Limitations of the Linear PI and Non-Linear IPR Models
The straight-line IPR (Q = J·ΔP) is a powerful but conditional model. Three conditions break it: reservoir pressure crossing the bubble point, high flow velocities generating non-Darcy turbulence, and production from gas wells. Knowing exactly when to switch models — and which model to use — is the mark of a competent production engineer.
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Lecture 2.4: When the Straight Line Lies — Non-Linear IPR Models for Real Wells
19:50
Begins with a field case where a straight-line PI test gave a 40% over-prediction of AOFP because the well was operating near its bubble point. Derives the physical reasons for non-linearity, presents Vogel's equation and its derivation basis, constructs composite IPR curves for partially saturated reservoirs, covers the back-pressure (Jones) equation for gas wells, and ends with a structured model-selection decision framework tested on four well scenarios.
Topics 2.1 through 2.3 developed the Productivity Index J and its normalised form SPI on the assumption that flow is single-phase, the reservoir is undersaturated (above the bubble point), and flow is in the Darcy (laminar) regime. Those conditions define the zone of validity of the linear model Q = J·(P̄ − Pwf). Understanding that zone of validity precisely and having replacement models ready when those conditions fail, is the subject of this topic.
The linear model fails in three distinct ways. First, as reservoir pressure depletes toward or below the bubble point, free gas evolves in the reservoir and near the wellbore. Gas saturation builds up, oil relative permeability kro decreases, and the IPR curves downward below the linear prediction. Vogel (1968) provided the most widely used empirical correction for this behaviour. Second, in high-rate wells (particularly gas wells and high-productivity oil wells), fluid velocity near the wellbore becomes large enough that inertial (turbulence) forces add to viscous pressure drop. This makes the pressure drop grow faster than linearly with rate, bending the IPR downward at high rates. Third, gas has strongly pressure-dependent properties (μg, z) that make the simple linear ΔP model structurally wrong; the back-pressure equation or pseudo-pressure formulation must be used instead.
This topic covers all three failure modes with their diagnostic signals, corrective models, and worked examples using the Karama Field KRM-4 scenario. The interactive simulator allows you to build composite IPR curves, compare linear vs Vogel predictions, and see when each matters for the production target assessment.
LEARNING OBJECTIVES
After completing this topic, you will be able to:
1. State the three physical conditions under which the linear PI model becomes invalid and explain the mechanism behind each. 2. Recognise diagnostic signals in well test data that indicate non-linear IPR behaviour. 3. Apply the Vogel equation to construct an IPR below the bubble point and calculate AOFP from a single-point well test. 4. Construct a composite IPR for a well that is above the bubble point at reservoir pressure but whose drawdown extends below the bubble point. 5. Use the back-pressure (Jones/LIT) equation for gas wells and explain why squared-pressure or pseudo-pressure formulations are needed. 6. Identify when non-Darcy (turbulence) effects are significant and apply the D-factor correction to the effective skin. 7. Select the appropriate IPR model for a given reservoir and operating condition using a structured decision framework. 8. Quantify the error introduced by using the linear PI model when the Vogel IPR should have been used.
PREREQUISITES
Topics 2.1, 2.2, and 2.3 are direct prerequisites. You must be comfortable with the PSS radial inflow equation, the PI definition J = Q/ΔP, skin S, the IPR straight line, and AOFP before studying the Vogel correction. You also need the concept of relative permeability kro from reservoir engineering (Module 03 preview).
PBL CONNECTION — KRM-4 PROBLEM SET
Sub-Problem 4 of the Karama Field KRM-4 problem set asks: given that reservoir pressure P̄ is expected to reach the bubble point (Pb = 3,650 psia) within 18 months, construct the Vogel IPR at P̄ = Pb and the composite IPR for an intermediate depletion state where P̄ = 4,200 psia. Determine whether the 1,200 STB/day production target can still be achieved with the current separator back-pressure of 3,100 psia, and if not, what Pwf is required.
Section 1 of 7
When Does the Linear PI Break Down?
Three physical mechanisms each produce a different pattern of deviation from the straight-line IPR. Identifying which mechanism is active, from well test data or operating history, determines which correction model to apply.
1.1 The Three Failure Modes — Overview
The linear PI assumes: (a) single-phase oil flow with constant k, μ, and B; (b) laminar (Darcy) flow throughout the drainage area; and (c) constant J throughout the drawdown range. Each failure mode violates one of these assumptions:
① Below-Bubble-Point Flow
When Pwf drops below Pb, gas exsolves. kro decreases with gas saturation. J is no longer constant, it falls at higher drawdown. IPR curves downward concavely. Model: Vogel or Composite IPR.
② Non-Darcy (Turbulent) Flow
At high rates near the wellbore, inertial pressure drop proportional to Q² adds to viscous ΔP proportional to Q. Effective skin increases with rate. IPR curves downward concavely. Model: Jones back-pressure equation.
③ Gas Well Properties
μg and z vary strongly with P. Linear ΔP form is structurally wrong. IPR may curve upward or downward depending on pressure regime. Model: Pseudo-pressure m(P) or back-pressure equation.
1.2 Failure Mode 1 — Below-Bubble-Point Physics
Above the bubble point, Pwf > Pb, the reservoir and near-wellbore zone contain single-phase undersaturated oil. The effective permeability to oil equals absolute permeability: ko = k. The PI is constant and the IPR is linear.
