The fundamental law governing fluid movement from a reservoir into a wellbore. Mastering radial Darcy flow and recognising when steady-state or pseudo-steady-state conditions apply, is the prerequisite for every PI, IPR, and Nodal Analysis calculation in this programme.
In 1856, French engineer Henry Darcy published results from sand-filter experiments on the fountains of Dijon. He observed that volumetric flow rate through a porous medium is directly proportional to the applied pressure gradient and inversely proportional to fluid viscosity. This deceptively simple empirical relationship, Darcy's Law, became the foundational equation of reservoir engineering and underpins every production rate calculation performed by a completions engineer today.
In a wellbore context, flow is not linear (like Darcy's sand columns) but radial: reservoir fluids converge from all directions toward the relatively tiny wellbore. This geometry creates a pressure distribution that is logarithmic in space, meaning that the largest pressure drops occur very close to the wellbore, a fact that makes near-wellbore conditions disproportionately important and explains why skin damage has such a dramatic effect on well productivity.
This topic develops the radial form of Darcy's law from first principles, derives the steady-state (SS) and pseudo-steady-state (PSS) inflow equations, and builds the conceptual foundation for the Productivity Index (PI) and Inflow Performance Relationship (IPR) covered in Topics 2.2 – 2.4.
▶
Lecture 2.1: From Darcy's Sand Column to a Producing Oil Well
18:40
Traces the logical path from linear Darcy flow → cylindrical shells → the radial inflow equation. Includes animated pressure traverse diagrams and a worked field example using North Sea reservoir data. Covers the physical meaning of each term, unit conversions, and the boundary-condition difference between SS and PSS regimes.
LEARNING OBJECTIVES
After completing this topic, you will be able to:
1. State Darcy's Law in both linear and radial forms, identifying each variable and its unit. 2. Derive the radial inflow equation for single-phase incompressible liquid under steady-state conditions. 3. Explain the difference between steady-state (constant outer pressure) and pseudo-steady-state (no-flow outer boundary) and identify which applies to a given reservoir situation. 4. Apply the steady-state and PSS inflow equations to calculate flow rate Q or bottomhole flowing pressure Pwf given reservoir and fluid properties. 5. Explain why the pressure gradient is logarithmically distributed in radial flow and why this makes near-wellbore conditions critical. 6. Use the equations to assess the impact of changing permeability, net pay, drainage radius, wellbore radius, or skin on well deliverability.
PREREQUISITE
Module 01 — Reservoir Fluid and Rock Properties should be completed first. You will need comfort with:
permeability (k, mD), viscosity (μ, cp), formation volume factor (B, RB/STB), porosity (ϕ), and fluid compressibility (ct). If you are uncertain about any of these, review Topic 1.1 – 1.3 before proceeding.
PBL CONNECTION — MODULE 02 PROBLEM SET
The Module 02 problem set centres on the Karama Field KRM-4 well, a 11,840 ft vertical oil producer in a moderate-permeability chalk reservoir (k = 18 mD). Sub-Problem 1 requires you to calculate the expected steady-state flow rate from first principles before any skin or completion effects are introduced. The radial inflow equation derived in this topic is the engine of that calculation. Every subsequent sub-problem builds on this baseline.
Section 1 of 6
Linear Darcy Flow — The Foundation
Before addressing the radial geometry of a producing well, we establish Darcy's law in its original linear form and identify every physical quantity it contains.
1.1 Darcy's Experiment (1856)
Darcy packed a vertical sand column of cross-sectional area A and length L and measured the volumetric flow rate Q of water passing through it under a fixed head differential. He found:
DARCY'S LAW — LINEAR FORM
q = (k · A · ΔP) / (μ · L)
q = volumetric flow rate (cm³/s in Darcy units; bbl/day in field units)
k = absolute permeability of the medium (Darcy; mD in practice)
A = cross-sectional area perpendicular to flow (cm²; ft²)
ΔP = pressure differential across the medium (atm; psi)
μ = fluid dynamic viscosity (cp)
L = length of the medium in the flow direction (cm; ft)
Physical meaning: Flow rate is linearly proportional to the driving pressure gradient (ΔP/L) and to the rock's ability to conduct fluid (k/μ). A higher viscosity fluid resists flow; a higher permeability rock conducts flow more easily.
