100 ft. Compare to the original Gbasic. Calculate the percentage change in Gbasic. What does this tell you about the sensitivity of AOF to well spacing assumptions? Compare this sensitivity to a ±1 unit change in skin S.
Guided Calculations

SP-4 Worked Answers

Tasks 4.1–4.4: Gtotal Full Assembly

Complete Gtotal Calculation: GK-22 Task 4.1 — Geometric ratio and ln: re/rw = 2,980 / 0.354 = 8,418.1 ln(8,418.1) = 9.038 ← natural log (base e), NOT log₁₀ Check: ln(8,000) = 8.987, ln(9,000) = 9.105 → 9.038 is plausible ✓ Compare: log₁₀(8,418) = 3.925, using this would be a 2.30× error Task 4.2 — Basic geometric factor: Gbasic = ln(re/rw) − 3/4 + S = 9.038 − 0.750 + 2.500 = 10.788 Skin contribution: 2.500 / 10.788 = 23.2% of Gbasic → Removing skin (S→0) reduces Gbasic from 10.788 to 8.288 = 23.2% reduction Task 4.3 — Non-Darcy apparent skin: D = 0.00035 d/Mscf, qg = 12,400 Mscfd S′ = D × qg = 0.00035 × 12,400 = 4.340 Unit check: (d/Mscf) × (Mscfd) = Mscf/Mscf × (d/d) = dimensionless ✓ Comparison: S′ = 4.34 vs. S = 2.50 Turbulence is 4.34/2.50 = 1.74× the mechanical skin — it DOMINATES Physical meaning: at 12.4 MMscfd the inertial pressure drop near the wellbore is equivalent to having an extra skin of +4.34 — in addition to the existing mechanical damage of +2.5 Task 4.4 — Total geometric factor: Gtotal = Gbasic + D·q = 10.788 + 4.340 = 15.128 ≈ 15.13 PI penalty from non-Darcy: Gtotal/Gbasic − 1 = 15.13/10.79 − 1 = 40.2% → Ignoring D·q overstates PI (and AOF) by 40% at the DST rate Component decomposition: ln(re/rw): 9.038 / 15.128 = 59.7% of Gtotal −3/4 PSS: −0.750 / 15.128 = −5.0% (reduces G, beneficial) Skin S: 2.500 / 15.128 = 16.5% of Gtotal Turb. D·q: 4.340 / 15.128 = 28.7% of Gtotal ← dominant skin term

Task 4.5: Stimulation vs. Rate Management

Option Comparison: PI Improvement Baseline: Gtotal = 15.13, PI ∝ 1/15.13 = 0.0661 (relative units) Option A — Acid stimulation (S: 2.5 → 0, D·q unchanged at 4.34): Gtotal,A = 9.038 − 0.750 + 0 + 4.340 = 12.628 PIA = 1/12.628 = 0.0792 PI improvement = (0.0792/0.0661) − 1 = 19.8% ← better option Option B — Rate restriction to 8,000 Mscfd (S unchanged at 2.5): D·qnew = 0.00035 × 8,000 = 2.800 Gtotal,B = 9.038 − 0.750 + 2.500 + 2.800 = 13.588 PIB = 1/13.588 = 0.0736 PI improvement = (0.0736/0.0661) − 1 = 11.3% ← lesser option BUT: rate restriction also directly cuts revenue (8,000 vs 12,400 Mscfd) Conclusion: Stimulation (Option A) gives +19.8% PI improvement vs. +11.3% for rate restriction, and stimulation preserves the production rate. The combination of both (stimulate + design for controlled max rate) would give Gtotal = 9.038 − 0.750 + 0 + 2.800 = 11.088 → PI improvement = 15.13/11.09 − 1 = 36.4% vs. baseline
This result illustrates why acid stimulation is typically the first engineering response to high-skin gas wells, even when non-Darcy dominates. Removing S provides permanent PI benefit. Rate restriction is a short-term operational trade-off that sacrifices revenue. The ideal solution is to stimulate and then set a maximum rate that keeps D·q at an acceptable level.

