Proppant Transport in Hydraulic Fractures

Overview

After pumping stops and the hydraulic pressure is released, the induced fracture closes under the in-situ stress. Without intervention, the fracture aperture returns to zero and the stimulation benefit is lost. Proppant -- granular material pumped as a slurry with the fracturing fluid -- fills the fracture and holds it open, creating a permanently conductive flow path from the reservoir to the wellbore.

The effectiveness of a fracture treatment depends critically on where proppant is placed. If proppant settles too quickly, it banks at the bottom of the fracture, leaving the upper portion unpropped. If the fluid cannot carry proppant to the fracture tip, only the near-wellbore region is propped, resulting in a shorter effective fracture than designed.

Understanding proppant settling behavior is therefore essential for:

• Fluid selection -- choosing viscosity and rheology to control settling
• Pump schedule design -- timing proppant stages for optimal placement
• Propped geometry prediction -- estimating the final conductive fracture dimensions
• Treatment optimization -- balancing proppant transport against other design constraints

---

Common Parameters

Common Proppant Types

---

Stokes Settling Velocity

Theory

A single spherical particle settling under gravity in an infinite, quiescent Newtonian fluid reaches a terminal velocity when the drag force balances the net gravitational force. At low Reynolds numbers ($Re_p < 0.1$), Stokes' law gives:

$$v_{s,Stokes} = \frac{d_p^2 (\rho_p - \rho_f) g}{18 \mu}$$

Where:

• $v_{s,Stokes}$ = Stokes settling velocity
• $d_p$ = particle diameter
• $\rho_p$ = particle density
• $\rho_f$ = fluid density
• $g$ = gravitational acceleration (32.17 ft/s$^2$)
• $\mu$ = dynamic viscosity

Particle Reynolds Number

The particle Reynolds number determines whether Stokes' law is applicable:

$$Re_p = \frac{d_p \, v_s \, \rho_f}{\mu}$$

Field-Corrected Settling Velocity

For intermediate Reynolds numbers typical of fracturing conditions, the Stokes velocity is corrected using empirical drag correlations. A common approach uses the Schiller-Naumann correction:

$$v_{s,field} = v_{s,Stokes} \times \frac{1}{1 + 0.15 Re_p^{0.687}}$$

This correction reduces the settling velocity relative to Stokes' law as inertial effects become significant. In practice, the field-corrected velocity accounts for the fact that proppant particles in fracturing slurries rarely settle in the pure Stokes regime.

---

Wall Effects (Faxen Correction)

Theory

In a hydraulic fracture, the narrow gap between fracture walls significantly retards particle settling. When the particle diameter is a significant fraction of the fracture width, hydrodynamic interaction with the walls creates additional drag.

The Faxen correction (also called the wall factor) modifies the settling velocity for a sphere settling between two parallel plates:

$$f_w = 1 - 2.104 \left(\frac{d_p}{w}\right) + 2.089 \left(\frac{d_p}{w}\right)^3 - 0.948 \left(\frac{d_p}{w}\right)^5$$

Where:

• $f_w$ = wall factor (dimensionless, $0 < f_w \leq 1$)
• $d_p$ = particle diameter
• $w$ = fracture width (gap between walls)

The corrected settling velocity becomes:

$$v_{s,wall} = v_s \cdot f_w$$

Physical Significance

Design implication: Fracture width should be at least 3 times the proppant diameter ($d_p/w < 0.33$) for adequate proppant placement. When $d_p/w > 0.5$, bridging occurs and proppant cannot enter the fracture.

---

Hindered Settling (Richardson-Zaki)

Theory

In concentrated proppant slurries, particle-particle interactions and the upward return flow of displaced fluid reduce the settling velocity. This is called hindered settling. Richardson and Zaki (1954) showed that the settling velocity of a suspension can be related to the single-particle velocity by:

$$v_{s,hindered} = v_s (1 - c_v)^n$$

Where:

• $c_v$ = volume fraction of solids (proppant concentration)
• $n$ = Richardson-Zaki exponent (depends on $Re_p$)

Richardson-Zaki Exponent

For most fracturing applications with viscous fluids and low $Re_p$, $n \approx 4.65$.

Effect of Concentration

Note: The approximate ppg (pounds of proppant per gallon of slurry) values assume sand proppant ($\rho_p = 2.65$ g/cm$^3$) in water-based fluid.

Design implication: At concentrations above 6--8 ppg ($c_v > 0.20$), hindered settling substantially reduces proppant settling, which can be advantageous for proppant placement but increases friction pressure during pumping.