1.2 Permeability — A Rock Property, Not a Fluid Property
Permeability k is purely a property of the pore structure of the rock. It is independent of the fluid flowing through it (provided the rock is fully saturated with a single, non-reactive fluid — the absolute permeability assumption). The Darcy unit is defined so that 1 Darcy = the permeability of a medium through which 1 cp fluid flows at 1 cm³/s across 1 cm² under 1 atm/cm gradient. In practice reservoir permeabilities are measured in milliDarcies (mD), where 1 D = 1000 mD.
Reservoir Type
Typical k Range (mD)
Commercial Significance
Tight gas / shale
0.001 – 0.1
Requires hydraulic fracturing
Low-permeability sandstone
0.1 – 10
May need stimulation
Moderate sandstone / chalk
10 – 100
Typical North Sea chalk reservoirs
High-permeability sandstone
100 – 1,000
Gulf of America, SE Asia turbidites
Gravel pack / fracture
> 1,000
Near-wellbore completions
1.3 Differential Form — The Foundation for Derivation
For a position-dependent pressure, Darcy's law is written in differential form. For flow in the x-direction:
DIFFERENTIAL DARCY'S LAW
u = −(k/μ) · (dP/dx)
u = Darcy flux/superficial velocity (volumetric flow per unit area, ft/day or cm/s)
dP/dx = local pressure gradient in the direction of flow
The negative sign ensures that flow is in the direction of decreasing pressure.
The Darcy flux u is NOT the actual pore velocity. It is the volumetric flow rate per unit bulk cross-sectional area. The true pore velocity is u/ϕ (where ϕ is porosity).
1.4 Assumptions Embedded in Darcy's Law
Darcy's law is valid under specific conditions. Understanding these assumptions is essential for knowing when to apply corrections or more complex models.
✔ Conditions Where Darcy's Law Applies
• Single-phase, fully saturating fluid
• Laminar (viscous-dominated) flow, Darcy or creeping flow regime
• Incompressible or slightly compressible fluid
• Homogeneous, isotropic porous medium
• No chemical interaction between fluid and rock
• Steady-state or quasi-steady conditions
✘ When Darcy's Law Breaks Down
• High-velocity / turbulent flow near the wellbore (non-Darcy effects → Forchheimer term needed)
• Multiphase flow (effective permeability and relative permeability corrections required)
• Gas flow at very low pressures (Klinkenberg slippage)
• Reactive fluids altering pore geometry
• Fracture-dominated flow (fracture aperture models replace matrix k)
IMPORTANT ASSUMPTION FOR THIS MODULE
Module 02 (Topics 2.1 – 2.4) deals exclusively with single-phase, undersaturated oil flow above the bubble point. Under these conditions the oil is a slightly compressible, single-phase liquid, and Darcy's law in its standard form applies directly. Once pressure drops below the bubble point and free gas evolves, two-phase IPR models (Vogel, Fetkovich) are required, these are covered in Module 04.
▶
Supplementary: Why Does Darcy's Law Work? — Pore-Scale Physics
9:15
An optional deep-dive into the Navier-Stokes equations at the pore scale and how volume-averaging yields the macroscopic Darcy equation. Recommended for engineers wanting a rigorous grounding before applying the empirical law.
Section 2 of 6
Radial Flow Geometry — Applying Darcy to a Wellbore
A producing well creates a pressure sink. Reservoir fluids flow inward radially from all directions. This geometry transforms Darcy's linear law into the radial inflow equation via a shell-balance derivation.
2.1 The Radial Flow Model
Consider a vertical well of radius rw penetrating a horizontal, homogeneous, isotropic reservoir of net pay thickness h. The well is produced at a constant sandface rate. We assume the reservoir is bounded by a circular outer boundary at radius re. At any radial distance r from the wellbore axis, the cross-sectional area available for radial flow is the lateral surface of a cylinder:
A(r) = 2π r hr = radial distance from wellbore centre (ft); h = net pay thickness (ft)
Figure 2.1.1 — Radial Flow into a Vertical Wellbore (Plan View)
2.2 Shell-Balance Derivation
Apply Darcy's law to an infinitesimally thin cylindrical shell at radius r. The Darcy flux u at this radius (positive inward, i.e., in the −r direction) is:
u(r) = −(k/μ) · (dP/dr)The minus sign: pressure decreases toward the well (dP/dr > 0 moving outward), so flow is inward (negative r direction).
