Fracture Geometry Models - PKN, KGD, and Radial

Overview

Classical 2D fracture models predict the geometry of a hydraulically induced fracture -- its width, length (or radius), net pressure, and volume -- as functions of injection rate, fluid properties, rock properties, and time. Three fundamental models serve as the foundation for fracture design:

All three models solve the coupled problem of:

1. Elasticity -- rock deformation under internal fluid pressure
2. Fluid mechanics -- viscous flow within the fracture
3. Material balance -- injected volume = fracture volume + leakoff volume
4. Fracture mechanics -- propagation at the fracture tip

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Common Parameters

Plane Strain Modulus

The plane strain modulus appears in all fracture width equations:

$$E' = \frac{E}{1 - \nu^2}$$

Typical values:

• Sandstone: $E = 1 - 6 \times 10^6$ psi, $\nu = 0.15 - 0.25$
• Shale: $E = 1 - 4 \times 10^6$ psi, $\nu = 0.25 - 0.35$
• Limestone: $E = 4 - 10 \times 10^6$ psi, $\nu = 0.25 - 0.30$

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PKN Model (Perkins-Kern-Nordgren)

Physical Description

The PKN model assumes:

• Confined height: fracture height $h_f$ is fixed by stress barriers above and below
• Vertical plane strain: each vertical cross-section deforms independently
• Length much greater than height: $x_f \gg h_f$
• Elliptical cross-section: width varies elliptically across the height at any position along the length
• Maximum width at wellbore: width decreases toward the fracture tip

    Plan View (top down)              Cross-Section at any x
    ┌─────────────────────┐           ┌────────────────┐
    │                     │           │    ◜      ◝    │ ▲
    │  ○                  │           │  ◜          ◝  │ │
    │ well  ──────────► tip           │◜     w(x)    ◝│ hf
    │  ○                  │           │  ◟          ◞  │ │
    │                     │           │    ◟      ◞    │ ▼
    └─────────────────────┘           └────────────────┘
    ◄────── xf ──────────►                ◄── w ──►
    Width decreases toward tip         Elliptical shape

Equations (No Leakoff)

Maximum width (at wellbore, $x = 0$):

$$w_{max} = 3.57 \left(\frac{Q \mu x_f}{E'}\right)^{1/4}$$

Where $w_{max}$ is in inches when $Q$ is in bbl/min, $\mu$ in cP, $x_f$ in ft, and $E'$ in psi.

Fracture half-length (no leakoff, $C_L = 0$):

$$x_f = 0.524 \left(\frac{E' Q^3}{\mu h_f^4}\right)^{1/5} t^{4/5}$$

Net pressure:

$$p_{net} = 2.24 \left(\frac{E'^3 \mu Q}{h_f^4}\right)^{1/5} t^{1/5}$$

Note: In the PKN model, net pressure increases with time as the fracture grows longer and friction losses increase. This is a distinguishing diagnostic feature.

Fracture volume (one wing, average width = $\pi/5 \times w_{max}$):

$$V_f = \frac{\pi}{5} w_{max} \cdot h_f \cdot x_f$$

Equations (With Carter Leakoff)

When leakoff is included, the half-length is reduced from the no-leakoff case. The Nordgren (1972) solution for fracture length with Carter leakoff uses the material balance:

$$x_f = \frac{Q t}{h_f \left(\bar{w} + 2 \pi C_L \sqrt{t} \cdot g(\eta)\right)}$$

Where $g(\eta)$ is a function of fluid efficiency that accounts for the time-dependent exposure of fracture faces, and $\bar{w}$ is the average fracture width.

For practical calculation, the length with leakoff is obtained iteratively from the material balance equation coupling width, length, and leakoff volume.

PKN Parameter Ranges

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KGD Model (Khristianovic-Geertsma-de Klerk)

Physical Description

The KGD model assumes:

• Confined height: fracture height $h_f$ is fixed
• Horizontal plane strain: deformation is uniform along the height
• Length comparable to or less than height: $x_f \leq h_f$
• Rectangular cross-section: width is constant along the height at any position
• Maximum width at center: width is maximum at $x = 0$ (center of length) and zero at the tips

    Plan View (top down)              Cross-Section at any x
    ┌─────────────────────┐           ┌────────────────┐
    │         tip         │           │                │ ▲
    │          │          │           │                │ │
    │  tip ── well ── tip │           │    w(x)        │ hf
    │          │          │           │   (uniform)    │ │
    │         tip         │           │                │ ▼
    └─────────────────────┘           └────────────────┘
    ◄────── xf ──────────►                ◄── w ──►
    Width max at center               Rectangular shape

Equations (No Leakoff)

Maximum width (at center of fracture, $x = 0$):

$$w_{max} = 3.22 \left(\frac{Q \mu x_f^2}{E' h_f}\right)^{1/4}$$

Fracture half-length (no leakoff):

$$x_f = 0.539 \left(\frac{E' Q^3}{\mu h_f^3}\right)^{1/6} t^{2/3}$$

Net pressure:

$$p_{net} = 1.09 \left(\frac{E' \mu^2 Q^2}{h_f^6}\right)^{1/6} t^{-1/3}$$

Note: In the KGD model, net pressure decreases with time. This is opposite to the PKN model and serves as a diagnostic indicator.

