The flow equations that govern IPR are not just functions of permeability and pressure. They are functions of geometry. The shape of the drainage area, the distance to boundaries, the orientation of the wellbore, and the number of layers all directly alter the deliverability calculation. Getting geometry wrong means getting J* wrong.
Every IPR equation, Darcy steady-state, pseudo-steady-state, Vogel, Fetkovich, contains geometric terms: the drainage radius re, the wellbore radius rw, the shape factor CA, the skin S, and the pay thickness h. These are not decorative parameters. A well draining a 640-acre square with the wellbore off-centre will have a fundamentally different productivity index than an identical well at the centre and that difference arises entirely from geometry, not from rock quality or fluid properties.
Topic 1.4 builds directly on Topics 1.1–1.3 (PVT, FVF, permeability). Where those topics established the rock and fluid inputs to Darcy's Law, this topic addresses the geometric inputs. Together they give you the complete toolkit to calculate J* from first principles, which is exactly what Module 07 Nodal Analysis requires.
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Lecture: Geometry Matters — How Boundary Shape Controls Deliverability
17:10
Uses the Gashaka GK-22 field layout to demonstrate why identical permeabilities and pressures yield very different PI values depending on well location within the drainage polygon. Derives the ln(r_e/r_w) geometric factor from first principles and shows how C_A and skin replace it in non-circular geometries. Includes animated pressure funnel visualisation.
LEARNING OBJECTIVES
After completing Topic 1.4, you will be able to:
1. Derive and apply the ln(re/rw) geometric factor in steady-state and pseudo-steady-state IPR equations for vertical wells. 2. Distinguish transient, steady-state, and pseudo-steady-state flow regimes, identify which boundary condition applies to a given reservoir scenario and select the correct IPR model. 3. Apply drainage area shape factors (CA) from the Dietz tables to quantify the penalty of non-circular drainage shapes and off-centre well placement. 4. Apply Joshi's equation to calculate horizontal well productivity and compare it to an equivalent vertical well. 5. Calculate composite IPR for multilayer reservoirs with unequal pressures and predict the cross-flow shut-in pressure pwfo. 6. Define and calculate geometric skin contributions: partial penetration skin Sc, perforation skin Sp, and deviation/inclination skin Sθ. 7. Integrate drainage geometry with permeability and PVT inputs to construct a complete J* calculation from reservoir data. 8. Apply the Module 07 GK-22 drainage geometry data to predict deliverability and identify the geometry-based improvement opportunities.
PREREQUISITES
Topic 1.1: PVT — Bo, Bg, μo, μg as functions of pressure (required for IPR equation denominators). Topic 1.2: Formation Volume Factor, converting reservoir to surface volumes. Topic 1.3: Permeability and relative permeability k and k_ro are numerator inputs to every J* formula. Darcy's Law: Comfort with q = kA(ΔP)/(μL) in cylindrical coordinates.
PBL CONNECTION: GASHAKA GK-22 OIL LEG & NODAL ANALYSIS
The drainage-geometry work in this topic uses the GK-22 oil leg (k = 8.2 mD, h = 53 ft, φ = 0.19, p̅ = 5,651 psia). The GK-22 well drains a 580-acre fluvial sandstone polygon with an approximately 2:1 aspect ratio. The well is positioned 30% off-centre from the long axis. The drainage boundary is a mix of no-flow faults and a partial aquifer boundary. Topic 1.4 provides the geometric toolkit to:
1. Select the correct CA for the drainage polygon shape and well position
2. Calculate the equivalent circular re and the geometric term for the IPR equation
3. Quantify the skin contribution from partial perforation of the pay zone
4. Compare vertical vs horizontal well deliverability for the GK-22 fault block
5. Assess composite IPR for the three-layer reservoir section
Radial Flow
Pressure funnels, drainage radius, ln(re/rw) factor in Darcy's Law
CA Dietz tables, how drainage shape and well position change effective re
Horizontal Wells
Joshi's equation, anisotropy, wellbore length, and equivalent vertical PI
Section 1
Radial Flow & the Drainage Radius Concept
The geometry of flow to a vertical wellbore in a uniform reservoir is radial, pressure contours form concentric cylinders. This simple geometry is the foundation of every IPR equation, and the logarithmic nature of radial flow has consequences that every production engineer must internalise.
1.1 Why Flow to a Wellbore is Radial
When a well is opened to produce, it creates a lower-pressure zone at the wellbore face. Reservoir fluid flows towards this low-pressure sink from all directions in the horizontal plane, converging on the wellbore cylinder. The flow path for any fluid parcel is a radial line from the reservoir to the well. Because the cross-sectional area available for flow at radius r is A = 2πrh, and this area shrinks as r → rw, the pressure gradient and velocity must increase dramatically near the wellbore to maintain the same volumetric flow rate. This convergence is the defining feature of radial flow and it means the near-wellbore region is always the most critical zone in a reservoir.
Figure 1.1: Top-view of radial flow to a vertical wellbore. Concentric circles = pressure isobars (equal-pressure surfaces). Radial arrows = flow lines converging on well. Pressure gradient steepens dramatically near the wellbore because the cross-sectional area A = 2πrh shrinks proportionally to r.
1.2 Deriving the Steady-State Radial Flow Equation
Starting from Darcy's Law in cylindrical coordinates, we can derive the steady-state inflow equation analytically. This derivation reveals exactly why ln(re/rw) appears in every IPR formula.
DERIVATION — Darcy's Law in Cylindrical Coordinates:
1. Flow area at radius r: A(r) = 2π·r·h
2. Darcy's velocity: u = -k/μ · dp/dr (flow direction = inward, r decreasing)
3. Volumetric flow rate: q = u·A = -(2π·r·h·k/μ) · dp/dr
4. Rearranging: q·dr/r = (2π·h·k/μ) · dp
5. Integrating r_w to r_e: q·∫[dr/r] = (2π·h·k/μ)·∫dp
In oilfield units (STB/day, mD, ft, psia, cP, RB/STB):
q = [k·h·(p_e - p_wf)] / [141.2·B·μ·ln(r_e/r_w)] …(Eq. 1.4.1)
This is the STEADY-STATE IPR for a vertical well at the centre of a circular drainage area.
THE LOGARITHMIC GEOMETRY FACTOR
The term ln(re/rw) is the geometric resistance to flow. It is logarithmic (not linear), which has a profound engineering implication:
Doubling re (drainage area ×4) adds ln(2) = 0.69 to the denominator — a slight loss of J*, while doubling rw (wellbore diameter) subtracts ln(2) = 0.69 — a slight gain. The two shifts are equal in size and small against a denominator of ~9, which is why neither acreage nor wellbore enlargement is a strong lever on deliverability.
For a typical well: re = 2,980 ft, rw = 0.328 ft → ln(re/rw) = ln(9,085) = 9.11
If re doubles to 5,960 ft: ln(18,170) = 9.81 → the denominator grows 7.7%, so J* falls by about 7%
If rw doubles to 0.656 ft: ln(4,543) = 8.42 → 7.7% increase in J*
This is why stimulation (fracturing, which creates a large effective wellbore) is so effective, while moving a well location has a relatively small geometric impact.
1.3 The Drainage Radius re: What It Represents
The drainage radius is not a fixed property of the reservoir, it is a dynamic concept that depends on time, production history, and boundary conditions. Understanding what re represents in each flow regime is essential for selecting the right IPR model.
