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Anisotropy in pore structure of porous media
IEOlU i01.0 ELSEVIER
Powder Technology 85 (1995) 143-151
Anisotropy in pore structure of porous media Carlos A. Grattoni, Richard A. D a w e * Earth Resources Engineering Department, Imperial College, London SW7 2AZ, UK Received 20 October 1994; revised 17 May 1995
Abstract
Realistic modelling of porous media is difficult. Different models are proposed depending on the property to be modelled, for example, porosity, permeability, electrical resistivity. Anisotropy is seldom considered. This paper examines the pore structure obtained for packs of spheres, with emphasis on the anisotropy. The structures can be reduced to an equivalent network which demonstrates that anisotropy of the pore structure is present even for homogeneous packs of spheres. The influence of anisotropy on the various porous media properties such as permeability, electrical resistivity and convective dispersion is highlighted. Keywords: Porous media; Regular packs; Pore structure; Anisotropy
1. Introduction
A porous medium is an intricate network of solid matrix and void space with channels that have irregular cross-section, transverse and longitudinal shapes. These channels can be interconnected and often allow fluids to move through the medium (i.e., are permeable). This morphology and topology is related to the origin of the porous medium and its evolution. Clearly the geometric characteristics will appear highly chaotic. The individual segments of the network are usually identified by size with the larger portions called pore bodies and are interconnected by throats. The complete pore structure contained within the porous medium is responsible for its physical properties such as permeability, electrical resistivity, convective dispersion, etc. To evaluate the average (macroscale) properties of the porous medium the exact description of the void space is probably unnecessary since the general characteristics of the void space can be embodied in the shape, transverse area and length of each segment of a network model. The alternative paths can be incorporated in the number of segments that converge in a node (coordination number) and finally the topology can be taken into account by the way the pores are connected. A sketch of a realistic representation of the pore space and its topological-morphological equivalent network is shown in Fig. 1. Anisotropy is the variation of a property (permeability, dispersion, resistivity, etc.) with the direction of measure* Corresponding author. 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved
SSD10032-5910(95)03016-3
O Pore body or node
m
Pore throat or bond
Fig. 1. (a) Bi-dimensional sketch of a porous medium; (b) its equivalent network.
ment [ 1-4]. Anisotropy is usually considered to be a macroscopic characteristic of the media but as will be shown later even homogeneous porous media can have a pore structure that is anisotropic and can give the physical parameters a directional quality. In this work a comprehensive description of the pore space for regular packs of spheres is presented which highlights the influence of the different pore structure elements. For the same particle material and size we can get different pore structures by arranging the grains in various ways and hence obtain a variation in the anisotropy effects. The description of the pore space can help to understand the role of the microscopic pore structure and its anisotropy on different processes, although some further information is needed to identify the physics of the process to be modelled [5]; e.g., fluid dispersion, fluid transport and electrical conductance. The pore space of the packs can be described macroscopically (pack characteristics and porosity) and microscopically (pores-throats shapes and sizes, number of interconnected
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pores, etc.) and we investigate how microscopic anisotropy is created by the internal pore structure (especially the topology) and can be present even for homogeneous cases. A methodology is developed to reduce the pore space to an equivalent network which is useful for developing a mathematical model to predict the behaviour of the porous medium. The equivalent network contains all the basic morphologictopologic characteristics of the pore space and the internal dimensions relevant to the process to be modelled including anisotropy effects.
2. General porous media models The complex network of pores within a solid matrix (grains of variable size) can be modelled by many alternative ways and thus there are many different types of models [3,4,6]. The complexity of a particular model depends on the processes to be modelled and the porous medium features considered. There is, as yet, no complete and absolute single model so a particular model is used to highlight particular features. The porous medium can be modelled at different scales: macroscopic (larger than cores), intermediate (thousands of pores) and microscopic (pore level). The models are used to estimate the behaviour of different processes but the pore level ones can also be used to model mechanisms [ 3 ] such as mass transfer, heat transfer, diffusion, etc. Each process depends in a different way on the pore structure characteristics and pore dimensions, so an appropriate model will vary accordingly. For instance, a pore space model for mercury injection may not be suitable for describing immiscible three-phase flow, but it may be satisfactory for describing electrical resistivity in a saturated porous medium. Each model is built to describe a particular behaviour but the model must not be confused with the morphology and topology of the real porous medium. For example, the bundle of capillary tubes to model mercury intrusion, where its geometry is only a representation compatible with the behaviour to be modelled but does not mean that the medium is in fact a bundle of capillary tubes. The term model will be applied to a representation of the pore space; we can have physical models (which can have simple idealized features), or a mathematical model which is a set of equations that represent the pore space and its behaviour. The most difficult part in modelling a porous medium is to select the correct geometric and pore structure characteristics which control the petrophysical properties such as permeability, electrical resistivity, porosity and multiphase flow. There have been many models suggested over the years in the literature [6] and most have considered the rocks to be macroscopically homogeneous and isotropic, but it is now necessary to re-examine them at the pore scale to understand how the pore structure and its anisotropy contributes to the underlying physics.
