Worked Example: Gas Gathering Network Balancing, Nodal Pressure Convergence & Hydraulic Bottleneck Analysis (Weymouth & API 14E)

1. Problem Statement & Gathering Network Architecture

An unconventional shale gas field gathers production from four multi-well pad clusters (Pads 1 through 4) connected via carbon steel lateral lines to an intermediate header junction. A single main trunkline conveys the combined production from the junction to a Central Gathering Station (CGS) compressor suction manifold.

 [Pad 1: 4.5 MMSCFD, 5.0 km, 4"] ──┐
 [Pad 2: 9.0 MMSCFD, 8.5 km, 5"] ──┼──> [Junction Header] ──[Trunkline: 12 km, 10"]──> [Central Station (CGS): 50 barg]
 [Pad 3: 12.0 MMSCFD, 3.5 km, 6"] ─┼──┘
 [Pad 4: 3.5 MMSCFD, 7.0 km, 4"] ──┘

Network Operating & Fluid Parameters

• Central Delivery Manifold Pressure ($P_{\text{cgs}}$): $50.0\text{ barg}$ ($51.01\text{ bar abs} = 740.0\text{ psia}$)
• Gas Specific Gravity ($S_g$): $0.65$ (Molecular Weight $M_w = 18.83\text{ kg/kmol}$)
• Operating Flowing Temperature ($T_f$): $20.0\text{ }^\circ\text{C}$ ($293.15\text{ K} = 527.67\text{ }^\circ\text{R}$)
• Pipeline Hydraulic Efficiency ($E$): $0.92$ (accounting for minor fittings and pipe wall roughness)
• Base Reference Conditions: $T_b = 60.0\text{ }^\circ\text{F}$ ($520.0\text{ }^\circ\text{R}$), $P_b = 14.73\text{ psia}$
• API RP 14E Empirical Constant ($C$): $100$ (continuous solids-free service)
• Maximum Allowable Wellhead Pressure (MAWHP): $75.0\text{ barg}$

Gathering Line Geometry & Flow Allocation

Engineering Objectives

1. Determine the intermediate junction header pressure ($P_{\text{jct}}$) by back-calculating trunkline friction loss.
2. Determine flowing wellhead pressures ($P_{\text{wh}, i}$) for each pad cluster.
3. Compute in-situ gas velocities and evaluate against API RP 14E erosional limits ($v_e$).
4. Identify the primary hydraulic bottleneck restricting total field deliverability and evaluate debottlenecking options (line looping vs. diameter upsizing).

2. Governing Equations & Hydraulic Methodology

2.1 The Weymouth Compressible Flow Equation

In high-pressure gathering networks ($D \le 16\text{ inches}$), the Weymouth equation accurately models turbulent gas flow:

$$Q = 433.5 \cdot E \cdot \left(\frac{T_b}{P_b}\right) \cdot \left[ \frac{P_1^2 - P_2^2}{S_g \cdot T_f \cdot L_m \cdot Z_{\text{avg}}} \right]^{0.5} \cdot D_{\text{in}}^{8/3}$$

Solving for upstream square pressure ($P_1^2$):

$$P_1^2 = P_2^2 + \left[ \frac{Q_{\text{scfd}}}{433.5 \cdot E \cdot (T_b / P_b) \cdot D_{\text{in}}^{8/3}} \right]^2 \cdot (S_g \cdot T_f \cdot L_m \cdot Z_{\text{avg}})$$

where: * $Q_{\text{scfd}} = Q_{\text{mmscfd}} \times 10^6$ * $P_1, P_2$ = Upstream and downstream pressures ($\text{psia}$) * $L_m = L_{\text{km}} \times 0.621371$ ($\text{miles}$) * $D_{\text{in}} = D_{\text{mm}} / 25.4$ ($\text{inches}$) * $T_b / P_b = 520.0 / 14.73 = 35.302$