Once Pwf drops below Pb, gas exsolves from solution in the pore space. The gas phase forms bubbles which coalesce into a continuous gas phase at the pore scale. This gas occupies pore space, reducing the space available for oil flow. The relative permeability to oil kro is now less than 1.0 and decreases as gas saturation Sg increases. The effective J becomes:
Japparent(Pwf) = 0.00708·k·kro(Sg)·h / [μo(P)·Bo(P)·denom]k_ro(S_g): falls from 1.0 at S_g=0 to 0 at residual oil saturation
S_g: increases as more gas exsolves below P_b
μ_o(P), B_o(P): also change with P below P_b (secondary effect)
→ J_apparent DECREASES as drawdown increases below P_b → IPR curves downward
Figure 2.4.1 — Effect of Bubble Point on IPR Shape
1.3 Diagnostic Signals for Non-Linear IPR
Field engineers can detect non-linearity before building a full IPR model by examining:
In the original Darcy's law derivation, flow is laminar (viscous forces dominate, inertial forces negligible). This holds at the Darcy velocity range typically encountered in reservoir matrix far from the wellbore. But near the wellbore, where the drainage area shrinks as flow converges radially, fluid velocity increases dramatically. At sufficiently high velocities, inertia-driven turbulence adds a pressure drop proportional to Q² (the Forchheimer term):
ΔPtotal = A·Q + B·Q²A = Darcy (viscous) coefficient [psi·day/STB], proportional to 1/J from laminar inflow equation
B = Non-Darcy (inertial) coefficient [psi·(day)²/STB²], depends on β (Forchheimer turbulence parameter)
For gas wells: A = [1,424T/(kh)] × [ln(r_e/r_w) − 0.75 + S] | B = [1,424Tβμ_g]/(kh²)
The B·Q² term becomes significant when B·Q > ~10% of A (i.e., at high rates)
Which wells are affected? Non-Darcy turbulence is most significant in: (1) high-rate gas wells (nearly always); (2) high-rate oil wells with J > 10–20 STB/d/psi; (3) gravel-pack completions with high flow concentration through the pack; (4) hydraulically fractured wells at the fracture face. For moderate-rate oil wells (J < 5 STB/d/psi, typical of Karama Field), the B·Q² term is usually <5% of total ΔP and the linear PI is adequate.
1.5 Failure Mode 3 — Gas Well IPR Non-Linearity
Gas viscosity μg and compressibility factor z vary substantially with pressure. At 5,000 psia in a typical dry gas reservoir, μg ≈ 0.025 cp and z ≈ 0.92; at 2,000 psia, μg ≈ 0.016 cp and z ≈ 0.82. These 35–56% changes in fluid properties mean the simple linear ΔP model significantly mis-represents inflow. The industry uses three progressively rigorous approaches:
Squared Pressure
Qg = C·(P̄²−Pwf²)n. The back-pressure equation. Empirical; n=1 is fully turbulent, n=0.5 is fully Darcy. Simple to use; adequate at P<2,000 psia where P² linearity holds. Covered in this topic.
Pseudo-Pressure m(P)
Qg ∝ [m(P̄)−m(Pwf)] where m(P) = ∫P_baseP 2P/(μz) dP. Rigorous. Valid at all pressures. Required for HPHT gas wells. Covered in Module 04.
Linear ΔP (avoid)
Qg ∝ ΔP directly. Only valid at very low pressure where μg and z change little. For most gas reservoir conditions this is wrong and should not be used in production engineering.
The dimensionless Forchheimer number Fo quantifies the ratio of inertial to viscous pressure drop:
Fo = β·ρ·v / μβ = Forchheimer turbulence parameter (m³/m or ft³/ft), function of k (higher k → lower β but higher v)
ρ = fluid density | v = Darcy velocity near wellbore (m/s or ft/s) | μ = viscosity
When F_o > 0.1: turbulence contributes >10% to ΔP, non-Darcy correction required
When F_o > 1.0: turbulence dominates; linear PI is completely invalid
For a typical gas well: v at rw = Q/(2π·rw·h) can reach 1–5 m/s at production rates of 10–50 MMscf/day. At these velocities, Fo typically ranges from 0.5 to 5.0, well into the turbulent regime. For a typical low-rate oil well (Q = 500 STB/day, rw = 0.354 ft, h = 95 ft), v at rw ≈ 0.0003 m/s and Fo ≈ 0.02, laminar, linear PI is valid.
Section 2 of 7
The Vogel IPR — Empirical Correction Below the Bubble Point
Vogel (1968) ran reservoir simulation on a wide range of dissolved-gas drive reservoirs and found that a single dimensionless equation describes IPR shape below the bubble point to within ±10% for most practical conditions. It remains the most widely used non-linear IPR model four decades later.
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Worked Tutorial: Constructing and Using the Vogel IPR — KRM-4 at Depletion
13:25
Derives the Vogel equation from its simulation basis, demonstrates the single-point test method for determining Qmax, constructs a full Vogel IPR for KRM-4 at P̄ = Pb = 3,650 psia, compares it to the linear extrapolation to show AOFP over-estimation, and assesses whether the 1,200 STB/day production target remains achievable.
2.1 The Vogel Equation
Vogel (1968) curve-fitted simulation results for dissolved-gas drive reservoirs in pseudo-steady state and found that normalising both flow rate and wellbore pressure by their maximum values collapsed all curves to a single dimensionless relationship:
VOGEL IPR EQUATION
Q/Qmax = 1 − 0.2·(Pwf/P̄) − 0.8·(Pwf/P̄)²Q = flow rate at P_wf (STB/day)
Q_max = maximum rate (AOFP) at P_wf = 0 (STB/day)
P_wf = bottomhole flowing pressure (psia)
P̄ = current average reservoir pressure (psia)
Valid when P̄ ≤ P_b (reservoir is at or below the bubble point)
Vogel's equation applies over the range 0 ≤ P_wf/P̄ ≤ 1.0
Rearranged to give Qmax from a single test point:
Qmax = Qtest / [1 − 0.2·(Pwf,test/P̄) − 0.8·(Pwf,test/P̄)²]A single stabilised (Q_test, P_wf,test) measurement with known P̄ is sufficient to determine Q_max and therefore the complete Vogel IPR.
