← Module 02 Hub  ·  Sub-Problem 1

SP-1 — Compute the Theoretical Productivity Index for KRM-4

Assemble the PSS radial inflow equation term-by-term and calculate Jideal (the undamaged well productivity potential). This is the ceiling value that a successful stimulation programme could recover, and the benchmark against which measured J will be compared in SP-2.

Topic 2.1 · Darcy Radial Flow
~35 min
Bloom: Apply & Analyse
Sub-Problem 1 of 4 · Topic 2.1

Context & Learning Goals

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Engineering Task: Theoretical Baseline
Before you can diagnose KRM-4, you must know what it should produce undamaged.

KRM-4 is producing approximately 35% below its pre-drill rate forecast. The first question is whether the pre-drill forecast was correct — i.e., does the reservoir have the capacity to deliver the target? To answer this, you must independently calculate Jideal from the PSS radial inflow equation, using core and log data alone, without reference to the well test result. This gives you an objective benchmark that is independent of any wellbore damage or stimulation.

Learning Goals for SP-1

  1. State the PSS form of the Darcy radial inflow equation and identify the physical meaning of each term.
  2. Apply the 0.00708 field-unit constant and explain its derivation from fundamental SI units.
  3. Calculate the geometric factor ln(re/rw) and the PSS denominator including the −0.75 correction.
  4. Assemble Jideal (S = 0) and convert it to a maximum rate prediction at the separator back-pressure.
  5. Identify which term in the denominator has the largest sensitivity (permeability k dominates; skin S has additive effect).
Knowledge Library Link
Before starting the tasks, review Topic 2.1 — Darcy Radial Flow & PSS Equation, specifically Sections 2 (equation derivation) and 3 (worked examples). Return here to apply the method to KRM-4.
Data Slice

SP-1 — Relevant Data for Theoretical J Calculation

The following subset of the KRM-4 Data Pack is needed for SP-1. Do not use well test J values in this sub-problem. You are calculating from reservoir parameters independently of any test result.

ParameterSymbolValueUnitsUse In
Permeabilityk18mDNumerator
Net pay thicknessh95ftNumerator
Oil viscosity at P̄μo1.4cpDenominator
Oil FVF at P̄Bo1.25RB/STBDenominator
Drainage radiusre1,320ftGeometric factor
Wellbore radiusrw0.354ftGeometric factor
Skin factor (ideal case)S0—Denominator (set to zero for ideal J)
Which Bo and μo?
Always evaluate at average reservoir pressure P̄ = 4,850 psia, not at bubble point or at surface conditions. The PSS equation describes flow from the reservoir to the wellbore at reservoir conditions; the FVF then converts reservoir volume to stock-tank volume. Using Bob = 1.28 would give a 2.4% error in J, small but worth avoiding.
Before You Calculate

KWL Planner — SP-1 Specific

K — Know

What do you already know about Darcy radial flow?

  • Darcy’s law: Q ∝ k·h·ΔP
  • PSS means drainage boundary is closed
  • The 0.75 correction comes from shape factor derivation
  • Higher k → higher J linearly
W — Want to know

Questions this SP should answer:

  • What Jideal does KRM-4’s reservoir support?
  • What rate is achievable at Pwf = 3,100 psia?
  • How sensitive is J to ±30% uncertainty in k?
  • What does the denominator actually mean physically?
L — Will learn

Fill this in after completing the tasks:

  • Jideal = ___ STB/d/psi
  • AOFP at Pwf=3,100 = ___ STB/day
  • Which denominator term is largest?
  • k sensitivity: ±30% k → J changes by ___
Guided Calculations

Task Sequence — Compute Jideal for KRM-4

Task 1: Write the PSS Radial Inflow Equation

The pseudo-steady-state form of Darcy’s radial inflow equation in field units is:

PSS Radial Inflow — Field UnitsJ = 0.00708 · k · h / [μ · B · (ln(r_e/r_w) − 0.75 + S)] where: J = Productivity Index [STB/day/psi] k = effective permeability to oil [mD] h = net pay thickness [ft] μ = oil viscosity at P̄ [cp] B = oil FVF at P̄ [res bbl/STB] r_e = drainage radius [ft] r_w = wellbore radius [ft] S = skin factor [dimensionless] The 0.00708 constant = 2π/141.2 — converts Darcy’s law from consistent SI units to oilfield units (mD, ft, cp, RB/STB, psi, STB/day).

Task 2: Calculate the Numerator

Compute the numerator of the PSS equation: 0.00708 × k × h

  1. Substitute values: k = 18 mD, h = 95 ft. Compute 0.00708 × 18 × 95. Show your working.
  2. Record your numerator: Write this number clearly — it is reused in Task 4.
  3. Physical interpretation: The numerator (k·h product) is sometimes called the flow capacity or transmissibility. What are its units? What does it tell you about the reservoir?
Expected numerator range
For KRM-4, the numerator should be in the range 12–14. If your answer is outside this range, recheck the 0.00708 constant and the unit of k (mD, not Darcy).

Task 3: Calculate the Geometric Factor

The geometric factor ln(re/rw) is the dominant term in the denominator for most wells.

  1. Compute the ratio: re/rw = 1,320/0.354. What is this ratio?
  2. Take the natural log: Compute ln(re/rw). Use your calculator. Record to 3 decimal places.
  3. Apply the PSS correction: The PSS denominator uses (ln(re/rw) − 0.75). Subtract 0.75 from your result.
  4. Context check: A typical range for ln(re/rw) on 160-acre spacing with 7-inch casing is 8.0–8.5. Does your answer fall in this range?

