measuring fracture length and width using PKN and KGD models for hydraulic fracturing?

Decoding Fracking: Cracking the Code on Fracture Length and Width

Hydraulic fracturing, or “fracking” as it’s more commonly known, is like giving an oil or gas well a supercharge. Think of it as a way to boost production, especially in those tricky reservoirs where oil and gas just don’t flow easily. The basic idea? We pump fluid down the well at crazy-high pressure to crack open the surrounding rock, creating fractures that let the good stuff flow more freely. But here’s the million-dollar question: how do we know how big these cracks are? That’s where understanding fracture length and width comes in – it’s absolutely crucial for getting the most bang for our buck. Several models help us estimate these fracture dimensions, and among the most popular are the PKN and KGD models. They’re not perfect, mind you, but they give us valuable insights under the right conditions.

Why Fracture Geometry Matters (A Lot!)

Imagine trying to water your garden with a hose that has a kink in it. Not very effective, right? Well, the same goes for fracking. The size and shape of the fractures we create directly impact how well the well produces. Fracture length dictates how much of the reservoir we’re actually tapping into, while fracture width determines how easily fluids can flow back to the well. Get these dimensions wrong, and you’re leaving money on the table. Accurate estimation is essential for designing efficient fracturing treatments, predicting well performance, and optimizing proppant placement.

PKN Model: When Fractures Go Long (and Tall)

The PKN model, named after Perkins, Kern, and Nordgren, is your go-to when you’re dealing with fractures that are long and thin, like a stretched-out rubber band. It’s a 2D model, meaning it simplifies things by assuming the fracture height stays pretty constant. This is typical in formations where you’ve got strong layers of rock above and below that prevent the fracture from growing upwards or downwards. In essence, it assumes that the fracture is much longer than it is tall. The PKN model also works best when we can assume the rock behaves in a predictable, elastic way, and when the fluid flows smoothly, without too much turbulence.

PKN Model: The Fine Print (Assumptions)

• Fracture length is much greater than fracture height
• The rock behaves predictably (linear isotropic elasticity)
• Fluid flows smoothly (laminar flow)
• The fracture stays the same height

Crunching the Numbers: PKN Style

Alright, let’s get a little technical, but I’ll keep it simple. The PKN model gives us formulas to estimate fracture half-length ($x_f$) and the maximum width at the wellbore ($w_{w,0}$). These formulas use things like the injection rate (how fast we’re pumping fluid), the rock’s properties (Young’s modulus and Poisson’s ratio), the fluid’s viscosity (how thick it is), the fracture height, and the pumping time.

• Fracture half-length ($x_f$):

  $\displaystyle x_f = \left( \frac{625}{512} \frac{Q_0^3 E’}{\mu h_f^4} t^4 \right)^{1/5}$

• Maximum width at the wellbore ($w_{w,0}$):

  $\displaystyle w_{w,0} = \left( \frac{512}{625 \pi^3} \frac{\mu Q_0^2 h_f}{E’^2} t \right)^{1/5}$

  Where:

  • $Q_0$ is the injection rate.
  • $E’$ is the plane strain modulus, defined as $E’ = E / (1 – \nu^2)$, where $E$ is Young’s modulus and $\nu$ is Poisson’s ratio.
  • $\mu$ is the fluid viscosity.
  • $h_f$ is the fracture height.
  • $t$ is the pumping time.

The average fracture width ($\bar{w}$) for a PKN fracture is related to the maximum width at the wellbore: $\bar{w}= (\pi/5) w_{w}$ .

KGD Model: When Height is King

Now, let’s flip the script. The KGD model, named after Khristianovich, Geertsma, and de Klerk, is best suited for fractures where the height is the dominant factor. Think of a tall, narrow fracture, almost like a crack in a wall. This model is often used when the fracture tends to spread out vertically, either because the rock layers aren’t strong enough to contain it, or because the stresses in the earth are aligned in a way that encourages upward growth. Like the PKN model, it assumes the rock behaves elastically and the fluid flows smoothly.

KGD Model: The Fine Print (Assumptions)

• Fracture height is much greater than fracture length
• The rock behaves predictably (linear isotropic elasticity)
• Fluid flows smoothly (laminar flow)

KGD: Calculating Fracture Dimensions

The KGD model also gives us ways to estimate fracture size, but the equations can get a bit hairy depending on the specific assumptions you make. A common approach involves using the Geertsma-de Klerk equation and Sneddon’s elasticity equation. For a static penny-shaped fracture without leakoff under constant net normal pressure $p_0$, the following equations apply :

• Fracture radius (R):

  $R = \left( \frac{3 E’ Q_0 t}{8 \sqrt{\pi} K_{Ic}} \right)^{0.4}$

• Wellbore net pressure ($p_0$):

  $p_0 = \frac{\sqrt{\pi} K_{Ic}}{2 \sqrt{R}}$

• Wellbore fracture width ($w_0$):

  $w_0 = \frac{8 p_0 R}{\pi E’}$

  Where:

  • $E’$ is the plane strain modulus.
  • $Q_0$ is the injection rate.
  • $t$ is the pumping time.
  • $K_{Ic}$ is the fracture toughness.

PKN or KGD: How to Choose?

So, which model do you use? It all boils down to the specific situation. If you’re dealing with long, confined fractures, PKN is your friend. If you’re seeing significant height growth, KGD is the way to go.

A Word of Caution: Models are Just Models

It’s crucial to remember that these models are simplifications. They’re based on assumptions that aren’t always true in the real world. Things like variations in the rock, weird fluid behavior, and existing cracks in the formation can all throw a wrench in the works.

That’s why more advanced models exist, like pseudo-3D (P3D) and full 3D models. These can handle more complex scenarios, but they also require more data and computing power. No matter which model you use, it’s always a good idea to calibrate it with real-world data, like microseismic monitoring.

The Bottom Line

The PKN and KGD models are valuable tools for understanding fracture geometry in hydraulic fracturing. They’re not perfect, but they provide a solid foundation for making informed decisions. By understanding their assumptions and limitations, and by combining them with real-world data and more advanced techniques, we can optimize our fracturing treatments and unlock the full potential of our reservoirs.