Reservoir Engineering: Steady State Flow
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Questions and Answers

How many types of fluids are reservoir fluids classified into?

  • Four
  • Five
  • Two
  • Three (correct)

    • Define incompressible fluids.

    Fluids whose volume (or density) does not change with pressure.

    Slightly compressible fluids exhibit small changes in __________, or density, with changes in pressure.

    volume

    Steady-state flow occurs when the pressure at every location in the reservoir changes with time.

    <p>False</p> Signup and view all the answers

    What is the fluid flowing condition when the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant?

    <p>Unsteady / Transient State Flow</p> Signup and view all the answers

    Define Pseudosteady-State Flow.

    <p>The pressure at any given location in the reservoir is declining linearly as a function of time.</p> Signup and view all the answers

    Which reservoir geometry shape has a significant effect on flow behavior?

    <p>Irregular</p> Signup and view all the answers

    Who introduced the concept of permeability?

    <p>Henry Darcy</p> Signup and view all the answers

    Darcy's Law states that for one-dimensional, horizontal flow through a porous medium, the flow rate is proportional to the pressure gradient and reciprocal to ___ viscosity.

    <p>fluid</p> Signup and view all the answers

    What is the formula for calculating the fluid velocity in Darcy's units?

    <p>v= apparent fluid velocity, cm/sec; k= permeability, darcy; μ= fluid viscosity, cp; p= pressure, atm; l= length, cm; ρ= fluid density, gm/cm3; g= Acceleration due to gravity, cm/sec2; and Z= elevation, cm.</p> Signup and view all the answers

    When is the fluid potential drop equal to the pressure drop according to Darcy's equation?

    <p>The fluid potential drop (Φ1 − Φ2) is equal to the pressure drop (p1 − p2) only when the flow system is horizontal.</p> Signup and view all the answers

    What is the relationship between pressure and volume for slightly compressible fluid?

    <p>The relationship is given by the equation V = Vref(1 + c(Pref − P)).</p> Signup and view all the answers

    What is the equation that results from substituting the relationship between pressure and volume into Darcy's equation?

    <p>q = qref(1 + c(Pref − P)).</p> Signup and view all the answers

    What is the equation for gas flow rate derived from the real-gas equation-of-state?

    <p>q = 0.001127(K / μ) dP / dx.</p> Signup and view all the answers

    What is the formula for the viscosity of the gas calculated using the Lee-Gonzales-Eakin method?

    <p>10^{-4}K exp[X(\frac{\rho_g}{62.4})Y]</p> Signup and view all the answers

    What is the value of the calculated viscosity of the gas?

    <p>0.0173 cp</p> Signup and view all the answers

    What is the formula to calculate the gas flow rate using equation (25)?

    <p>\frac{0.111924(Ak p_1^2 - p_2^2)}{TZ\mu_g L}</p> Signup and view all the answers

    What is the calculated gas flow rate using the given formula and data in the text?

    <p>1,224,242 scf/day</p> Signup and view all the answers

    Which of the following are outcomes of the lesson discussed in the text?

    <p>All of the above</p> Signup and view all the answers

    What is the main characteristic of radial flow of incompressible fluids in the reservoir?

    <p>Fluids move toward the producing well from all directions</p> Signup and view all the answers

    What is the formula for the approximate drainage area of a single well, Aw?

    <p>Aw = AT / qT</p> Signup and view all the answers

    In the context of radial flow equations, what does Qo represent?

    <p>Qo represents the oil flow rate in STB/day.</p> Signup and view all the answers

    What is the equation used to calculate the oil flow rate of slightly compressible fluid?

    <p>Qo = 0.00708 kh ln(1 + c(pwf - pe))</p> Signup and view all the answers

    What does the term 'm(p)' or 'ψ' represent in the radial flow equations?

    <p>m(p) or ψ represents the real gas potential or real gas pseudo-pressure.</p> Signup and view all the answers

    Why is Zg considered constant over a pressure range in gas flow rate approximation?

    <p>Zg is considered constant due to its negligible variation over the pressure range.</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    This quiz covers primary reservoir characteristics, linear and radial flow behavior of reservoir fluids in porous media, and mathematical relationships describing flow behavior.

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