Fracture volume (one wing, average width = $\pi/4 \times w_{max}$):

$$V_f = \frac{\pi}{4} w_{max} \cdot h_f \cdot x_f$$

KGD Parameter Ranges

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Radial (Penny-Shaped) Model

Physical Description

The radial model assumes:

• No height confinement: fracture propagates radially from the wellbore
• Circular (penny-shaped) geometry: fracture is a flat disk
• Axial symmetry: width depends only on radial distance from center
• Maximum width at center: width is maximum at the wellbore and zero at the circular tip

This model applies when stress barriers are absent or when the fracture has not yet reached confining layers (early time behavior).

    Plan View (top down)              Side View (cross-section)
         ╱────────╲
       ╱            ╲                 ────────────────────────
     ╱                ╲                  ◜                ◝
    │        ●         │              ◜         w(r)        ◝
     ╲      well      ╱                  ◟                ◞
       ╲            ╱                 ────────────────────────
         ╲────────╱
    ◄──────── R ────────►             ◄────────── R ──────────►
       Circular front                   Ellipsoidal opening

Equations (No Leakoff)

Maximum width (at wellbore center):

$$w_{max} = 3.65 \left(\frac{\mu Q R}{E'}\right)^{1/4}$$

Fracture radius (no leakoff):

$$R = 0.572 \left(\frac{E' Q^3}{\mu}\right)^{1/9} t^{4/9}$$

Net pressure:

$$p_{net} = 2.51 \left(\frac{E'^3 \mu^5 Q^5}{R^{13}}\right)^{1/9}$$

Or expressed in terms of time:

$$p_{net} \propto t^{-1/3}$$

Note: Like the KGD model, net pressure in the radial model decreases with time.

Fracture volume (average width = $8/(3\pi) \times w_{max}$):

$$V_f = \frac{8}{3\pi} w_{max} \cdot \pi R^2 = \frac{8}{3} w_{max} R^2$$

Radial Model Parameter Ranges

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Carter Leakoff Model

Theory

Carter (1957) described fluid loss from a fracture face with a time-dependent velocity:

$$u_L(t, \tau) = \frac{C_L}{\sqrt{t - \tau}}$$

Where:

• $u_L$ = leakoff velocity normal to fracture face, ft/min
• $C_L$ = leakoff coefficient, ft/min$^{1/2}$
• $t$ = current time, min
• $\tau$ = time when that fracture face element was first exposed

Leakoff Coefficient

The total leakoff coefficient combines three resistances in series:

$$\frac{1}{C_L} = \frac{1}{C_I} + \frac{1}{C_{eff}}$$

Where:

$$C_{eff} = \frac{C_{II} C_{III}}{C_{II} + C_{III}} \left(1 + \sqrt{1 + \left(\frac{2C_{II}C_{III}}{C_{II}^2 + C_{III}^2}\right)}\right) \div 2$$

In practice, the overall $C_L$ is often determined directly from a mini-frac (diagnostic fracture injection test).

Typical Leakoff Coefficients

Cumulative Leakoff Volume

The total volume leaked off from one face of one wing over the fracture area:

$$V_L = 2 h_f C_L \int_0^{x_f} \sqrt{t - \tau(x)} \, dx$$

For a fracture with constant-rate propagation, this simplifies using the spurt loss and the leakoff area integral.

Fluid Efficiency

The fluid efficiency at any time is:

$$\eta = \frac{V_{frac}}{V_{total}} = \frac{V_{frac}}{V_{frac} + V_{leakoff}}$$

Where $V_{total} = Q \cdot t$ (total pumped volume).

Material balance for the fracture at time $t$:

$$Q t = V_{frac} + V_{leakoff} + V_{spurt}$$

Rearranging:

$$\eta = 1 - \frac{V_{leakoff} + V_{spurt}}{Q t}$$

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Model Comparison

Assumptions Summary

Scaling Laws (No Leakoff)

When to Use Each Model

Net Pressure Diagnostic

The trend of net pressure during pumping indicates which model governs:

log(p_net)
    │
    │     ╱                PKN: slope = +1/5
    │   ╱
    │ ╱
    │╱─────────────────    Transition
    │╲
    │  ╲                   KGD/Radial: slope = -1/3
    │    ╲
    └──────────────────→ log(t)

• Rising net pressure: PKN behavior (fracture lengthening, increasing friction)
• Falling net pressure: KGD or Radial behavior (fracture widening, decreasing resistance)
• Flat net pressure: Equilibrium or height growth (not captured by 2D models)

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Related Topics

• Hydraulic Fracturing Overview -- Model selection guide and workflow
• Proppant Transport -- Settling velocity and placement design
• Well Flow Overview -- Post-frac productivity and equivalent skin
• Productivity Index -- Fractured well productivity calculations

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References

1. Perkins, T.K. and Kern, L.R. (1961). "Widths of Hydraulic Fractures." Journal of Petroleum Technology, 13(9), pp. 937-949. SPE-89-PA.

2. Nordgren, R.P. (1972). "Propagation of a Vertical Hydraulic Fracture." Society of Petroleum Engineers Journal, 12(4), pp. 306-314. SPE-3009-PA.

3. Geertsma, J. and de Klerk, F. (1969). "A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures." Journal of Petroleum Technology, 21(12), pp. 1571-1581. SPE-2458-PA.

4. Carter, R.D. (1957). "Derivation of the General Equation for Estimating the Extent of the Fractured Area." Appendix to: Howard, G.C. and Fast, C.R., "Optimum Fluid Characteristics for Fracture Extension." Drilling and Production Practice, API, pp. 261-270.

5. Sneddon, I.N. (1946). "The Distribution of Stress in the Neighbourhood of a Crack in an Elastic Solid." Proceedings of the Royal Society A, 187(1009), pp. 229-260.