Task 4.6: re Sensitivity

Drainage Radius Sensitivity: The Logarithm Damping Effect Original: re = 2,980 ft → ln(2,980/0.354) = ln(8,418) = 9.038 Gbasic = 9.038 − 0.750 + 2.500 = 10.788 Reduced: re = 2,100 ft → ln(2,100/0.354) = ln(5,932) = 8.688 Gbasic = 8.688 − 0.750 + 2.500 = 10.438 Change in Gbasic: (10.438 − 10.788)/10.788 = −3.2% Change in AOF (∝ 1/G): ≈ +3.3% Comparison: re changes by 30% → G changes by only 3.2% → AOF changes 3.3% Compare to ±1 unit in skin S: Gbasic with S = 3.5: 9.038 − 0.750 + 3.500 = 11.788 → AOF decline: 8.5% Gbasic with S = 1.5: 9.038 − 0.750 + 1.500 = 9.788 → AOF increase: 9.3% → A ±1 unit change in skin has 2.5–3× more impact on AOF than a 30% change in re. → Engineers should focus more on controlling skin than on well spacing for PI optimisation.
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Think-Pair-Share: The Non-Darcy Problem in Context
Discuss before closing SP-4

GK-22 at 12.4 MMscfd has D·q = 4.34: the turbulence skin exceeds the mechanical skin by 74%. For a horizontal well with 800 ft of open hole (effective rw,effective enlarged dramatically), the near-wellbore velocity would be far lower and D would be smaller. Discuss: if GK-22 had been drilled as a horizontal well, how would the denominator change? Which of the four components (ln term, PSS correction, S, D·q) would be most affected and in which direction?

KWL Reflection

Update Your KWL Table: SP-4

K: Confirmed
  • Gbasic = ln(re/rw) − ¾ + S for PSS circular drainage
  • Must use ln (natural log), NOT log10
  • Skin adds directly to G, units cancelled to dimensionless
  • re uncertainty is damped by the logarithm
W: Questions Answered
  • D·q = 4.34 at DST rate: 74% larger than mechanical skin S = 2.5
  • Gtotal = 15.13; ignoring D·q overstates AOF by 40%
  • Stimulation (+19.8% PI) beats rate restriction (+11.3%)
  • 30% re change → only 3.2% change in G
  • ±1 skin unit → ±8–9% AOF, 3× more impact than re uncertainty
L: New Learning
  • Write your own entry
  • Example: "D·q is rate-dependent — the deliverability equation is nonlinear in q for gas wells; iterating to find true AOF requires solving Gtotal(q) = f(q)"
Just-in-Time Resources

Just-in-Time Resources

Reach for these the moment you get stuck on this sub-problem. They map one-to-one onto the physics and the calculation you are doing here.

Study

Module 01 · Topic 4: Drainage Geometry & Radial Flow
Watch
  • Lecture 1.4A — Geometry Matters: How Boundary Shape Controls Deliverability
  • Lecture 1.4B — Radial Flow to Steady-State IPR: Step by Step
Self-check

geometry_calculator.py, verified calculator; reproduce your numbers.
Knowledge Check

SP-4: Five MCQ

Q1: ln(re/rw) Calculation

For GK-22 with re = 2,980 ft and rw = 0.354 ft, the correct value of ln(re/rw) is:

A. 3.925, using log10(2,980/0.354) instead of the natural logarithm
B. 9.038, the natural logarithm (ln, base e = 2.718) of re/rw = 8,418
C. 8,418, the ratio re/rw itself, before taking the logarithm
D. 7.69, computing ln(re) − ln(rw) but using ft/m unit conversion error
Explanation
re/rw = 2,980/0.354 = 8,418. ln(8,418) = 9.038. This must be the natural logarithm (ln, base e) — the Darcy radial flow equation derives from integration of 1/r in cylindrical coordinates, which gives ln(r). Using log10 underestimates by a factor of 1/2.303 = 0.434, giving 3.925, a factor-of-2.3 error in the geometric term. Option D shows a unit inconsistency error. Always verify: ln(re/rw) should be in the range 7–11 for typical oil and gas wells.

Q2: What the −¾ Term Represents

In the geometric factor G = ln(re/rw) − ¾ + S, the −¾ correction:

A. Accounts for wellbore storage during the DST buildup period
B. Converts the pseudo-steady-state equation from using outer boundary pressure pe to using average reservoir pressure p̅: it equals −½ − ¼ where −½ is the PSS boundary correction and an additional −¼ arises because p̅ is lower than pe
C. A safety factor of 75% applied to the geometric term to account for non-circular drainage shape
D. Converts from Darcy units to field units: the 0.75 is an empirical unit conversion factor
Explanation
The derivation of PSS radial flow gives ln(re/rw) − ½ when the outer boundary pressure pe is used. Using the average reservoir pressure p̅ (which is always lower than pe in PSS) requires an additional correction of −¼, giving −½ − ¼ = −¾. For steady-state flow (constant pe boundary), the denominator is ln(re/rw) − 0. The distinction between PSS (p̅) and SS (pe) matters practically because well tests measure p̅ (from pressure buildup), not pe.