---

Power-Law Fluid Corrections

Theory

Fracturing fluids (crosslinked gels, guar-based fluids) are non-Newtonian and exhibit shear-thinning (power-law) behavior:

$$\tau = K \dot{\gamma}^{n'}$$

Where:

• $\tau$ = shear stress
• $K$ = consistency index (higher $K$ = more viscous)
• $\dot{\gamma}$ = shear rate
• $n'$ = flow behavior index ($n' < 1$ for shear-thinning fluids)

For a sphere settling in a power-law fluid at low Reynolds numbers, the modified Stokes velocity is:

$$v_{s,PL} = \frac{d_p^{n'+1} (\rho_p - \rho_f)^{n'} g^{n'}}{18^{n'} K} \cdot X(n')$$

Where $X(n')$ is a correction factor that depends on the flow behavior index. For practical calculations, the general drag coefficient correlation for power-law fluids gives:

$$C_D = \frac{24}{Re_{PL}} \cdot Y(n')$$

Where the power-law Reynolds number is:

$$Re_{PL} = \frac{d_p^{n'} v_s^{2-n'} \rho_f}{K}$$

Practical Significance

Crosslinked fluids with low $n'$ and high $K$ dramatically reduce settling velocity, which is one of the primary reasons they are used in conventional fracturing -- to maintain proppant in suspension during placement.

---

Settling Distance

Definition

The settling distance ($d_s$) is the vertical distance a proppant particle falls while being transported horizontally through the fracture over a given time period or over a given horizontal distance:

$$d_s = v_s \cdot t_{transport}$$

Where $t_{transport}$ is the time for fluid to travel from the wellbore to the fracture tip:

$$t_{transport} = \frac{x_f}{v_{fluid}}$$

The horizontal fluid velocity within the fracture is approximately:

$$v_{fluid} = \frac{Q}{h_f \cdot w}$$

Combining:

$$d_s = v_s \cdot \frac{x_f \cdot h_f \cdot w}{Q}$$

Settling Criterion

For adequate proppant placement, the settling distance should be small relative to the fracture height:

$$\frac{d_s}{h_f} \ll 1 \quad \text{(good transport)}$$

---

Transport Ratio

Definition

The transport ratio ($T_R$) compares the horizontal proppant transport velocity to the vertical settling velocity:

$$T_R = \frac{v_{horizontal}}{v_{settling}} = \frac{Q / (h_f \cdot w)}{v_s}$$

A high transport ratio indicates that proppant moves horizontally much faster than it settles vertically, resulting in better fracture-tip proppant placement.

Interpretation

Design Strategies to Improve Transport

---

Combined Settling Velocity

In practice, all effects act simultaneously. The effective settling velocity in a fracture combines Stokes settling, field correction, wall effects, hindered settling, and fluid rheology:

$$v_{s,eff} = v_{s,base} \cdot f_{wall} \cdot (1 - c_v)^n$$

Where $v_{s,base}$ already incorporates the fluid rheology (Newtonian or power-law) and any Reynolds number corrections.

Calculation Procedure

Step 1: Calculate Stokes settling velocity
        v_Stokes = dp^2 (rho_p - rho_f) g / (18 mu)

Step 2: Check Reynolds number
        Re_p = dp * v_Stokes * rho_f / mu
        If Re_p > 1, apply field correction

Step 3: Apply wall effect
        fw = 1 - 2.104(dp/w) + 2.089(dp/w)^3 - 0.948(dp/w)^5

Step 4: Apply hindered settling
        fh = (1 - cv)^n    where n = 4.65 for Stokes regime

Step 5: Combine
        v_eff = v_corrected * fw * fh

Step 6: Calculate settling distance and transport ratio
        ds = v_eff * xf * hf * w / Q
        TR = Q / (hf * w * v_eff)

---

Applicability and Limitations

Model Validity

Typical Fracturing Conditions

---

Related Topics

• Hydraulic Fracturing Overview -- Fracturing fundamentals and workflow
• Fracture Geometry Models -- PKN, KGD, Radial width and length equations
• Well Flow Overview -- Post-frac well productivity and skin
• PVT Overview -- Fluid density and viscosity for calculations

---

References

1. Stokes, G.G. (1851). "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums." Transactions of the Cambridge Philosophical Society, 9, pp. 8-106.

2. Richardson, J.F. and Zaki, W.N. (1954). "Sedimentation and Fluidisation: Part I." Transactions of the Institution of Chemical Engineers, 32, pp. 35-53.

3. Economides, M.J. and Nolte, K.G. (2000). Reservoir Stimulation, 3rd Edition. John Wiley & Sons. Chapter 6: Proppant Transport.

4. Novotny, E.J. (1977). "Proppant Transport." SPE-6813-MS, presented at the 52nd Annual Fall Technical Conference, Denver, Colorado, 9-12 October.

5. Shah, S.N. (1986). "Proppant Settling Correlations for Non-Newtonian Fluids." SPE Production Engineering, 1(6), pp. 446-448. SPE-13835-PA.