The skin value from a well test is not purely formation damage. At high flow rates, turbulent inertial forces add a rate-dependent component that inflates the apparent skin and curves the IPR. Engineers who ignore this misdiagnose well performance and design unnecessary treatments.
In Topic 3.1 you learned that the skin factor S is the single engineerable variable in the Darcy inflow equation, and that it captures all near-wellbore deviations from ideal radial flow. The GK-22 well returned a skin of +14 from its pressure build-up test, costing 1,564 stb/d of production. But before you can design a treatment, you must answer a prior question: is that +14 entirely formation damage, or does it include a rate-dependent turbulence contribution?
In practice, the skin measured in a well test, called total skin S′, is a superposition of two physically distinct contributions:
S′ = S + D·q
The first term, S (the Darcy skin), is rate-independent. It arises from formation damage, geometric restrictions, and completion imperfections. Acid, re-perforating, or hydraulic fracturing can reduce it.
The second term, Dq (the non-Darcy skin), is strictly rate-dependent. It arises from inertial turbulent pressure losses as high-velocity fluids deviate from laminar Darcy behaviour in the near-wellbore zone and perforation tunnels. Acid cannot fix it. You must manage it through rate control, perforation optimisation, or gravel-pack design.
For the GK-22 oil well at 782 stb/d in 85 md rock, we will demonstrate that Dq is effectively zero — confirming S′ = S = +14 is pure formation damage and the entire treatment budget should target acid and workover. This distinction is not always so clean, particularly in gas wells, where turbulence can dominate the measured skin.
▶
Lecture 3.2a: Total Skin — Breaking the Well Test Number Apart
14:20
Introduces the S′ = S + Dq decomposition with field examples from the GK-22 oil well and FK-7 gas well. Covers why classical reservoir engineering texts used only S, the velocity regimes near perforations that generate turbulence, and how to identify non-Darcy behaviour from multi-rate test shape. The Forchheimer equation is derived step-by-step with physical explanation.
▶
Lecture 3.2b: Calculating D — Theory, Correlations, and Field Practice
11:45
Derives the non-Darcy coefficient D from the inertial coefficient β. Explains the Katz and Dake β vs permeability correlations, the critical sensitivity of D to perforation geometry (D ∝ 1/hp²), and demonstrates the multi-rate graphical method for separating S from Dq. Full worked example with FK-7 DST data.
LEARNING OBJECTIVES
After completing this topic, you will be able to:
1. State the total skin equation S′ = S + Dq and explain the physical meaning of each term, distinguishing treatable (S) from non-treatable (Dq) components. 2. Describe the Darcy–Forchheimer flow regime transition and identify the velocity threshold at which laminar flow breaks down near perforations. 3. Calculate the non-Darcy coefficient D from the inertial resistance coefficient β using the Katz and Dake correlations. 4. Calculate Dq at a given flow rate and assess its magnitude relative to the Darcy skin S, using the GK-22 oil well as a reference case. 5. Explain why the gas well IPR is quadratic (not linear) and how the B coefficient in the Forchheimer deliverability equation captures turbulence effects. 6. Separate S and Dq from multi-rate well test data using a Forchheimer diagnostic plot (Δp/q vs q). 7. Diagnose whether a high measured S′ is primarily formation damage (treatable) or turbulence (not treatable by stimulation alone) and recommend the appropriate intervention strategy.
PREREQUISITE
Topic 3.1 must be completed first. The physical meaning of skin, Hawkins' formula, and the PI ratio equation are required foundations. The GK-22 canonical data set established in Topic 3.1 is used throughout this topic and all remaining Module 03 topics.
PBL CONNECTION — GK-22 & FK-7
Two wells anchor the Module 03 problem set and both require total skin decomposition:
GK-22 (oil, S′ = +14, q = 782 stb/d): At this rate with k = 85 md, Dq is negligible. This topic quantifies why, confirming that the full +14 is formation damage. All six Module 03 topics build on this confirmed starting point.
FK-7 (gas well, S′ = +18.4 at 18 MMscfd): Multi-rate DST data shows S′ increasing with rate. You must separate S from Dq, determine whether the non-Darcy component is perforation-driven or formation-driven, and recommend whether acid, reperforation, or both is the correct intervention. FK-7 data is introduced in this topic.
Section 1
Darcy vs Non-Darcy Flow — When Laminar Flow Breaks Down
Darcy's law assumes slow, laminar, viscous-dominated flow. Near the wellbore and through perforations, velocities can be orders of magnitude higher. The physics changes fundamentally — and so does the pressure drop behaviour.
1.1 Darcy's Law and Its Fundamental Assumption
Darcy's law states that the pressure gradient driving flow is directly proportional to the fluid velocity — a linear, viscous-dominated relationship:
−dP/dr = (μ/k) × u [Darcy: viscous-dominated, laminar]
This is valid when flow is laminar and viscous forces dominate inertial forces. In a large reservoir, pores are small, velocities are typically 0.001–0.1 ft/day, and the porous-media Reynolds number is well below unity. Darcy's law is an excellent approximation everywhere except the near-wellbore region.
But as fluid converges radially toward the wellbore (r → rw), continuity demands velocity ∝ 1/r. By the time flow enters a perforation tunnel with a 0.4-inch diameter and ~0.1 in² cross-section, a well producing 10 MMscfd of gas is driving flow at velocities that generate significant inertial forces. Darcy's proportionality breaks down — and pressure drop grows faster than linearly with rate.