2.2 Physical Basis and Range of Validity
Vogel’s equation is an empirical fit to simulation results for solution-gas drive reservoirs. Its theoretical basis is the combination of Darcy’s law for two-phase flow with pressure-dependent kro(Sg) and the material balance for a closed reservoir. The equation is not theoretically derivable from first principles without those additional relationships, but its accuracy was validated against a wide range of reservoir properties (k, h, PVT properties, relative permeability curves) and found to be robust within ±10-20% of actual rates for most conditions.
Vogel’s equation is valid when:
✓
Reservoir pressure P̄ is at or below the bubble point Pb (fully saturated reservoir)
✓
Production is from a solution-gas drive or combined drive reservoir (not pure water drive)
✓
The reservoir is producing in pseudo-steady state
✓
No water or gas coning is occurring
When Vogel is less accurate: In water-drive reservoirs (high water saturation reduces relative permeability relationships), in wells with non-Darcy turbulence (combined effects not captured), or in HPHT reservoirs where PVT properties deviate significantly from the Vogel simulation basis. For these cases, the modified Vogel equation with an efficiency factor, or a Standing correction, may be more appropriate.
2.3 Building the Vogel IPR Table — Step-by-Step
1
Obtain P̄ from a recent build-up or material balance. Vogel requires knowing current average reservoir pressure.
2
Run a single stabilised flow test at rate Qtest. Measure stabilised Pwf,test.
3
Calculate Qmax using the rearranged Vogel equation above.
4
Tabulate the IPR: For Pwf = P̄, 0.9P̄, 0.8P̄, … down to 0, calculate Q = Qmax·[1−0.2(Pwf/P̄)−0.8(Pwf/P̄)²] at each Pwf.
5
Verify consistency: Q at Pwf = P̄ should equal 0; Q at Pwf = 0 should equal Qmax. The mid-point Q at Pwf = 0.5P̄ should be 0.70·Qmax (check: 1 − 0.2×0.5 − 0.8×0.5² = 0.700).
WORKED EXAMPLE 1KRM-4 Vogel IPR at P̄ = Pb = 3,650 psia
Context: KRM-4 reaches bubble point P̄ = Pb = 3,650 psia. A single-point well test at this depletion state gives: Qtest = 900 STB/day at Pwf,test = 2,500 psia.
Compare to linear PI prediction (WRONG at P̄ = Pb):
Linear J = 0.60 STB/d/psi (from Topic 2.2 test, measured above P_b)
Q_linear at P_wf=3,100 = 0.60 × (3,650 − 3,100) = 330 STB/day
→ Linear model also misses target but slightly differently.
Linear AOFP = J × P̄ = 0.60 × 3,650 = 2,190 STB/day (OVER-estimated vs Vogel 1,845)
Error in AOFP = (2,190−1,845)/1,845 = +18.7% over-prediction by linear model
2.4 Standing’s Modification of Vogel — Accounting for Damage
Vogel’s original equation implicitly assumed zero skin (undamaged well). Standing (1970) introduced an efficiency factor EF (equivalent to Flow Efficiency FE from Topic 2.2) to extend Vogel to damaged wells:
Q/Qmax = 1 − 0.2·EF·(Pwf/P̄) − 0.8·(EF)²·(Pwf/P̄)²EF = Flow Efficiency = J_actual/J_ideal (same as FE from Topic 2.2, <1.0 for damaged wells, >1.0 for stimulated)
When EF=1.0: reduces exactly to the standard Vogel equation.
When EF<1.0: curves shift downward (lower Q at same P_wf), damaged well.
When EF>1.0: curves shift upward (higher Q at same P_wf), stimulated well.
If a well test is unavailable, Qmax for the Vogel IPR can be estimated from reservoir parameters using Fetkovitch’s (1973) relationship between Qmax and the linear PI J (measured above Pb) at the bubble point:
Qmax = J·Pb/1.8J = linear PI measured (or calculated) above P_b (STB/d/psi)
P_b = bubble point pressure (psia)
This relationship comes from evaluating the derivative of the Vogel equation at P_wf=P_b
and matching it to the linear PI slope at that pressure.
KRM-4 check: J = 0.60 STB/d/psi, Pb = 3,650 psia. Qmax,Fetkovitch = 0.60×3,650/1.8 = 1,217 STB/day. This is slightly lower than the 1,845 STB/day from the Vogel single-point test because the well test was at a more effective operating condition. The Fetkovitch method is a pre-depletion estimate; the well test value is preferred when available.
Section 3 of 7
The Composite IPR — Above and Below the Bubble Point
Most producing wells operate with P̄ above Pb but Pwf below Pb. A composite IPR stitches together the linear PI (above Pb) and the Vogel curve (below Pb) to cover the full operating range.
3.1 The Composite Model Concept
When P̄ > Pb but Pwf < Pb, the inflow regime transitions partway down the drawdown range:
From Pwf = P̄ down to Pwf = Pb: linear PI applies (single-phase oil, constant J). This portion of the IPR is a straight line.
From Pwf = Pb down to Pwf = 0: Vogel IPR applies (two-phase flow, declining J with increasing gas saturation). This portion is the Vogel curve anchored at the bubble point.
The two segments must be joined continuously at Pwf = Pb (same Q and same dQ/dPwf at the junction, though in practice only continuity of Q is enforced; the slopes will differ slightly).
3.2 Constructing the Composite IPR — The Procedure
1
Identify P̄ and Pb. Confirm P̄ > Pb (composite model applies; if P̄ ≤ Pb, use pure Vogel).
2
Calculate Q at the bubble point using the linear PI: Qb = J·(P̄ − Pb). This is the flow rate at the junction Pwf = Pb.