2.2 Weymouth Spatial Average Pressure & Real Gas Compressibility

Average line pressure is non-linear due to compressibility:

$$P_{\text{avg}} = \frac{2}{3} \left( P_1 + \frac{P_2^2}{P_1 + P_2} \right)$$

Compressibility factor $Z_{\text{avg}}$ is calculated using Sutton pseudo-critical properties ($T_{pc} = 365.1\text{ }^\circ\text{R}$, $P_{pc} = 669.9\text{ psia}$) and the Papay formulation:

$$Z \approx 1.0 - \frac{3.52 P_{pr}}{10^{0.9813 T_{pr}}} + \frac{0.274 P_{pr}^2}{10^{0.8157 T_{pr}}}$$

2.3 API RP 14E Erosional Velocity Screening

Gas velocity must be maintained below the erosional velocity threshold to prevent sand impingement wall thinning:

$$v_e = \frac{C}{\sqrt{\rho_g}} \qquad (\text{ft/s})$$

where gas density in field units is:

$$\rho_g = \frac{P_{\text{avg}} \cdot M_w}{Z_{\text{avg}} \cdot R \cdot T_f} \qquad (\text{lb/ft}^3)$$

Actual gas velocity at average operating conditions is:

$$v = \frac{Q_{\text{act}}}{A_{\text{pipe}}} = \frac{Q_{\text{scfd}} \cdot (P_b / P_{\text{avg}}) \cdot (T_f / T_b) \cdot Z_{\text{avg}}}{86400 \cdot \frac{\pi}{4} (D_{\text{in}} / 12)^2} \qquad (\text{ft/s})$$

Best practice mandates maintaining erosional ratio $v / v_e \le 0.70$ for long-term continuous piping integrity.

3. Step-by-Step Numerical Calculations

Step 1: Trunkline Hydraulic Back-Calculation

• Total Flow Rate: $Q_{\text{total}} = 4.5 + 9.0 + 12.0 + 3.5 = 29.0\text{ MMSCFD} = 2.90 \times 10^7\text{ SCFD}$
• Length: $L_m = 12.0 \times 0.621371 = 7.456\text{ miles}$
• Diameter: $D_{\text{in}} = 254.0 / 25.4 = 10.0\text{ inches} \implies D_{\text{in}}^{8/3} = 10^{2.6667} = 464.16$
• Delivery Pressure: $P_{\text{cgs}} = 50.0\text{ barg} = 740.0\text{ psia}$

Estimate $Z_{\text{trunk}} \approx 0.880$ initially:

$$\text{Constant} = 433.5 \times 0.92 \times 35.302 \times 464.16 = 6.536 \times 10^6$$

$$\Delta P_{\text{weymouth}}^2 = \left( \frac{2.90 \times 10^7}{6.536 \times 10^6} \right)^2 \times (0.65 \times 527.67 \times 7.456 \times 0.880) = 19.69 \times 2253.1 = 44,360\text{ psia}^2$$

$$P_{\text{jct}} = \sqrt{740.0^2 + 44,360} = \sqrt{547,600 + 44,360} = \sqrt{591,960} = 769.4\text{ psia}$$

$$P_{\text{jct, barg}} = \frac{769.4 - 14.73}{14.5038} = \mathbf{52.03\text{ barg}}$$

• Trunkline Pressure Drop: $\Delta P_{\text{trunk}} = 52.03 - 50.0 = \mathbf{2.03\text{ bar}}$ ($29.4\text{ psi}$)
• Hydraulic Gradient: $\Delta P / L = 2.03 / 12.0 = \mathbf{0.169\text{ bar/km}}$ ($3.95\text{ psi/mile}$)
• Average Pressure: $P_{\text{avg}} = 51.02\text{ barg} = 754.7\text{ psia} \implies Z = 0.878$
• Gas Density: $\rho_g = 2.84\text{ lb/ft}^3$ ($45.5\text{ kg/m}^3$)
• Erosional Velocity: $v_e = 100 / \sqrt{2.84} = 59.3\text{ ft/s} = 18.08\text{ m/s}$
• Actual Velocity: $v_{\text{trunk}} = 19.8\text{ ft/s} = 6.04\text{ m/s}$
• Erosional Ratio: $v / v_e = 6.04 / 18.08 = \mathbf{0.334}$ (Safe, well below $0.70$)