1.2 The Forchheimer Equation
Forchheimer (1901) extended Darcy's law to account for inertial pressure losses by adding a quadratic velocity term:
Pressure drop proportional to velocity. Dominates at low flow rates. The only term in classical Darcy analysis. Gives a linear relationship between Δp and q. Used for all reservoir flow away from the wellbore.
βρu² — Inertial Term
Pressure drop proportional to velocity squared. Dominates at high flow rates. β is the inertial resistance coefficient (ft²). Gives a quadratic relationship between Δp and q. Significant only near the wellbore and in perforations.
β — Inertial Coefficient
The key material property linking rock structure to turbulence intensity. Strong inverse function of permeability: high-k rock has low β; damaged, low-k rock has very high β. Not a measurable PVT property — must be estimated from correlations or lab tests.
1.3 Velocity Cascade — Where Turbulence Acts
Turbulent effects are most severe where velocities are highest: the perforation tunnels. For a producing well, flow velocity increases by orders of magnitude from the reservoir to the wellbore:
Figure 3.2.1 — Velocity cascade from far reservoir to wellbore. Turbulence is concentrated at the perforation entry point where the entire well's flow must pass through a small number of small-diameter tunnels. In gas wells producing at high rates, this is the dominant source of non-Darcy skin.
1.4 When Does Non-Darcy Flow Become Significant?
The transition from Darcy to non-Darcy flow is characterised by the porous-media Reynolds number Rep. Non-Darcy effects become significant when Rep > 0.1–1.0. In practical terms:
Well Type
Typical u at rw
Non-Darcy Significance
Engineering Action
Low-rate oil well (q < 500 stb/d, k > 50 md)
< 1 ft/d
Negligible
Use S only — Dq ≈ 0. GK-22 at 782 stb/d falls here.
High-rate oil well (q > 5,000 stb/d)
5–50 ft/d
Low to moderate
Check if Dq > 10% of S′
Gas well (< 10 MMscfd)
50–500 ft/d at perf entry
Moderate to high
Always separate S and Dq from multi-rate test
High-rate gas well (> 50 MMscfd)
> 1,000 ft/d at perf entry
Dominant
Dq may far exceed S. Perf design critical.
KEY INSIGHT
Classical reservoir engineering texts assume S′ ≈ S because they were developed for low-rate oil wells where turbulence is negligible. Generally, the guidelines explicitly state: "in high rate and high GLR oil wells and in gas wells, the effects of high flow velocities must be included as an equivalent, rate dependent skin (Dq), so that S′ = S + Dq." Never apply a single-rate well test skin value to a gas well without first checking for rate-dependent effects.
Consider pushing water through a garden hose at progressively higher rates. At low flow, the water moves in smooth, parallel layers (laminar flow) and the pressure required increases proportionally with the flow rate: 2× the rate = 2× the pressure. This is Darcy behaviour.
Open the tap fully and the flow becomes turbulent: chaotic eddies develop, fluid elements mix and collide, and energy is dissipated as heat. The pressure drop now increases roughly as the square of velocity: 10× the velocity requires ~100× the pressure drop. This is Forchheimer behaviour.
In a perforation tunnel — 0.4 inch diameter, 12 inches long — a gas well at 10 MMscfd is pushing gas through each perforation at velocities exceeding 100 ft/s in high-rate completions. The gas cannot flow smoothly; it jets, eddies, and loses kinetic energy. That energy loss appears as an extra pressure drop at the wellbore — exactly what the Dq term represents in S′.
Key implication: Acid cannot dissolve turbulence. Acid can restore permeability (reducing β and therefore D) but cannot eliminate the fundamental inertial physics. The only engineering controls on Dq are: flow rate management, perforation shot density (D ∝ 1/hp²), perforation diameter, and gravel-pack conductivity.
The GK-22 well (ko = 85 md, q = 782 stb/d, hp = 42 ft) sits firmly in the low-rate oil well category. We can estimate the near-wellbore velocity:
At rw = 0.35 ft: u = q × Bo / (2π × rw × h) = 782 × 1.32 / (2π × 0.35 × 42) = 1033 / 92.2 ≈ 11.2 res bbl/d per unit area ≈ 0.6 ft/day
This is well below the threshold where inertial effects become significant in a 85 md formation. For comparison, a typical gas well at 10 MMscfd with the same perf geometry would see velocities 500–5,000 times higher through the perforations.
This pre-analysis confirms that the Module 03 diagnosis for GK-22 can proceed without a multi-rate test: S′ = S = +14 is entirely formation damage. All treatment design in Topics 3.3–3.6 works from this confirmed baseline.
Section 2
Total Skin S′ = S + Dq — The Complete Equation
Understanding how D and q combine to produce rate-dependent skin transforms the way you read well test results, design multi-rate programmes, and choose between stimulation options.
2.1 The Total Skin Equation — Decomposed
S′ = S + D · q
S' — Total Skin, Dimensionless. What the well test actually measures via Horner or MDH analysis. This is the apparent skin at the specific rate q at which the test was conducted. It always contains both components unless q is very small.
S — Darcy Skin. Dimensionless; rate-independent. Formation damage + geometric effects + completion skin. The only component treatable by stimulation, reperforation, or workover. Constant regardless of flow rate. Topics 3.3–3.6 decompose this further.