3
Calculate Qmax for the Vogel portion: Use Fetkovitch’s relationship anchored at the bubble point: Qmax = Qb + J·Pb/1.8. This ensures continuity and correct slope matching at the junction.
4
Linear portion (Pwf = P̄ to Pb): Q = J·(P̄ − Pwf). Straight line.
5
Vogel portion (Pwf = Pb to 0): Q = Qb + (Qmax−Qb)·[1−0.2(Pwf/Pb)−0.8(Pwf/Pb)²]. The Vogel equation is applied with Pb as the reference pressure for the below-bubble-point segment.
6
Combine and plot. The composite IPR consists of the linear segment above Pb and the Vogel segment below Pb, joined continuously at (Qb, Pb).
COMPOSITE IPR FORMULAE
Qb = J · (P̄ − Pb)Q at the bubble point junction
Qmax = Qb + J · Pb / 1.8Total AOFP of the composite IPR
WORKED EXAMPLE 2Composite IPR — KRM-4 at P̄ = 4,200 psia (Intermediate Depletion)
Given: J = 0.60 STB/d/psi (from well test), P̄ = 4,200 psia, Pb = 3,650 psia. Separator back-pressure imposes Pwf = 3,100 psia. Which J? This is the current-well composite, so it uses the measured Jmeas = 0.60 (giving Qb = 330 STB/day). For the post-stimulation scenario (skin removed, S = 0) you would instead use Jideal = 0.926, which gives Qb = 0.926×550 = 509 STB/day and a correspondingly higher composite Qmax. Match J to the scenario you are modelling.
3.3 Error Analysis — What Happens if You Ignore the Composite Model?
Using the linear PI to predict rate at Pwf = 3,100 psia (below Pb) gives:
Linear PI (wrong): Q = J × (P̄ − P_wf) = 0.60 × (4,200 − 3,100) = 660 STB/day
Composite IPR (correct): Q = 638 STB/day
Error = (660 − 638)/638 = +3.4% over-prediction at this operating condition.
For AOFP comparison:
Linear AOFP = J × P̄ = 0.60 × 4,200 = 2,520 STB/day
Composite AOFP = 1,547 STB/day
Error = (2,520 − 1,547)/1,547 = +62.9% over-prediction by linear model!
Critical finding: The AOFP error from using the linear PI (when the composite IPR should be used) grows dramatically as P̄ approaches and then falls below Pb. At moderate drawdowns (e.g., Pwf ≈ 0.8Pb), the error may be small (<10%). But the AOFP, which is used for artificial lift sizing and reserves estimation, can be over-estimated by 40–80%. Always use the composite or Vogel IPR when any part of the Pwf operating range is expected to fall below Pb.
Section 4 of 7
Non-Darcy Flow and Gas Well IPR
Gas wells almost universally require non-linear IPR models. This section covers the laminar-inertial-turbulent (LIT / Jones) approach, the back-pressure (Rawlins-Schellhardt) equation, and introduces the pseudo-pressure formulation that rigorously handles pressure-dependent gas properties.
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Gas Well IPR — From Back-Pressure Tests to LIT Analysis
15:15
Demonstrates a three-point back-pressure test analysis on a chalk gas well: plotting ΔP² vs Q on log-log (Rawlins-Schellhardt), determining n and C, calculating AOFP, and comparing to the LIT approach where A and B are separated. Shows how to detect turbulence-dominated vs laminar-dominated flow and what each implies for stimulation vs compression strategy.
4.1 The Rawlins-Schellhardt Back-Pressure Equation for Gas Wells
The earliest empirical gas well IPR model is the back-pressure equation (Rawlins and Schellhardt, 1935), also called the Deliverability Equation:
Qg = C · (P̄² − Pwf²)nQ_g = gas flow rate (Mscf/day)
C = deliverability coefficient (field-specific constant)
P̄ = average reservoir pressure (psia)
P_wf = bottomhole flowing pressure (psia)
n = deliverability exponent: n=1.0 (pure laminar Darcy flow), n=0.5 (fully turbulent flow)
In practice: 0.5 ≤ n ≤ 1.0. Most gas wells: n = 0.7–0.9
4.2 Determining C and n — The Back-Pressure Test
At least two stabilised flow points at different rates are needed. Taking logarithms:
log(Qg) = log(C) + n · log(P̄² − Pwf²)This is a straight line on a log-log plot of Q_g vs (P̄²−P_wf²).
Slope = n (the deliverability exponent)
y-intercept at (P̄²−P_wf²) = 1 gives log(C)
Run a 3–4 point back-pressure test: Produce at four different rates Q1,...,Q4, each stabilised. Measure Pwf at each. Know P̄ from a prior build-up.
Calculate ΔP² = P̄²−Pwf² for each rate.
Plot log(Q) vs log(ΔP²) on Cartesian axes of log paper. Fit a straight line. Its slope is n.
Calculate C from any test point: C = Qi / (ΔPi²)n.
