2 4 22 LOG PERMEABILITY ( m )

- 2 0 - 1 8 - 1 6 - 1 4 - 1 2 - 1 0 SAND SANDSTONE ^ ...: -I CLAY 1 iliiW lilliiilii SILTSTONE SHALE ^--- ^ 1...r...'...1 VOLCANIC ' I f ' -.. LIMESTONE - DOLOMITE GRANITE 1 " ' ■ 1 METAMORPHIC

Figure 2.6: Typical crustal rocks types and their permeabilities, measured at a nominal effective pressure o f 10 MPa at room temperature (reproduced from Brace et al. 1968).

2.4 Approaches to Modelling Permeability

Since permeability is intrinsically a difficult property to measure in the field much attention has been focused on attempts to independently predict permeability from theoretical and empirical relationships between permeability and other more readily measurable rock properties.

There are a number of theoretical approaches used to describe permeability. These include effective medium theory (David et al. 1990), network modelling (Zhu et al. 1995), and physical statistical models and percolation theory (Gueguen and Palciauskas 1992; Gueguen and Dienes 1989). Further, permeability may be derived indirectly by the use of a number of empirical physical relationships, such as those described by the Carman-Kozeny type relations (Carman 1952; Paterson

1983).

i Network models

Network theory is based on the analogy between the flow of a permeating medium within a porous network (Darcy’s law) and the principles of electrical flow (Ohm’s law), where the flow rate is proportional to the conduit permeability (conductivity) and the difference in pressure (voltage) between the end points. The pore network is replaced by considering a regular network of variable elements, i.e. nodes and bonds, where the conductance is located in the bonds. In a network model, nodes and conducting bonds are physical analogues of nodal pores and throats (Zhu and Wong 1996). Network modelling has been used successfully to describe the permeability evolution in

a Effective M edium Theories - Regular Topology at Sm all Scale

The aim of effective medium theory (EMT) (Kirkpatrick 1973) is to infer an average physical parameter from the statistics of local conducting (or permeable) elements. A permeable medium is approximated by a 3D network with regular topology, defined and calculated from minivariables, which are averages of the grain scale physical variables. From microvariables effective properties are defined and calculated, for example pore dimensions (Doyen 1988), and porosity and specific surface area (Berryman and Blair 1986). Hence in EMT the real discontinuous medium is replaced by an effective, regular continuous medium, and is the homogenous equivalent regular network for which the macroscopic conductance (permeability) is the same as for the heterogeneous system and physical properties are described in terms of an average or effective property. EMT computations rely on the accurate statistical representative data on pore morphology, including 3-D pore geometry characterisation (Lin and Cohen 1982), although characterisation of pore connectivity is more critical (David et al. 1990). EMT is applicable where the medium is statistically homogenous, but

heterogeneous on a small scale. However, David et al. (1990) found that EMT solutions

underestimated permeability in highly heterogeneous media.

Hi Physical Models ~ Disordered Topology

A number of microstructural models have been proposed to relate microstructural parameters to

permeability. These include statistical (Gueguen and Dienes 1989) and geometrical models

(Paterson 1983). The ‘microstructure’ refers to the microstructure of the pore network, which is described by a set of statistical distributions of microstructural parameters describing the pore

geometry and pore distribution. Geometrical models approximate real systems by describing

permeability in terms of networks of capillaries or cracks of different sizes but simple shapes (Gueguen and Dienes 1989). Capillary models assume that transport is by means of 1-d randomly distributed capillaries of variable length (d), radius (r) and spacing (1) (Figure 2.7) and crack models assume transport through the connection of 2-d cracks of aperture (2w), diameter (2c) and distribution (1). Permeability is controlled by the distribution of the respective microvariables. Capillary models relate best to highly porous sedimentary rocks, and crack models relate best to crystalline materials.

Permeability is calculated using statistics (Gueguen and Dienes 1989) to relate the three independent microvariables; average pipe (crack) length, average pipe radius (crack aperture) and average pipe (crack) spacing. The microvariables which control permeability are crack aperture or pipe radii, where small changes in either variable result in large changes in permeability.

Gueguen et al. (1987) successfully applied the crack model to determine permeability changes with stress for a low porosity sandstone (Fontainebleau) and a North Sea chalk. In the case of crack nucléation and crack growth, permeability changes are proportional to changes in crack

length as long as the network is above the percolation threshold. In the case of crack closure the key microvariable is crack spacing (1) since crack closure results in a decrease in crack density.

2w n

Figure 2.7: a) Pipe and b) crack models of porous media (reproduced from Gueguen and Palciauskas 1992).

IV Percolation Theory

Consider the following problem: we have a granular solid and create porosity and permeability by removing grains or creating cracks (grain boundary or transgranular) randomly. Initially these pore and cracks will be isolated, but at higher densities of pores and cracks there is a finite probability that some pores and cracks will intersect to form clusters although permeability is still zero; at still higher densities some clusters will be even larger, and at some critical density (the percolation threshold) pores and cracks will form an infinite cluster, spanning the porous body (Figure 2.8). At this point the body becomes permeable. Percolation theory attempts to address this problem (Gueguen and Palciauskas 1992; Gueguen and Dienes 1989).

Percolation threshold effects occur which cannot be anticipated through effective medium computations, for example in relation to permeability and electrical conductivity near the percolation threshold (Gueguen and Dienes 1989). Using the crack model of Gueguen and Dienes (1989) percolation theory has been used to predict permeability changes with stress (Gueguen et al. 1987).