The volumetric flow rate at radius r (in reservoir conditions) is the flux times the cylindrical surface area:
q_res = u(r) · A(r) = −(k/μ) · (dP/dr) · 2πrh
Under steady-state conditions, q_res is constant at every radius (mass conservation, what flows in at re must equal what flows out at rw). Separating variables and integrating from rw to re:
INTEGRATION — DERIVATION STEPS
1
Rearrange: dP = −[qμ / (2πkh)] · dr/r
2
Integrate left side from Pwf to Pe: ∫dP = Pe − Pwf
3
Integrate right side from rw to re: ∫dr/r = ln(re/rw)
4
Combine and rearrange for q: q_res = [2πkh(Pe−Pwf)] / [μ · ln(re/rw)]
5
Convert to surface conditions by dividing by B (formation volume factor) and apply unit conversion constant for oilfield units.
2.3 Why the Logarithm Matters — The Pressure Distribution
Solving the steady-state radial flow equation for the local pressure profile P(r) gives a logarithmic distribution:
P(r) = P_wf + [qμ / (2πkh)] · ln(r/r_w)
Key insight: The pressure rises logarithmically with distance from the wellbore. Moving from rw = 0.33 ft to 3.3 ft (10× increase) consumes the same pressure drop as moving from 3.3 ft to 33 ft (another 10× increase), and again from 33 ft to 330 ft. Half the total drawdown is consumed in the first few feet around the wellbore. This is why near-wellbore damage (skin) has such a disproportionate effect on well productivity and why near-wellbore stimulation (acidising, frac & pack) is so impactful.
Figure 2.1.2 — Logarithmic Pressure Profile: P vs ln(r/r_w)
2.4 The Role of ln(re/rw)
The denominator of the steady-state radial inflow equation contains the natural logarithm of the drainage-to-wellbore radius ratio. Because this ratio is typically 1000:1 or greater (re ≈ 1000–3000 ft; rw ≈ 0.25–0.5 ft), ln(re/rw) ≈ 7–8 in most situations. Crucially, this logarithmic dependence means that large changes in drainage area have a modest impact on flow rate, while small changes in near-wellbore conditions have a large impact.
INSIGHTSensitivity of ln(r_e/r_w)
For rw = 0.33 ft:
r_e = 500 ft → ln(500/0.33) = ln(1515) = 7.32
r_e = 1000 ft → ln(1000/0.33) = ln(3030) = 8.02 (+9.5% vs 500 ft)
r_e = 2000 ft → ln(2000/0.33) = ln(6060) = 8.71 (+19% vs 500 ft)
Doubling drainage radius increases ln(r_e/r_w) by only ln(2) = 0.69 — roughly 8%.
This means DOUBLING the drainage area increases flow rate by only ~8%.
Compare: reducing r_w from 0.33 ft to 0.10 ft (fracture half-length effect):
ln(r_e/0.10) vs ln(r_e/0.33) = increase of ln(0.33/0.10) = ln(3.3) = 1.19 → +16% flow rate
(And effective wellbore radius from fracturing can be much larger still.)
Section 3 of 6
Steady-State Inflow Equation
When a constant-pressure source (aquifer or gas cap) maintains the outer boundary pressure, reservoir pressure at every radius is constant in time. This steady-state condition yields the simplest form of the radial inflow equation.
3.1 Steady-State Conditions
Steady-state (SS) flow exists when the pressure at every point in the reservoir is constant with time. This requires a pressure-maintenance mechanism at the outer boundary, typically a strong aquifer, an active water injection programme, or a gas cap in pressure equilibrium. The reservoir is said to have a constant pressure outer boundary.
Steady-State Characteristics
• Outer boundary pressure Pe is constant
• Pressure profile does not change with time
• Flow rate at every radius is equal and constant
• Typical of aquifer-supported or water-injected reservoirs
• Rate prediction is straightforward: solve equation directly
Typical Field Examples
• Chalk reservoirs with active underlying aquifer
• Offshore fields under pattern water injection
• Early production from strong edge-water drives
STEADY-STATE RADIAL INFLOW EQUATION — OILFIELD UNITS
Q = 0.00708 · k · h · (P_R − P_wf) / [μ · B · (ln(r_e/r_w) + S)]
Q = oil flow rate at surface conditions (STB/day, stock tank barrels per day)
k = effective permeability to oil (mD, millidarcies)
h = net pay thickness (ft)
P_R = average reservoir pressure at drainage boundary (psia)
P_wf = bottomhole flowing pressure at sandface (psia)
μ = oil viscosity at reservoir conditions (cp, centipoise)
B = oil formation volume factor (RB/STB, reservoir barrels per stock tank barrel)
r_e = drainage radius (ft)
r_w = wellbore radius (open-hole radius or casing ID at depth) (ft)
S = dimensionless skin factor (positive = damage; negative = stimulation)
0.00708 = unit conversion constant (Darcy to oilfield units)
Note on the constant 0.00708: This arises from converting permeability (mD→D: ÷1000), area (ft²→cm²: ×929), length (ft→cm: ×30.48), pressure (psi→atm: ÷14.696), viscosity (cp is already consistent), volume (bbl→cm³: ×158,987), and time (day→s: ×86,400). The constant also incorporates the 2π from the cylindrical geometry.