WORKED EXAMPLE 3Back-Pressure Test Analysis — Chalk Gas Well
Test data: P̄ = 3,800 psia (from build-up). Three stabilised flow points:
Test Point
Q (Mscf/day)
P_wf (psia)
ΔP² = P̄²−P_wf² (psia²)
1
2,800
3,400
14,440,000 − 11,560,000 = 2,880,000
2
5,100
2,900
14,440,000 − 8,410,000 = 6,030,000
3
7,600
2,100
14,440,000 − 4,410,000 = 10,030,000
Step 1: Determine n from log-log plot (two-point calculation using points 1 and 3)
n = log(Q_3/Q_1) / log(ΔP_3²/ΔP_1²)
= log(7,600/2,800) / log(10,030,000/2,880,000)
= log(2.714) / log(3.482)
= 0.4336 / 0.5419 = 0.800
Step 2: Calculate C from test point 1
C = Q_1 / (ΔP_1²)^n = 2,800 / (2,880,000)^0.800
(2,880,000)^0.800 = exp(0.800 × ln(2,880,000)) = exp(0.800 × 14.874) = exp(11.899) = 146,900
C = 2,800 / 146,900 = 0.01906 Mscf/day/psia^(2n)
Step 3: Validate at test point 2
Q_pred = 0.01906 × (6,030,000)^0.800
(6,030,000)^0.800 = exp(0.800 × 15.612) = exp(12.490) = 266,100
Q_pred = 0.01906 × 266,100 = 5,073 Mscf/day (actual 5,100 → error <0.5% ✓)
Step 4: AOFP at P_wf = 14.7 psia
ΔP² = 3,800² − 14.7² = 14,440,000 − 216 ≈ 14,439,784
Q_AOFP = 0.01906 × (14,439,784)^0.800
(14,439,784)^0.800 = exp(0.800 × 16.485) = exp(13.188) = 534,000
Q_AOFP = 0.01906 × 534,000 = 10,178 Mscf/day ≈ 10.2 MMscf/day
4.3 The LIT (Laminar-Inertial-Turbulent) Jones Equation
The LIT approach separates the Darcy (A·Q) and non-Darcy (B·Q²) components explicitly, allowing the engineer to quantify how much pressure drop comes from turbulence vs laminar flow:
(P̄² − Pwf²) / Qg = A + B·QgA = Darcy coefficient (psia²/Mscf/day), function of k, h, T, geometry, S
B = Non-Darcy coefficient (psia²/(Mscf/day)²), function of β, k, h, T, gas properties
Plotting ΔP²/Q vs Q on Cartesian axes gives a straight line: intercept = A, slope = B
The D-factor (non-Darcy skin) is: D = B/A (units: day/Mscf)
4.4 Rate-Dependent (D-Factor) Skin in LIT Analysis
The non-Darcy contribution can be expressed as an equivalent rate-dependent skin S′ added to the mechanical skin S:
Stotal = S + D·QgS = mechanical skin (damage, stimulation — rate-independent)
D = non-Darcy flow coefficient (day/Mscf; also called Dq, high-velocity skin coefficient)
D·Q_g = turbulence contribution to skin (increases with rate)
Total effective skin S_total increases with Q → J_apparent decreases at higher rates → IPR curves downward
Engineering implication: In a well where turbulence is significant (high B in LIT analysis), stimulation that increases near-wellbore k can actually reduce non-Darcy pressure drop because higher k means lower velocity for the same rate. Conversely, adding a gravel pack (which can concentrate flow) may increase B even if it removes the formation skin. LIT analysis is essential for optimising high-rate well completions.
4.5 The Pseudo-Pressure Formulation — The Rigorous Approach
The most rigorous gas well IPR formulation uses the real-gas pseudo-pressure function m(P):
m(P) = ∫P_baseP 2P / (μg·z) dP (psia²/cp)Pseudo-pressure m(P) is tabulated by evaluating the integral numerically at each pressure step.
The gas well inflow equation becomes:
Q_g = kh·[m(P̄)−m(P_wf)] / [1,424·T·(ln(r_e/r_w)−0.75+S+DQ_g)]
This is exact for laminar gas flow with no approximation for μ_g or z variation.
Approach
Valid Pressure Range
Complexity
Accuracy
Use Case
Squared pressure (ΔP²)
P < 2,000 psia
Low
Good at low P
Low-P gas reservoirs
Back-pressure (C, n)
Empirical — any P
Low
Good with 3+ test pts
Field deliverability tests
Linear ΔP
Very low P (<400 psia)
Very low
Poor at high P
Avoid
Pseudo-pressure m(P)
All pressures
High (needs PVT)
Excellent
HPHT wells, tight gas, full analysis
LIT (Jones)
All pressures (with ΔP² or m(P))
Medium
Excellent + turbulence split
High-rate wells, turbulence quantification
m(P) is computed by numerical integration from a base pressure Pbase (usually 14.7 psia). At each pressure step, the integrand 2P/(μg·z) is evaluated using PVT correlations (Hall-Yarborough, Lee-Kesler, etc.):
P (psia)
μg (cp)
z
2P/(μg·z) (psia/cp)
m(P) (psia²/cp)
500
0.0125
0.940
84,960
21,900,000
1,000
0.0144
0.870
159,500
113,000,000
2,000
0.0180
0.820
271,000
395,000,000
3,000
0.0220
0.840
325,000
749,000,000
4,000
0.0264
0.900
336,000
1,103,000,000
The table shows that m(P) is very non-linear with P. Using [m(P̄)−m(Pwf)] instead of (P̄²−Pwf²) can change the calculated rate by 15–40% at high pressures. In tight gas and HPHT reservoirs, pseudo-pressure is mandatory for rigorous IPR construction. Module 04 covers this in full detail with worked numerical examples.
Section 5 of 7
IPR Model Selection — A Decision Framework
Choosing the right IPR model is an engineering decision, not a guessing game. This section provides a structured decision framework and applies it to four well scenarios at different depletion states and fluid types.
5.1 The Selection Framework — Four Questions
Before choosing an IPR model, answer four questions in sequence:
Is the well an oil well or a gas well? If gas: use back-pressure equation (C, n) or LIT equation. If oil: proceed to Question 2.
Is P̄ above, at, or below the bubble point Pb?
• P̄ > Pb and entire operating range (Pwf) is also > Pb: Linear PI
• P̄ > Pb but Pwf is expected to drop below Pb: Composite IPR
• P̄ ≤ Pb (reservoir fully saturated): Vogel IPR
Are flow rates high enough for non-Darcy effects? Check: if J > 10–20 STB/d/psi for oil, or rate > 10 MMscf/day for gas, use LIT analysis or add D-factor to skin.
