8.1 Volumetric Method: Static and Dynamic Gas Migration Control

During well control operations, conventional constant bottomhole pressure (CBHP) techniques such as the Driller's Method or the Wait-and-Weight Method require continuous, unobstructed fluid circulation down the drillstring and up the annulus. In operational scenarios where circulation is impossible, wellsite supervisors must deploy non-circulating contingency protocols. The Volumetric Method is the primary technique used to control wellbore pressures when a gas influx migrates up a shut-in wellbore.

Indications for the Volumetric Method

The volumetric method can control pressure from migrating free gas when suitable circulation is unavailable. Situations requiring assessment include:

• Drill Bit Off Bottom: The drillstring is high off bottom during a trip, or the well kicks while pulling out of the hole (POOH) or running in the hole (RIH).
• Completely Plugged Drillstring: Drill bit nozzles are plugged by lost circulation material (LCM), a flapper-type float valve has jammed shut, or the drillpipe is packed off by formation sloughing.
• Parted or Severed Drillstring: Mechanical failure has severed the string above the bit, preventing circulation to the kick depth.
• Complete Rig Pump / Power Plant Failure: Total loss of mud pumps, prime movers, or surface circulating manifolds prevents pump-driven circulation.
• Pipe Out of Hole: The well is shut in on blind or shear rams with an empty hole, and gas is percolating toward surface.

---

The Physics of Gas Migration in a Closed Wellbore

Hydrocarbon gas (primarily methane, $\text{CH}_4$) has a density substantially lower than drilling fluid, typically ranging between $1.0\text{ and }2.5\text{ ppg}$ ($0.05\text{ to }0.13\text{ psi/ft}$) under downhole conditions, compared to weighted drilling muds ranging from $9.0\text{ to }18.0+\text{ ppg}$ ($0.468\text{ to }0.936+\text{ psi/ft}$). Because of this density contrast, gravitational buoyancy forces the gas bubble to percolate upward through the liquid mud column.

Gas migration speed depends on mud rheology, gas distribution, geometry and inclination. Use the observed pressure-rise rate and the applicable model when estimating migration; do not select a universal speed from well depth alone.

The Closed Wellbore Pressure Escalation Hazard

When a gas kick enters at bottomhole, its initial pressure is approximately equal to reservoir pore pressure ($P_{\text{gas, initial}} \approx P_{\text{form}}$). If a well is shut in and gas migrates toward the surface without being permitted to expand ($V = \text{constant}$), the gas retains its initial reservoir pressure as it ascends (disregarding minor real gas temperature and compressibility adjustments):

$P_{\text{gas}}(t) \approx P_{\text{gas, initial}} \approx P_{\text{form}}$

As the bubble rises by a vertical height $h_{\text{migration}}$, liquid mud transfers from above the bubble to below it. The column of mud beneath the bubble increases, while the bubble's internal pressure remains near $P_{\text{form}}$. Consequently, bottomhole pressure ($P_{\text{BHP}}$) rises continuously by the hydrostatic head of the mud column accumulating below the bubble:

$P_{\text{BHP}} = P_{\text{form}} + G_{\text{mud}} \times h_{\text{migration}} = P_{\text{form}} + (0.052 \times \rho_{\text{mud}} \times h_{\text{migration}})$

Where:

• $\rho_{\text{mud}}$ = Drilling fluid density in pounds per gallon ($\text{ppg}$)
• $G_{\text{mud}}$ = Mud hydrostatic pressure gradient ($\text{psi/ft}$)
• $h_{\text{migration}}$ = Vertical distance the gas has migrated upward from its point of origin ($\text{ft}$)

Simultaneously, surface shut-in casing pressure ($\text{SICP}$) increases at the exact same rate:

$\Delta \text{SICP} = G_{\text{mud}} \times h_{\text{migration}} = 0.052 \times \rho_{\text{mud}} \times h_{\text{migration}}$

If gas reaches the surface in a completely shut-in well without expansion, the surface casing pressure will equal original reservoir pore pressure minus the negligible hydrostatic head of the gas column ($P_{\text{surf}} \approx P_{\text{form}}$). This pressure rise can exceed formation or equipment limits. Calculate the current pressure at the controlling formation and compare it with the applicable integrity limit, and compare surface pressure with its current-profile limit; neither fracture nor an underground blowout follows automatically from every migration pressure rise.

