Reservoir Engineering: Steady State Flow

Steady-State Flow in Reservoir Engineering

• Flow in porous media is complex and cannot be explicitly described like flow through pipes or conduits.
• The primary reservoir characteristics that must be considered are:
  • Types of fluids in the reservoir
  • Flow regimes
  • Reservoir geometry
  • Number of flowing fluids in the reservoir

Primary Reservoir Characteristics

• Types of fluids:
  • Incompressible fluids
  • Slightly compressible fluids
  • Compressible fluids

Types of Fluids

• Isothermal compressibility coefficient (c) is the controlling factor in identifying the type of reservoir fluid.
• c is described mathematically by two equivalent expressions:
  • In terms of fluid volume: c = - (1/V) * (dV/dP)T
  • In terms of fluid density: c = (1/ρ) * (dρ/dP)T

Incompressible Fluids

• Defined as fluids whose volume (or density) does not change with pressure: (dV/dP) = 0 and (dρ/dP) = 0

Slightly Compressible Fluids

• Exhibit small changes in volume or density with changes in pressure.
• Most crude oils fall in this category.
• Changes in volumetric behavior can be described by separating variables and integrating Equation (1): -c ∫ dP = ∫ (dV/V)

Slightly Compressible Fluids (Continued)

• Mathematical description: V = Vref * e^(c(Pref - P))
• Exponent can be represented by a series expansion: e^x = 1 + x + x^2/2! + x^3/3! + ...
• Approximation: e^x = 1 + x
• Combining equations: V = Vref * (1 + c(Pref - P)) and ρ = ρref * (1 - c(Pref - P))

Compressible Fluids

• Experience large changes in volume as a function of pressure.
• All gases are considered compressible fluids.
• Isothermal compressibility is described by: cg = - (1/P) * (dZ/dP)T

Flow Regimes

• Three types of flow regimes:
  • Steady-state flow
  • Unsteady-state flow
  • Pseudosteady-state flow

Steady-State Flow

• The pressure at every location in the reservoir remains constant and does not change with time: (dP/dt) = 0
• Occurs when the reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations.### Unsteady / Transient State Flow
• The fluid flowing condition where the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant.
• The pressure derivative with respect to time is essentially a function of both position and time.

Pseudosteady-State Flow

• The pressure at any given location in the reservoir is declining linearly as a function of time.
• The pressure derivative with respect to time is constant.

Characteristic of Fluids

• The shape of a reservoir has a significant effect on its flow behavior.
• Most reservoirs have irregular boundaries.
• Rigorous mathematical description of geometry is often possible only with the use of numerical simulators.

Reservoir Geometry

• The actual flow geometry may be represented by one of the following flow geometries: Linear flow, Radial flow, Spherical flow, or Hemispherical flow.

Radial Flow

• Flow into or away from a wellbore will follow radial flow lines from a substantial distance from the wellbore.
• In the absence of severe reservoir heterogeneities, fluids move toward the well from all directions and converge at the wellbore.
• Ideal radial flow into a wellbore can be represented by a radial-cylindrical flow model.

Linear Flow

• When flow paths are parallel and the fluid flows in a single direction (i.e., no flow perpendicular to or angular to the main flow streams).
• The cross-sectional area to flow must be constant.
• A common application of linear flow equations is the fluid flow into vertical hydraulic fractures.

Ideal Linear Flow into Vertical Fracture

• The fluid flow into a vertical hydraulic fracture can be represented by a linear flow model.

Spherical Flow

• A well which is only perforated in the central part of the formation and not in the entire pay zone could result in spherical flow in the vicinity of the perforations.
• The kind of limited perforation in the central part of the pay zone could be beneficial if there is a possibility of water coning from the bottom and gas encroachment from the top.

Hemispherical Flow

• A well which only partially penetrates the pay zone not in the entire pay zone could result in hemispherical flow.
• This kind of partial penetration could be beneficial if there is a possibility of water coning from the bottom but no chance of gas encroachment from the top.

Number of Flowing Fluids in the Reservoir

• Single-phase flow (oil, water, or gas)
• Two-phase flow (oil-water, oil-gas, or gas-water)
• Three-phase flow (oil, water, and gas)

Henry Darcy

• A 19th-century French engineer who introduced the concept of permeability (unit: mD).
• Studied vertical flow of water through packed sand.

Darcy’s Law

• States that the flow rate is proportional to the pressure gradient and reciprocal to viscosity.
• The parameters that affect fluid flow are permeability, cross-sectional area, viscosity, and pressure gradient.

Conditions for Darcy’s Law

• Laminar (viscous) flow
• Steady-state flow
• Incompressible fluids
• Homogeneous formation

Steady-State Flow

• Represents the condition that exists when the pressure throughout the reservoir does not change with time.
• The applications of the steady-state flow include linear flow of incompressible fluids, radial flow of incompressible fluids, and multiphase flow.

Linear Flow of Incompressible Fluids

• The flow occurs through a constant cross-sectional area A.

• Both ends are entirely open to flow.

• No flow crosses the sides, top, or bottom.

• The flow behavior in this system can be expressed by integrating the differential form of Darcy’s equation within limits of injection end (x=0, P=P1) and producing end (x=L, P=P2).### Fluid Potential and Darcy's Equation

• Fluid potential at any point in the reservoir is the pressure at that point plus the pressure exerted by a fluid head extending to an arbitrarily assigned datum level.

• Δzi is the vertical distance from a point i in the reservoir to the datum level.

• Fluid potential (Φ) can be calculated using Darcy's equation: v = -k/μ * (dP/dl) + ρg * dZ/dl

Darcy's Units

• In terms of Darcy units, the apparent fluid velocity (v) is related to permeability (k), fluid viscosity (μ), pressure (p), length (l), fluid density (ρ), and acceleration due to gravity (g).
• v = -k/μ * (dP/dl) + ρg * dZ/dl

Field Units

• In field units, fluid potential (Φ) is calculated using: Φi = pi + ρ * γ * Δzi
• v = -A/B * k/μ * (dP/dl) + ρg * dZ/dl

Linear Flow of Slightly Compressible Fluids

• Relationship between pressure and volume for slightly compressible fluids: V = Vref * (1 + c * (Pref - P))
• Modified to write in terms of flow rate: q = qref * (1 + c * (Pref - P))
• Substituting into Darcy's equation: q = -k/μ * (dP/dl) + ρg * dZ/dl

Compressible Fluids (Gases)

• Real-gas equation-of-state: PV = nZRT
• At standard conditions, the volume occupied by n moles is: Vsc = nZscRTsc/Psc
• Combining the two expressions: PV = ZT/PscVsc
• Equivalently, in terms of flow rate: q = Qsc * Psc/P * ZT/Tsc
• Rearranging: q = -0.001127 * K * dP/dx / (A * μ)

Gas Flow Rate

• Separating variables and integrating: Qsc = 0.003164 * Tsc/Ak * (p12 - p22) / (PscTZμg L)
• Simplification: Qsc = 0.111924 * Ak * (p12 - p22) / (TZμg L)

Example

• Calculating gas flow rate in scf/day for a linear system with a specific gravity of 0.72, flowing at 140°F, with upstream and downstream pressures of 2100 psi and 1894.73 psi, respectively.