Is there significant formation damage (skin > 0)? If Vogel applies and FE < 1.0: use Standing’s modified Vogel with EF = FE to account for damage in the Vogel equation.
Scenario
P̄ vs Pb
Pwf vs Pb
High Rate?
Fluid
Correct Model
KRM-4 early life
P̄ = 4,850 > Pb = 3,650
Pwf = 3,100 < Pb
No
Oil
Composite IPR
KRM-4 at Pb
P̄ = Pb = 3,650
Pwf < Pb
No
Oil
Vogel IPR
High-rate GoM well
P̄ = 6,500 > Pb = 2,800
Pwf = 1,200 < Pb
Yes (J>15)
Oil
Composite + D-factor
Chalk gas well
P̄ = 3,800 (gas)
n/a (gas)
Yes (always)
Gas
Back-pressure (C,n) or LIT
Tight gas HPHT
P̄ = 8,500 (gas)
n/a
Yes
Gas
Pseudo-pressure LIT
Fully undersaturated oil
P̄ = 5,200 >> Pb = 1,800
Pwf = 3,000 > Pb
No
Oil
Linear PI
5.2 Error Quantification — When Does the Wrong Model Matter?
Not every deviation from the true IPR is operationally significant. The following table shows the approximate error in rate prediction and AOFP from using the wrong model, as a function of how far below the bubble point the operating Pwf is:
Pwf/Pb ratio
Rate error (Linear vs Composite, at Pwf)
AOFP error (Linear vs Composite)
Engineering significance
> 0.95 (Pwf barely below Pb)
< 5%
10–20%
Minor at operating point; significant for reserves
0.75–0.95
5–15%
20–40%
Notable — use composite IPR
0.50–0.75
10–25%
40–70%
Significant — always use composite/Vogel
< 0.50 (Pwf << Pb)
> 25%
> 60%
Severe — linear PI completely invalid
5.3 Practical Workflow for Field Application
For a New Well (First Production)
1. Check PVT report for Pb. Note initial P̄ from DST.
2. If P̄ >> Pb and expected Pwf also > Pb: start with Linear PI.
3. As P̄ depletes toward Pb, transition to composite IPR (calculate annually).
4. After P̄ crosses Pb: switch to Vogel IPR.
5. Monitor GOR: rising GOR is the real-time signal that gas is evolving — update to Vogel.
For an Established Well
1. Run a two-rate PI test. If J from the two rates is consistent: linear IPR still valid.
2. If Jlow-rate > Jhigh-rate: non-linearity detected. Identify cause (bubble point vs turbulence).
3. Check GOR vs history: increasing GOR → below-bubble-point (Vogel). Constant GOR but J declining with rate → turbulence (LIT).
4. Build Vogel or LIT IPR from latest test data.
FIELD CASEError Consequence: North Sea Producer — Wrong IPR Leading to Over-Specified Pump
A North Sea chalk producer with P̄ = 3,800 psia and Pb = 3,400 psia was producing at Pwf = 2,500 psia (Pwf/Pb = 0.74). The completion engineer used the linear PI (J = 1.8 STB/d/psi) to design an ESP, predicting the well would produce 2,340 STB/day at the target Pwf = 1,800 psia.
Actual result after pump installation: The well produced 1,680 STB/day — 28% below the linear prediction. The composite IPR (built after the fact) predicted 1,720 STB/day — accurate to within 2%. The ESP was specified for 2,340 STB/day and was grossly over-engineered for the actual rate, running at 72% capacity. A smaller, cheaper pump would have been adequate.
Root cause: Using linear PI when the composite IPR was required. At Pwf/Pb = 0.74, the error from the linear model was ~25% in rate prediction — sufficient to cause a costly completion error.
Vogel’s equation was derived for vertical wells producing by solution-gas drive only. For wells producing at high GOR with significant gravity override, or for wells with horizontal components (near-horizontal wells in thick formations), Klins and Clark (1993) proposed modifications to the Vogel coefficients:
Q/Qmax = 1 − C1·(Pwf/P̄) − C2·(Pwf/P̄)²C_1 and C_2 are modified coefficients (vs Vogel's 0.2 and 0.8) that account for specific drive mechanism (solution gas, water drive, combined drive) and producing GOR. For most solution-gas drive wells: C_1 ≈ 0.20, C_2 ≈ 0.80 (original Vogel). For wells with high GOR: C_1 may range 0.15–0.25, C_2 from 0.75–0.85.
In practice, Vogel’s original coefficients are adequate for most oil production engineering calculations. The Standing modification (EF factor for damage/stimulation) is more commonly applied in the field than the Klins-Clark adjustment.
Section 6 of 7 — Interactive
Interactive Simulators — Non-Linear IPR Models
Three tools: (1) a Vogel IPR builder from a single well test point; (2) a Composite IPR builder that stitches linear and Vogel segments and compares to the (wrong) linear extrapolation; (3) a back-pressure gas well IPR calculator. Use these to prepare for Sub-Problem 4 of the KRM-4 problem set.
Experiments: Set defaults (KRM-4 at P̄=Pb=3,650 psia) — observe Q at separator Pwf=3,100 psia vs target 1,200 STB/day. Raise EF above 1.0 (stimulated) — watch Q_max grow. Set EF=0.65 (damaged) — compare to Topic 2.2 KRM-4 FE value. The Vogel curve shape changes with EF.
Simulator 2 — Composite IPR Builder (Linear + Vogel) vs Linear Extrapolation INTERACTIVE
Initialising…
Experiments: Default = KRM-4 at intermediate depletion (P̄=4,200; Pb=3,650). Compare composite Q at Pwf=3,100 (638 STB/day) to linear prediction. Reduce P̄ toward Pb — watch the linear segment shrink and the AOFP error grow. Set P̄ = Pb — the composite becomes pure Vogel.
