Información Necesaria para la Notificación

Under natural conditions, a porous medium volume at some depth in a ground water aquifer or in an oil reservoir is subjected to an internal stress or hydrostatic pressure of the fluid saturation the medium, which is a hydrostatic pressure that has the same values at different direction, and to an external stress exerted by the formation in which the particular volume is surrounded and may have different value at different directions. The external stress of the formation can lead to the compaction of the porous medium that is a function of the formation depth. Krumbein and Sluss (1951) showed that porosity of the sedimentary rocks is a function of the degree of compaction of the rock (Figure 2-17).

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Figure 2-17 – Porosity Reduction as an Effect of Compaction Increment by Depth

Compaction effect on the porosity that leads to porosity reduction is principally due to the packing rearrangement after compaction. The porosity of shales is greatly reduced by compaction largely because “bridging’ is eliminated by the greater forces. Addition to the effect of compaction on the grain arrangement, rocks also are compressible. Three kinds of compressibility must be distinguished in rocks:  Rock matrix compressibility

 Rock bulk compressibility  Pore compressibility

Rock matrix compressibility is the fractional change in volume of the solid rock materials (grains) with a unit change in pressure. Rock bulk compressibility is the change in volume of the bulk volume of the rock with a unit change in pressure. Pore compressibility is the fractional change in the pore volume of the rock with a unit change in pressure.

The depletion of fluids from the pore space of a reservoir rock results in a change in the internal pressure in the rock while the external pressure in constant, thus results a change in the net pressure. This change in the net stress could leads to a change in grain, pore and bulk volume of the rock. Pore volume change is an interesting subject to the reservoir engineer. Bulk volume change is an important subject in the areas that surface subsidence could cause appreciable property damage. Volume change under the pressure effect can be expressed as compressibility coefficient. The coefficient of solid matrix compressibility, pore compressibility and bulk compressibility are defined for of a saturated porous medium as the fractional change in the volume with a unit change in the pressure:

(2-9) (2-10) (2-11) 0 10 20 30 40 50 60 0 1000 2000 3000 4000 5000 6000 Por o si ty (% ) Depth of burial, ft Shales Sandstones

35 The value of (in some literature mentioned as as rock compressibility) can be determined by saturating the rock with a fluid, immersing the rock in a pressure vessel containing the saturating fluid, then imposing a hydrostatic pressure on the fluid and observing the change in the volume (or ) of

the rock sample. The compressibility of solid matrix ( or ) is considered for most rock to be

independent of the imposed pressure.

But reservoir rocks are under other conditioning of loading than this experiment. A rock buried at depth is subjected to an overburden load due to the overlying sediments which is in general greater than the internal hydrostatic pressure of the formation fluids. (Figure 2-18.a) shows an experimental apparatus that simulate this condition for a sample rock. A core sample is enclosed in a copper jacket which is then immersed in a pressure vessel and connected to a Jurguson sight glass gauge. The hydraulic pressure system is arranged so that a saturated core can be subjected to variable internal (or pore) pressure and external (or overburden) pressure. The resulting internal volume change is indicated by the position of the mercury slug level in the sight glass. Typical curve are obtained shown if (Figure 2-18.b). The ordinate is the reduction in pore space resulting from a change in overburden pressure. The slop of the curve is the compressibility of the form

( ) (

) (2-12) It may be noted that the slop of the curves can be considered constant over most of the pressure range above 1000 psi. Hall (1953) ran some similar tests. He designate the compressibility term (2-12) as formation compaction component as total rock compressibility and develop a correlation of this function with porosity (Figure 2-19.a). Also he investigated ( ) ( ) at constant overburden pressure. This he designated as effective rock compressibility and correlated with porosity (Figure 2-19.b). In Figure 2-19.a and b, it may be noted that compressibility decreases as the porosity increases. The value of can be determined by measuring the change in the bulk volume of a jacketed sample by varying the external hydrostatic pressure while maintaining a constant internal pressure. For sandstones and shale it can be shown that:

(2-13)

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(a) (b)

Figure 2-18 – a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility b) Rock compressibility test result6

(a) (b)

Figure 2-19 – a) Formation Compaction Component of Total Rock Compressibility b) Effective reservoir rock compressibility7

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Carpenter and Spencer (1940)

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37 eq. (2-14) provided that Cr is much less than CB. Therefore

(2-15)

This equation states that total change in volume is equal to the change in the pore volume.

Geertsma (1957) stated that in reservoir only the vertical component of overburden pressure is constant and the stress components in horizontal plane are characterized by boundary condition that there is no bulk deformation in those directions. For these boundary conditions he developed the following approximation for sandstone:

(2-16)

Thus, the effective pore compressibility for reservoir rocks under the depletion of pore pressure is only one-half of that determined by present methods in the laboratory.