2.1. Mathematical networks
The pioneering studies on porous media structure as networks were used to study immiscible fluid displacements [ 7,8 ]. These models considered that the pore system has two kinds of voids, Fig. 1 (b). The larger voids (pores or nodes) are connected to the smaller ones (tubes or bonds), which can have circular, ellipse or square shape and a random size distribution. Commonly, the regular networks have a unique coordination number (Z), which is the number of branches converging to a pore or node. Several types of network have been used as models of porous media and differ mainly in the tube sizes, transverse section shape and connectivity. The network tube radius distribution can be arbitrary specified or qualitatively obtained from experiments such as mercury porosimetry (which already assumes a model) or from thin sections. The tubes can have constant lengths or they can be proportional to their diameter (e.g., the wider tubes can be shorter) and the nodes (pores) can have volume or not [6].
3. Packs of spheres Regular packs of spheres are a 3-D representation of natural porous rocks albeit in simple form. They have the advantage that there is full control over the geometry, since the shape and size of pores and throats are defined by the pack characteristics and the size of the spheres [9]. Also the pore structure and internal dimensions can be accurately visualized by pore casts and the dimensions calculated and, as will be shown, used to construct the equivalent network of the pore space and mathematical model. Packs of spheres have been used as a model of unconsolidated or poorly consolidated porous media previously. Graton and Fraser [ 10] in their pioneering work presented many pore structures to study the effects of packing characteristics on porosity and permeability. Rose [ 11 ] studied the effect of pack structure on the surface area and volume of pendular rings. Smith [ 12-14] presented several experimental and theoretical papers relating to different aspects of packs of spheres, such as structure of random packs, fluid distribution, liquid capillary rise and capillary retention. Sundberg [ 15] calculated the electrical resistivity and porosity of different packs. However, none of these earlier workers attempted the description of the pore space as a mathematical network of pores and throats mainly because of the computational difficulties. Nevertheless they made a very detailed description of the physical pore space and the principles involved in pore scale modelling. Most pore structures, as we shall see, are anisotropic, therefore their properties (permeability, electrical resistivity) will depend on the direction in which they are measured. Thus these properties cannot be in reality, true scalar quantities. However, there are (macroscopic) scalar quantities (porosity, water saturation) which are by definition an average value within the pore structure and will not fully reflect the porous
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145
media composition or the directional nature of the internal pore structure.
4. Description of the packs of spheres studied Regular packs of spheres (also called systematic packs) are used in this work to study the relationships between the void space at the pore scale and its vectorial characteristics. Six cases of simple packing are analysed and their void space described by an equivalent network that contains all the relevant information about the pore space (pore and throat dimensions, coordination number and spatial interconnections). The arrangements of spheres within the packs will be described using the layer as the basic element of the pack. The void space within the spheres gives the principal characteristics of the pore volume. The description of packs of spheres used in this work is developed from Ref. [ 10]. Some definitions will be introduced for clarification. A row is an arrangement of solid spheres of uniform diameter, d, with their centres located on a straight line; layers are a group of rows in contact with each other forming an angle between 60 ° and 90 ° as shown in Fig. 2. A systematic pack is an arrangement of spheres formed by layers on top of each other. The layer is the basic building block and is used here to obtain different packs by stacking the layers on top of each other. Several different types of structure can be obtained by displacing adjacent layers. Three main ways of stacking layers one upon another to form a pack will be considered and are shown in Fig. 3: (i) in each layer each sphere centre is directly on top of the one below i.e., the upper layer is on top of the one below; packs 1 and 4. (ii) The upper layer is shifted a distance d/2 in one direction but the rows are directly on top of the ones below; packs 2 and 5. (iii) The upper layer is shifted so the rows and the spheres are not overlying the one below (not directly on top of each other), and the distance between the centres of the layers is minimum; packs 3 and 6. The packs generate a 3-D interconnected porous network in which the shape, size of pores and throats are defined by each pack and for some processes by the size of the spheres. In order to visualise fully the pore morphology the six packs
(a)
(b)
(c)
Fig. 2. Plan structure of a layer. (a) Rows at 90 °, square layer; (b) rows at 750; (c) rows at 60 °, rhombie layer.