D — Non-Darcy Coefficient, d/Mscf (gas) or d/stb (oil). Inertial resistance per unit flow rate. Depends on β, permeability, gas/oil gravity, viscosity, and perforation geometry. Determined from multi-rate tests or calculated from correlations.
q — Flow Rate, Mscf/d (gas) or stb/d (oil). The rate at which the well was tested. For gas wells, D uses Mscf/d to keep units consistent with the 1422T Darcy group. For oil, stb/d is used with the appropriate D units.
2.2 The Critical Consequence — Rate-Dependent S′
The most important practical consequence is that a single-rate well test gives you S′ at that specific rate, not S itself. Using S′ as if it were the rate-independent S in a deliverability model produces systematic errors:
Error 1: Overestimate damage from high-rate test
Test conducted at high rate → large Dq → S′ appears very high. If you use S′ as S, you overestimate formation damage, over-design the acid treatment, and spend money treating turbulence that acid cannot fix.
Error 2: Optimistic forecast from low-rate test
Test conducted at low rate (small Dq) → S′ ≈ S. If you then produce at high rate, the actual S′ will be higher than tested because Dq adds significantly. Production forecast will be over-optimistic.
The solution is to conduct multi-rate tests and use the Forchheimer plot to separate S and D independently. This is mandatory for all gas wells and high-rate oil wells before designing any intervention.
2.3 The Forchheimer Diagnostic Plot
Rearranging the inflow equation to group rate-dependent terms isolates the key diagnostic relationship. For an oil well:
(Δp) / q = A + B·q where A = 141.2μB/(kh) × [ln(0.472re/rw) + S] and B = 141.2μB/(kh) × D
For gas wells using pseudo-pressure (the technically correct approach for all pR > 1,500 psi):
[m(p̄R) − m(pwf)] / qg = A + B·qg
Figure 3.2.2 — The Forchheimer diagnostic plot. Plot Δp/q vs q from multi-rate tests. A straight line through the points confirms Forchheimer behaviour. The y-intercept (A) captures the Darcy skin S and reservoir properties. The slope (B) captures the non-Darcy coefficient D. A curved upward-bowing line indicates additional effects such as two-phase flow near the wellbore or pressure depletion between test periods.
KEY CONCEPTTwo-Rate Separation Method
With exactly two well tests at different rates, S and D can be separated directly:
D = (S′2 − S′1) / (q2 − q1)
S = S′1 − D × q1 (verify: should also equal S′2 − D × q2)
This is the simplest field method. With more rates, fit a straight line to the Δp/q vs q plot; the slope gives D and the y-intercept gives A (from which S is extracted knowing k, h, T, re/rw).
WORKED EXAMPLEFK-7 Gas Well — Two-Rate Skin Separation
FK-7 DST data: Two flow periods at stabilised conditions:
Rate 1: q₁ = 8 MMscfd → S′₁ = +11.2 | Rate 2: q₂ = 18 MMscfd → S′₂ = +18.4
Step 1: Non-Darcy coefficient D
D = (S'₂ - S'₁) / (q₂ - q₁)
= (18.4 - 11.2) / (18 - 8)
= 7.2 / 10
= 0.720 d/MMscf (= 7.20 x 10⁻⁴ d/Mscf)
Step 2: Darcy skin S (from Rate 1)
S = S'₁ - D x q₁ = 11.2 - 0.720 x 8 = 11.2 - 5.76 = +5.44
Verification (Rate 2):
S = S'₂ - D x q₂ = 18.4 - 0.720 x 18 = 18.4 - 12.96 = +5.44 ✓
Step 3: Non-Darcy skin at each tested rate
At q = 8 MMscfd: Dq = 0.720 x 8 = 5.76 (51% of S' = 11.2)
At q = 18 MMscfd: Dq = 0.720 x 18 = 12.96 (70% of S' = 18.4)
At design production rate (q = 25 MMscfd):
Dq = 0.720 x 25 = 18.0 (predicted at full production rate)
Interpretation: At FK-7's design production rate, turbulence accounts for the majority of total skin. A matrix acid job addressing only S = +5.4 should be the primary chemical intervention. However, reperforation to increase hp (reducing D) is likely the higher-value intervention for long-term productivity — explored quantitatively in the D Coefficient section.