Simulator 3 — Gas Well Back-Pressure IPR (C, n method) INTERACTIVE
Initialising…
Defaults match the chalk gas well from Worked Example 3 (C=0.01906, n=0.80, P̄=3,800). Change n from 0.80 to 0.50 (fully turbulent) — observe how the curve bends downward at high rates. Set n=1.0 (pure Darcy) — the gas IPR becomes a straight line in ΔP² terms.
PBL EXERCISE — SUB-PROBLEM 4 PREPARATION
Using Simulator 2 (Composite IPR): Set J = 0.60 STB/d/psi (your Sub-Problem 2 result), Pb = 3,650 psia, and P̄ = 4,200 psia (intermediate depletion). Record:
(a) Qb (rate at bubble point junction);
(b) Qmax (composite AOFP);
(c) Q at Pwf = 3,100 psia;
(d) Whether the 1,200 STB/day target is achievable at current back-pressure;
(e) Pwf required to achieve 1,200 STB/day (if target not achievable naturally).
Then set P̄ = Pb = 3,650 psia and use Simulator 1 (Vogel). Run the single-point calculation with Qtest = 900 STB/day at Pwf,test = 2,500 psia. Record Qmax and the operating Q at Pwf = 3,100 psia. These are the inputs to Sub-Problem 4.
Assessment — Section 7 of 7
Knowledge Check
Ten questions at problem-set difficulty covering Vogel equation, composite IPR construction, model selection, error quantification, non-Darcy flow, and gas IPR. Complete worked explanations provided for all questions.
1. A well produces 1,400 STB/day at Pwf = 2,200 psia with P̄ = Pb = 3,600 psia. Using the Vogel equation, what is Qmax?
P_wf/P̄ = 2,200/3,600 = 0.6111. Vogel factor = 1 − 0.2(0.6111) − 0.8(0.6111)² = 1 − 0.1222 − 0.8×0.3734 = 1 − 0.1222 − 0.2987 = 0.5791. Q_max = Q_test / factor = 1,400 / 0.5791 = 2,418 STB/day (option B). The method: compute the Vogel factor at the test-point P_wf/P̄ ratio, then divide the test rate by that factor.
2. At what value of Pwf/P̄ does the Vogel equation predict Q = 0.5 × Qmax?
Set Q/Q_max = 0.5: 1 − 0.2x − 0.8x² = 0.5, so 0.8x² + 0.2x − 0.5 = 0. Quadratic formula: x = [−0.2 + sqrt(0.04 + 4×0.8×0.5)] / (2×0.8) = [−0.2 + sqrt(1.64)] / 1.6 = [−0.2 + 1.2806] / 1.6 = 0.675 (option C). Note that P_wf/P̄ = 0.5 does NOT give Q = 0.5 Q_max (it gives 0.70 Q_max) — this is the key non-linearity of the Vogel model. The IPR is concave, so equal pressure reductions give diminishing rate gains at lower P_wf. Option A (the linear assumption that half-pressure gives half-rate) is wrong for Vogel.
3. For the composite IPR construction, what formula gives Q at the bubble point (the linear/Vogel junction), and what formula gives the total AOFP?
Q_b is simply the flow rate from the linear PI segment at P_wf = P_b: Q_b = J×(P̄−P_b). The Vogel portion adds additional production from P_wf = P_b down to P_wf = 0; its AOFP contribution is J×P_b/1.8 (from Fetkovitch’s matching of the Vogel derivative to the linear PI slope at the bubble point). Total Q_max = Q_b + J×P_b/1.8. Option A is the correct pair. Option B gives the wrong junction Q (J×P_b would be the AOFP of a well with P̄=P_b, not the junction rate). Options C and D are invented formulas with no physical basis.
4. J = 1.2 STB/d/psi, P̄ = 4,500 psia, P_b = 3,200 psia. A separator imposes P_wf = 2,000 psia. What is the composite IPR flow rate?
Since P_wf = 2,000 < P_b = 3,200, use the Vogel segment. Q_b = 1.2×(4,500−3,200) = 1.2×1,300 = 1,560. Q_max = 1,560 + 1.2×3,200/1.8 = 1,560 + 2,133 = 3,693. Vogel factor at P_wf=2,000: P_wf/P_b = 2,000/3,200 = 0.625; factor = 1−0.2(0.625)−0.8(0.625)² = 1−0.125−0.3125 = 0.5625. Q = Q_b + (Q_max−Q_b)×0.5625 = 1,560 + 2,133×0.5625 = 1,560 + 1,200 = 2,760 STB/day (option B). Option A (3,000) is the linear PI value — an over-estimate because P_wf is below P_b.
5. A linear PI test gives J = 2.5 STB/d/psi at P̄ = 5,000 psia (well above P_b = 2,500 psia). The engineer calculates linear AOFP = J×P̄ = 12,500 STB/day. What is the correct composite AOFP using Fetkovitch's approach?
Q_b = J×(P̄−P_b) = 2.5×(5,000−2,500) = 2.5×2,500 = 6,250. Q_max_Vogel = Q_b + J×P_b/1.8 = 6,250 + 2.5×2,500/1.8 = 6,250 + 3,472 = 9,722 ≈ 9,700 STB/day (Option B). The linear AOFP over-estimates by (12,500−9,722)/9,722 = +28.6%. This is a massive error for an artificial lift system that might be sized to produce at 80% AOFP (i.e., 10,000 STB/day from linear vs the real 7,778 STB/day). Even when P̄ is well above P_b, the composite AOFP is substantially lower than the linear extrapolation because the Vogel segment below P_b produces significantly less than the linear model predicts.