Pack I
Pack 2
Pack 4
Pack 5
Pack 3
Pack 6
Fig. 3. Regular packs and their void space, Packs 1, 2 and 3 have square layers. Packs 3, 5 and 6 have rhombic layers. The packs have two types of throats: concave-square, pack 1; or concave-triangular, pack 6.
described in the next section were physically constructed using plastic balls of 20 mm of diameter with four balls per basic layer and 2 or 3 layers [ 16]. Plaster casts of the pore space were made and cut up in different directions to understand their spatial relationships and studied in detail to identify the influence of the internal pore structure on anisotropy.
4.1. Packs: arrangement of spheres In this work the packs with square (90 °) and rhombic (60 ° ) layers are considered which when combined with the three ways of stacking produce the six types of packs studied. A schematic representation of the six packs and their unit pore volume (described later) are shown in Fig. 3. Pack 1 (simple cubic). This pack has successive square layers stacked so that each sphere is above the one in the bottom layer. The planes that pass through the centres of the spheres form a cube because the square layer structure is present in the three orthogonal directions. This pack has the maximum porosity (with the spheres having contact with each other) of 47.64%. Pack 2 (orthorhombic). Here, as in pack 1, the basic structure is the square layer but the successive layers are displaced a distance d/2 in the direction of one of the rows. Thus the angle between successive layers is 60°; i.e., the second layer is in a lower position than for pack 1, The porosity is 39.54%. Pack 3 (rhombohedral or face-centered). The square layer is also present in this pack but the successive layers are located over the central point of the diagonal of the lower layer (point x in Fig. 2 ( a ) ) . Thus the vertical distance between layers results in 0.707 d giving the minimum porosity of the packs with a square layer (25.95%). Pack 4 (orthorhombic). The basic structure in this pack is the rhombic layer and the spheres in the successive layers are
C.A, Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
146
(a)
(b)
Pack I
Pack 2
structure is present in all directions. It is important to notice that if a sphere is in position y, Fig. 2(c), the position z will remain empty (a sphere will not fit in it because it will be surrounded by spheres in rhombic arrangement) which creates two types of path; one interrupted by the second layer and another that passes through the layers (see Fig. 3). This composite nature with the minimum porosity, 25.95%, is of special importance because it has the same porosity of pack 3 but with a different pore structure.
(c)
Pack 3
(d)
4. 2. Packs: pore space
Fig. 4, Normalization of the unit cell for pack 2. (a) Original cell; (b) cut sector to make it perpendicular to the base; (c) normalized cell; (d) norrealized cells for the packs with square layers.
The following description of pore space is based on our unit volume cell for each pack. The basic structure (minimum portion) of a systematic pack gives a complete representation of the pack and the void contained in it (Fig. 3). The unit volume cell is bounded by the planes which intersect the centres of the adjacent spheres, forming a paraUelepiped with a square or rhombic base (Fig. 4 ( a ) ) . The main characteristics of the packs studied are presented in Table 1. In these packs of spheres the porosity is proportional to the distance between layers and the voids present are interconnected so producing a succession of throats and pores. The systems with square layers have a larger porosity than the ones with rhombic layers for the same way of stacking (compare packs 2 and 5). Packs 1 to 3 (square layer) have the same throat shape (Fig. 3) but as the distance between layers
stacked above the ones in the lower layer. This pack has the same solid structure as pack 2 but rotated 90°; i.e., interchanging rows/layers. The porosity is 39.54%. Pack 5 (tetragonal). This pack, like pack 4, has rhombic layers and the row in successive layers are displaced a distance d/2. The layers form an angle of 60 ° and generate a rhombic structure in two planes and a square structure in the third. The distance between layers is 0.866 d as in pack 2. The porosity is 30.19%. Pack 6 (rhombohedral or hexagonal). The rhombic layer is also the base of this pack but the second layer is placed so that the spheres are located above position y (Fig. 2 ( c ) ) , being the distance between layers=0.816 d. The rhombic Table 1 Basic characteristics of the systematic pack of spheres Pack 1
Pack 2
Pack 3
Pack 4
Pack 5
Pack 6
Type of layer
square
square
square
rhombic
rhombic
rhombic
Spacing of layers
d
dX/r~
dr/2
d
dv/~
dv~
Volume of unit cell
1.00 d 3
0.87 d 3
0.71 d 3
0.87 d 3
0.75 d 3
0.71 d 3
Porosity
47.64
39.54
25.95
39.54
30.19
25.95
(%) Table 2 Comparison of different parameters for packs of spheres Packs
PAR
Z
Throat
~b
L
A
F
1'-2"
j, c
1" d
s-s
~,
,L
=
1 '-3'
~,
? f
~ = =
,[, L ,I, $
1"
J, 1" 1"
,[, J, = =
=
4'-5' 2'-2" b 3'-Y'
s-s t-t s-t s-t
= J, ~
1" ,[
5'-5"
=
=
t-t
=
=
=
1"
" ' perpendicular to the layer. b . parallel to the layer. c ~ decrease in the value. 1" increase in the value. • = no change in the value. f ? not comparable.