2.4 GK-22 Oil Well — Confirming S′ = S
For the Module 03 PBL anchor well (GK-22), we apply the same framework to confirm the non-Darcy term is negligible at the production rate of 782 stb/d:
GK-22 skin-audit case — canonical data: ko = 85 md · h = 42 ft · μo = 1.8 cp · Bo = 1.32 rb/stb · p̄R = 4,200 psi · pwf = 2,500 psi · rw = 0.35 ft · re = 1,650 ft · q = 782 stb/d · S′ = +14
WORKED EXAMPLEGK-22 — Estimating Dq to Confirm It Is Negligible
For oil wells, the non-Darcy term uses an analogous D formulation. We estimate D using the Katz β correlation with GK-22 data:
Step 1: Estimate β (Katz correlation, k_p = 85 md assuming no perf damage)
β = 4.11 x 10¹⁰ / k_p^1.33
= 4.11 x 10¹⁰ / 85^1.33
= 4.11 x 10¹⁰ / 265.3
= 1.55 x 10⁸ ft⁻¹
Step 2: Estimate D for oil (analogous form, oil gravity γ_o = 0.85)
D_oil = 2.22 x 10⁻¹⁵ x β x γ_o x k_o x h / (h_p² x r_w x μ_o)
= 2.22 x 10⁻¹⁵ x 1.55x10⁸ x 0.85 x 85 x 42 / (42² x 0.35 x 1.8)
= 2.22 x 10⁻¹⁵ x 4.72 x 10¹¹ / (1764 x 0.63)
= 1.048 x 10⁻³ / 1111.3
= 9.43 x 10⁻⁷ d/stb
Step 3: Non-Darcy skin Dq at q = 782 stb/d
Dq = D x q = 9.43 x 10⁻⁷ x 782 = 7.37 x 10⁻⁴ ≈ 0.001
As fraction of total skin:
Dq / S' = 0.001 / 14 = 0.007% → Effectively zero
Conclusion: The non-Darcy contribution at GK-22's current production rate is 0.001 skin units — analytically indistinguishable from zero. This confirms that for all Module 03 calculations: S′ = S = +14, all formation damage, all treatable. Topics 3.3–3.6 proceed on this basis. No multi-rate test is required to justify the treatment design.
Section 3
The Non-Darcy Coefficient D — Theory and Calculation
D can be calculated from rock properties and perforation geometry, enabling pre-drill prediction and completion design optimisation long before a well test is available.
3.1 Derivation of D from Formation Properties
Integrating the Forchheimer equation through the near-wellbore region and expressing the result in the dimensionless skin framework gives the non-Darcy coefficient D:
D = 2.22×10²⁻ · β · γg · kg · h / (hp² · rw · μwf) (Mscf/d)²
Symbol
Parameter
Units
Notes for FK-7 / GK-22
β
Inertial resistance coefficient
ft²
KEY variable — strong function of k; use Katz or Dake correlation
γg
Gas gravity (air = 1.0)
—
FK-7: 0.62; For oil wells use γo
kg
Effective gas permeability in perforated zone
md
FK-7: 12 md; if damage present, kp < k
h
Net pay thickness
ft
Total pay, not just perforated interval
hp
Effective perforated interval
ft
APPEARS SQUARED in denominator — most sensitive parameter
rw
Wellbore radius
ft
GK-22 and FK-7: 0.35 ft
μwf
Gas viscosity at pwf
cp
Evaluated at wellbore flowing conditions
3.2 The Inertial Coefficient β — Katz and Dake Correlations
β is the key unknown. It is a strong, inverse function of permeability. Two correlations are:
Katz Correlation (Eq. 4a.23)
β = 4.11×10¹⁰ / kp1.33
kp is the effective permeability in the perforated zone (md). Commonly used in North Sea practice. Gives higher β at low permeability relative to Dake. Conservative choice for design.
Dake Correlation (Eq. 4a.24)
β = 2.73×10¹⁰ / kp1.10
Alternative correlation; gives lower β at the same permeability. The difference between Katz and Dake illustrates the inherent uncertainty in β — can vary 2–5× for the same rock type. Run both and bracket the answer.
UNCERTAINTY ALERT — EXPLICIT WARNING
"The difference between these correlations illustrates the high degree of uncertainty over the β factor even in undamaged core material. However, the β factor required in these calculations is that in the immediate wellbore area where permeability may have been affected by completion operations."
This means: formation damage reduces kp dramatically, which increases β (via the power-law), which increases D, which increases Dq. Stimulating a damaged gas well is doubly effective: it reduces S (by restoring near-wellbore kp) AND reduces D (by increasing kp in the β correlation). This double benefit makes acid treatments far more valuable in damaged gas wells than the Darcy skin reduction alone suggests.
3.3 The hp² Rule — Perforation Geometry Dominates D
The most practically actionable insight in the D formula is the hp² dependence in the denominator: halving the effective perforated interval quadruples D. This means perforation design decisions made at completion time have a lasting, quadratic impact on turbulence-related production losses throughout the well's life.
Low Shot Density
Fewer perforations → each perf carries more flow → higher velocity per tunnel → higher Dq. Moving from 8 spf to 2 spf (same interval hp but 4× fewer effective tunnels) approximately quadruples D per unit interval.
Plugged Perforations
If 50% of perforations are plugged by debris, sand, or cement contamination, effective hp halves → D quadruples. This is why underbalanced TCP perforating dramatically reduces turbulence skin in gas wells — clean perforations stay open.
Increased Shot Density
Moving from 4 spf to 16 spf in the same perforated interval quadruples hp and reduces D by 16×. In high-rate gas wells, maximising shot density is the most cost-effective non-Darcy skin reduction strategy available.