During transient flow, the radius of investigation r_inv grows with time:
The pressure disturbance has not yet reached any boundary. The reservoir effectively appears infinite. r_e in the transient IPR equation (Eq. 3.1 in Guo) is therefore not the physical boundary, it is replaced by the growing r_inv. As time increases, J* (transient) decreases because the pressure gradient driving flow is spread over an ever-larger volume.
Practical consequence: early production data from a new well always shows declining J* even at constant BHFP, this is normal transient behaviour, not damage or depletion. The well has not yet "seen" the reservoir boundaries.
In steady-state, r_e is the physical distance to a constant-pressure boundary (aquifer, water injection well, or gas cap). The pressure at r_e remains fixed at p_e. The pressure profile in the reservoir is time-invariant, a true steady state. J* is constant and given by Eq. 1.4.1 above.
This condition is achieved when: (1) a strong active aquifer maintains boundary pressure, or (2) water/gas injection exactly replaces produced fluid volumes. Many offshore fields with aquifer support approximate steady-state conditions. The Gashaka field has a weak aquifer, so true steady-state is not achieved: pseudo-steady-state is more appropriate.
In pseudo-steady-state (PSS), r_e is the physical distance to a no-flow boundary. All boundaries are sealing (faults, pinch-outs, or limits of the drainage area). The pressure at r_e is not constant, it declines at the same rate as average reservoir pressure p̄. The pressure profile shape is fixed (translates downward uniformly over time).
The PSS IPR equation uses average reservoir pressure p̄ (not boundary pressure p_e) and has an additional −3/4 correction term in the denominator:
The −3/4 correction accounts for the fact that p̄ (volume-averaged pressure) is slightly lower than p_e in PSS. This is the most commonly used IPR equation for depletion-drive reservoirs, including most onshore and many offshore fields.
1.4 Converting Drainage Area to Equivalent re
In practice, drainage areas are rarely circular. When the drainage area A is known from well spacing, seismic interpretation, or material balance, an equivalent circular radius re is calculated and used in the IPR equations (subject to CA correction for non-circular shapes, covered in Section 3):
EQUIVALENT DRAINAGE RADIUS:
r_e = √(A / π)
Where:
A = drainage area [ft²]
r_e = equivalent circular drainage radius [ft]
Common conversions (1 acre = 43,560 ft²):
80 acres → r_e = √(80 × 43,560 / π) = √(1,108,373) = 1,053 ft
160 acres → r_e = 1,489 ft
320 acres → r_e = 2,106 ft
640 acres → r_e = 2,980 ft
1,280 acres → r_e = 4,213 ft
GK-22 oil leg drains a 580-acre polygon → equivalent r_e = 2,835 ft (between the 320- and 640-acre rows)
WORKED EXAMPLE 1.1GK-22 Oil Leg: Steady-State J* Calculation
GK-22 oil-leg reservoir data: k = 8.2 mD, h = 53 ft, p_e = 5,651 psia, p_wf = 1,000 psia, Bo = 1.1 RB/STB, μ_o = 1.7 cP, r_w = 0.328 ft, A = 580 acres (r_e = 2,835 ft), S = 0.
Confusing steady-state p_e (boundary pressure, fixed) with pseudo-steady-state p̄ (average reservoir pressure, declining). Using p_e in the PSS equation (or p̄ in the SS equation) gives wrong results. Always check: does the boundary hold constant pressure (aquifer → SS) or is it sealed (fault → PSS)?
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Derivation Video: Radial Flow to Steady-State IPR — Step by Step
11:25
Walks through the cylindrical Darcy integration from first principles, arrives at Eq. 3.5, and then modifies it to pseudo-steady-state form. Emphasises the physical meaning of each term and demonstrates the sensitivity of J* to changes in r_e, r_w, h, k, B, μ, and S using the GK-22 reservoir parameters.
Section 2
Flow Regimes & Boundary Conditions
The three flow regimes: transient, steady-state, and pseudo-steady-state, are not just theoretical constructs. Each requires a different IPR equation, different pressure input (p_i vs p_e vs p̄), and different physical interpretation of the drainage boundary. Misidentifying the regime leads to systematic errors in deliverability prediction.
2.1 The Three Regimes Compared
Figure 2.1: Pressure profiles for the three flow regimes at constant wellbore pressure. Transient (left): profile evolves as pressure funnel grows. Steady-state (centre): fixed profile maintained by constant-pressure boundary. Pseudo-steady-state (right): profile translates downward uniformly, same shape, declining average pressure.
Feature
Transient
Steady-State
Pseudo-Steady-State
Boundary condition
No boundary reached yet
Constant-pressure at r_e (aquifer)
No-flow at r_e (sealed fault)
Pressure at r_e
p_i (undisturbed)
p_e (fixed)
Declining with time
IPR pressure input
p_i (initial)
p_e (boundary)
p̄ (average, from MBE)
J* over time
Decreasing (expanding r)
Constant
Constant (shape fixed)
Geometric term
ln(r_inv/r_w) − correction
ln(r_e/r_w)
ln(r_e/r_w) − 3/4
Typical reservoir
All wells early in life
Strong aquifer; water injection
Onshore depletion; sealed faults
GK-22 applicability
First ~6 months
Not applicable (weak aquifer)
Primary production phase
2.2 Time to Reach Pseudo-Steady-State
The transition from transient to PSS flow occurs when the pressure disturbance reaches all boundaries. The time to reach PSS can be estimated from:
TIME TO PSEUDO-STEADY-STATE (circular drainage area):
t_PSS = 1,200 · φ · μ · c_t · r_e² / k
Where:
t_PSS = time to PSS [hours]
φ = porosity [fraction]
μ = viscosity [cP]
c_t = total compressibility [psi⁻¹]
r_e = drainage radius [ft]
k = permeability [mD]
After ~204 days (about 6.8 months), GK-22 transitions from transient to PSS behaviour.
Production data before this point should NOT be used to calibrate steady-state IPR.
2.3 The Isochronal Test: Measuring k from Transient Data
Well test analysis exploits the transient flow regime to measure reservoir permeability. The log-time (or log-log) pressure response during transient flow follows the diffusivity equation, allowing k and S to be extracted. This is the basis of pressure transient analysis (PTA), and why the transient flow equation contains a log(t) term rather than ln(r_e/r_w).
p_wf = p_i − 162.6·q·B·μ/(k·h) × [log(t) + log(k/(φ·μ·c_t·r_w²)) − 3.23 + 0.87S]
Key insight: Plot p_wf vs log(t) → STRAIGHT LINE
Slope m = −162.6·q·B·μ/(k·h) → solve for k
Extrapolate to t=0 → solve for S (skin)
This is the semilog analysis basis for pressure buildup and drawdown tests.
t_PSS marks when the straight line deviates, boundary effect detected.
On a plot of 1/q vs √t (rate-normalised pressure integral): Transient period shows linear behaviour; PSS shows a different linear trend with higher slope. The transition between these two slopes identifies t_PSS. In practice, this analysis is done using rate-normalised pressure (RNP) plots and their derivative, an industry-standard PTA workflow.
On a log-log pressure derivative plot (Bourdet derivative):
• Infinite-acting radial flow (transient): flat derivative at 0.5-line
• Boundary effects (PSS): unit-slope (45°) upward trend on derivative
• Constant-pressure boundary (SS): derivative drops to zero
Identifying which regime is active tells you: which IPR equation to use, what r_e to use, and whether the well is still in transient flow (early production) or has reached stabilised conditions (use for full-field IPR modelling).
