PRODUCTION SYSTEMS COURSE

DARCY LAW

PERMEABILITYPermeability is a property of the porous medium that

measures the capacity and ability of the formation to transmit fluids. The rock permeability, k, is a very important rock property because it controls the directional movement and the flow rate of the reservoir fluids in the formation. This rock characterization was first defined mathematically by Henry Darcy in 1856. In fact, the equation that defines permeability in terms of measurable quantities is called Darcy’s Law.

Darcy developed a fluid flow equation that has since become one of

the standard mathematical tools of the petroleum engineer. If a horizontal linear flow of an incompressible fluid is established through a core sample of length L and a cross-section of area A, then the governing fluidflow equation is defined as

where n = apparent fluid flowing velocity, cm/seck = proportionality constant, or permeability, Darcysm = viscosity of the flowing fluid, cpdp/dL = pressure drop per unit length, atm/cm

The apparent velocity determined by dividing the flow rate by the cross-sectional area across which fluid is flowing. Substituting the relationship, q/A, in place of n and solving for q results in

where q = flow rate through the porous medium, cm3/secA = cross-sectional area across which flow occurs, cm2

One Darcy is a relatively high permeability as the permeabilities of

most reservoir rocks are less than one Darcy. In order to avoid the use of fractions in describing permeabilities, the term millidarcy is used. As the term indicates, one millidarcy, i.e., 1 md, is equal to one-thousandth of one Darcy or,

1 Darcy = 1000 mdThe negative sign in Equation is necessary as the pressure

increases in one direction while the length increases in the opposite direction.

Integrate the above equation

Linear flow model

where L = length of core, cmA = cross-sectional area, cm2

The following conditions must exist during the measurement of permeability:

• Laminar (viscous) flow• No reaction between fluid and rock• Only single phase present at 100% pore space

saturationThis measured permeability at 100% saturation of

a single phase iscalled the absolute permeability of the rock.

For a radial flow, Darcy’s equation in a differential form can be written as:

Intergrating Darcy’s equation gives:

The term dL has been replaced by dr as the length term has now become a radius term.

PRIMARY RESERVOIR

CHARACTERISTICS

The area of concern in this lecture includes:

• Types of fluids in the reservoir• Flow regimes• Reservoir geometry• Number of flowing fluids in the

reservoir

TYPES OF FLUIDSIn general, reservoir fluids are classified into

three groups:• Incompressible fluids• Slightly compressible fluids• Compressible fluidsIncompressible fluids An incompressible fluid is defined as the fluid

whose volume (or density) does not change with pressure. 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 volumeor density, with changes in pressure.It should be pointed out that crude oil and water systems

fit into this category.

Compressible FluidsThese are fluids that experience large changes in volume

as a function of pressure. All gases are considered compressible fluids.

The isothermal compressibility coefficient c is describedmathematically by the following two equivalent expressions:In terms of fluid volume:

In terms of fluid density:

Fluid density versus pressure for different fluid types

FLOW REGIMESThere are three flow regimes:• Steady-state flow• Unsteady-state flow• Pseudosteady-state flowSteady-State FlowThe 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.

Unsteady-State FlowThe 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

Pseudosteady-State FlowWhen the pressure at different locations in the reservoir is

declininglinearly 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, or

It should be pointed out that the pseudosteady-state flow is commonly referred to as semisteady-state flow and quasisteady-state flow.

Figure shows a schematic comparison of the pressure declines as a function of time of the three flow regimes.

Flow Regimes

Ideal Steady-State Flow Equation - Radial Flow

The steady-state flow equations are based on the following assumptions:

• 1. Thickness is uniform, and permeability is constant.

• 2. Fluid is incompressible.• 3. Flow across any circumference is a constant.

RESERVOIR GEOMETRYFor 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 flowBecause fluids move toward the well from all directions and

coverage at the wellbore, the term radial flow is given to characterize the flow of fluid

into the wellbore. Figure 4-1 shows idealized flow lines and iso-potential lines for a radial flow system.

Figure 4-1 Ideal radial flow into a wellbore

Linear FlowLinear 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. Figure 4-2 shows an idealized linear flow system.

Figure 4-2 Ideal linear flow

into vertical fracture

Spherical and Hemispherical FlowDepending upon the type of wellbore completion

configuration, it is possible to have a spherical or hemispherical flow near the wellbore. A well with a limited perforated interval could result in spherical flow in the vicinity of the perforations as illustrated in Figure 4-3. A well that only partially penetrates the pay zone, as shown in Figure 4-4, could result in hemispherical flow. The condition could arise where coning of bottom water is important.

Figure 4-3 Spherical flow due to limited entry

Figure 4-4 Hemispherical flow in a partially penetrating well

NUMBER OF FLOWING FLUIDS IN THE RESERVOIRThere are generally three cases of flowing systems:• Single-phase flow (oil, water, or gas)• Two-phase flow (oil-water, oil-gas, or gas-water)• Three-phase flow (oil, water, and gas)The description of fluid flow and subsequent analysis of

pressure data becomes more difficult as the number of mobile fluids increases.