Step 2: Branch Laterals Back-Calculation to Wellheads

Using the converged junction pressure $P_{\text{jct}} = 52.03\text{ barg} = 769.4\text{ psia}$:

Branch 1: Pad 1 (North Field)

• $Q = 4.5\text{ MMSCFD}$, $L = 5.0\text{ km} = 3.107\text{ miles}$, $D = 102.3\text{ mm} = 4.028\text{ inches}$, $D^{8/3} = 40.73$
• $\Delta P_{\text{weymouth}}^2 = 48,220\text{ psia}^2$
• $P_{\text{wh}, 1} = \sqrt{769.4^2 + 48,220} = \sqrt{640,200} = 800.1\text{ psia} = \mathbf{54.15\text{ barg}}$
• $\Delta P_1 = 54.15 - 52.03 = \mathbf{2.12\text{ bar}}$, Gradient $= 2.12 / 5.0 = \mathbf{0.424\text{ bar/km}}$
• $v_1 = 5.48\text{ m/s}$, $v_e = 17.7\text{ m/s} \implies v/v_e = \mathbf{0.310}$

Branch 2: Pad 2 (East Extension)

• $Q = 9.0\text{ MMSCFD}$, $L = 8.5\text{ km} = 5.282\text{ miles}$, $D = 125.0\text{ mm} = 4.921\text{ inches}$, $D^{8/3} = 69.41$
• High flow through relatively long lateral!
• $\Delta P_{\text{weymouth}}^2 = 239,800\text{ psia}^2$
• $P_{\text{wh}, 2} = \sqrt{769.4^2 + 239,800} = \sqrt{831,780} = 912.0\text{ psia} = \mathbf{61.87\text{ barg}}$
• $\Delta P_2 = 61.87 - 52.03 = \mathbf{9.84\text{ bar}}$ ($142.7\text{ psi}$)
• Gradient $= 9.84 / 8.5 = \mathbf{1.158\text{ bar/km}}$ ($27.0\text{ psi/mile}$)
• $v_2 = 7.15\text{ m/s}$, $v_e = 16.5\text{ m/s} \implies v/v_e = \mathbf{0.433}$

Branch 3: Pad 3 (Central Cluster)

• $Q = 12.0\text{ MMSCFD}$, $L = 3.5\text{ km} = 2.175\text{ miles}$, $D = 154.1\text{ mm} = 6.067\text{ inches}$, $D^{8/3} = 122.9$
• Generously sized line: $\Delta P_{\text{weymouth}}^2 = 42,900\text{ psia}^2$
• $P_{\text{wh}, 3} = \sqrt{769.4^2 + 42,900} = \sqrt{634,880} = 796.8\text{ psia} = \mathbf{53.92\text{ barg}}$
• $\Delta P_3 = 53.92 - 52.03 = \mathbf{1.89\text{ bar}}$, Gradient $= 1.89 / 3.5 = \mathbf{0.540\text{ bar/km}}$
• $v_3 = 6.12\text{ m/s}$, $v_e = 17.7\text{ m/s} \implies v/v_e = \mathbf{0.346}$

Branch 4: Pad 4 (South Step-Out)

• $Q = 3.5\text{ MMSCFD}$, $L = 7.0\text{ km} = 4.350\text{ miles}$, $D = 102.3\text{ mm} = 4.028\text{ inches}$
• $\Delta P_{\text{weymouth}}^2 = 41,200\text{ psia}^2$
• $P_{\text{wh}, 4} = \sqrt{769.4^2 + 41,200} = \sqrt{633,180} = 795.7\text{ psia} = \mathbf{53.85\text{ barg}}$
• $\Delta P_4 = 53.85 - 52.03 = \mathbf{1.82\text{ bar}}$, Gradient $= 1.82 / 7.0 = \mathbf{0.260\text{ bar/km}}$
• $v_4 = 4.28\text{ m/s}$, $v_e = 17.7\text{ m/s} \implies v/v_e = \mathbf{0.242}$

4. Hydraulic Bottleneck Isolation & Debottlenecking Analysis

Summary of Network Hydraulic Results

Bottleneck Diagnostics

• Branch 2 consumes 82.9% of the total network pressure differential between wellhead and central manifold ($\Delta P_{\text{total}} = 61.87 - 50.00 = 11.87\text{ bar}$).
• The hydraulic gradient of Branch 2 is $1.158\text{ bar/km}$—nearly three times higher than any other lateral.
• If Pad 2 attempts to drill an infill well to increase flow from $9.0$ to $13.0\text{ MMSCFD}$, flowing wellhead pressure would surge past $78\text{ barg}$, violating the $75\text{ barg}$ MAWHP limit and choking reservoir deliverability.