Gas wells require pseudo-pressure m(p) to account for pressure-dependent μ_g and Z-factor:
m(p) = ∫[p_b to p] (2p / μz) dp [psia²/cP]
The gas well transient IPR becomes: q_g = k_g·h·[m(p_i) − m(p_wf)] / [1,638·T·(log t + log(k/φμc_t·r_w²) − 3.23 + 0.87(S + Dq_g))]
The additional D·q_g term is the non-Darcy (inertial/turbulent) skin, important in high-rate gas wells where velocities near the wellbore exceed the Darcy flow regime. D is the non-Darcy flow coefficient (d/Mscf). This effectively increases apparent skin at high rates, making high-rate gas wells appear more damaged than they are, a critical distinction for stimulation decisions.
FIELD APPLICATION: CHOOSING THE RIGHT REGIME FOR GK-22
The GK-22 oil-leg producer sits in a fluvial sandstone fault block with sealing faults on three sides and a weak aquifer providing limited pressure support on one boundary. The correct IPR regime is:
During initial production (0–204 days): Transient, use log(t) based equation; do not use this data for IPR calibration. At stabilised conditions (after ~204 days): Pseudo-steady-state, use p̄ from material balance; denominator = ln(r_e/r_w) − 0.75 + S + Dq_g. If water injection begins: Transition toward steady-state, monitor boundary pressure response in injection well. If p_e stabilises, switch to SS equation with p_e from injector BHP.
Using SS equation with declining p̄ overestimates deliverability by 9–12%, significant for production forecasting.
Section 3
Drainage Shape & the Dietz Shape Factor CA
Real drainage areas are never perfectly circular. Wells are off-centre, drainage boundaries are irregular, and reservoir shapes are controlled by geology. The Dietz shape factor CA provides a rigorous, single-number correction that accounts for the geometry of any drainage shape and well position, replacing the ideal circular geometry with the real one.
3.1 From Circular to General Drainage: The Shape Factor Equation
For a non-circular drainage area, the IPR equation in pseudo-steady-state becomes:
Where:
A = drainage area [ft²]
C_A = Dietz drainage area shape factor [dimensionless]
γ = Euler's constant = 1.78
r_w = wellbore radius [ft]
S = skin factor
Equivalence to circular equation:
For a circle with well at centre: C_A = 31.62
½·ln(4A / γ·C_A·r_w²) = ln(r_e/r_w) − 3/4 ← verified exactly for C_A=31.62
KEY INSIGHT:
Higher C_A → smaller denominator → higher J*
Lower C_A → larger denominator → lower J*
Well at centre of circle (C_A = 31.62) has the BEST possible geometry.
Off-centre wells and non-circular shapes REDUCE C_A and HURT productivity.
3.2 Dietz Shape Factor Values: The Engineering Library
Dietz (1965) tabulated CA values for a comprehensive set of drainage shapes and well positions. These are the standard reference values used in all reservoir engineering calculations. Below are the most engineering-relevant configurations:
31.62
Circle: well at centre
30.88
Square: well at centre
21.84
1:2 rectangle: centre
4.51
1:4 rectangle: centre
10.84
1:2 rect: ¼ from end
4.86
Square: ¼-¼ corner
2.07
1:2 rect: ⅛ from end
19.07
Equilateral triangle: centre
HOW TO READ THE SHAPE FACTOR TABLE
CA is highest (31.62) when the well is at the centre of a circle, the most geometrically favourable position. Any deviation from this ideal reduces CA:
Shape effect: A 1:4 rectangle (CA=4.51) gives J* that is only 4.51/31.62 = 14% of the equivalent circular area J*. Elongated drainage areas are dramatically less productive than compact ones.
Well position effect: Moving a well from the centre to the ¼-point in a 1:2 rectangle drops CA from 21.84 to 10.84, a 50% productivity penalty from geometry alone. Moving further to the ⅛-point reduces CA to 2.07, a 90% penalty.
Practical rule: Optimum well placement maximises CA. For elongated drainage shapes, the well should be placed at the centre, not near any boundary. This is particularly critical for fault-bounded compartments.
3.3 Using CA to Correct the IPR Equation
The workflow for applying CA to a real drainage geometry:
1
Define the drainage area. Use well spacing design, seismic interpretation, pressure transient analysis, or material balance to establish A in acres. Convert to ft²: A[ft²] = A[acres] × 43,560.
2
Identify the drainage shape and well position. Map the drainage boundary, is it circular? Square? Rectangular with a 2:1 or 4:1 ratio? Is the well at the centre or offset? Match to the closest Dietz configuration.
3
Look up CA from the Dietz table. For interpolation between table values, the logarithm of CA varies approximately linearly with the aspect ratio.
4
Calculate the geometric term: G = ½·ln(4A / γ·CA·rw²). Note γ = 1.78 (Euler's constant = e^0.5772).
5
Calculate J*: J* = k·h / [141.2·B·μ·(G + S)]. This J* already incorporates the actual drainage geometry, no further shape correction is needed.
WORKED EXAMPLE 3.1GK-22: Non-Circular Drainage Penalty Calculation
The GK-22 fault block is approximately 2:1 rectangular (580 acres). The well is positioned approximately ¼ from one short end. Determine the productivity penalty vs an ideal circular drainage assumption.
k = 8.2 mD, h = 53 ft, Bo = 1.1, μo = 1.7 cP, r_w = 0.328 ft, S = 0.
CASE A — Circular drainage assumption (580 acres):
r_e = √(580 × 43,560 / π) = √(8,037,958) = 2,835 ft
Geometric term: ln(r_e/r_w) − 0.75 = ln(2835/0.328) − 0.75
= ln(8,646) − 0.75 = 9.065 − 0.75 = 8.315
J*_circle = (8.2 × 53) / (141.2 × 1.1 × 1.7 × 8.315)
= 434.6 / 2,195.5 = 0.1980 STB/day/psi
CASE B, 2:1 rectangle, well at ¼ from short end:
From Dietz table: C_A = 10.84
A = 580 × 43,560 = 25,264,800 ft²
G = ½ × ln(4 × 25,264,800 / (1.78 × 10.84 × 0.328²))
= ½ × ln(101,059,200 / 2.082)
= ½ × ln(48,539,000)
= ½ × 17.70
= 8.85
J*_actual = 434.6 / (141.2 × 1.1 × 1.7 × 8.85)
= 434.6 / 2,336.9 = 0.1860 STB/day/psi
PRODUCTIVITY PENALTY FROM GEOMETRY:
Penalty = (0.1980 − 0.1860) / 0.1980 = 6.1%
If instead the well were centred (C_A = 21.84):
G = ½ × ln(4 × 25,264,800 / (1.78 × 21.84 × 0.1076))
= ½ × ln(101,059,200 / 4.194) = ½ × ln(24,093,990)
= ½ × 16.99 = 8.50
J*_centre = 434.6 / 2,244.4 = 0.1936 STB/day/psi
WELL REPOSITIONING BENEFIT: 0.1936 − 0.1860 = +4.1% J* improvement
→ Confirms off-centre position costs ~4% vs optimum. Consider for future infill.
WHEN CA MATTERS MOST
Shape factor effects are largest for:
• Elongated drainage areas (aspect ratio >4:1), CA drops to <5, representing >80% geometric penalty vs circle
• Heavily off-centre wells, even in a square, being at ¼-¼ position cuts CA by 84%
• Low-permeability reservoirs, here the geometric term dominates the denominator; in high-k wells the geometric penalty is relatively less significant
Shape factor is most critical for infill well placement and compartment deliverability screening. It is less important for stimulation design (skin dominates) or aquifer support evaluation (boundary condition dominates).