WORKED EXAMPLEFK-7 Gas Well — Calculating D and Diagnosing Plugged Perforations
FK-7 data: kg = 12 md, h = 65 ft, hp = 52 ft (80% of the 65 ft net pay perforated, 4 spf), rw = 0.35 ft, γg = 0.62, μwf = 0.022 cp, T = 620°R
Step 1: β using Katz correlation (assume k_p = k_g = 12 md, no perf damage)
β = 4.11 x 10¹⁰ / 12^1.33
= 4.11 x 10¹⁰ / 24.8
= 1.66 x 10⁹ ft⁻¹
Step 2: D using nominal perforation geometry
D = 2.22x10⁻¹⁵ x 1.66x10⁹ x 0.62 x 12 x 65
------------------------------------------------
52² x 0.35 x 0.022
Numerator = 2.22x10⁻¹⁵ x 1.66x10⁹ x 485.6 = 1.790x10⁻³
Denominator = 2704 x 0.35 x 0.022 = 20.82
D_calc = 1.790x10⁻³ / 20.82 = 8.60x10⁻⁵ d/Mscf = 0.086 d/MMscf
Step 3: Compare with measured D from two-rate test
D_measured = 0.720 d/MMscf (from Section 2 worked example)
D_calc = 0.086 d/MMscf
Ratio = 0.720 / 0.086 = 8.4x → Measured is ~8 times theoretical
Step 4: Back-calculate effective h_p
D ∝ 1/h_p², so: h_p,eff = h_p,nominal / √(D_meas/D_calc)
= 52 / √8.4 = 52 / 2.90 ≈ 18 ft
Conclusion: Only ~18 ft of the nominal 52 ft perforated interval
is effectively open (35% of nominal shot count contributing flow).
Engineering Action: TCP reperforation with minimum 1,000 psi underbalance is indicated to clean all 52 ft of perforations. Post-workover expected D: 0.086 d/MMscf. At q = 18 MMscfd, Dq drops from 12.96 to 1.55 — a reduction of 11.4 skin units from perforation cleanup alone, with no acid required.
3.4 Comparing A and B from Well Tests
The equations give explicit expressions for A and B in the gas well deliverability equation:
A = 1422T[ln(0.472re/rw) + S] / (kgh) psia²/cp per Mscf/d
B = 1422T · D / (kgh) psia²/cp per (Mscf/d)²
These relationships allow you to extract D directly from B once kh and T are known from the build-up analysis, providing an independent check on the graphical method.
Consider FK-7 with drilling damage reducing near-wellbore kp from 12 md to 2 md in the first 3 ft around the wellbore. Two things happen simultaneously:
Effect 1 (Darcy skin S increases): Using Hawkins' formula, Sd = (k/kp − 1) × ln(rs/rw) = (12/2 − 1) × ln(3.35/0.35) = 5 × 2.26 = +11.3. A large positive skin addition from the damage itself.
Effect 2 (D increases because β increases): With kp = 2 md instead of 12 md, β = 4.11×10¹⁰/21.33 = 1.5×10¹⁰ vs 1.66×10⁹ at k=12 md — a β increase of ~9×. This feeds directly into D, multiplying the turbulence penalty at the same flow rate.
This compound effect explains why stimulating a damaged gas well delivers far greater production improvement than treating the same well with no initial damage: you simultaneously collapse both S and D.
Section 4
Gas Well IPR — The Quadratic Inflow Equation
Gas well inflow is inherently quadratic because of turbulence. Understanding the A–B form of the deliverability equation is essential for designing gas well completions and interpreting multirate test data.
4.1 The Gas Deliverability Equation
For gas wells, the standard pseudo-pressure form of the deliverability equation is:
m(p̄R) − m(pwf) = A·qg + B·qg² (psia²/cp)
Δm(p) = A·qg + B·qg²
A — Darcy Coefficient, psia²/cp per Mscf/d.
A = 1422T[ln(0.472re/rw) + S]/(kgh). Contains the treatable Darcy skin S. Acid and stimulation reduce A. Proportional to q — linear IPR component.
B — Non-Darcy Coefficient, psia²/cp per (Mscf/d)²
B = 1422T·D/(kgh). Contains turbulence coefficient D. Perforation optimisation and rate management reduce B. Proportional to q² — quadratic IPR component.
Solving for qg at a given pwf:
qg = [−A + √(A² + 4B·Δm(p))] / (2B) (Mscf/d)
This quadratic solution confirms the gas well IPR is always a concave-downward curve (when plotted as pwf vs q), with increasing curvature as B grows relative to A. The AOF (Absolute Open Flow, when pwf = 0) is correspondingly reduced by turbulence.
4.2 How Turbulence Shapes the Gas IPR
Figure 3.2.3 — Effect of turbulence coefficient B on gas well IPR shape. All three wells have identical reservoirs and Darcy skin (same A). The high-B well has its AOF reduced to ~61% of the B=0 case purely from turbulence. The increasing curvature with B is the visual signature of non-Darcy flow on the IPR plot.
4.3 Rawlins–Schellhardt Back-Pressure Equation
The classical empirical back-pressure equation, still widely used in the field:
qg = C · (p̄R² − pwf²)n (Mscf/d)
The delivery exponent n runs from n = 1.0 (pure Darcy laminar flow) to n = 0.5 (fully turbulence-dominated flow). The guidelines state: "n is an indicator of the flow regime within the gas reservoir." A measured n < 0.8 is a strong indicator that turbulence is significant and D must be quantified before designing a treatment.
n value
Flow Regime
Turbulence Status
Recommended Response
n = 1.0
Pure Darcy laminar
None — Dq = 0
Single-rate skin = S. Standard treatment design.
0.85–1.0
Near-Darcy
Minor
Confirm with two rates. Dq likely < 15% of S′.
0.65–0.85
Mixed regime
Moderate
Multi-rate test mandatory. Separate S and Dq before treatment.
