Rod String Analysis

Overview

The sucker rod string transmits motion from the surface pumping unit to the downhole pump. It is the most critical component in a rod pump system — rod failures account for the majority of rod pump workovers. Proper rod string design requires calculating loads, stress, stretch, and dynamic effects.

---

Rod Properties

Standard Rod Sizes

Rod Weight Calculations

Weight in air:

$$W_{air} = w_r \times L$$

Buoyant weight in fluid:

$$W_{buoy} = W_{air}(1 - \gamma_f / \gamma_s)$$

Where:

• $w_r$ = rod weight per foot (lbs/ft)
• $L$ = rod string length (ft)
• $\gamma_f$ = fluid specific gravity
• $\gamma_s$ = steel specific gravity (≈ 7.85)

The buoyancy factor $(1 - \gamma_f/\gamma_s)$ is typically 0.87-0.93 for oilfield fluids.

---

Rod Stretch

Fluid Load Stretch

The weight of fluid above the pump stretches the rods:

$$\delta_f = \frac{W_f \cdot L}{A_r \cdot E} = \frac{62.4 \gamma_f L^2 A_p}{144 A_r E}$$

Rod Weight Stretch

The rod string stretches under its own weight:

$$\delta_r = \frac{w_r L^2}{2 A_r E} = \frac{\gamma_s L^2}{2E}$$

Total Static Stretch

$$\delta_{total} = \delta_f + \delta_r$$

Tapered Rod Strings

For tapered strings with $n$ sections:

$$\delta_f = \frac{W_f}{E} \sum_{i=1}^{n} \frac{L_i}{A_{r,i}}$$

$$\delta_r = \frac{1}{2E} \sum_{i=1}^{n} \frac{w_{r,i} L_i^2}{A_{r,i}} + \frac{1}{E} \sum_{i=1}^{n} \frac{W_{below,i} \cdot L_i}{A_{r,i}}$$

Where $W_{below,i}$ is the weight of all rod sections below section $i$.

---

Polished Rod Loads

Fluid Load

$$W_f = 0.433 \times \gamma_f \times L \times A_p$$

This is the hydrostatic weight of the fluid column acting on the plunger area.

API 11L Dynamic Method

The API 11L method uses dimensionless acceleration factors that depend on the ratio $N/N_0$:

$$N_0 = \frac{v_s}{4L}$$

Where $v_s$ = speed of sound in steel ≈ 16,300 ft/s, and $N_0$ is the natural frequency of the rod string (in SPM after unit conversion).

Peak Polished Rod Load:

$$PPRL = W_f + W_{r,buoy}(1 + F_1)$$

Minimum Polished Rod Load:

$$MPRL = W_{r,buoy}(1 - F_2) - W_f \cdot F_2$$

The factors $F_1$ and $F_2$ increase with pumping speed ratio $N/N_0$.

Simplified Method (Without Dynamic Factors)

For quick estimates:

$$PPRL = W_f + W_{r,buoy}$$ $$MPRL = W_{r,buoy}$$

These ignore dynamic effects and are conservative for PPRL but non-conservative for rod stress range.

Load Range

$$\Delta W = PPRL - MPRL$$

The load range determines:

• Rod stress range (fatigue life)
• Counterbalance requirements
• Surface unit sizing

---

Stress Analysis

Maximum and Minimum Stress

$$\sigma_{max} = \frac{PPRL}{A_r}$$

$$\sigma_{min} = \frac{MPRL}{A_r}$$

Stress Range

$$\Delta\sigma = \sigma_{max} - \sigma_{min} = \frac{PPRL - MPRL}{A_r}$$

Modified Goodman Diagram

Rod fatigue life is assessed using the modified Goodman criterion:

$$\frac{S_a}{S_e} + \frac{S_m}{S_u} \le 1$$

Where:

• $S_a$ = alternating stress = $\Delta\sigma / 2$
• $S_m$ = mean stress = $(\sigma_{max} + \sigma_{min}) / 2$
• $S_e$ = endurance limit
• $S_u$ = ultimate tensile strength

Allowable Stress (API 11L)

---

Counterbalance and Torque

Ideal Counterbalance Effect

$$CBE_{ideal} = \frac{PPRL + MPRL}{2}$$

Peak Torque

$$T_{peak} = \frac{(PPRL - CBE) \times S}{4 \times 12}$$

The factor of 12 converts inches to feet for torque in ft-lbs.

Polished Rod Horsepower

$$PRHP = \frac{\Delta W \times S \times N}{33,000 \times 12}$$

Where 33,000 ft-lbs/min = 1 HP.

---

Design Guidelines

---

Related Topics

• Rod Pump Overview — System design workflow and components
• Pump Displacement — How stretch affects production rate
• ESP Overview — Alternative for higher rates

---

References

1. API Recommended Practice 11L (2008). "Design Calculations for Sucker Rod Pumping Systems (Conventional Units)." American Petroleum Institute.

2. Takacs, G. (2015). Sucker-Rod Pumping Handbook. Gulf Professional Publishing.

3. Gibbs, S.G. (1963). "Predicting the Behavior of Sucker-Rod Pumping Systems." Journal of Petroleum Technology, 15(7), 769-778. SPE-588-PA.

4. Gipson, F.W. and Swaim, H.W. (1988). "The Beam Pumping Design Chain." In Petroleum Engineering Handbook. SPE.

5. Brown, K.E. (1980). The Technology of Artificial Lift Methods, Vol. 2a. PennWell Books.