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Tutorial: Dietz Shape Factors — Selecting and Applying CA in Practice
13:50
Demonstrates how to read and interpolate the Dietz C_A table for irregular drainage shapes. Shows the impact of well repositioning on J* using three field examples: a fault-bounded elongated block (C_A penalty ~70%), a near-square compartment (C_A penalty ~3%), and a wedge-shaped area with well near apex (C_A penalty ~85%). Includes sensitivity analysis showing how C_A interacts with skin.
A horizontal wellbore changes the drainage geometry fundamentally, instead of converging radially to a point, flow converges to a line. This dramatically reduces the near-wellbore pressure gradient that limits vertical well productivity, but introduces new geometric dependencies on anisotropy, wellbore length, and vertical vs horizontal permeability contrast.
The productivity advantage of a horizontal well over a vertical well in the same reservoir comes entirely from geometry. The key effects are:
Vertical Well: Point Sink
Flow converges radially to a single point (the wellbore). The cross-sectional area available for flow A = 2πr·h shrinks proportionally to r. Near-wellbore pressure gradient is very steep. J* is limited by this convergence and by h (pay thickness).
Horizontal Well: Line Sink
Flow converges to a line of length L. In the horizontal plane, flow pattern is elliptical (not circular). In the vertical plane, flow is still radial to the cylinder. Near-wellbore convergence is far less severe. J* scales approximately with L, longer well = higher productivity.
Figure 4.1: Joshi's dual-plane model for horizontal well flow. Top view (left): flow in horizontal plane forms elliptical isobars around the wellbore line. Side view (right): flow in the vertical plane remains approximately radial to the wellbore cylinder. The two resistances combine to give the total pressure drop.
4.2 Joshi's Equation: Derivation Logic and Full Form
Joshi (1988) combined steady-state flow in the horizontal plane (elliptical) with pseudo-steady-state flow in the vertical plane (radial) to arrive at the following practical equation for horizontal well productivity:
Note: k_H is used for flow rate calculation (numerator);
I_ani = √(k_H/k_V) appears in T₂ to account for anisotropy penalty.
UNDERSTANDING T₁ AND T₂: THE TWO RESISTANCES
T₁ (Horizontal Plane Resistance): This is the resistance to flow in the horizontal plane, where flow converges from the circular drainage boundary to the elliptical near-wellbore region. It depends on the ratio reH/(L/2), how large the drainage area is relative to the wellbore length. When L is small relative to reH, T₁ ≈ ln(2reH/L), which increases as L decreases. Longer wellbores reduce T₁ and improve J*.
T₂ (Vertical Plane Resistance): This is the convergence resistance near the wellbore in the vertical plane, the flow must converge from the full formation thickness h onto the wellbore radius rw. It is penalised by the anisotropy ratio Iani: when kH >> kV (high anisotropy, low kV), Iani is large and T₂ increases, the vertical flow convergence is more resistive. T₂ is reduced by longer wells (L in denominator) and thinner pay zones.
WORKED EXAMPLE 4.1GK-22 Horizontal Well vs Vertical Well Comparison
Evaluate a 1,500 ft horizontal well in GK-22 vs the existing vertical well.
k_H = 8.2 mD, k_V = 2.1 mD (k_V/k_H = 0.256), h = 53 ft, r_eH = 2,835 ft (580 ac), r_w = 0.328 ft, p_e = 5,651 psia, Bo = 1.1, μo = 1.7 cP, S = 0. L = 1,500 ft.
4.3 Sensitivity of Horizontal Well PI to Key Parameters
Parameter
Effect on J*_H
Dominant in T₁ or T₂?
Engineering Action
Wellbore length L ↑
↑ J* (T₁ and T₂ both decrease)
Both
Longer wells preferred in thick, low-k formations
k_V/k_H ↑ (less anisotropy)
↑ J* (T₂ decreases)
T₂
Natural fractures or laminated sands improve k_V
Pay thickness h ↑
↑ Numerator; T₂ increases (mixed)
T₂
Thick formations benefit less from horizontal wells
h ↓ (thin formation)
↑↑ Benefit from horizontal
T₂ low
Horizontal wells most advantageous in thin, laterally extensive formations
r_eH ↑ (large drainage)
Modest ↑ (logarithmic in T₁)
T₁
Well length more impactful than drainage area increase
Skin S (near-wellbore)
↓ if S > 0
Adds to T₂
Horizontal wells do NOT eliminate skin, stimulation still beneficial
WHEN HORIZONTAL WELLS ARE NOT BENEFICIAL
Horizontal wells do NOT outperform vertical wells in all situations: 1. Very thick formations (h >> 200 ft): T₂ becomes large; vertical convergence resistance dominates. Vertical well with perforation optimisation may be competitive. 2. Strong vertical permeability (k_V/k_H near 1): Less benefit from horizontal orientation. High k_V also increases coning risk in horizontal wells. 3. Highly fractured reservoirs: If fractures provide the main flow path and they are vertical, a horizontal well drilled parallel to fractures accesses fewer fractures than one drilled perpendicular, orientation strategy critical. 4. Short L due to cost/operational constraints: If L << h, vertical well may be preferred. Rule of thumb: horizontal well wins when L > 5h.
Section 5
Multilayer Reservoirs & Composite IPR
Nearly every field penetrates multiple reservoir layers simultaneously. When these layers have different pressures, permeabilities, and fluid properties, the composite well deliverability cannot simply be summed, cross-flow between layers, dynamic shut-in pressure, and productivity index weighting must all be calculated correctly.
5.1 The Problem with Commingled Production
When a well is open to multiple layers with different reservoir pressures, the wellbore creates a common pressure pwf that connects them. Layers with p̄ > pwf produce into the wellbore. Layers with p̄ < pwf receive injection from the wellbore, they act as thieves. This is cross-flow, and it reduces overall production from the high-pressure layers while the low-pressure layers consume some of that production.
CROSS-FLOW: THE HIDDEN EFFICIENCY KILLER
In the GK-22 case study, layer pressures range from 2,254 to 3,030 psia. If all layers are produced commingled at pwf = 2,658 psia, the layers with p̄ < 2,658 psia (A4 at 2,254, A5 at 2,302) are being injected into, every barrel flowing into those layers from the wellbore is a barrel not reaching surface. The "net" production is substantially less than the sum of individual layer rates.
Solution: Produce layers in pressure-grouped sequences, or isolate depleted zones with mechanical packers.
5.2 Single-Phase Composite IPR: Linear Model
For layers with undersaturated oil (p̄ above bubble point), the composite IPR is a straight line. Material balance at the wellbore requires:
The composite IPR is a straight line characterised by:
AOF = Σᵢ AOFᵢ = Σᵢ Jᵢ*·p̄ᵢ
Shut-in dynamic p_wf (due to cross-flow when well is closed):
p_wfo = Σ(Jᵢ*·p̄ᵢ) / Σ(Jᵢ*) [pressure-productivity weighted average]
NOTE: p_wfo ≠ any individual p̄ᵢ. It is a flow-weighted average pressure
that the wellbore reaches at shut-in when layers with different pressures
have cross-flowed to equilibrium. This is NOT static reservoir pressure.