¢
1"
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C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
and the porosity decreases the pore structure (path) becomes more tortuous because of its increasing complexity. In packs 4 to 6 the porosity and the distance between layers follow the same tendency but the paths are more complicated. For example, pack 6 has two types of paths in parallel, one passing readily through the layers and the other enclosed by spheres. It must be noticed that the unit cell of pack 4 has the same basic structure and internal distribution as pack 2, but rotated 90 °. The throat planes (which separate adjacent pores) also intercept the centres of adjacent spheres and therefore the throats are located on the faces of the unit volume cell. There are two types of throat: (i) concave-square formed by four adjacent spheres and (ii) concave-triangle formed by three adjacent spheres. For example, the rhombic layers form a double concave-triangular throat connected by the corners and the square layer form a concave-square throat, see Fig. 3. The same throat shapes are produced by the overlaying of layers. Thus all the six packs have a combination of concavetriangular and concave-square throat shapes in different directions. For example pack 2 has concave-square throats perpendicular to the layer plane but concave-triangle throats in the other directions; this is an indication of the anisotropy of this pack. A close inspection of the void space shows that packs 2-6 all have anisotropy in different degrees but the symmetry in all directions of pack 1 makes it unusually isotropic. The pack have a systematic variation but the pore spaces themselves do not have such a systematic variation and therefore the pore space topology and morphology is difficult to catalogue and interpret. The underlying porous media structure (morphology and topology) which are necessary ingredients for the description of the internal void space affects the physical macroscopic processes (permeability, electrical resistivity, etc.). With this knowledge we can now reduce the pore structure to a network to model the process.
it easier for the determination of the equivalent network and the construction of the experimental packs (e.g., the unit cell of packs 1, 2 and 3 will have a similar shape, as shown in Fig. 4 ( d ) ) . The networks reported later are based on the normalized unit cells. The algorithm to reduce the porous space to the equivalent network consists of three main steps, as follows. 5.1. Reduction o f pore space
If a sphere is allowed to grow at each point on the pore wall until the surface of the sphere touches other points on the pore wall then the centres of those spheres form surfaces which run through the pore space called 'stripwork', which is a connected assembly of strips. This stripwork has some general properties. (i) Each point has associated with it the radius of the tangent sphere to the solid and the number of contacts that it has made with the pore walls. (ii) Double contact points define the surface of the strips. Triple contact points define continuous lines (intersection of surfaces) that bound the strips. Quadruple contact points are usually isolated and are the intersection of triple contact points. (iii) The corners on the pore walls and edges with angles less than 180° are part of the outer boundary of the stripwork. A simple example is shown in Fig. 5, where two triangular tubes converge to form a square section; when reduced to its stripwork each triangular tube will be reduced to three fiat strips, Fig. 5(b) and the square section to four equal strips.
5
5. Representation of morphology and topology through an equivalent network Networks are a good way of representing different processes within the pore space [ 17-19], but there is not yet an available algorithm to reduce the pore space to a general network. However, as will be shown, each physical process needs to be considered separately and have its special characteristics recognized. An equivalent network is a representation of the pore space that contains the topologic and relevant morphologic characteristics of a determined porous space. In this work all the unit volume cells were normalized before reducing the void space to a network. This normalization consists in passing four planes (perpendicular to the layers) around the base as shown for pack 2 in Fig. 4 ( a ) (c). This rearrangement of planes and volumes does not alter the structure or the size of the unit volume cell due to the special characteristics of the systematic packs, but it makes
2
bond
4
3
x-x
node
(c) Fig. 5. Reduction of a pore space to an equivalent network. (a) Original pore spaceand reduction to a stripworkin sectionx-x. 1= edge, 2 = double contact points, 3 =triple contact point, 4 =traces of the strip, 5 =pore wall; (b) reduced stripwork; (c) equivalentnetwork.