Task 4: Assemble the Full Denominator and Calculate Jideal

  1. Assemble the μ·B product: Compute μo × Bo = 1.4 × 1.25.
  2. Full denominator: Compute μ·B × (ln(re/rw) − 0.75 + S) with S = 0. Show all steps.
  3. Compute Jideal: Jideal = numerator / denominator. This is your undamaged benchmark.
  4. AOFP at Pwf = 3,100 psia: Compute Q = Jideal × (P̄ − Pwf) = Jideal × (4,850 − 3,100). Does this meet the 1,200 STB/day target?

Task 5: Sensitivity Analysis

Permeability k is the most uncertain parameter (±30% is typical for core plug measurements). Compute Jideal at k = 12.6 mD (P10) and k = 23.4 mD (P90).

Casek (mD)NumeratorDenominatorJidealQ at Pwf=3,100
P10 (−30%)12.6——calculatecalculate
Base (P50)18.0——calculatecalculate
P90 (+30%)23.4——calculatecalculate

SP-1 Deliverable

Record Jideal (S=0) in STB/day/psi to 3 decimal places. This is your carry-forward value for SP-2. Also record: the full denominator value, the AOFP at current back-pressure, and the P10/P90 sensitivity range. If working in a team, compare results and resolve discrepancies before proceeding.

Just-in-Time Resources

Just-in-Time Resources

Targeted Module 02 assets for this sub-problem. Use them to refresh the method, watch the relevant lecture, and check your own numbers.

Study Course topic page — Topic 2.1 — Darcy Radial Flow & the PSS Inflow Equation (equation derivation and worked radial-flow examples).
Watch Lecture 2.1 — From Darcy’s Sand Column to a Producing Oil Well.
Self-check radial_inflow.py — verified calculator: reproduce your numbers.
Assessment

Knowledge Check — SP-1

Five auto-marked questions based on the KRM-4 data. Attempt each before revealing the answer.

Question 1

The 0.00708 constant in the field-unit PI equation comes from:

A. An empirical fit to well test data from the Gulf of Mexico
B. 2π divided by 141.2, where 141.2 is the unit conversion factor between Darcy and oilfield units
C. The Peaceman well productivity correction for numerical simulators
D. The product of Darcy’s proportionality constant and the Lewis & Horner shape factor
✓ Correct. 0.00708 = 2π/141.2. The factor 141.2 converts from Darcy’s law in consistent SI units (m, Pa, m², m³/s) to oilfield units (ft, psi, mD, bbl/day). This ensures dimensional consistency without requiring SI inputs.

Question 2

Using KRM-4 data, the geometric factor ln(re/rw) equals (to 2 decimal places):

A. 6.52
B. 7.47
C. 8.22
D. 9.80
✓ re/rw = 1,320/0.354 = 3,729. ln(3,729) = 8.224. The PSS denominator term is 8.224 − 0.75 = 7.474 (at S=0). If you got 7.47, you correctly computed the denominator term but were asked for ln(re/rw) before subtracting 0.75.

Question 3

Jideal for KRM-4 (S=0, using k=18 mD) is closest to:

A. 0.55 STB/day/psi
B. 0.75 STB/day/psi
C. 0.926 STB/day/psi
D. 1.24 STB/day/psi
✓ Numerator = 0.00708 × 18 × 95 = 12.10. Denominator = 1.4 × 1.25 × 7.474 = 13.08. Jideal = 12.10/13.08 = 0.926 STB/day/psi. This is the undamaged ceiling value for KRM-4. Compare to measured J (from SP-2) to quantify the damage effect.

Question 4

The maximum rate KRM-4 can produce at Pwf = 3,100 psia, assuming S = 0 (ideal), is:

A. 926 STB/day
B. 1,621 STB/day
C. 1,200 STB/day
D. 1,850 STB/day
✓ Q = Jideal × (P̄ − Pwf) = 0.926 × (4,850 − 3,100) = 0.926 × 1,750 = 1,621 STB/day. This comfortably exceeds the 1,200 STB/day target — so the reservoir has sufficient capacity. The underperformance must be a wellbore issue (damage), not a reservoir issue. This insight bridges SP-1 to SP-2.

Question 5

If permeability k were 50% higher (27 mD instead of 18 mD), Jideal would:

A. Increase by exactly 50%, to approximately 1.389 STB/day/psi
B. Increase by less than 50% because of denominator non-linearity
C. Increase by more than 50% because k appears twice in the equation
D. Not change because k only affects AOFP, not J
✓ J is linearly proportional to k (k appears only in the numerator). Therefore a 50% increase in k gives exactly a 50% increase in J: 0.926 × 1.5 = 1.389 STB/day/psi. This linearity is the reason core plug permeability uncertainty directly translates 1:1 to J uncertainty.
Output

Carry-Forward Values — Record Before Proceeding to SP-2

Jideal (S=0)
0.926
STB/day/psi
k·h product
1,710
mD·ft
ln(re/rw)−0.75
7.474
PSS denom (S=0)
AOFP (Pwf=3,100)
1,621
STB/day (ideal)
Interpretation
Jideal = 0.926 STB/day/psi confirms the reservoir is capable of delivering well above the 1,200 STB/day target at current back-pressure (ideal rate = 1,621 STB/day). The actual measured J (SP-2) will be lower due to skin. The difference quantifies the value of stimulation.