---

The Governing Principle: Controlled Volumetric Expansion

The volumetric method allows controlled gas expansion while keeping BHP within a planned pressure band. In the idealized model below, use a uniform vertical annulus, incompressible mud, negligible gas head/temperature change, a discrete free-gas influx and no losses or additional influx.

Choose safety and working increments only after checking the available pressure window. At the influx depth: $P_{\text{floor}}=P_{\text{form}}+\Delta P_{\text{safety}}$ $P_{\text{peak}}=P_{\text{floor}}+\Delta P_{\text{working}}$ The peak must also keep pressure at each controlling formation below its applicable integrity limit. Calculate that formation's pressure from the actual fluid layers. Compare surface casing pressure with the current-profile surface limit and equipment ratings; do not compare BHP directly with a shoe-depth fracture pressure or a surface MAASP.

During the closed build step, migration raises BHP by the working increment. During the constant-casing-pressure liquid bleed, gas expansion removes mud head and lowers BHP by that increment, returning it to the safety floor. The method controls a band; BHP is not constant throughout each batch.

At the influx-depth datum:
  Cycle peak  = formation pressure + safety + working increment
  Cycle floor = formation pressure + safety
  Formation pressure lies below the floor.
Separately verify pressure at the weak zone and surface equipment limits.

---

Mathematical Derivations and Working Parameters

Executing the volumetric method requires establishing three foundational operational values:

1. Safety Margin Pressure Increment ($\Delta P_{\text{safety}}$)

Before initiating fluid bleed-off, the supervisor allows casing pressure to build above initial stabilized $\text{SICP}$ by a predetermined safety margin, typically $50\text{ to }100\text{ psi}$ ($3.5\text{ to }7.0\text{ bar}$) according to operator well control policies:

• This initial build adds an intentional hydrostatic overbalance to the wellbore: $P_{\text{BHP}} = P_{\text{form}} + \Delta P_{\text{safety}}$
• It establishes the lower operational pressure datum (the overbalance floor). During all subsequent expansion and bleed cycles, bottomhole pressure is never permitted to drop below this floor, protecting against secondary reservoir influxes caused by normal choke manipulation fluctuations or measurement inaccuracies.

2. Working Pressure Increment ($\Delta P_{\text{working}}$)

The operator selects a discrete working pressure increment ($\Delta P_{\text{working}}$), typically $50\text{ to }100\text{ psi}$ ($3.5\text{ to }7.0\text{ bar}$) per operator guidelines, to govern each expansion cycle:

• Smaller increments ($50\text{ psi}$ / $3.5\text{ bar}$) maintain tighter bottomhole pressure control but require frequent choke operations.
• Larger increments ($100\text{ psi}$ / $7.0\text{ bar}$) reduce choke manifold wear and crew fatigue, provided the casing shoe fracture margin (MAASP) accommodates the higher peak pressures.
• While $50\text{ to }100\text{ psi}$ is standard industry training practice, the exact value is an operational guideline tailored to well geometry, choke precision, and formation breakdown limits, not an inflexible statutory absolute.

3. Mud Bleed Volume Increment ($\Delta V_{\text{bleed}}$)

To offset the working pressure increment ($\Delta P_{\text{working}}$), a precise volumetric batch of drilling mud ($\Delta V_{\text{bleed}}$) must be bled from the annulus into a calibrated trip tank. The relationship is derived directly from the hydrostatic equation:

$\Delta P_{\text{working}} = G_{\text{mud}} \times \Delta h_{\text{mud}}$

Where $\Delta h_{\text{mud}}$ is the equivalent vertical height of mud displaced by the expanding gas downhole:

$\Delta h_{\text{mud}} = \frac{\Delta V_{\text{bleed}}}{C_{\text{ann}}}$

Substituting $\Delta h_{\text{mud}}$ into the hydrostatic equation:

$\Delta P_{\text{working}} = G_{\text{mud}} \times \left( \frac{\Delta V_{\text{bleed}}}{C_{\text{ann}}} \right)$

Solving for the required mud bleed volume ($\Delta V_{\text{bleed}}$):

Field Units (IWCF Formula 23):