6. The producing GOR of a well has increased steadily from 400 scf/STB to 900 scf/STB over 6 months while reservoir pressure has declined from 4,100 to 3,800 psia. The bubble point is 3,600 psia. What does this imply for the IPR model, and why?
P̄ = 3,800 psia is above P_b = 3,600 psia, but P_wf is likely dropping below P_b during production (typical for wells producing at significant drawdown). When P_wf < P_b, gas exsolves from oil in the near-wellbore zone. This dissolved gas comes out of solution and is produced with the oil, causing the GOR to rise. The rising GOR is the field diagnostic for approaching or crossing the bubble point at P_wf. The composite IPR should be applied immediately — the linear segment covers P_wf = 4,100 down to P_wf = 3,600 (P_b), and the Vogel segment covers below P_b. Option A is wrong (a gas cap would produce reservoir gas at roughly constant GOR, not solution gas increasing progressively).
7. In the LIT (Jones) equation (ΔP²/Q = A + B·Q), what does a large B coefficient relative to A indicate, and what completion action would most effectively reduce it?
B is the non-Darcy (inertial/turbulent) coefficient. A large B/A ratio means that turbulent pressure drop (B×Q) contributes more than viscous pressure drop (A) to the total ΔP². Turbulence is driven by high velocity near the wellbore, and velocity = Q/(2π×r×h×porosity). Increasing near-wellbore permeability k (via fracturing) allows the same rate Q to flow at lower velocity, reducing turbulence. Stimulation that reduces the skin S (acid) reduces the A coefficient (Darcy term) but does not directly reduce B. Option A is partially correct (skin reduction reduces A) but does not address the B term. Option C correctly identifies the mechanism and the appropriate remedy.
8. A gas well back-pressure test gives n = 0.65 and C = 0.0050 (Mscf/day/(psia²)n). P̄ = 4,000 psia. What is the expected flow rate at P_wf = 2,500 psia?
ΔP² = P̄² − P_wf² = 4,000² − 2,500² = 16,000,000 − 6,250,000 = 9,750,000 psia². Q_g = C×(ΔP²)^n = 0.0050 × (9,750,000)^0.65. (9,750,000)^0.65: log10(9,750,000) = 6.989; 0.65×6.989 = 4.543; 10^4.543 ≈ 34,900. Q_g = 0.0050 × 34,900 = 175 Mscf/day (option A). The method: compute ΔP², raise to the power n, then multiply by C. The small C value here (0.0050) gives a modest rate; always verify the units of C and n before calculating.
9. The Standing modification of the Vogel equation uses an Efficiency Factor EF. If a well has FE = 0.60 (from Flow Efficiency analysis in Topic 2.2), what is the Standing-modified Vogel prediction for Q/Q_max at P_wf/P̄ = 0.70?
Standing’s modified Vogel: Q/Q_max = 1 − 0.2×EF×(P_wf/P̄) − 0.8×(EF)²×(P_wf/P̄)². With EF=0.60, P_wf/P̄=0.70: first term = 0.2×0.60×0.70 = 0.084; second term = 0.8×(0.60)²×(0.70)² = 0.8×0.36×0.49 = 0.141. Q/Q_max = 1 − 0.084 − 0.141 = 0.775 (option A). Note this exceeds the unmodified Vogel value at x=0.70 (0.468): because EF<1 shrinks the coefficients on the P_wf/P̄ terms, less is subtracted. The EF factor lowers the damaged well’s absolute Q_max, while the normalised Q/Q_max ratio rises — remember the comparison is meaningful only at equal Q_max.
10. A well is producing with P̄ = 4,800 psia, P_b = 4,600 psia, and the separator imposes P_wf = 3,200 psia. The engineer uses the linear PI (J = 0.9 STB/d/psi) and predicts Q = 1,440 STB/day. Is this an over-estimate, under-estimate, or about right, and why?
P̄ = 4,800 > P_b = 4,600 (only 200 psi above bubble point), but P_wf = 3,200 < P_b = 4,600. The linear PI is valid only for P_wf ≥ P_b = 4,600. Since the operating P_wf = 3,200 is 1,400 psi below P_b, a large portion of the IPR falls in the Vogel (non-linear, downward-curving) zone. The composite model gives a lower rate at P_wf = 3,200 than the linear extrapolation. Check with composite: Q_b = 0.9×(4,800−4,600) = 0.9×200 = 180 STB/day. Q_max = 180 + 0.9×4,600/1.8 = 180 + 2,300 = 2,480. Vogel at P_wf=3,200: x=3,200/4,600=0.696; factor=1−0.139−0.388=0.473. Q=180+2,300×0.473=180+1,088=1,268 STB/day. Linear: 0.9×(4,800−3,200)=1,440. So linear over-estimates by (1,440−1,268)/1,268=+13.6%. Option A is correct. Option B is wrong because even though P̄>P_b, the linear PI is only valid when P_wf is also above P_b.
TOPIC COMPLETE — NEXT STEPS
You have completed Topic 2.4: Limitations of the Linear PI and Non-Linear IPR Models. You are now equipped to handle the full range of single-phase and two-phase inflow modelling scenarios. Next topics:
Topic 2.5 — Skin Factor Components: Decomposes total skin into mechanical damage, perforation, completion, and partial penetration components — essential for designing targeted stimulation treatments.
Module 04 — Multiphase Inflow Performance: Extends Vogel, composite IPR, and pseudo-pressure gas IPR to the full multiphase flow regime, including water breakthrough and GOR management.
KRM-4 Sub-Problem 4: Construct the composite IPR at P̄ = 4,200 psia and the Vogel IPR at P̄ = Pb = 3,650 psia. Determine artificial lift requirements to maintain the 1,200 STB/day target through depletion.