3.2 Including Skin in the Equation
The skin factor S captures any deviation from ideal radial flow caused by near-wellbore alteration. It appears as an additive term to ln(re/rw) in the denominator. A positive skin increases the denominator, reducing Q. A negative skin (stimulation) decreases it, increasing Q. In terms of effective wellbore radius rw', the skin defines rw' = rw · e−S.
WORKED EXAMPLE 1Karama Field KRM-4 — Steady-State Q
Given data:
Reservoir:
Net pay (h) = 95 ft
Permeability (k) = 18 mD
Average reservoir P_R = 4,850 psia
Drainage radius (r_e) = 1,320 ft (160-acre spacing)
Wellbore radius (r_w) = 0.354 ft (8.5-in hole → ID/2 = 4.25 in = 0.354 ft)
Skin factor (S) = 0 (undamaged, for now)
Fluid:
Oil viscosity (μ) = 1.4 cp
FVF (B_o) = 1.25 RB/STB
Flowing BHP (P_wf) = 3,100 psia (target drawdown)
Result: KRM-4 is expected to produce approximately 1,472 STB/day under undamaged steady-state conditions with a 1,750 psi drawdown.
3.3 Effect of Skin on Steady-State Production
Now examine how damage (positive skin) and stimulation (negative skin) alter the KRM-4 well's productivity at the same Pwf = 3,100 psia:
Skin S
Condition
ln(r_e/r_w) + S
Q (STB/day)
Change vs S=0
+20
Severe damage (scale, fines)
28.22
429
−71%
+10
Moderate damage
18.22
664
−55%
+5
Mild damage
13.22
916
−38%
0
Undamaged
8.22
1,472
baseline
−2
Stimulated (acid wash)
6.22
1,944
+32%
−5
Matrix acid job
3.22
3,752
+155%
Key takeaway: A skin of +20 halves production. A matrix acid job achieving S = −5 more than doubles it. Skin is the most impactful variable under the engineer's control in this equation, which is why Module 03 dedicates an entire topic to deconstructing the skin factor into its mechanical, damage, and stimulation components.
Starting from SI/Darcy units: q [cm³/s] = (k [D] × A [cm²] × ΔP [atm]) / (μ [cp] × L [cm])
Convert to oilfield: Q [STB/day] = q [cm³/s] × (86400 s/day) / (158,987 cm³/bbl) / B
Permeability: k [mD] / 1000 = k [D]
Area at r: A = 2πrh requires the radial integration → yields ln(r_e/r_w) in denominator
Note: Some references use 0.00708 with psi; others use 141.2 in the denominator (reciprocal form). Both are equivalent.
The derivation above assumed a perfectly circular drainage area. In practice, drainage areas are rarely circular. They may be rectangular (due to well spacing patterns), elongated (fault-bounded compartments), or irregular. Note that PROSPER uses the Dietz shape factor CA and drainage area A rather than re directly.
For a well at the centre of a circular drainage area, CA = 31.6 and the equation reduces to the standard ln(re/rw) form.
For a well at the centre of a square: CA = 30.9 (very similar).
For off-centre or rectangular drainage, CA can differ significantly.
A complete table of CA values is in Module 01 Topic 1.4.
For this topic, we use the circular approximation. Topic 2.2 introduces the productivity index concept, where A and CA appear in the PSS form.
Section 4 of 6
Pseudo-Steady-State Inflow
Most reservoir depletion scenarios operate under pseudo-steady-state (PSS) conditions, not true steady-state. Understanding the difference (and using the correct equation) is essential for accurate rate and pressure prediction.