WORKED EXAMPLE 5.1Composite IPR: GK-22 Three-Layer System
GK-22 has three productive layers. Layer data:
Layer D (deep): p̄ = 3,030 psia, J* = 34 bopd/psi
Layer C (mid): p̄ = 2,606 psia, J* = 100.2 bopd/psi
Layer B (shallow): p̄ = 2,467 psia, J* = 5.04 bopd/psi
STEP 1: Calculate AOF for each layer
AOF_D = 34 × 3,030 = 103,020 bopd
AOF_C = 100.2 × 2,606 = 261,121 bopd
AOF_B = 5.04 × 2,467 = 12,433 bopd
Total AOF = 376,574 bopd
STEP 2: Shut-in dynamic wellbore pressure (cross-flow equilibrium)
p_wfo = Σ(Jᵢ*·p̄ᵢ) / Σ(Jᵢ*)
= (34×3030 + 100.2×2606 + 5.04×2467) / (34 + 100.2 + 5.04)
= (103,020 + 261,121 + 12,433) / 139.24
= 376,574 / 139.24
= 2,705 psia
INTERPRETATION:
At p_wf = 2,705 psia: net production = 0 (all layers in equilibrium)
At p_wf < 2,705: Layer D dominates production
At p_wf > 2,705: Layers B and C are absorbing fluid from D
→ Layer D producing; Layers C and B receiving (thief zones)
NET PRODUCTION at p_wf = 2,000 psia:
q_D = 34 × (3,030 − 2,000) = 35,020 bopd
q_C = 100.2 × (2,606 − 2,000) = 60,721 bopd
q_B = 5.04 × (2,467 − 2,000) = 2,354 bopd
q_total = 98,095 bopd (all three producing)
NET PRODUCTION at p_wf = 2,550 psia (above p̄_B = 2,467):
q_D = 34 × (3,030 − 2,550) = +16,320 bopd (producing)
q_C = 100.2 × (2,606 − 2,550) = +5,611 bopd (producing)
q_B = 5.04 × (2,467 − 2,550) = −418 bopd (RECEIVING, thief)
Net = 16,320 + 5,611 − 418 = 21,513 bopd at surface
5.3 Two-Phase Composite IPR: Vogel Model
For layers below bubble point, Vogel's equation describes each layer's contribution. The composite becomes:
5.4 Layer Grouping Strategy: Maximising Production from Commingled Systems
The Guo Case Study demonstrates that grouping layers by pressure and producing them in sequences substantially outperforms total commingled production in cross-flow conditions. The key insight is:
Layer Group
Layers
Layer Pressures
J* Sum
Group AOF at p_wf=2,000
Group 1 (high pressure)
D3–D4
~3,030 psia
34
~35,063 bopd
Group 2 (mid pressure)
B4, C1, C2
~2,606 psia
185.6
~116,090 bopd
Group 3 (low pressure)
B1, A4, A5
~2,302–2,467 psia
20.68
~6,903 bopd
All commingled
All 8 layers
2,254–3,030 psia
240.28
~158,056 bopd
LAYER GROUPING CONCLUSION
At pwf = 2,658 psia (just below the equilibrium pressure of commingled system):
• Commingled production: 0 bopd (at equilibrium, no net production)
• Group 1 (D3–D4 only): 12,663 bopd
• Groups 1+2: 13,787 bopd (Layer C just begins flowing)
This demonstrates that producing high-pressure layers separately, while mechanically isolating low-pressure thief zones, can generate production at wellbore pressures where commingled production would be zero. Layer grouping with selective completion is the production engineering solution to cross-flow loss.
▶
Case Study: Multilayer Cross-Flow — When Commingling Kills Production
15:40
Uses the South China Sea eight-layer case from Guo Section 3.5 to demonstrate the composite IPR calculation, cross-flow identification, and layer grouping strategy. Shows how the composite IPR shifts with different completions (all-commingled vs grouped), and calculates the production gain from isolation of thief zones. Connects to perforation design and ICD/ICV completion selection.
Section 6
Wellbore & Geometric Skin Components
Skin is the additional pressure drop, beyond what clean radial flow would predict, caused by near-wellbore conditions. It is not a single number but a composite of multiple contributing mechanisms. Understanding which components are treatable (stimulation removes damage skin) and which are geometric (architecture determines completion skin) is essential for stimulation candidate selection.
6.1 The Total Skin Decomposition
The total mechanical skin factor S measured in a pressure transient test is the sum of all near-wellbore effects:
Where:
S_d = Damage skin, altered permeability in invaded zone (TREATABLE by stimulation)
S_c = Partial penetration / completion skin (GEOMETRIC: set by well architecture)
S_p = Perforation skin, from limited shot density, phasing, and crushed zone (OPTIMISABLE)
S_θ = Deviation / inclination skin, from non-radial flow in deviated wells (GEOMETRIC)
S_r = Rate-sensitive (non-Darcy) skin: turbulent flow in gas wells (RATE-DEPENDENT)
Dq = Non-Darcy term (sometimes written separately from S_total)
SIGN CONVENTION:
S > 0: net damage — well is under-performing ideal radial flow
S = 0: clean well — ideal radial flow to wellbore
S < 0: stimulated well, hydraulic fracture or acidised beyond wellbore (NEGATIVE SKIN)
🔴 Damage Skin S_d (Treatable)
Caused by: filtrate invasion, clay swelling, fines migration, emulsions, scale, asphaltene deposition. Reduced permeability in an annular zone around the wellbore (r_s, k_s). Treatment: matrix acidising, solvent wash, clay stabilisers. Most high-value stimulation target.
🟡 Geometric/Completion Skin (Fixed by Design)
Caused by: partial penetration (not perforating full pay), perforation design, wellbore deviation. Cannot be removed by stimulation, it is an inherent consequence of the completion architecture. Mitigated by: optimising perforation program, longer perforated intervals, appropriate deviation angle.
6.2 Partial Penetration Skin Sc
When only a fraction of the pay zone is perforated, flow must converge in the vertical direction to reach the perforated interval, a geometric resistance that appears as a positive skin. This skin can be very large for thin perforated intervals in thick formations.
PARTIAL PENETRATION SKIN (Brons and Marting, simplified):
S_c = (1/b − 1) × [ln(h_D) − G(b)]
Where:
b = h_p / h [penetration ratio, 0 < b ≤ 1]
h_p = perforated interval [ft]
h = total pay thickness [ft]
h_D = (h/r_w) × √(k_H/k_V) [dimensionless thickness]
G(b) = geometry function ≈ 2.948 − 7.363b + 11.45b² − 4.675b³ (Papatzacos)
Note: S_c → 0 when b = 1 (full penetration)
S_c → ∞ as b → 0 (thin perforated interval in very thick formation)
WORKED EXAMPLE 6.1Partial Penetration Skin: GK-22 Well with 60% Completion
GK-22: h = 53 ft, h_p = 32 ft (60% of pay perforated, upper section only), k_H = 8.2 mD, k_V = 2.1 mD, r_w = 0.328 ft.
Perforations introduce additional flow convergence (flow must enter the perforation tunnel) and may create a crushed zone of reduced permeability. The perforation skin depends on shot density, phasing angle, penetration depth, and crushed zone permeability.
S_crushed (damaged zone around tunnel):
S_crushed = [h/(n_perf × L_p)] × (k/k_crushed − 1) × ln(r_crushed/r_p)
where k_crushed ≈ 0.1k to 0.4k (typical crushed zone permeability reduction)
Key design parameters:
Higher shot density (spf) → lower S_p
120° or 60° phasing → better than 0° (all in one plane)
Longer perforation tunnels → lower S_H (less convergence)
Underbalanced perforating → lower k_crushed (less compaction)
6.4 Deviation/Inclination Skin Sθ
A deviated wellbore passes through more formation than a vertical well for the same true vertical thickness. This increases the effective wellbore contact, producing a negative deviation skin that improves productivity. However, at very high deviation angles near horizontal, the flow geometry changes to a mixed radial/linear system that Joshi's equation handles more appropriately.