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C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
The strips meet at lines which are formed by the triple contact points and these lines join at a quadruple contact point (equidistant point to the surfaces where the triangular and square tubes join). The external borders of the strip are the boundaries of the stripwork. The same stripwork could be obtained by growing the solid, normal to its surface at a constant rate, and stopping the solid growth at the points where two nonadjacent pore-wall surfaces meet.
5.2. Primitive network The stripwork is shrunk by moving its boundary points along the shortest possible distance on the curved surface (geodesics orthogonal to the strip). Two possibilities can arise. (i) Two boundary points may meet each other. In this case one of the points disappears and the remaining point is not moved any more, remains fixed and forms part of the primitive (basic) network. (ii) Two or more strips can join at a line (higher contact line). In this case the boundary point may merge on the line and disappear as a boundary (no further track of the boundary point is then kept). The primitive network is obtained when the system cannot be reduced any further. Each point of this network has associated with it several quantities to model the process (such as radius, transverse area, etc.). Some definitions for the primitive network are: nodes: the positions of local maxima of the equivalent radius of the network and branches; throats: the positions of local minima of the network. Each point of the network has associated with itself the value of the pore level descriptors needed for the pore level laws. For example, the primitive network of the pore space discussed in Fig. 5 (a) reduces to three straight branches joining in a node, shown in Fig. 5(c).
5.3. Working network This is obtained from the primitive network but incorporates some additional geometrical or process criteria to take into account or eliminate some details, e.g., dead end pores could be eliminated because they do not play any role in the flow. The equivalent network does not assume any pre-determined characteristics for the bonds. Therefore, the lines connecting two nodes or pore bodies is only a representation of the topology and does not mean that these connections are straight or they have any special cross-section shape. Then the algorithm used to obtain the equivalent network does not induce any anisotropy effects. Each branch or bond of the network has associated with it the main quantities (characteristics) needed to model the process, and they will be described in the next section.
6. Pore level laws and descriptions The processes to be modelled by a mathematical network requires the pore space to be treated as an assemblage of bonds or branches. The transport of material, fluid, electricity through these segments are incorporated by using the pore level laws (mechanisms--equations). The pore level laws determine how fluids, tracers and ions behave during the transport processes and are used to calculate the average (macroscopic) transport properties. A variety of pore structure descriptors are needed for each process and have to be considered during the construction of the working network. The pore level laws include the descriptors needed ; for example
6.1. Permeability If it is assumed (as is often done) that each segment or branch follows the Hagen-Poiseuille equation: A pTl'rh4
q=
8/xl
(1)
where q is the volumetric flow rate, AP is the pressure difference across the segment,/~ is the viscosity of the fluid, l is the length of the segment and rh is the hydraulic radius [ 5 ], which is defined as the minimum cross-section of flow (q) divided by the wetted perimeter (Wp). Then for each segment the length and mean hydraulic radius are used as descriptors to model the permeability. The total permeability of the porous medium is obtained by solving the equations from all the bonds and nodes. The individual bond characteristics have an influence on the total behaviour of the medium.
6.2. Microscopic dispersion Here the concentration changes have to be monitored and the mixing of the streams entering in the pore body need to be considered and therefore the velocities in each pore segment (v =q/a) and the volume of the pore body (al) have to be obtained as pore descriptors.
6.3. Electrical resistivity The law used is based on the definition of the resistivity itself: Rb=Rwl/am
(2)
where Rb is the resistance of the segment or branch, Rw is the resistivity of the fluid and am is the mean transverse area of the segment. Then the pore level descriptors needed are the mean transverse area perpendicular to the segment direction and its length [ 16].
CA. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
(a)
(b)
(c)
Fig. 6. Reductionof the unit volumecellforpack 1. (a) Originalporespace; (b) stripwork; (c) equivalentnetworkoftbe cell.
Fig. 7. Representation of the transverse area m different positions along an
elementof pack 1. '-
Pack I
Pack 4
Pack 2
Pack 5
Pack 3
Pack 6
Fig. 8. Equivalentnetworkforthe six packs.The numberin the intersections representthe coordinationnumberof the porebody.