0.50–0.65
Turbulence-dominated
Severe
Perf redesign is priority. Acid alone will not restore productivity.
n = 0.5
Fully turbulent
Maximum
AOF severely constrained by inertia. Rate management and perf optimisation critical.
The pressure-squared approach (p̄R² − pwf²) assumes μZ = constant over the pressure range, a reasonable approximation only when pR < 1,500 psi or when the pressure changes involved are small relative to the mean pressure.
The pseudo-pressure m(p) approach is always correct. It incorporates the actual μZ variation with pressure through numerical integration (the trapezoid summation). "The pseudo-pressure approach is valid for all pressure ranges, and given that most test analyses and deliverability estimates are today conducted using computer software, there is no real reason not to use this correct approach."
Field rule: Use pseudo-pressure for all deliverability work when pR > 1,500 psi (which covers virtually most North Sea wells). The pressure-squared form is acceptable only for low-pressure shallow gas reservoirs or quick screening calculations.
Low-permeability gas wells can take days or weeks to stabilise at each rate. Isochronal tests avoid this by flowing at each rate for the same fixed time period (so the same drainage radius is investigated at each rate) and shutting in between rates to restore pressure.
The procedure: unstabilised points at the end of each isochronal flow period are plotted to define the slope B. A single extended production period at a final flowrate until pseudo-steady state provides the stabilised point. A line through this stabilised point drawn parallel to the unstabilised line gives A from its y-intercept. This method is practical for k < 5 md gas wells where full stabilisation takes weeks.
For FK-7 (k = 12 md), stabilisation time at each rate is typically 1–3 days, so a conventional multi-rate test with 2–3 flow periods is feasible and preferred over the isochronal approach.
Section 5
Separating S and Dq from Well Test Data
The diagnostic workflow for decomposing total skin into treatable and non-treatable components, the core skill for Module 03 problem-set completion.
5.1 Four-Step Diagnostic Framework
Step 1: Screen by well type and rate
Oil well, q < 3,000 stb/d, k > 20 md: Dq very likely negligible. Treat S′ ≈ S and proceed with damage diagnosis (GK-22 confirmed this path).
Gas well or high-rate oil: Always assume Dq may be significant. Proceed to Step 2.
Step 2: Multi-rate test available?
Yes (preferred): Plot S′ vs q or Δp/q vs q. If S′ increases linearly with q, turbulence is confirmed. Slope = D, intercept = S.
No (single-rate only): Calculate D theoretically from Katz or Dake β correlation. Estimate Dq and subtract from S′ to estimate S.
Step 3: Compare Dmeasured vs Dtheoretical
If Dmeasured >> Dtheoretical (e.g. > 3×), suspect plugged or partially effective perforations (hp,eff << hp,nominal). Re-perforating under underbalance is the priority action over acid. (FK-7 showed 8.4× discrepancy → only 35% of perfs open.)
Step 4: Assess S separately for treatment
Once D is known, S = S′ − Dq at the test rate. If S > +2: formation damage present → acid or workover. If S ≈ 0 to +2: damage minor, turbulence is the main problem → completion optimisation is the priority.
5.2 S′ vs q Rate-Sensitivity Diagnostic Plot
When multiple flow periods at different rates are available, the most powerful single diagnostic is the S′ vs q plot:
Figure 3.2.4 — Rate-sensitivity plot for skin decomposition. Case A (straight line, positive slope) = clean S + Dq separation, ideal. Case B (flat line) = pure formation damage, no turbulence. Case C (upward-curving) = non-linear effects such as two-phase flow near the wellbore, condensate blockage, or depletion between test periods, requires more complex analysis beyond the simple linear decomposition.
5.3 Module 03 Summary Table — GK-22 vs FK-7
Bringing together the GK-22 and FK-7 analyses to illustrate the contrast between oil well (turbulence negligible) and gas well (turbulence dominant) skin decomposition:
GK-22 Canonical Data (locked for all Module 03 topics):
ko = 85 md · h = 42 ft · μo = 1.8 cp · Bo = 1.32 rb/stb ·
p̄R = 4,200 psi · pwf = 2,500 psi · rw = 0.35 ft ·
re = 1,650 ft · q = 782 stb/d · S′ = +14
Parameter
GK-22 Oil (782 stb/d)
FK-7 Gas (8 MMscfd)
FK-7 Gas (18 MMscfd)
FK-7 Gas (25 MMscfd design)
Measured S′
+14.0
+11.2
+18.4
(+23.4 predicted)
Dq (turbulence skin)
0.001 (negligible)
+5.76
+12.96
+18.0 (predicted)
S (Darcy damage skin)
+14.0 (confirmed)
+5.44
+5.44
+5.44
Turbulence % of S′
≈0%
51%
70%
77%
Non-Darcy coefficient D
9.4×10²⁻ d/stb
0.720 d/MMscf (from two-rate test)
Priority intervention
Matrix acid (Sd analysis Topics 3.3–3.4)
Acid + perf optimisation
TCP reperforation + acid
TCP reperforation (primary) + acid
Risk of misdiagnosis (single rate)
Low, Dq negligible
High
Very high
Extreme
Many exploration and early development DSTs are single-rate. When multi-rate data is unavailable, the following protocol minimises misdiagnosis risk:
1. Calculate D theoretically using the Katz correlation with the nominal perforated interval and the well test's kh estimate. Use both Katz and Dake to bracket the result.
