Compressibility of Natural Gases

For a liquid phase, the compressibility is small and usually assumed to be constant. For a gas phase, the compressibility is neither small nor constant.

By definition, the isothermal gas compressibility is the change in volume per unit volume for a unit change in pressure or, in equation form:

where cg = isothermal gas compressibility, 1/psi

Incompressible Fluids

An incompressible fluid is defined as the fluid whose volume (or density) does not change with pressure, i.e.:

Incompressible fluids do not exist; this behavior, however, may be assumed in some cases to simplify the derivation and the final form of many flow equations.

Ahmed (2001)

Isothermal Compressibility Coefficient of Crude Oil

Isothermal compressibility coefficients are required in solving many reservoir engineering problems, including transient fluid flow problems, and they are also required in the determination of the physical properties of the undersaturated crude oil.

By definition, the isothermal compressibility of a substance is defined mathematically by the following expressions:

• In terms of fluid volume:

• In terms of fluid density:

where V and ρ are the volume and density of the fluid, respectively.

For a crude oil system, the isothermal compressibility coefficient of the oil phase co is defined for pressures above the bubble-point by one of the following equivalent expressions:

where

• co = isothermal compressibility, 1/psi
• ρo = oil density lb/ft^3
• Bo = oil formation volume factor, bbl/STB

At pressures below the bubble-point pressure, the oil compressibility is defined as:

where Bg = gas formation volume factor, bbl/scf

Ahmed (2001)

Pseudosteady-State Flow

When the pressure at different locations in the reservoir is declining linearly as a function of time, i.e., at a constant declining rate, the flowing condition is characterized as the pseudosteady-state flow. Mathematically, this definition states that the rate of change of pressure with respect
to time at every position is constant, orIt should be pointed out that the pseudosteady-state flow is commonly referred to as semisteady-state flow and quasisteady-state flow.

Ahmed (2001)

Unsteady-State (Transient) Flow

The unsteady-state flow (frequently called transient flow) is defined as the fluid flowing condition at which the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant. This definition suggests that the pressure derivative with respect to time is essentially a function of both position i and time t, thus
Ahmed (2001)

Steady-State Flow

The flow regime is identified as a steady-state flow if the pressure at every location in the reservoir remains constant, i.e., does not change with time. Mathematically, this condition is expressed as:The above equation states that the rate of change of pressure p with respect to time t at any location i is zero. In reservoirs, the steady-state flow condition can only occur when the reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations.

Ahmed (2001)

Flow Regimes

There are basically three types of flow regimes that must be recognized in order to describe the fluid flow behavior and reservoir pressure distribution as a function of time. There are three flow regimes:

• Steady-state flow


• Unsteady-state flow


• Pseudosteady-state flow

Ahmed (2001)

Productivity Index

A commonly used measure of the ability of the well to produce is the Productivity Index. Defined by the symbol J, the productivity index is the ratio of the total liquid flow rate to the pressure drawdown. For a water-free oil production, the productivity index is given by:
where

• Qo = oil flow rate, STB/day
• J = productivity index, STB/day/psi
• avg(Pr) = volumetric average drainage area pressure (static pressure)
• pwf = bottom-hole flowing pressure
• Deltap = drawdown, psi

The productivity index is generally measured during a production test on the well. The well is shut-in until the static reservoir pressure is reached. The well is then allowed to produce at a constant flow rate of Q and a stabilized bottom-hole flow pressure of pwf. Since a stabilized pressure at surface does not necessarily indicate a stabilized pwf, the bottom-hole flowing pressure should be recorded continuously from the time the well is to flow. The productivity index is then calculated from the above Equation.

It is important to note that the productivity index is a valid measure of the well productivity potential only if the well is flowing at pseudosteadystate conditions. Therefore, in order to accurately measure the productivity index of a well, it is essential that the well is allowed to flow at a constant flow rate for a sufficient amount of time to reach the pseudosteady-state as illustrated in the Figure. The figure indicates that during the transient flow period, the calculated values of the productivity index will vary depending upon the time at which the measurements of pwf are made.

Since most of the well life is spent in a flow regime that is approximating the pseudosteady-state, the productivity index is a valuable methodology for predicting the future performance of wells. Further, by monitoring the productivity index during the life of a well, it is possible to determine if the well has become damaged due to completion, workover, production, injection operations, or mechanical problems. If a measured J has an unexpected decline, one of the indicated problems should be investigated.

Ahmed (2001)

Cricondentherm, Cricondenbar and Critical Point

• Cricondentherm (Tct)—The Cricondentherm is defined as the maximum temperature above which liquid cannot be formed regardless of pressure (point E at Pressure-Temperature Diagram). The corresponding pressure is termed the Cricondentherm pressure pct.
• Cricondenbar (pcb)—The Cricondenbar is the maximum pressure above which no gas can be formed regardless of temperature (point D at Pressure-Temperature Diagram). The corresponding temperature is called the Cricondenbar temperature Tcb.
• Critical point—The critical point for a multicomponent mixture is referred to as the state of pressure and temperature at which all intensive properties of the gas and liquid phases are equal (point C at Pressure-Temperature Diagram). At the critical point, the corresponding pressure and temperature are called the critical pressure pc and critical temperature Tc of the mixture.

Ahmed (2001)

References

Ahmed, T., Reservoir Engineering Handbook, Second Edition, Gulf Professional Publishing, 2001

Pressure-Temperature Diagram

The conditions under which oil or gas phases exist are a matter of considerable practical importance. The experimental or the mathematical determinations of these conditions are conveniently expressed in different types of diagrams commonly called phase diagrams. One such diagram is called the pressure-temperature diagram.

Figure shows a typical pressure-temperature diagram of a multicomponent system with a specific overall composition.

These multicomponent pressure-temperature diagrams are essentially used to:

• Classify reservoirs
• Classify the naturally occurring hydrocarbon systems
• Describe the phase behavior of the reservoir fluid

Ahmed (2001)