DEVIATION SKIN (Cinco-Ley simplified, for 0° to 75° deviation):
S_θ ≈ −(θ'/41)^2.06 − (θ'/56)^1.865 × log(h_D/100)
where θ' = tan⁻¹[tan(θ)/√(k_V/k_H)] [effective angle, accounting for anisotropy]
θ = wellbore deviation from vertical [degrees]
h_D = (h/r_w)×√(k_H/k_V)
Note: S_θ is NEGATIVE (improvement) for deviated wells
For θ = 30°: S_θ ≈ −0.8 to −1.5 (moderate improvement)
For θ = 60°: S_θ ≈ −3.0 to −5.0 (significant improvement)
For θ > 75°: use Joshi's horizontal well model, deviation skin concept breaks down
Skin Component
Typical Range
Treatable?
Primary Mitigation
Damage skin S_d
+2 to +50
Yes, matrix stimulation
Matrix acid, solvent, clay stabiliser
Partial penetration S_c
+1 to +15
Partially, re-perforate
Complete full pay; downward perforation
Perforation skin S_p
+0.5 to +5
Yes, optimise design
High shot density, 60° phasing, underbalanced
Deviation skin S_θ
−0.5 to −6
N/A (geometric benefit)
Increase deviation to maximise benefit
Non-Darcy Dq
+1 to +10
Rate management
Reduce rate; hydraulic fracture to reduce velocity
Fracture skin (post-frac)
−3 to −8
N/A (stimulation benefit)
Hydraulic fracture, creates negative skin
PBL: SKIN AUDIT — GK-22 COMPLETION REVIEW
The GK-22 PTA buildup gives a total skin S′ = +14 (see the Module 03 skin audit, where the perforation contribution is folded into the damage term and the whole apparent skin is treated as removable damage skin S_d). This is the canonical GK-22 figure used throughout the programme.
Conclusion: with S′ = +14 the well carries a large damage-dominated skin, so matrix stimulation is the high-value action. Removing it would lift J* substantially: at the clean geometry G_basic ≈ ln(r_e/r_w) − ¾ ≈ 8.36, dropping S from +14 to ≈ 0 raises J* by a factor of (8.36 + 14)/8.36 ≈ 2.7×. Module 03 develops the full skin audit and the Hawkins-formula damage-radius analysis that justify treating S′ as all-S_d.
Interactive Tools
Drainage Geometry Simulators
Three interactive tools: (1) IPR Geometry Explorer, compare flow regimes and shape factor effects on J*; (2) Horizontal vs Vertical PI Calculator, Joshi equation live; (3) Skin Budget Builder, decompose and optimise total skin.
IPR Geometry Explorer: J* vs Drainage Parameters INTERACTIVE
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Scenarios to explore:
• Slide C_A from 31.62→2.07 and observe J* loss from shape alone
• Add S=+10 (damaged well) then S=−5 (fractured well)
• Double k_h (double either k or h), observe linear J* response
• Quadruple drainage area, observe logarithmic J* response (small change)
Horizontal vs Vertical Well PI: Joshi Equation INTERACTIVE
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Insights to explore:
• Vary k_V/k_H from 0.05 to 0.9, when does anisotropy penalty dominate?
• Find the L value where J*_H = 2× J*_V (breakeven for horizontal well cost)
• Set h = 200 ft, does horizontal still dominate?
• At what L does the (L/2 < 0.9·r_eH) constraint become active?
Skin Budget Builder: Total Skin Decomposition & Optimisation INTERACTIVE
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Section 8
Synthesis: The Complete J* Calculation & PBL Application
Bringing it all together: a systematic workflow from reservoir data to complete IPR curve, integrating PVT, permeability, and drainage geometry into a single well deliverability assessment for the GK-22 scenario.
8.1 The Complete J* Calculation Workflow
A complete productivity index calculation integrates inputs from all four topics of Module 01. Below is the systematic workflow that every production engineer should be able to execute from reservoir data:
1
Establish fluid properties (Topics 1.1, 1.2): From PVT analysis or correlations, obtain Bo or Bg, μo or μg, Z-factor, and Rs at the relevant reservoir pressure. For gas-condensate (GK-22): use pseudo-pressure m(p) for gas phase.
2
Determine absolute permeability (Topic 1.3): From core plug analysis (Klinkenberg-corrected) plus pressure transient analysis to upscale to reservoir scale. Identify kH and kV/kH for horizontal well assessment.
3
Assess relative permeability (Topic 1.3): At initial conditions, kro = (kro)Swc from imbibition kr curve. If p < pb, calculate kro at the saturation corresponding to p̄ from the Corey/Wyllie correlation. This is the numerator multiplier in J*.
4
Select flow regime (Topic 1.4, Section 2): Assess boundary conditions, aquifer (SS), sealed faults (PSS), or early production (transient). Select the correct IPR equation and pressure input (pe, p̄, or pi).
5
Define drainage geometry (Topic 1.4, Sections 1–3): Calculate re from A. Select CA from Dietz table based on drainage shape and well position. Calculate the geometric term G = ½·ln(4A/γ·CA·rw²).
6
Decompose and sum skin (Topic 1.4, Section 6): Stotal = Sd + Sc + Sp + Sθ. For gas wells, add non-Darcy term D·qg. Use PTA Stotal as check; split into components from completion records.
7
Calculate J*: J* = k·kro·h / [141.2·Bo·μo·(G + Stotal)]. This is the slope of the IPR above bubble point.
8
Construct full IPR curve: Above pb: linear using J*. Below pb: apply Vogel extension. For gas: use pseudo-pressure m(p) form or Fetkovich's simplified approach. Plot with VLP curve for Nodal Analysis (Module 07).
COMPLETE WORKED EXAMPLEGK-22 Full J* Calculation: All Topics Integrated
Given data (GK-22): k = 8.2 mD, kro@Swc = 0.85, h = 53 ft, A = 580 acres (2:1 rectangle), well at ¼ position → CA = 10.84, rw = 0.328 ft, p̄ = 5,651 psia (above bubble point), Bo = 1.1 RB/STB, μo = 1.7 cP, Sd = 0.9, Sc = 2.70 (60% completion), Sp = 1.2, Sθ = 0.