In summary it can be stated that the pore volume compressibilities of consolidated sandstones are of the order of reciprocal psi.

PERMEABILITY

Permeability 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 D’ Arcy in 1856. By analogy with electrical conductors, permeability represents reciprocal of residence which porous medium offers to fluid flow. Poiseuille’s equation for viscous flow in a cylindrical tube is a well-known equation

(2-17)

Where:

 = fluid velocity, cm/sec  = tube diameter, cm

 = pressure loss over length L,  = fluid viscosity, centipoise

 = length over which pressure loss is measured, cm A more convenient form of Poiseuille’s equation is

(2-18)

If assume that the rock is consist of a lot of tube in different group with different radius, total flow rate from this system by using the equation (2-18)

38 ∑

= number of tubes of radius

= number of groups of tubes of different radii

Example 2-4

Derive Poiseuille’s equation for viscous flow in a horizontal cylindrical tube.

Solution

Consider a horizontal flow in a circular pipe. Assume a disc shape element of the fluid in the middle of the cylinder that is concentric with the tube and with radius equal to rw and length equal to . The forces on the disc are due to the pressure on the upstream and downstream face of the disc and shear force over the rim of the element Figure 2-20.

Figure 2-20 – Flow through a Pipe

According to the steady state condition:

(2-19) After simplification (2-20) (2-21) (2-22)

According to the shear stress definition and using equation (2-21) and then rearranged and integration on both side (2-23) Maximum velocity at r=0 so (2-24)

For viscous flow in the cylindrical tube

( ) ̅ (2-25) From (2-21), (2-24) and (2-25) P (P+P) Flow r rw 

39 ̅ (2-26) d = 2rw , so after rearrangement : Poiseuille’s equation

A cast of the flow channel in a rock formation is shown in (Figure 2-21). It is seen that the flow channels are of varying sizes and shapes and are randomly connected. So it is not correct to use the Poiseuille’s equation for flow in the porous media.

Figure 2-21 – Metallic cast of pore space in a consolidated sand8

In 1856, D’ Arcy (commonly known as Darcy) developed a fluid flow equation that has since become one of the standard mathematical tools of the petroleum engineer. He investigated the flow of water through sand filter for water purification. His experimental apparatus showed schematically in (Figure 2-22)

Figure 2-22 – Schematic Drawing of Darcy Experiment of Flow of Water through Sand9

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40 Darcy interpreted his observation so as to yield result essentially as following equation:

(2-27)

Here Q represents the volume rate of flow of water downward through the cylindrical sand pack of cross sectional area A and height h and K is a proportionality constant. The sand pack was assumed to be fully saturated with water. Later investigator found that Darcy’s law could be extended to other fluid as well as water and that the constant of proportionality K could be written as . The final form of the Darcy equation for horizontal linear flow of an incompressible fluid is established through a core sample of length L and a cross-section of area A is

(2-28)

Where:

K = Proportionality constant or permeability, Darcy’s

= Viscosity of flowing fluid, cp

= Pressure drop per unit length,

= Volumetric flow rate,

This is a linear law, similar to Newton’s law of viscosity, Ohm’s law of electricity, Fourier ‘s law of heat conduction, and Fick’s law of diffusion.

Substituting the relationship  = Q/A, in equation (2-28) results in

(2-29)

The velocity, , in Equation 2-29 is not the actual velocity of the flowing fluid but is the apparent velocity determined by dividing the flow rate by the cross-sectional area across which fluid is flowing.

“Darcy” is a practical unit of permeability (in honor of Henry Darcy). A porous material has permeability equal to 1 Darcy if a pressure difference of 1 atm will produce a flow rate of 1 of a fluid with 1 cP viscosity through a cube having side 1 cm in length. Thus

(

)

( ₎ (2-30)

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. The negative sign in equation (2-29) is necessary as the pressure increases in one direction while the length increases in the opposite direction.

Permeability correlations Radial Version of Darcy’s Law Permeability and conductivity

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Example 2-5

Find the Darcy’s equation for isothermal flow of ideal gas.

Solution

Multiply both side of equation (2-28) by density so we have a mass flow equation:

But because of isothermal flow, which “b” is as a “base condition”. For ideal gas at constant temperature: . Therefore

(2-31)

Separating variable and integrating

(2-32)

Define ̅ as and ̅ as flow rate at ̅. Then: ̅̅̅̅ . Substituting in equation (2-32) ̅

(2-33)

There for flow rate of ideal gas could be found from the Darcy’s law for the incompressible fluid when the flow rate defined at the algebraic mean pressure.