4.1 What is Pseudo-Steady-State?
In most reservoirs, there is no strong aquifer or pressure-maintenance mechanism. As the well produces, reservoir fluids expand slightly and the reservoir pressure declines at a uniform rate everywhere. At any instant, the pressure profile still follows the logarithmic shape, but the entire profile is shifting downward over time. This is called pseudo-steady-state (PSS), also called semi-steady-state or depletion drive.
PSS applies from the moment the pressure transient created by the start of production reaches the outer boundary (no-flow boundary). Before that moment, the well is in transient flow, that regime is covered in Module 03 (Skin and Well Testing). Here we focus on PSS as the reservoir deliverability model for production engineering design.
Transient Flow
Pressure wave still expanding. Outer boundary not yet felt. Pressure at well drops rapidly with time. Well test analysis domain.
Pseudo-Steady State
Outer boundary felt. Entire reservoir depleting uniformly. Pressure everywhere declining at same rate. Production engineering design domain.
Steady State
Outer boundary maintained at constant P by aquifer or injection. Pressure profile static. Relatively rare without active pressure support.
4.2 The PSS Inflow Equation
The PSS form replaces the outer boundary pressure Pe with the current average reservoir pressure P̄ (mean pressure over the entire drainage volume). The geometric correction converts the boundary condition: instead of ln(re/rw), the PSS equation uses ln(re/rw) − 0.75 (for a circular drainage area). This −0.75 accounts for the different pressure distribution compared to steady state.
PSEUDO-STEADY-STATE INFLOW EQUATION — OILFIELD UNITS
Q = 0.00708 · k · h · (P̄ − P_wf) / [μ · B · (ln(r_e/r_w) − 0.75 + S)]
P̄ = current average reservoir pressure (psia) — changes with time as reservoir depletes
−0.75 = geometric correction for PSS boundary condition (circular drainage area)
All other symbols identical to the steady-state form
Alternative notation: ln(re/rw) − 0.75 = ln(0.472 · re/rw)
(because ln(0.472) ≈ −0.75; the 0.472 multiplier on re
4.3 Comparing SS and PSS
For the same reservoir with the same current pressure conditions, the PSS equation predicts a slightly higher flow rate than the steady-state equation (because −0.75 reduces the denominator). This makes physical sense: in PSS, the pressure difference driving flow comes from the entire reservoir depleting, which is a more energetic driving mechanism than the steady-state case where Pe is maintained at the boundary but pressure at intermediate radii is also lower.
Parameter
Steady State (SS)
Pseudo-Steady State (PSS)
Outer boundary condition
Constant Pe
No-flow (q = 0 at re)
Reference pressure in ΔP
Boundary pressure Pe
Average reservoir P̄
Denominator term
ln(re/rw) + S
ln(re/rw) − 0.75 + S
Pressure profile
Static with time
Declining uniformly with time
When applicable
Active aquifer / injection
Most depletion-drive reservoirs
Material balance needed?
No
Yes — to track P̄ over time
WORKED EXAMPLE 2KRM-4 at PSS — Effect of Reservoir Depletion
Using the same KRM-4 parameters, compare PSS production at three stages of reservoir depletion. S = 0; Pwf = 3,100 psia (fixed by separator backpressure).
Observation: As P̄ declines from 4,850 to 3,500 psia, production falls from 1,620 to 371 STB/day, a 77% decline. Maintaining Pwf at 3,100 psia through artificial lift installation becomes critical to sustain economic rates as the reservoir depletes.
4.4 Transient Flow — The Third Regime (Overview)
Before PSS establishes, the well is in transient (infinite-acting radial) flow. In this period the pressure transient is still propagating outward and the apparent productivity appears high but is declining. The transient inflow equation takes a different form, incorporating elapsed time t, porosity ϕ, and total compressibility ct — this is the foundation of well test analysis. The key equation is:
Q = kh(P_i − P_wf) / [162.6μB(log t + log(k/ϕμc_t r_w²) − 3.23 + 0.87S)]
P_i = initial reservoir pressure; t = time (hours); ϕ = porosity; c_t = total compressibility (psi⁻¹)
This form is used in pressure transient analysis, not in production engineering flow rate design.
For PI and IPR calculations, always use the PSS or SS forms above.
The PSS equation is derived by applying a no-flow outer boundary condition (dP/dr = 0 at r = re) rather than the constant-pressure condition (P = Pe at r = re) used for steady state.