$\Delta V_{\text{bleed}} (\text{bbl}) = \frac{\Delta P_{\text{working}} (\text{psi}) \times C_{\text{ann}} (\text{bbl/ft})}{G_{\text{mud}} (\text{psi/ft})} = \frac{\Delta P_{\text{working}} (\text{psi}) \times C_{\text{ann}} (\text{bbl/ft})}{0.052 \times \rho_{\text{mud}} (\text{ppg})}$

Where:

• $\Delta V_{\text{bleed}}$ = Batch volume of mud to bleed into the trip tank ($\text{bbl}$)
• $\Delta P_{\text{working}}$ = Selected working pressure increment ($\text{psi}$)
• $C_{\text{ann}}$ = Capacity of the uniform vertical annular interval used in this model ($\text{bbl/ft}$)
• $\rho_{\text{mud}}$ = Drilling fluid density in the annulus ($\text{ppg}$)
• $G_{\text{mud}} = 0.052 \times \rho_{\text{mud}}$ = Mud hydrostatic pressure gradient ($\text{psi/ft}$)

SI Metric Units:

$\Delta V_{\text{bleed}} (\text{litres}) = \frac{\Delta P_{\text{working}} (\text{bar}) \times C_{\text{ann}} (\text{litres/m})}{G_{\text{mud}} (\text{bar/m})}$

Where the mud hydrostatic gradient in bar per metre is: $G_{\text{mud}} (\text{bar/m}) = \frac{\rho_{\text{mud}} (\text{kg/m}^3) \times 9.81}{100,000} = \frac{\text{SG} \times 9.81}{100} \approx \frac{\text{SG}}{10.2}$

Substituting $G_{\text{mud}} = \frac{\text{SG}}{10.2}$ yields: $\Delta V_{\text{bleed}} (\text{litres}) = \frac{\Delta P_{\text{working}} (\text{bar}) \times C_{\text{ann}} (\text{litres/m}) \times 10.2}{\text{SG}}$

Or in cubic metres: $\Delta V_{\text{bleed}} (\text{m}^3) = \frac{\Delta P_{\text{working}} (\text{bar}) \times C_{\text{ann}} (\text{m}^3/\text{m}) \times 10.2}{\text{SG}}$

---

Step-by-Step Volumetric Cycle Execution

The standard static volumetric method follows a rigorous cyclic protocol executed at the remote hydraulic choke console:

+------------------------------------------------------------------------+
| Phase 1: Build Initial Safety Margin                                   |
|   - Well shut in. Allow SICP to rise by ΔP_safety (50-100 psi / 3.5-7 bar).|
|   - Establishes overbalance floor: BHP = P_form + ΔP_safety.           |
+------------------------------------------------------------------------+
                                   |
                                   v
+------------------------------------------------------------------------+
| Phase 2: Allow Pressure to Build Working Increment                     |
|   - Choke closed. Allow SICP to rise by ΔP_working (e.g., 50 psi).     |
|   - Casing pressure reaches: P_target = P_current + ΔP_working.        |
+------------------------------------------------------------------------+
                                   |
                                   v
+------------------------------------------------------------------------+
| Phase 3: Bleed Calculated Mud Increment at Constant Pressure           |
|   - Open choke to bleed mud into calibrated trip tank.                 |
|   - Hold casing pressure strictly constant at P_target during bleed.   |
|   - Stop bleed when exactly ΔV_bleed has entered the trip tank.        |
+------------------------------------------------------------------------+
                                   |
                                   v
+------------------------------------------------------------------------+
| Phase 4: Close Choke and Monitor                                       |
|   - Close choke. Bottomhole pressure is restored to overbalanced state.|
|   - Reset volume counters and repeat Phase 2 through Phase 4.          |
+------------------------------------------------------------------------+

The Cardinal Choke Rule of the Volumetric Method

A critical failure mode on well control simulators is opening the choke and bleeding pressure down.

---

Terminal Condition: Gas Arrival at Surface

The volumetric method can only function while liquid drilling mud is available to be bled from the annulus. When the gas bubble reaches the surface under the blowout preventer, the volumetric method reaches its terminal operational limit.

Recognize Gas at the Bleed Point

Gas-cut/foamy liquid or free gas at the choke shows that discharge can no longer be counted as pure liquid volume. Pressure-rise rate and choke cooling alone do not locate the gas or prove arrival.

