The isothermal compressibility coefficient is essentially the controlling factor in identifying the type of the reservoir fluid. In general, reservoir fluids are classified into three groups:

- Incompressible fluids
- Slightly compressible fluids
- Compressible fluids

The isothermal compressibility coefficient c is described mathematically by the following two equivalent expressions.

In terms of fluid volume.

C = - 1/ V (ƏV/ ƏP)

In terms of fluid density.

C = 1/ρ (Əρ / ƏP)

Where V and ρ are volume and density respectively.

**Incompressible fluids**

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

ƏV/ ƏP = 0

Əρ / ƏP = 0

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.

**Slightly compressible fluids**

These “slightly” compressible fluids exhibit small changes in volume, or density, with changes in pressure.

**Compressible fluid**

These are fluids that experience large changes in volume as a function of pressure. All gases are considered compressible fluids. The isothermal compressibility of any compressible fluid is described by the following expression:

Cg = 1/ p – 1/Z (ƏZ / ƏP) at constant temp…

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
- Pseudo 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 (ƏP / Ət) at any position remain zero.

The above equation states that the rate of change of pressure p with respect to time t at any location 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.

**Unsteady-State 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

(ƏP / Ət) = f (i, t)

**Pseudo steady-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 pseudo steady-state flow. Mathematically, this definition states that the rate of change of pressure with respect to time at every position is constant, or It should be pointed out that the pseudo steady-state flow is commonly referred to as semi steady-state flow and quasisteady-state flow.

(ƏP / Ət)= Constant

**Reservoir geometry**

The shape of a reservoir has a significant effect on its flow behavior. Most reservoirs have irregular boundaries and a rigorous mathematical description of geometry is often possible only with the use of numerical simulators. For many engineering purposes, however, the actual flow geometry may be represented by one of the following flow geometries:

- Radial flow
- Linear flow
- Spherical and hemispherical flow

**Radial Flow**

In the absence of severe reservoir heterogeneities, flow into or away from a well bore will follow radial flow lines from a substantial distance from the well bore. Because fluids move toward the well from all directions and coverage at the well bore, the term *radial flow *is given to characterize the flow of fluid into the well bore.

**Linear Flow**

Linear flow occurs when flow paths are parallel and the fluid flows in a single direction. In addition, the cross sectional area to flow must be constant. A common application of linear flow equations is the fluid flow into vertical hydraulic fractures.

**Spherical and Hemispherical Flow**

Depending upon the type of well bore completion configuration, it is possible to have a spherical or hemispherical flow near the well bore. A well with a limited perforated interval could result in spherical flow in the vicinity of the perforations. A well that only partially penetrates the pay zone, could result in hemispherical flow. The condition could arise where coning of bottom water is important.

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