6.4. Other petrophysical parameters Porosity, surface area per unit volume, pore-throats dimensions and distribution can be obtained as a consequence of the three previous parts (permeability, microscopic dispersion and electrical resistivity). The variations of the pore level descriptors (a, am, l, Wp) for a segment of pack 1 can be inferred from Figs. 6 and 7. The descriptors needed for each segment are scalar quanrifles and therefore the anisotropy will not be generated by them but will appear due to the topology of the pore structure (e.g., coordination number and connectivity of the network) and the potential externally applied (for example: pressure, concentration or electrical voltage). Therefore, the anisotropic characteristics of a porous medium will be reflected in different processes, but the magnitude of the effects of the porous anisotropy will depend on the individual process.
7. Packs: equivalent n e t w o r k
The methodology to obtain an equivalent network and the pore level descriptors needed to model the pore level laws have been described and now the equivalent network for our
149
six packs is described and analysed. Pack 1 will be explained in some detail to give the methodology while for the other packs the final results will be shown. As described earlier, pack 1 consists of square layers of overlying spheres and the pore walls are convex surfaces forming six concave-square throats (Fig. 6 ( a ) ) . The stripwork consists of orthogonal planes of width d, as shown in Fig. 6(b), and its edges are the points where a pair of spheres touches each other. When the stripwork is reduced to the primitive network the points (edges of the stripwork) grow forming geodesic circles, Fig. 6(b), and due to the orthogonal position of the spheres the circles meet each other at orthogonal lines (normal to each other at one point, as shown in Fig. 6(c) ). Thus the equivalent network is a repetitive (systematic) structure with coordination number (Z) equal to six. Pores are located where the six branches intersect and the throats (and throat planes) are at half length between pores. The equivalent dimensions of the pores and throats are 0.732 d and 0.414 d respectively, where d is the diameter of the solid sphere. From this the ratio of pore and throat transverse areas (PAR) is 3.12 and the ratio of the maximum and minimum length in the throat plane is 2.41 (Fig. 7). Fig. 8 shows the equivalent networks for the six packs. The topology and structure of the unit void and its equivalent network can show the isotropic or anisotropic nature of the packs and give an indication of the directional behaviour of these packs. The following analysis is based on the void space and the networks of Figs. 3 and 8 respectively. Pack 1 will show an isotropic behaviour because it is symmetric and has only one type of path which is the same in each orthogonal direction. Each node has two symmetric connections with respect to the pore on each orthogonal plane, so producing a coordination number equal to six. Packs 2 and 4 will show similar anisotropy behaviour since they have the same morphology and topology (rotated 90 ° ) having two types of path. The coordination number is equal to seven, with five branches on one plane and the other two perpendicular to it. Therefore, these two packs will be anisotropic in two of the three main directions. Packs 3, 5 and 6 also show anisotropy behaviour. They have very complex topologies with two types of pores in each void cell: one of large size and highly interconnected ( Z = 7 or 8) and another smaller and less interconnected (Z= 3, 6, and 4 respectively). Additionally pack 6 also has two types of path perpendicular to the layer plane, one directly connected to the upper and lower layers and the other restricted within the layer. This suggests that due to the anisotropic structure of the pore space any process considered will be dependent on the direction of measurement. This analysis shows that microscopic anisotropy can occur due to the internal pore structure of the porous media (mainly topology), even though the packs themselves are homogeneous on a bulk scale. For example, different degrees of electrical resistivity anisotropy [4] were experimentally measured in these packs demonstrating that anisotropy can occur even for spherical particles with the same size. Finally,
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the effect of the coordination number on anisotropy is not always clear since there are some parameters that cannot be changed independently such as mean transverse area and length, and their relationship in natural porous media still needs to be identified.
these techniques should bring a better understanding of the influence of pore scale structure upon on different processes; i.e., fluid dispersion, permeabilities, capillary pressures and electrical resistivity.