2. Calculate Dq at the test rate: Dq = D × qtest
3. Estimate S: S = S′ − Dq. If Dq / S′ < 15%, proceed with S = S′ as a reasonable approximation. If Dq / S′ > 30%, flag that multi-rate testing is required before committing to treatment design.
4. Sensitivity check: Re-run with both Katz and Dake β values and with hp,eff = 50% of nominal (to account for partial perforation effectiveness). Present S as a range.
5. Plan a multi-rate test at the earliest opportunity, ideally the first clean-up production period, to confirm the decomposition. Decisions based on single-rate S′ for gas wells carry a formal uncertainty flag in the well delivery report.
MODULE 03 PBL TASK
GK-22 (your primary well): Having confirmed that S′ = S = +14 with negligible turbulence, you now proceed to Topic 3.3 to decompose the Darcy skin S into its formation damage component Sd using Hawkins' formula. The question becomes: how much of S = +14 is from drilling filtrate invasion vs perforation geometry vs other factors?
FK-7 (supplementary analysis): Using the FK-7 two-rate DST data provided in the SP-2 data pack, complete the S/Dq separation worksheet. Calculate the shot density upgrade required (changing spf from current 4 spf to target value) to reduce Dq below +3.0 at the design production rate of 25 MMscfd.
Interactive Tools
Total Skin & Non-Darcy Simulators
Three interactive tools to build intuition for S + Dq decomposition, gas well IPR curvature, and the effect of perforation design on turbulence. GK-22 and FK-7 presets are loaded by default.
FK-7 baseline: k=12, h=65, hp=52, q=18 MMscfd
Key experiments:
► Halve hp (26 ft): D quadruples → Dq x4
► Reduce k to 2 md (damage): β spikes ~9×
► Increase hp to 65 ft: D drops significantly
► Compare FK-7 with GK-22 (k=85, h=42)
Assessment
Knowledge Check — Total Skin S′ and Non-Darcy Flow
Ten questions covering the S + Dq decomposition, physical meaning of D, gas well IPR, test interpretation, and the GK-22/FK-7 PBL context. Score ≥ 8/10 before proceeding to Topic 3.3.
1. The total skin S′ measured in a well test is defined as S′ = S + Dq. What does the Darcy skin S represent, and what is the key engineering distinction from Dq?
D is the complete and correct answer. S is the Darcy (rate-independent) skin arising from formation damage, geometric restrictions, and completion effects — all of which can be reduced by acid, reperforation, or hydraulic fracturing. Dq is the non-Darcy (rate-dependent) turbulence component that grows linearly with flow rate. Acid can partially reduce D (by improving near-wellbore permeability, which lowers β) but cannot eliminate turbulence caused by the fundamental geometry of convergent flow through perforations.
2. Using the GK-22 canonical data (ko = 85 md, h = 42 ft, μo = 1.8 cp, Bo = 1.32 rb/stb, rw = 0.35 ft, q = 782 stb/d, S′ = +14), the calculated Dq is approximately 0.001. What does this confirm?
C is correct. Dq = 0.001 is 0.007% of S′ = +14 — analytically negligible. This confirms that for GK-22 at 782 stb/d in 85 md rock, the entire measured skin is formation damage. The diagnostic protocol is complete: S = S′ = +14, and Topics 3.3–3.6 will decompose this into its damage sub-components (Sd, Sp, Sc) to design the correct intervention. No multi-rate test is required for GK-22.
3. FK-7 gas well DST data: S′₁ = +11.2 at q₁ = 8 MMscfd; S′₂ = +18.4 at q₂ = 18 MMscfd. Calculate D and the Darcy skin S.
B is correct. D = (S′₂ − S′₁)/(q₂ − q₁) = (18.4 − 11.2)/(18 − 8) = 7.2/10 = 0.72 d/MMscfd. Then S = S′₁ − D×q₁ = 11.2 − 0.72×8 = 11.2 − 5.76 = +5.44. Verification: S′₂ − D×q₂ = 18.4 − 0.72×18 = 18.4 − 12.96 = +5.44 ✓. At 18 MMscfd, Dq = 12.96 accounts for 70% of S′ = 18.4. Acid addressing S = +5.44 is correct, but perforation optimisation is the larger-value intervention.
4. In the Forchheimer equation (−dP/dr = μu/k + βρu²), the inertial coefficient β is primarily a function of:
A is correct. β is primarily a function of permeability via the Katz correlation: β = 4.11×10¹⁰ / k1.33 and Dake: β = 2.73×10¹⁰ / k1.10 — both strong inverse power-law relationships. This is why formation damage (reducing near-wellbore kp) is doubly harmful: it increases S (Darcy skin) AND increases β (which increases D, amplifying Dq). Gas viscosity and density appear in the D formula as separate terms but do not affect β directly.
5. The non-Darcy coefficient D contains hp² in the denominator. If the effective perforated interval hp is halved due to 50% of perforations being plugged, how does D change?
C is correct. D ∝ 1/hp². If hp halves, D increases by 1/(0.5)² = 4×. This explains the FK-7 FK-7 discrepancy (Dmeasured = 8.4× Dtheoretical): back-calculating hp,eff = 52/√8.4 ≈ 18 ft, meaning only 35% of nominal perforations are open. This hp² rule is the most practically important formula in the D expression — it means perforation cleanup has a quadratic, not linear, benefit on turbulence reduction.