Stop the liquid-volume bleed calculation and secure/reassess the well. If gas is at the top and circulation remains unavailable, prepare lubricate-and-bleed with suitable pumping and gas-handling equipment. If a safe circulation path has been restored, use the selected circulating procedure instead. Gas discharge must use the designed gas-handling route, not an unprotected liquid measuring tank.

---

Dynamic Volumetric Method During Stripping Operations

When stripping is selected, track pipe displacement, migration and changes in influx height separately. Closed-end pipe entering liquid displaces its external volume; an open, freely filling bore displaces metal volume. Combine the appropriate displacement accounting with the approved migration-control increments.

Pipe entering a gas interval can increase gas-column height and reduce annular mud head. The standard pressure correction is: $\Delta P_{\text{casing}}=\Delta h_{\text{influx}}(G_{\text{mud}}-G_{\text{influx}})$ The planned casing reference can therefore rise during gas entry even when no additional influx occurs. Use the stripping sheet and actual geometry; do not hold one casing pressure and bleed external pipe volume indiscriminately through every phase. Re-establish safe circulation when the assessed plan permits.

---

Worked Field Problem: Full Volumetric Calculation Schedule

Field Units Problem:

• Well Depth: $11,500\text{ ft TVD}$
• Drillstring: $5.0\text{ in}$ drillpipe, bit pulled to $6,500\text{ ft TVD}$
• Casing: $9\text{-}5/8\text{ in}$, $40.0\text{ lb/ft}$ ($D_{\text{in}} = 8.835\text{ in}$)
• Mud Density ($\rho_{\text{mud}}$): $12.0\text{ ppg}$
• Initial Stabilized $\text{SICP}$: $600\text{ psi}$
• Gas Influx Location: Inside the $9\text{-}5/8\text{ in}$ casing opposite $5.0\text{ in}$ drillpipe
• Planned Safety Margin ($\Delta P_{\text{safety}}$): $100\text{ psi}$
• Working Pressure Increment ($\Delta P_{\text{working}}$): $50\text{ psi}$

Step 1: Calculate Annular Capacity at Gas Influx Location

$C_{\text{ann}} = \frac{D_{\text{casing, ID}}^2 - D_{\text{pipe, OD}}^2}{1029.4} = \frac{(8.835)^2 - (5.0)^2}{1029.4} = \frac{78.057 - 25.0}{1029.4} = \frac{53.057}{1029.4} = 0.05154\text{ bbl/ft}$

Step 2: Calculate Mud Hydrostatic Gradient

$G_{\text{mud}} = 0.052 \times 12.0 = 0.624\text{ psi/ft}$

Step 3: Calculate Required Mud Bleed Volume ($\Delta V_{\text{bleed}}$)

$\Delta V_{\text{bleed}} = \frac{\Delta P_{\text{working}} \times C_{\text{ann}}}{0.052 \times \rho_{\text{mud}}} = \frac{50 \times 0.05154}{0.624} = \frac{2.577}{0.624} = \mathbf{4.13\text{ bbl}}$

Step 4: Operational Execution Schedule

---

SI Metric Worked Example:

• Mud Density ($\text{SG}$): $1.45\text{ SG}$ ($1,450\text{ kg/m}^3$)
• Annular Capacity at Gas Influx: $26.8\text{ litres/m}$
• Selected Working Pressure Increment: $\Delta P_{\text{working}} = 5.0\text{ bar}$

Step 1: Calculate Mud Hydrostatic Gradient

$G_{\text{mud}} = \frac{1.45}{10.2} = 0.14216\text{ bar/m}$

Step 2: Calculate Required Mud Bleed Volume

$\Delta V_{\text{bleed}} = \frac{\Delta P_{\text{working}} \times C_{\text{ann}}}{G_{\text{mud}}} = \frac{5.0\text{ bar} \times 26.8\text{ L/m} \times 10.2}{1.45} = \mathbf{942.6\text{ litres}} \quad (0.943\text{ m}^3)$

(Notice that without the erroneous $/100$ factor, the bleed volume correctly matches physical hydrostatic displacement: $942.6\text{ litres}$ of $1.45\text{ SG}$ mud corresponds to a vertical height of $942.6 / 26.8 = 35.17\text{ m}$, exerting $35.17\text{ m} \times 0.14216\text{ bar/m} = 5.0\text{ bar}$, exactly balancing the working pressure increment).