9. List of symbols 8. Discussion a
The pore spaces of six systematic packs of spheres have been analysed to assess the effects of pore structure on anisotropy of porous media. Their pore space have been reduced to an equivalent network that considers all the relevant parameters needed to model their behaviour: coordination number, morphology, topology, length of connections and mean equivalent area (which includes pore-to-throat aspect ratio, pore and throat shape variations and throat dimensions). The systematic pack of spheres are simple porous media but even they give complex pore spaces. The relationships between pore space structure and the processes occurring in natural or artificial porous media can be modelled by using equivalent networks if they include the main internal dimensions of the porous medium. This can help in the understanding of the basic process physics. The packs analysed have shown the existence of microscopic anisotropy (due to the internal pore structure) even if the medium is macroscopically homogeneous and is composed of uniform size spherical particles. It has been shown that reducing the pore space (at a microscopic scale) to an equivalent network is a valid and useful technique for modelling the porous system behaviour. Irregular packs of spheres of the same diameter are a combination of the packs described in this paper and the macroscopic properties could be obtained from the networks of Fig. 8. For example, using the unit volume cells presented in Fig. 4 an irregular pack can be built as follows: rotating pack 3 ( 180° in the vertical direction) stacking pack 1 over it and pack 2 over pack 1. Therefore the macroscopic properties of this compound pack can be easily calculated from the networks, pore level laws and descriptors presented in this paper. The methodology presented can also be applied to more complex porous media (random packs with grains of different sizes) providing that the shape of the grains is known and the positions of the grains can be inferred. For example, by using the grain consolidation model [6] where spheres of different diameters are randomly packed, then if the spheres are allowed to grow evenly in all directions, or preferentially in one direction, and the growing process is stopped when a desired porosity is reached, depending on the conditions imposed, different natural and artificial porous media can be simulated. In a similar way other grain shapes can be used to represent other pore spaces. Once the matrix structure is defined the methodology presented in this paper can be readily applied. Such applications to natural porous medium (e.g. rocks) will be feasible in the near future when 3-D pore space reconstruction, see Ref. [20], using computer image analysis and thin sections, becomes more available. The merging of
am
A d F l L AP
PAR q rh
Rb Ro Rw s t u
% Z
minimum cross section of flow mean transverse area of the segment or branch total transverse area of the unit cell diameter of the sphere formation factor, Ro/Rw length of the segment or branch total length of the unit cell pressure difference across the segment or branch ratio of the pore and throat transverse areas volumetric flow rate hydraulic radius, a/Wp resistance of a segment or branch resistivity of the network resistivity of the fluid square throat triangular throat velocity in a pore segment or branch, q/a wetted perimeter coordination number, number of branches converging to a pore
Greek letters /x ~b
viscosity of the fluid porosity
Acknowledgements We would like to thank Deminex UK Oil and Gas Ltd., the EU under the Joule II programme and PSTI for financial support.
References [ 1] Y. Bernab~, Fault Mechanics and Transport Properties of Rocks, Academic Press, London, 1992, Ch. 6. [2l W.K. Sawyer, C.I. Pierce and R.B. Lowe, Rep. No. 7519, US Department of the Interior, Bureau of Mines, Washington, DC, 1971. [3] F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure Academic Press, San Diego, CA, 2nd edn., 1992. [4] C.A. Grattoni and R.A. Dawe, Int. Symp. Society of Core Analysts, Stavanger, Norway, Sept. 1994, p. 173. [5] J.W. Amyx, D.M. Bass Jr. and R.L. Whiting, Petroleum Reservoir Engineering - - Physical Properties, McGraw-Hill, New York, 1960. [6] J. Van Brakel, Powder Techaol., 11 (1975) 205. [7] J.E. Owen, Trans. AIME, 195 (1952) 169. [8] I. Fatt, Trans. AIME, 207 (1956) 144. [9] R.A. Dawe, Sch. Sci. Rev., 74 (1992) 79.
C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151 [ 10] L.C. Graton and H.J. Fraser, J. Geol., XLII (1935) 785. [11] W. Rose, J. Appl. Phys., 29 (1958) 687. [ 12] W.O. Smith, P.D. Foote and P.F. Busang, Phys. Rev., 34 (1929) 1271. [ 13] W.O. Smith, P.D. Foote and P.F. Busang, Phys. Rev., 36 (1930) 524. [ 14] W.O. Smith, Physics, 4 (1933) 425. [ 15] K. Sundberg, Geophys. Prospect. AIME (1932) 367.
151
[ 16] C.A. Grattoni, Ph.D. Thesis, University of London, UK, 1994. [ 17] P.D. Yale, Ph.D. Thesis, University of Stanford, USA, 1984. [18] X.D. Jing and J.S. Archer, 32nd SPWLA Annu. Logging Symp., Midland, TX, USA, June 1991. [ 19] K.K. Mohanty and S.J. Salter, Prepr. AIChEAnnu. Fall Meet., 1983. [20] C. Lin and M.H. Cohen, J. Appl. Phys., 53 (1982) 4152.