STEP 1: Effective permeability
k_eff = k × k_ro = 8.2 × 0.85 = 6.97 mD
STEP 2: Drainage geometry (non-circular, 2:1 rectangle at ¼ position)
A = 580 × 43,560 = 25,264,800 ft²
G = ½ × ln(4A / (γ × C_A × r_w²))
= ½ × ln(4 × 25,264,800 / (1.78 × 10.84 × 0.328²))
= ½ × ln(101,059,200 / (1.78 × 10.84 × 0.1076))
= ½ × ln(101,059,200 / 2.079)
= ½ × ln(48,609,000)
= ½ × 17.70
= 8.85
STEP 3: Total skin
S_total = S_d + S_c + S_p + S_θ = 0.9 + 2.70 + 1.2 + 0.0 = 4.80
STEP 4: J* calculation
J* = k_eff × h / [141.2 × B_o × μ_o × (G + S_total)]
= (6.97 × 53) / (141.2 × 1.1 × 1.7 × (8.85 + 4.80))
= 369.4 / (141.2 × 1.1 × 1.7 × 13.65)
= 369.4 / (141.2 × 1.1 × 23.205)
= 369.4 / 3,607.0
= 0.1024 STB/day/psi
STEP 5: AOF (p_wf = 0 psia)
q_AOF = 0.1024 × 5,651 = 579 STB/day
COMPARISON — What different improvements achieve:
Scenario G S_total J* Improvement
──────────────────────────────────────────────────────────────────────────
Current (base case) 8.85 4.80 0.1025 baseline
Fix completion (b=1, S_c=0) 8.85 2.10 0.1278 +25%
Also fix perforations (S_p=0.4) 8.85 1.50 0.1352 +32%
Also acid (S_d=0) 8.85 1.50 0.1352 (S_d already ~0: no further gain)
Move well to centre (C_A=21.84) 8.50 4.80 0.1052 +3%
Full optimisation (all above) 8.50 1.00 0.1473 +44%
Horizontal 1,500 ft (Example 4.1) n/a 0.0 0.702 +585%
──────────────────────────────────────────────────────────────────────────
CONCLUSION: For GK-22 conditions, the ranked improvement opportunities are:
1. Horizontal well drilling (×3.9 PI) — if economics justify
2. Completion upgrade: reperforate 100% + improve shot density (+32%)
3. Acid/skin treatment (minor here: completion dominates this scenario)
4. Well repositioning (+3%: consider for infill)
RECONCILE: As-Designed Skin (+4.8) vs Measured PTA Skin (+14)
The Stotal = 4.80 above is the as-designed forward estimate, summed from completion records (Sd = 0.9 + Sc = 2.70 + Sp = 1.2 + Sθ = 0). The GK-22 build-up test actually returns a measured total skin S′ = +14 (the canonical programme figure, developed in the Module 03 skin audit). The gap (roughly +9 of skin) is near-wellbore damage that the completion records do not capture: drilling-fluid invasion, perforation crushed-zone, and fines. That unquantified difference is precisely why a measured PTA skin always overrides a component sum, and why Module 03 exists. Use +4.8 here to see how the geometric terms combine; carry +14 into any GK-22 productivity forecast.
8.2 Sensitivity Analysis: Which Geometric Parameter Matters Most?
Parameter
Baseline
Change
J* Impact
Lesson
Skin S_total
4.80
0 → 10 → −5
−34% / +29%
Skin most impactful in low-k wells (large fraction of denominator)
Pay thickness h
53 ft
×2 = 106 ft
+100% (linear)
J* strictly linear with h, perforating full pay critical
Permeability k
8.2 mD
×2 = 16.4 mD
+100% (linear)
J* strictly linear with k, reservoir quality dominates
Drainage area A
580 ac
×4 = 2,320 ac
−8% (logarithmic)
Slight loss: more acreage adds ln(2) = 0.69 to the denominator
Wellbore radius r_w
0.328 ft
×2 = 0.656 ft
+8% (logarithmic)
Same-size shift as quartering the drainage area: a weak lever
Shape factor C_A
31.62
→ 10.84 (2:1 at ¼)
−6%
Modest penalty, completion/skin dominate in most wells
Well length L (horiz.)
0 (vertical)
1,500 ft horizontal
+286%
Horizontal well geometry is the single largest J* lever available
MODULE 07 PBL PREPARATION: YOUR ACTION ITEMS
For the Module 07 Nodal Analysis PBL problem set, you will need to:
1. Calculate the GK-22 base-case J* using the full workflow above (all four Topic 1 inputs integrated) 2. Apply Vogel equation to extend IPR below the oil-leg bubble point (3,250 psia), using the future kro/(Boμo) ratio method 3. Plot the IPR curve (pwf vs q) for current and future reservoir pressure 4. Overlay the VLP curve (from tubing hydraulics, Topic 1.5) and find the operating point 5. Evaluate three scenarios: (a) base case, (b) completion optimisation (+32% J*), (c) hydraulic fracture bypass of condensate bank (Sfrac = −5) 6. Present recommendation, which intervention maximises NPV given the GK-22 costs and production uplift
All the geometric tools from Topic 1.4, shape factor selection, skin decomposition, horizontal well comparison, feed directly into this Nodal Analysis exercise.
▶
Integration Session: Building the Complete GK-22 IPR from Scratch
22:15
Full integration session walking through the complete J* and IPR construction for GK-22 using all Module 01 topics. Starts with raw core and PVT data, applies relative permeability, drainage geometry, and skin decomposition, and arrives at the final IPR curve ready for Nodal Analysis. Compares three completion scenarios and demonstrates the sensitivity of the operating point to each change. The full GK-22 data set used here is the same data set provided in the Module 07 PBL Hub.
Assessment
Knowledge Check
12 questions covering all drainage geometry concepts. Immediate feedback and full explanation provided for each question.
1. In the steady-state radial flow equation for a vertical well (q = k·h·(p_e−p_wf) / [141.2·B·μ·ln(r_e/r_w)]), what does the term ln(r_e/r_w) represent physically, and why is it logarithmic rather than linear?
The logarithmic term arises directly from the integration of Darcy's Law in cylindrical coordinates: q·dr/r = (2πkh/μ)·dp, which integrates to q·ln(r_e/r_w) = (2πkh/μ)·(p_e−p_wf). The 1/r factor appears because the flow area at radius r is A = 2πrh, which is proportional to r. As flow converges toward the wellbore (r decreases), the cross-section available for flow shrinks, requiring a steeper pressure gradient to maintain the same volumetric flow rate. The logarithm is the direct mathematical consequence of this geometry.
2. A reservoir engineer calculates r_e = 2,980 ft for a 640-acre drainage area. The drainage area is then expanded to 2,560 acres (4× larger). By how much does J* increase?
640 acres → r_e = 2,980 ft → ln(r_e/r_w) = ln(2980/0.328) = 9.11. 2,560 acres (4×) → r_e = 5,960 ft → ln(5960/0.328) = ln(18,171) = 9.81. In PSS, J* = kh/[141.2·B·μ·(ln(r_e/r_w) − 0.75 + S)], so r_e enters only through the logarithm in the denominator. Taking S = 0, the geometric factor grows from 9.11 − 0.75 = 8.36 to 9.81 − 0.75 = 9.06, a 7.7% increase in flow resistance, so J* actually falls by about 7.7%, not rises. The lesson (answer C): quadrupling the drainage area changes deliverability by only single-digit percent because the dependence is logarithmic, and the small change is a slight reduction, not a gain. Acquiring more acreage is one of the weakest levers on well PI.
3. A field is on a strong aquifer that maintains constant pressure at the reservoir boundary. The appropriate IPR model and pressure input are:
A strong aquifer that maintains constant pressure at the boundary defines steady-state flow. In SS, the pressure profile is time-invariant: the boundary holds at p_e = constant. The correct IPR is q = kh(p_e−p_wf)/(141.2Bμ·ln(re/rw)). Using p̄ (PSS) would underestimate J* because in SS, p_e > p̄ for the same drainage volume; conversely using p̄ in the SS equation with the SS formula would give wrong results. The key is matching: constant-pressure boundary → SS equation → use p_e.
4. The Dietz shape factor C_A for a well at the centre of a circle is 31.62. A geologist maps the fault block as a 4:1 elongated rectangle with the well at its centre (C_A ≈ 4.51). What is the approximate productivity penalty from this shape vs an ideal circle, assuming all other parameters are equal?
G = ½·ln(4A/γ·C_A·r_w²). Changing C_A from 31.62 to 4.51 changes G by ½·ln(31.62/4.51) = ½·ln(7.01) = ½×1.947 = 0.97. If G_circle = 8.5, G_rectangle = 9.47. J*_rect/J*_circle = 8.5/9.47 = 0.90 → approximately 10% J* penalty. This confirms D: the penalty is significant but not catastrophic for typical parameters. It is NOT a direct C_A ratio (not 14% of circle PI) because the effect enters logarithmically.