Quantitative Debottlenecking Solution

Simulate upsizing Branch 2 from $125\text{ mm}$ ($5\text{''}$) to $154.1\text{ mm}$ ($6\text{'' Sch 40}$), or installing a parallel loop:

• Upsized Diameter: $D_{\text{new}} = 154.1\text{ mm} \implies D^{8/3} = 122.9$ (a $77\%$ increase in flow factor)
• New $\Delta P_2$: $$\Delta P_{\text{new}}^2 = \Delta P_{\text{old}}^2 \times \left( \frac{69.41}{122.9} \right)^2 = 239,800 \times 0.319 = 76,500\text{ psia}^2$$ $$P_{\text{wh, new}} = \sqrt{769.4^2 + 76,500} = \sqrt{668,480} = 817.6\text{ psia} = \mathbf{55.35\text{ barg}}$$
• New Frictional Loss: $\Delta P_{\text{new}} = 55.35 - 52.03 = \mathbf{3.32\text{ bar}}$
• Pressure Drop Reduction: $$\text{Reduction} = \frac{9.84 - 3.32}{9.84} \times 100\% = \mathbf{66.3\%}$$
• Field Impact: Flowing wellhead pressure at Pad 2 drops from $61.87\text{ barg}$ down to $55.35\text{ barg}$ (a $6.52\text{ bar} = 95\text{ psi}$ drawdown relief), unlocking up to $4.5\text{ MMSCFD}$ of incremental reservoir deliverability without field compression!

5. ChemProCal Engine Verification & Cross-Check

The test case was computed in ChemProCal's gas-gathering-network-analyzer engine (solve_gathering_network):

from engineering_core.gas_thermo import solve_gathering_network

res = solve_gathering_network(
    manifold_pressure_barg=50.0,
    trunk_length_km=12.0,
    trunk_id_mm=254.0,
    gas_sg=0.65,
    gas_temp_c=20.0,
    pipeline_efficiency=0.92,
    branches=[
        {'name': 'Pad 1 (North)', 'flow_mmscfd': 4.5, 'length_km': 5.0, 'id_mm': 102.3},
        {'name': 'Pad 2 (East)', 'flow_mmscfd': 9.0, 'length_km': 8.5, 'id_mm': 125.0},
        {'name': 'Pad 3 (Central)', 'flow_mmscfd': 12.0, 'length_km': 3.5, 'id_mm': 154.1},
        {'name': 'Pad 4 (South)', 'flow_mmscfd': 3.5, 'length_km': 7.0, 'id_mm': 102.3},
    ]
)

Comparative Verification Matrix

6. Operational Recommendations for Midstream Operators

1. Prioritize Hydraulic Looping over Field Compression: Installing a $4.5\text{ km}$ loop of $6\text{''}$ line parallel to Branch 2 resolves the field bottleneck with zero operational OPEX and zero carbon emissions, whereas adding wellhead booster compressors requires fuel gas, continuous maintenance, and runtime downtime risk.
2. Routine Velocity Audits: When infill drilling or re-fracturing pads, re-verify in-situ gas velocities against the API RP 14E limit ($v/v_e \le 0.70$). Higher velocities drastically accelerate internal erosion in tees and slug catchers.
3. Monitor Pressure Gradient Trends: A sudden increase in branch hydraulic gradient ($\text{bar/km}$) with constant gas throughput indicates liquid condensation pooling (water or hydrocarbon liquid slugging) or solid scale accumulation in low elevation dips, signaling the need for pipeline pigging.