Solving the continuity equation (which, for a slightly compressible fluid under PSS, gives a linear pressure decline in time everywhere) yields a pressure profile P(r,t). Integrating this over the drainage volume to obtain P̄ and then expressing the wellbore pressure Pwf in terms of P̄ introduces the −0.75 correction.
Mathematically: when the boundary is a no-flow circle, the average pressure P̄ relates to the boundary pressure Pe (at the same instant) by: P̄ = Pe − [qμB / (2πkh)] × 0.75. This is why substituting P̄ for Pe in the denominator requires subtracting 0.75 from ln(re/rw).
Physical meaning of 0.75: It is ½ln(γ) = ½ln(e^{1.5}) where γ = e^{0.5772} is the exponential of the Euler–Mascheroni constant. It appears in the solution to the diffusivity equation for a cylindrical drainage volume and reflects how average pressure sits below the boundary pressure in a closed system.
Section 5 of 6 — Interactive
Interactive Simulators
Use these tools to build intuition for how each parameter in the radial inflow equation affects well deliverability. Adjust sliders and observe the changes in flow rate and the logarithmic pressure profile in real time.
Scenarios to try:
• Set S = +15 → observe production collapse
• Set S = −5 → observe stimulation benefit
• Increase k from 18 → 180 mD (same chalk, stimulated)
• Compare SS vs PSS at same conditions
• Reduce P̄ progressively to simulate depletion
Notice: The steeper the curve near the wellbore (left), the larger the pressure drop consumed in the near-wellbore zone. Adding skin (positive S) creates an additional vertical pressure step right at the wellbore — this is the skin pressure drop Δp_skin = 0.869 × S × m, where m is the slope of the Horner plot.
The grey region represents pressure consumed by the formation. The red region represents additional skin pressure drop.
PBL EXERCISE
Using Simulator 1, find the minimum permeability threshold (for the KRM-4 reservoir parameters) below which Q drops under 500 STB/day at the same drawdown. This threshold defines whether the Karama Field KRM-4 well would require stimulation to be commercially viable. Record your finding and bring it to Sub-Problem 1 of the Module 02 problem set.
Assessment — Section 6 of 6
Knowledge Check
Ten questions covering Darcy's law, radial flow geometry, steady-state and PSS inflow equations, and their engineering implications. All questions are at the level expected in the Module 02 problem set.
1. In Darcy's Law (q = kAΔP/μL), what does the permeability k physically represent?
Permeability is exclusively a property of the rock's pore geometry — its size, connectivity, and tortuosity. It is independent of the fluid (provided single-phase flow and no rock-fluid reactions). This is why we must separately account for fluid properties (μ, B) and relative permeability effects (multiphase) in the flow equations.
2. Why does the pressure drop per unit radial distance (dP/dr) increase dramatically as you approach the wellbore?
From the Darcy equation in differential form: dP/dr = −qμ/(2πkhr). As r decreases (approaching the wellbore), the denominator decreases, so dP/dr increases in magnitude. The constant flow rate q must pass through progressively smaller cylindrical surfaces — convergent flow — requiring steeper pressure gradients. This geometric effect is independent of rock properties.
3. A well produces 1,200 STB/day from a reservoir with k = 50 mD, h = 80 ft, P̄ = 5,000 psia, μ = 0.9 cp, B = 1.18 RB/STB, r_e = 1,000 ft, r_w = 0.33 ft, S = 0 under PSS conditions. What is P_wf? (Use ln(1000/0.33) = 8.02)
Rearranging the PSS equation for drawdown: ΔP = Q × μ × B × (ln(re/rw) − 0.75) / (0.00708 × k × h). Numerator: 1200 × 0.9 × 1.18 = 1,274.4; × (8.02 − 0.75) = 1,274.4 × 7.27 = 9,264.6. Denominator: 0.00708 × 50 × 80 = 28.32. ΔP = 9,264.6 / 28.32 ≈ 327 psi. Therefore P_wf = P̄ − ΔP = 5,000 − 327 = 4,673 psia (option A). The well is highly productive (kh = 4,000 mD·ft, low-viscosity oil), so 1,200 STB/day requires only ~327 psi of drawdown. Always write out numerator and denominator separately before dividing.
