Lecture Outline

  • 2.1. Parameters of porous medium
  • 2.2. Parameters of fractured medium

Parameters of porous media

From the point of view of filtration theory, firm skeleton of rock plays geometrical role. It means that the skeleton of rock limits that area of space in which the liquid goes. Only in special cases it is necessary to consider interaction forces between skeleton and adjacent liquid. Therefore, the properties of rocks in the theory of a filtration are described by some set of geometrical characteristics.

The major characteristic is porosity mo, equal to the relation of porous volume V to total amount of element V:

m^Vp/V. (2.1)

Figure 2.1. Cross-sectional view of a porous media (white areas are pore space, pattern areas are grains)

It is within these pore spaces that the oil, gas and/or water reside. Therefore a primary application of porosity is to quantify the storage capacity of the rock, and to define subsequently the volume of hydrocarbons available to be produced. From a drilling perspective, the rate of penetration and the volume of drilling fluid lost to a formation by invasion are related to porosity.

Porosity is an average property defined over the representative elementary volume. That is, as shown in Figure 2.2, the porous media volume must be larger than file size of a single pore so that adding more pores will maintain a meaningful statistical average; but it must be smaller than the non-uniformity of the entire flow domain. Thus the representative elementary volume provides a uniform porosity value for the domain of the porous media.

Several types of porosity have been defined based on the degree of connectivity or the time of pore development. Total porosity is the ratio of the total pore space of the media to the total bulk volume. Effective porosity is the ratio of interconnected pore space to the bulk volume of the rock.

Figure 2.3 is an example of total vs. effective porosity in a vuggy rock. Notice the pathway for fluid to migrate in connected pores and the isolated nature of others. Production of hydrocarbons is dependent upon the fluid to flow in the porous media.

Subsequently, it is the effective porosity, which is of importance to reservoir engineering. Isolated pores are of little value to the recovery of hydrocarbons.


“ homogeneous


Figure 2.2. Definition of porosity and representative elementary volume

Bulk volume measurements. Bulk volume measurements are classified into two types: linear measurement and displacement methods. Linear measurement is simply physically measuring the sample with a vernier caliper and then applying the appropriate geometric formula. This method is quick and easy, but is subject to human error and measurement error if the sample is irregularly shaped. Displacement methods rely on measuring either volumetrically or gravimetrically the fluid displaced by the sample. Gravimetric methods observe the loss in weight of the sample when immersed in a fluid, or observe the change in weight of a pycnometer filled with mercury and with mercury and the sample. Volumetric methods measure the change in volume when the sample is immersed in fluid. For all displacement methods, the fluid is prevented from penetrating into the pore space by coating the rock surface with paraffin, saturating the rock with the same fluid, or using mercury as the displacing fluid.

Pore Volume Measurements. Several methods have been developed to measure the pore volume of a sample. The original mercury injection methods such as Washbum-Bunting and Kobe are obsolete and seldom used. Their elimination was due to the destructive nature of mercury and the lack of accurate results. A second method is called the fluid resaturation method. A clean and dried sample is weighted, saturated with a liquid of known density, and then reweighed. The weight change divided by the density of the fluid results in the pore volume.

This technique also yields effective porosity; however, complete saturation is seldom obtained and therefore porosity is commonly lower than that determined from the Boyle’s Law method. Furthermore, if the sample is water sensitive then oil should be used as the saturating fluid. The procedure is slow, however numerous samples can be run simultaneously.

Alongside with full porosity a concept of dynamic porosity is frequently used. Full porosity is described by (2.1), and dynamic one is described by the formula:

m = V JV, mob. 7

where Vmob is the volume occupied with a mobile liquid.

Further on, by porosity we shall mean dynamic porosity, except for special cases.

The porosity of firm materials (sand, bauxites, etc.) varies in a small extent even at large changes of pressure, but the porosity of clay is very susceptible to compression. So the porosity of clay slate at usual pressure is equal to 0,4 - 0,5. For gas and oil reservoirs in most cases m = 15 - 22 %, but it can vary over a wide range: from some percent up to 52 %.

For a fictitious ground we have mQ = 0,259 when a = 60° and m = 0,476 when a = 90°. The clearance of a fictitious ground ms = 0,0931 when a = 60° and m = 0,2146 when a = 90°.

Flow of a liquid goes through a surface so it is necessary to introduce the parameter connected to the area. Such geometrical parameter is called clearance (wj and it is defined as the relation of the area of pores in a cross-section F l to all area of cross-section of a sample F:

m=Fd/F. (2.2)

In most cases clearance is set to be equal to porosity.

Porosity and clearance of a fictitious ground do not depend on diameter of spherical particles, and they depend only on a degree of stacking. For real media the factor of porosity depends on density of stacking of particles and their size - the lesser the size of grains, the greater is its porosity.

To describe a real medium, we introduce average size of porous channel (5) and average size of diameter of an individual grain of a porous skeleton (d).