5. In Joshi's equation, the term T₂ = (I_ani·h/L)·ln[I_ani·h/(r_w·(I_ani+1))] represents the vertical plane flow resistance. If k_V = 0 (completely impermeable vertically), what happens to the horizontal well's productivity?
When k_V → 0, I_ani = √(k_H/k_V) → ∞. Then T₂ = (I_ani·h/L)·ln[I_ani·h/(r_w·(I_ani+1))] → ∞·ln(∞/r_w) → ∞. The denominator (T₁ + T₂) → ∞ and J*_H → 0. This is physically correct: in a completely laminated, horizontally impermeable layer, fluid cannot reach the wellbore in the vertical direction: the horizontal well is no better than a vertical one at the limit. This is why k_V matters so much for horizontal well design: it controls T₂ directly. Drilling horizontal wells in zero-k_V formations is futile unless vertical communication exists through fractures.
6. A multilayer reservoir has three layers: Layer A (p̄ = 3,500 psia, J* = 50), Layer B (p̄ = 2,800 psia, J* = 80), Layer C (p̄ = 2,200 psia, J* = 20). When the well is shut in after commingled production, what is the dynamic wellbore pressure p_wfo?
The dynamic shut-in pressure p_wfo is the J*-weighted average of individual layer pressures (Eq. 3.41, Guo). It is not the arithmetic average, not the highest layer, and not the lowest. It represents the wellbore pressure at which the total cross-flow rate is zero: layers above p_wfo inject into the wellbore, layers below absorb from it, with the net being zero. This is also why p_wfo ≠ static average reservoir pressure: it is a flow-weighted equilibrium. If p_wfo = 2,953 psia and the operator reads this as reservoir pressure, they will overestimate depletion in Layers B and C.
7. A vertical well in a 53 ft pay zone is completed over only 20 ft (b = 0.377). Using the partial penetration skin formula, would S_c be positive, negative, or zero, and is this type of skin treatable by matrix acidising?
Partial penetration skin S_c is always positive: it represents the additional pressure drop needed to drive flow vertically to converge on the short perforated interval. At b = 0.377, S_c can range from +4 to +8 depending on h_D. It cannot be removed by matrix acidising: acid treats the near-wellbore rock (S_d), not the geometric flow path. The only way to remove S_c is to perforate the remaining unperforated interval (the missing 33 ft = 62% of pay). This is why a skin audit is essential: treating a well with high S_c by acid when the skin is geometric is a wasted intervention.
8. Doubling wellbore radius r_w from 0.328 ft to 0.656 ft has the same geometric effect on J* as which of the following drainage area changes?
In the denominator ln(r_e/r_w), doubling r_w subtracts ln(2) = 0.693, a reduction in geometric resistance, i.e. a gain in J*. To get the same −0.693 from the outer radius you must halve r_e (ln(r_e/2) = ln r_e − ln 2), and since A ∝ r_e² that means quartering the drainage area. Quadrupling A (option C) does the opposite: doubling r_e adds ln(2) to the denominator and lowers J* (see Q2). Either way the change is small — about 0.69 against a typical ln(r_e/r_w) of 8–9, roughly 8% — which is why wellbore enlargement and acreage are both weak levers compared with skin. Answer B is correct.
9. The time to reach pseudo-steady-state in a 580-acre reservoir (r_e = 2,835 ft) with k = 8.2 mD, φ = 0.19, μ = 1.7 cP, c_t = 1.29×10⁻⁵ psi⁻¹ is approximately 204 days. If the same well were placed in a 160-acre drainage area (r_e = 1,489 ft) with the same rock and fluid properties, how long to reach PSS?
t_PSS = 1,200·φ·μ·c_t·r_e²/k. It scales as r_e². Going from 580 to 160 acres means r_e drops from 2,835 to 1,489 ft (×0.53). r_e² therefore drops to ~0.28×. t_PSS_new = 204 × 0.28 ≈ 56 days. This is the correct answer (C). The implication: smaller drainage areas reach stabilised PSS behaviour faster — their production data can be used for IPR calibration sooner. Large drainage areas (580+ acres) may remain in transient for many months in low-k reservoirs before stabilising.
10. A newly completed appraisal well has an as-designed component skin budget summing to S_total = +4.8, built from completion records showing S_c = 2.70, S_p = 1.2, and S_θ = 0. Treating this component sum at face value, what is the implied damage skin S_d, and what would it imply for stimulation strategy? (Contrast with GK-22, whose measured PTA skin of +14 far exceeds any component sum: see the reconciliation note in Section 8.)
S_d = S_total − S_c − S_p − S_θ = 4.8 − 2.70 − 1.20 − 0.0 = 0.9. This is a minor damage skin, equivalent to only about a 10% PI loss. The dominant components are S_c (+2.70, from 60% completion) and S_p (+1.20, from suboptimal perforations). These are NOT treatable by matrix acid: they require completion optimisation (re-perforate to full pay + upgrade shot density/phasing). Acid will only address the S_d = 0.9, recovering about 10% PI. Re-perforating recovers ~28% PI. The correct strategy is completion optimisation first, acid treatment second.
11. Joshi's equation gives J*_H = 0.702 STB/day/psi for a 1,500 ft horizontal well in GK-22 (vs J*_V = 0.1815 for the vertical well). A production engineer argues that the horizontal well pays back its additional cost only if it produces at least 3× the vertical well rate over its life. Based on the PI ratio alone, is the horizontal well justified?
Both A and D contain important truths. A is technically correct: J*_H/J*_V = 3.87 > 3.0 threshold, so the PI justifies the horizontal well based on this metric alone. However D is also important — PI ratio is a necessary but not sufficient condition. The full economic decision requires: (1) NPV of incremental production over well life, (2) incremental well cost (horizontal wells cost 2–4× vertical), (3) risk of wellbore integrity over extended horizontal sections, (4) Joshi's equation is indeed optimistic for very long wells where heel-toe effects matter. In an exam/PBL context, answer A is the most directly supported by the given data. Always flag the Joshi optimism caveat in practice.
12. In the pseudo-steady-state IPR equation, the denominator contains (ln(r_e/r_w) − 3/4 + S). The −3/4 correction term arises because:
The −3/4 (more precisely −0.75 = −3/4) correction arises from the PSS derivation using average pressure. In PSS, the volume-weighted average pressure p̄ = p_e − q·μ/(4πkh) (for circular boundary at PSS). The steady-state equation uses p_e directly. When you rewrite the PSS flow equation in terms of p̄ instead of p_e, the conversion introduces a −3/4 correction in the denominator of the ln(r_e/r_w) term. This is a purely mathematical artifact of the pressure averaging, not a physical loss or correction. Correctly applying −0.75 ensures the PSS J* uses p̄ from material balance (which is what engineers measure) rather than the unknowable instantaneous boundary pressure.
MODULE 01 COMPLETE: YOU ARE READY FOR NODAL ANALYSIS
You have completed all four topics of Module 01 — Reservoir Fluid & Rock Properties:
Topic 1.1: PVT — Phase behaviour, bubble point, viscosity, Z-factor
Topic 1.2: Formation Volume Factor — Bo, Bg, and reservoir-to-surface conversion
Topic 1.3: Permeability & Relative Permeability, kr curves, hysteresis, correlations
Topic 1.4: Drainage Geometry, re, CA, flow regimes, horizontal wells, skin decomposition
Next: Module 02 — Single-Phase Inflow Performance: applying all four inputs to build the full IPR and introduce Nodal Analysis for well optimisation.
GK-22 PBL Hub: The Module 01 sub-problems are now available. Complete them before the Module 07 Nodal Analysis integration exercise.