P E R M E A B I L I T Y T H R E S H O L D D F R A G M E N T A T I O N

t

t

Figure 2.8: Homogeneous and Inhomogeneous cracking with the development o f two levels o f percolation threshold (reproduced from Gueguen and Palciauskas 1994).

V Empirical Relationships and the Carman-Kozeny Type Relationships

Based on an ‘equivalent channel model’ (Paterson 1983), where the pore space is replaced by a single, cylindrical, tortuous, non-intersecting channel or pipe of constant cross-sectional area, Carman-Kozeny type relations have been used to relate permeability to relatively easily measured rock properties. The Carman-Kozeny relations vary depending on variations in porosity, tortuosity and particle size distributions (Berryman and Blair 1986). Carman-Kozeny relations have been derived by constraining microstructural parameters using quantitative image analysis (Doyen 1988; Gueguen and Dienes 1989; Lin and Cohen 1982), porosity and mercury injection measurements (Carman 1956; Jacquin 1964; Bourbie and Zinszner 1985), intergranular surface area (Carman 1956), and formation factor, F, derived from electrical conductivity measurements (Heard and Page 1982; Brace era/. 1965, 1968; Archie 1942; Katz and Thompson 1986).

A general expression of the Kozeny formula is;

where (]) is the porosity, R is the hydraulic radius, defined as the ratio of pore volume to the pore surface area. C is a numerical constant dependent upon the shape of the cross-sectional area of the pores, which varies between 3/5 for high aspect ratio pores and 1/3 for slits, analogous to cracks (Wyllie and Spangler 1952). Hence, R, the hydraulic radius, is the characteristic cross-sectional dimension of the equivalent channel for determining the bulk flow rate. Since natural flow paths are tortuous, equation may be adapted to take tortuosity into account via F ’,

k = C R- / F ’ 2.24

Where

F ’ = (1, /1) - / (j) 2.25

1„ is the actual path length and 1 the length of the porous body. Both hydraulic radius, R, and tortuosity, F ’, can be derived from electrical conductivity measurements, where F, the formation resistivity factor or formation factor, F, relates the electrical conductivity of a pore fluid (Vjif) to the

effective conductivity of a rock through

^.fu. - p

where F depends primarily on the pore microstructure and incorporates tortuosity.

Brace et al. (1968) found a consistent relationship between F and (j) over a porosity range of 0.1 - 0.001, suggesting a linear relationship between porosity and permeability for intact Westerly

granite. Models using hydraulic radius (from formation factor) based on microstructural

observations made at atmospheric pressure may underestimate changes in crack and pore dimensions due to applied stress and hence may no be used reliably to predict changes in permeability with stress.

Vi Permeability : Porosity Relations

Generally, there is no simple relationship between permeability and porosity (Schneidegger 1974), although in some materials a clear relationship or several families of relationships exist where grain size is constant with porosity (Jacquin 1964) over some range of experimental conditions (Bourbie and Zinszner 1985). No simple relationship exists between the two properties since permeability is a topological property and porosity simply expresses a ratio of void volume to rock volume and makes no inference of pore interconnectivity.

Summary to Approaches to Modelling Permeability

Simple empirical and theoretical relationships are limited in that they are not based an a physical understanding of the effects of factors which control connectivity and hence permeability, such as applied stress. Limitations of empirical and theoretical relationships emphasise the importance of the empirical measurement of permeability as an independent property.

2.5 Specific Storage: the Role of Rock and Fluid Compressibility

In order to investigate fluid flow it is convenient to consider two simple end-member cases: • steady state flow

• flow resulting from a transient change in pressure or head.

In nature, more commonly transient changes in fluid pressure generate a pore pressure pulse which decays with time. In this type of flow, the compressibilities of the rock matrix and of the pore fluid become significant since they help to control the rate of restoration of fluid equilibrium (Figure 2.9). The compressibility properties of the rock and of the pore fluid are introduced through the concept of specific storage (S,„,.;.). P r e s s u r e = x P r e s s u r e = x + A ) S T E A D Y S T A T E F L O W T I M E B ) T R A N S I E N T T Y P E F L O W

Figure 2.9 : Steady state and transient types o f flu id flow. Steady state flow occurs only where a fixed pressure gradient exists, otherwise flow is zero; transient flow occurs more commonly in nature and is affected by the elastic properties o f the rock and pore fluid.

Because of the elasticity of a rock matrix and of a pore fluid, changes in stress and pore pressure cause changes of the mass or pressure of the fluid in any given volume of rock. The specific storage of a saturated porous media is defined as the volume of pore fluid which is expelled or taken into the pore volume (or storage) per unit change in hydraulic head, h. Changes in fluid storage are due to the compressibility of the rock matrix and the compressibility of the pore fluid.

Considering the change in fluid storage owing to compaction of the porous medium, the volume of water expelled from the unit volume of the porous medium is equal to the reduction in volume of the

unit volume of the porous medium. The volume reduction of the porous network, AV^is negative and the volume of the pore fluid released, AV/;„ is positive (Freeze and Cherry 1979), so that

A = - h V j = C„„„ 2.27

where the effective stress ( = g Ah), where is the density of water [M/L^], g is acceleration due

to gravity [L/T"], and is the compressibility of the porous network [1/(M/LT")]. For a unit reduction in hydraulic head. Ah = -1 we have

AV/,„ = C ,„ „ P j,^g 2.28

Considering the volume of pore fluid produced by the expansion of the fluid

2.29

The volume of water Yflu in the total unit volume is (|)V^, Cjiu is fluid compressibility, and ^ is the porosity. Hence 2.29 becomes

2.30