6. In the gas well deliverability equation Δm(p) = A·qᵛ + B·qᵛ², the delivery exponent n in the Rawlins-Schellhardt equation (q = C(Δp²)n) takes the value n = 0.5 when:
C is correct. When inertia completely dominates (B·q² >> A·q), then Δm(p) ≈ B·q², so q ≈ (Δm(p)/B)¹² — a square-root relationship equivalent to n = 0.5 in the Rawlins-Schellhardt form. Conversely, n = 1.0 means pure laminar Darcy flow (A·q dominates). Real wells fall between 0.5 and 1.0. A measured n < 0.8 from a multi-rate test is a strong signal that turbulence is significant and D must be quantified before treatment design.
7. A gas well measured Dmeasured = 1.20 d/MMscfd from a multi-rate test. Theoretical Dcalc using Katz correlation and nominal hp = 60 ft gives Dcalc = 0.075 d/MMscfd. What is the most likely explanation and recommended action?
C is correct. Ratio = 1.20/0.075 = 16. Since D ∝ 1/hp²: hp,eff = 60/√16 = 60/4 = 15 ft. Only 15/60 = 25% of nominal perforations are effectively contributing to flow. The Katz uncertainty is typically 2–3×, not 16×, so correlation error (Option A) cannot explain this discrepancy. TCP reperforation with 1,000+ psi underbalance to clean all 60 ft of perforations is the clear priority. The expected D post-workover would drop from 1.20 to 0.075 d/MMscfd, reducing Dq at 20 MMscfd from 24.0 to 1.5 — a transformational improvement.
8. An acid treatment is designed for a gas well based on a single-rate well test giving S′ = +20. The actual Darcy skin S was +2, with Dq = +18 (turbulence dominates at the test rate). After acid successfully reduces S from +2 to −1, what is the new S′ at the same production rate?
B is the best answer. The acid removes the Darcy damage skin (S: +2 → −1, a gain of 3 units). Dq at the same production rate will not change significantly because the flow velocity and perforation geometry are unchanged. New S′ = S_new + Dq = (−1) + 18 = +17. The well improved by only 3 skin units despite the full acid treatment because turbulence dominated and was not addressed. If the same 3 skin-unit improvement had been achieved by increasing shot density (reducing D and thus Dq by 3 units), the relative benefit at high rates would be much larger because D reduction is multiplied by q. This case illustrates why Option C (slight improvement from better k reducing β) is acknowledged but B is the primary answer — the β improvement from S → −1 is secondary to the fundamental hp² problem.
9. Which statement about the Forchheimer diagnostic plot (Δp/q vs q) is CORRECT?
B is correct. The Forchheimer plot (Δp/q vs q) gives a straight line when Forchheimer behaviour holds: y-intercept = A (Darcy coefficient, contains S and reservoir properties) and slope = B (non-Darcy coefficient, contains D). A curved upward-bowing line (Option A) does indicate non-linear effects, but these could include two-phase flow, condensate blockage, or depletion between test periods — not necessarily two-phase flow specifically. A flat line means no rate-dependent skin (pure Darcy, Dq = 0), but the y-intercept (A) still encodes the formation damage skin S — so a flat line does NOT mean no damage (Option C is wrong). The plot works for oil wells too, using Δp/q instead of Δm(p)/q (Option D wrong).
10. GK-22 (S′ = +14, Dq ≈ 0) and FK-7 (S′ = +18.4 at 18 MMscfd, S = +5.44, Dq = +12.96) are both candidates for intervention. Rank the following four actions by expected benefit to EACH well (most benefit first), using the J ratio = (7 + S′old)/(7 + S′new):
C is correct. GK-22 has S = S′ = +14 all from damage — acid is clearly the primary intervention. FK-7 analysis: TCP reperforation reduces Dq from 12.96 to ~1.5 (hp clean-up); new S′ = 5.44 + 1.5 = +6.94. J ratio for reperforation alone = (7+18.4)/(7+6.94) = 25.4/13.94 = 1.82×. Subsequent acid reducing S from +5.44 to 0: new S′ = 0 + 1.5 = +1.5. J ratio vs original = (7+18.4)/(7+1.5) = 25.4/8.5 = 2.99× total improvement. For FK-7, reperforation alone delivers 1.82×, acid alone delivers only (25.4/11.44) = 2.22×, but the combination delivers 2.99× — and the order matters: reperforation first cleans the tunnels so acid can penetrate effectively.
NEXT STEPS
Topic 3.2 complete. You can now decompose total skin into rate-dependent (Dq) and rate-independent (S) components, calculate D theoretically from Katz/Dake β correlations, interpret the quadratic gas well IPR, and apply the four-step diagnostic framework to both GK-22 and FK-7.
Proceed to Topic 3.3: Formation Damage Skin (Sd) — which uses Hawkins' formula to quantify the physical restriction caused by mud filtrate invasion, cement filtrate, and other near-wellbore permeability alteration. For GK-22, you will calculate Sd from estimated invasion depth and permeability ratio, building the full skin audit.
All GK-22 canonical data used in this topic is locked for consistent use across Topics 3.3–3.6 and the Module 03 PBL package.