4. What is the key difference between the steady-state (SS) and pseudo-steady-state (PSS) inflow equations?
This is a common confusion. Steady-state uses P_e (the constant boundary pressure) in ΔP and ln(r_e/r_w) in the denominator — the outer boundary maintains constant pressure (aquifer or injection). PSS uses P̄ (average reservoir pressure, which declines with time) and ln(r_e/r_w) − 0.75 in the denominator — the outer boundary is no-flow and the whole reservoir depletes uniformly. Option A has the labels switched — the correct statement is the reverse: PSS uses P̄ and has −0.75; SS uses P_e and has no −0.75 correction.
5. A well has a skin factor S = +12. Which statement best describes its physical meaning?
Skin is a dimensionless number representing a concentrated pressure drop at the wellbore. Physically, S = +12 means the effective wellbore radius is r_w' = r_w × e⁻¹² — an extremely small effective radius. In the inflow equation, ln(r_e/r_w) + S becomes ln(r_e/r_w) + 12, dramatically increasing the denominator and reducing Q. It does not reduce permeability in absolute terms (permeability reduction near the wellbore manifests as positive skin, via the Hawkins formula, but skin itself is a lumped, dimensionless quantity).
6. Doubling the drainage radius r_e from 1,000 ft to 2,000 ft (all else equal, SS conditions) increases Q by approximately:
This is the logarithmic insensitivity to drainage radius. If r_w = 0.33 ft: ln(1000/0.33) = 8.02; ln(2000/0.33) = 8.71. Increase = 0.69 out of 8.02 = 8.6%. Doubling drainage radius (quadrupling drainage area) increases flow rate by less than 9%. This fundamental result explains why infill drilling for rate acceleration gives diminishing returns per additional well — the incremental drainage radius benefit is logarithmically small.
7. The Karama Field KRM-4 well is acidised and the skin reduces from +8 to −2. Net pay h = 95 ft, k = 18 mD, ΔP = 1,750 psia, μ = 1.4 cp, B = 1.25, ln(re/rw) = 8.22. By what factor does flow rate increase? (PSS conditions)
Before acid job: denominator = 8.22 − 0.75 + 8 = 15.47. After acid job: denominator = 8.22 − 0.75 − 2 = 5.47. Flow rate ratio = 15.47/5.47 = 2.83×. The closest answer is C (~2.5×, 150% increase) — the actual ratio is even higher at 2.83×. This demonstrates the enormous leverage of achieving negative skin. A skin improvement of 10 units (from +8 to −2) increases production by ~183% because the denominator is reduced from 15.47 to 5.47.
8. Which of the following is NOT an assumption of the basic radial Darcy inflow equation?
Option D violates the assumptions — Darcy's law as derived here assumes single-phase, undersaturated oil flow above the bubble point. Once pressure drops below the bubble point, free gas evolves and creates a two-phase mixture. The relative permeability to oil decreases (gas occupies pore space), viscosity changes, and the simple linear IPR breaks down. Two-phase models (Vogel, Fetkovich, or mechanistic multiphase) are required — covered in Module 04.
9. The radial inflow equation uses ln(r_e/r_w) in the denominator. Under what conditions would this be replaced with ln(0.472 r_e/r_w)?
ln(0.472 × re/rw) = ln(re/rw) + ln(0.472) = ln(re/rw) − 0.75. This is precisely the PSS denominator correction. Some references write the PSS equation as ln(0.472 re/rw) rather than ln(re/rw) − 0.75 to emphasise that it is the standard SS equation with an adjusted effective drainage radius.
10. Why does the Formation Volume Factor B_o appear in the radial inflow equation denominator?
Darcy's law operates on actual fluid volumes in the reservoir at reservoir pressure and temperature — reservoir barrels (RB). Production is measured and reported at surface (stock tank) conditions — STB. Because oil dissolves gas at reservoir conditions and the gas exsolves at surface, a reservoir barrel of oil produces less than one stock tank barrel (B_o > 1.0 typically). Dividing by B_o converts reservoir flow rate to surface rate. This is why Q in the inflow equation is in STB/day while the Darcy flux operates on RB/day in the reservoir.
TOPIC COMPLETE — NEXT STEPS
You have completed Topic 2.1: Darcy's Law for Radial Flow. You are now ready to:
Topic 2.2 — The Productivity Index (PI): Simplify the radial inflow equation into the PI relationship Q = J(P̄ − P_wf), understand when J is a constant, and use it to design production targets.
Module 02 Sub-Problem 1: Apply the steady-state and PSS inflow equations to the Karama Field KRM-4 well to establish a baseline flow rate. This is the first quantitative deliverable in the Module 02 problem set.