The simplest geometrical characteristic of the porous medium is effective diameter of particles. Effective diameter def, of a real porous medium is such diameter of the spheres forming an equivalent fictitious ground at which the hydraulic resistance of a real ground and equivalent fictitious ground are the same. Effective diameter can be determined by granulometric composition (figure 2.4) by the formula:

Figure 2.4. The histogram of distribution ofparticles in the sizes

d, = 3i

«7 A

here d. is average diameter of the ith fractions; n. is a mass part of the i‘h fraction.

To put into compliance the diameter determined by sieve or by microscopic methods, with the diameter determined through resistance of a ground to a flow, the given diameter is multiplied by a factor of the hydraulic shape. If the diameters are determined by hydrodynamic methods they do not require this correction.

Effective diameter is important, but it is not an exhaustive characteristic of the porous medium because it does not give information about stacking of particles, about their form, etc. Two samples of the ground with identical effective diameters, but with various form of particles and structure of stacking, have various filtration characteristics.

Thus, to define geometrical structure of the porous medium we need additional objective parameters besides porosity and effective diameter. One of them is the hydraulic radius of pores (R). For an ideal ground there is a relationship between radius of pores and the diameter of particles of a fictitious ground:

R=md / [12(1-m)].

Variations of filtration flow are determined basically by friction of a fluid on a skeleton of reservoirs which depends on the area of surface of rock particles. In this connection one of the major parameters is specific surface sspec, i.e. the total area of a surface of the particles contained in a unit of volume. For a fictitious ground:

6(1 -m)

°spec ,

The specific surface of oil containing rocks with sufficient accuracy is determined by the formula:

Я =7,0 IO5 spec ’ // ’


here к is permeability.

Average value of Sspec for oil containing rocks is large and it changes within the limits of40 000 - 230 000 m2/m3. Rocks with a specific surface greater than 230 000 m2/m3 are impenetrable or weakly penetrable (clay, clay sand, etc.).

The key characteristic of filtration properties of rock is permeability. Permeability is a measure of the ability of a porous material to allow fluids to pass through it. It is a critical property in defining the flow capacity of a rock sample. The unit of measurement is the Darcy, named after the French scientist who discovered the phenomenon. Absolute permeability characterizes physical properties of rock and is determined at the presence of only one phase which is chemically inert in relation to the rock.

Permeability is measured in m2 (in SI system), in Darcy (D) (in technical system) and ID is equal to 1,02 mkm2 = l,02-1012m2.

The physical sense of permeability к reflects the fact that permeability characterizes the area of section of channels of the porous medium through which a filtration goes.

Factors affecting permeability. Numerous factors affect the magnitude and/or direction of permeability.

1. Textural properties.

a. Pore size/ grain size

b. Grain size distribution

c. Shape of grains

d. Packing of grains

  • 2. Gas slippage.
  • 3. Amount, distribution, and type of clays.
  • 4. Type and amount of secondary porosity.
  • 5. Overburden pressure.
  • 6. Reactive fluids.
  • 7. High velocity flow effects.

Let us begin by investigating the role of textural properties on the permeability. Experimental evidence has shown that к is proportional to cd2, where c is a characteristic of the rock properties and d is the grain diameter. The dimension of permeability is L2, which is directly related to the cross-sectional area of the pore throats. Therefore as grain size increases, so will the pore throat size and a subsequent increase in permeability occurs. In Figure 2.5, an artificial mixing of sands illustrates the significant effect of grain size on permeability. As can be seen, an approximate 25:1 increase in permeability occurs from coarse to very fine grains.

The effect of sorting on the permeability is also shown in Figure 2.5. It is not as dramatic as grain size; however, the illustration does show an increase in sorting (better or well sorted) will improve the permeability. This is why in gravel pack operations the selection of the gravel is important, both from a size and sorting viewpoint.

Figure 2.5. Effect of grain size and sorting on permeability

The effect of shape and packing on permeability can be seen in Figure 2.6.

Figure 2.6. Textural parameters and permeability

Notice in these examples, the more angular the grains or the flatter the grain shape, a more pronounced anisotropy develops.

Permeability of rocks varies over a wide range: coarse-grained sandstone has permeability 1 - 0,1 D; dense sandstones - 0,01 - 0,00ID.

The size of permeability depends on the size of pores for model of an ideal ground with tubes of radius R:

к = m7?2/8, here R, micron; k, D.

For real mediums the radius of pores is linked with permeability through the formula:

7 Ю5 V m

here k, D; R, m ф is a structural factor = 0,5035/m1,1 for granular mediums).

Since the radius of pores is linked with a specific surface, then the permeability is also linked with it:

S =2m/k.


Permeability of rocks varies over a wide range: coarse-grained sandstones have a permeability of 1 - 0,1D; dense sandstones, accordingly, 0,01-0,00 ID.

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