Q3: Non-Darcy Apparent Skin S′

At qg = 12,400 Mscfd with D = 0.00035 d/Mscf, the non-Darcy apparent skin S′ is:

A. 0.434, using q in MMscfd (12.4) instead of converting to Mscfd (12,400)
B. 4.34: D [d/Mscf] × q [Mscf/d] = 0.00035 × 12,400 = 4.34 (dimensionless); the units (d/Mscf) × (Mscf/d) cancel to give a dimensionless skin
C. 43.4 — the correct formula uses q in scfd, not Mscfd
D. 2.50: D·q equals the mechanical skin for this well by coincidence
Explanation
S′ = D[d/Mscf] × q[Mscfd] = 0.00035 × 12,400 = 4.34. The unit check: (d/Mscf) × (Mscf/d) = 1 (dimensionless) ✓. Option A uses q = 12.4 MMscfd as-is without converting, giving 0.00035 × 12.4 = 0.00434, which is off by a factor of 1,000. This is the most common unit error with the D coefficient: always confirm that q and D use consistent Mscf/d and d/Mscf units.

Q4: Gtotal and AOF Penalty

Gtotal for GK-22 at the DST rate of 12.4 MMscfd is:

A. 10.79, omitting the non-Darcy term D·q from Gbasic
B. 15.13: Gbasic + D·q = 10.788 + 4.340 = 15.128 ≈ 15.13; omitting D·q overestimates AOF by 15.13/10.79 − 1 = 40%
C. 12.63, adding D·q to ln(re/rw) alone without including −¾ and S
D. 13.29, adding S′ = D·q to ln(re/rw) and −¾ but omitting mechanical skin S
Explanation
Gtotal = ln(re/rw) − ¾ + S + D·q = 9.038 − 0.750 + 2.500 + 4.340 = 15.128. All four components must be included. Option A (Gbasic = 10.79) omits D·q, the error that overestimates AOF by 40%. This is a significant overestimate that could result in undersizing the FPSO separator by 40%. Option C omits S and −¾. Option D omits mechanical skin S = 2.5.

Q5: re Sensitivity via Logarithm Damping

If re is reduced from 2,980 to 2,100 ft (a 30% reduction), Gbasic changes by approximately:

A. 30%: Gbasic is proportional to re so a 30% reduction in re gives a 30% change in G
B. 15% — re appears as ln(re) so the sensitivity is approximately halved
C. Only ~3%: because re appears inside a natural logarithm, a 30% change in re produces only a ~0.35 change in ln(re/rw) out of a Gbasic of ~10.8, far less than a ±1-unit change in skin S which changes G by ~10%
D. Gbasic does not change, only wellbore radius rw affects the geometric factor, not drainage radius
Explanation
ln(2,100/0.354) = ln(5,932) = 8.688. Gbasic,new = 8.688 − 0.750 + 2.500 = 10.438. Change = (10.438 − 10.788)/10.788 = −3.2%. Compare to ±1 unit in S: changing S from 2.5 to 3.5 gives G = 11.788, a change of 9.3%. The ±1-skin-unit sensitivity is 3× larger than the 30%-re sensitivity. This is the key insight from the logarithm: drainage area uncertainty (which can be ±50% in an appraisal well) has surprisingly little effect on PI: engineers should focus far more on controlling skin and turbulence than on well spacing.
SP-4 Complete

Carry-Forward to SP-5

SP-4 Carry-Forward Values: Record These Now

CF-4a, ln(re/rw): 9.038 (natural log of 8,418; re/rw = 2,980/0.354)
CF-4b — Gbasic: 10.788 (= 9.038 − 0.750 + 2.500)
CF-4c — Non-Darcy skin at DST rate: S′ = D·q = 0.00035 × 12,400 = 4.340
CF-4d — Gtotal at DST rate (12.4 MMscfd): 15.128 ≈ 15.13
CF-4e — D·q at AOF rate (for SP-5 iteration): recalculate D·q when AOF is known; AOF will be higher than 12.4 MMscfd so Gtotal at AOF > 15.13
CF-4f — Stimulation benefit: removing S reduces Gtotal from 15.13 → 12.63 (+19.8% PI improvement)
CF-4g, re sensitivity note: 30% re change → only 3% AOF change; skin dominates PI sensitivity