Powder Technology 85 (1995) 143-151
Anisotropy in pore structure of porous media Carlos A. Grattoni, Richard A. D a w e * Earth Resources Engineering Department, Imperial College, London SW7 2AZ, UK Received 20 October 1994; revised 17 May 1995
Abstract
Realistic modelling of porous media is difficult. Different models are proposed depending on the property to be modelled, for example, porosity, permeability, electrical resistivity. Anisotropy is seldom considered. This paper examines the pore structure obtained for packs of spheres, with emphasis on the anisotropy. The structures can be reduced to an equivalent network which demonstrates that anisotropy of the pore structure is present even for homogeneous packs of spheres. The influence of anisotropy on the various porous media properties such as permeability, electrical resistivity and convective dispersion is highlighted. Keywords: Porous media; Regular packs; Pore structure; Anisotropy
1. Introduction
A porous medium is an intricate network of solid matrix and void space with channels that have irregular cross-section, transverse and longitudinal shapes. These channels can be interconnected and often allow fluids to move through the medium (i.e., are permeable). This morphology and topology is related to the origin of the porous medium and its evolution. Clearly the geometric characteristics will appear highly chaotic. The individual segments of the network are usually identified by size with the larger portions called pore bodies and are interconnected by throats. The complete pore structure contained within the porous medium is responsible for its physical properties such as permeability, electrical resistivity, convective dispersion, etc. To evaluate the average (macroscale) properties of the porous medium the exact description of the void space is probably unnecessary since the general characteristics of the void space can be embodied in the shape, transverse area and length of each segment of a network model. The alternative paths can be incorporated in the number of segments that converge in a node (coordination number) and finally the topology can be taken into account by the way the pores are connected. A sketch of a realistic representation of the pore space and its topological-morphological equivalent network is shown in Fig. 1. Anisotropy is the variation of a property (permeability, dispersion, resistivity, etc.) with the direction of measure* Corresponding author. 0032-5910/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved
SSD10032-5910(95)03016-3
O Pore body or node
m
Pore throat or bond
Fig. 1. (a) Bi-dimensional sketch of a porous medium; (b) its equivalent network.
ment [ 1-4]. Anisotropy is usually considered to be a macroscopic characteristic of the media but as will be shown later even homogeneous porous media can have a pore structure that is anisotropic and can give the physical parameters a directional quality. In this work a comprehensive description of the pore space for regular packs of spheres is presented which highlights the influence of the different pore structure elements. For the same particle material and size we can get different pore structures by arranging the grains in various ways and hence obtain a variation in the anisotropy effects. The description of the pore space can help to understand the role of the microscopic pore structure and its anisotropy on different processes, although some further information is needed to identify the physics of the process to be modelled [5]; e.g., fluid dispersion, fluid transport and electrical conductance. The pore space of the packs can be described macroscopically (pack characteristics and porosity) and microscopically (pores-throats shapes and sizes, number of interconnected
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pores, etc.) and we investigate how microscopic anisotropy is created by the internal pore structure (especially the topology) and can be present even for homogeneous cases. A methodology is developed to reduce the pore space to an equivalent network which is useful for developing a mathematical model to predict the behaviour of the porous medium. The equivalent network contains all the basic morphologictopologic characteristics of the pore space and the internal dimensions relevant to the process to be modelled including anisotropy effects.
2. General porous media models The complex network of pores within a solid matrix (grains of variable size) can be modelled by many alternative ways and thus there are many different types of models [3,4,6]. The complexity of a particular model depends on the processes to be modelled and the porous medium features considered. There is, as yet, no complete and absolute single model so a particular model is used to highlight particular features. The porous medium can be modelled at different scales: macroscopic (larger than cores), intermediate (thousands of pores) and microscopic (pore level). The models are used to estimate the behaviour of different processes but the pore level ones can also be used to model mechanisms [ 3 ] such as mass transfer, heat transfer, diffusion, etc. Each process depends in a different way on the pore structure characteristics and pore dimensions, so an appropriate model will vary accordingly. For instance, a pore space model for mercury injection may not be suitable for describing immiscible three-phase flow, but it may be satisfactory for describing electrical resistivity in a saturated porous medium. Each model is built to describe a particular behaviour but the model must not be confused with the morphology and topology of the real porous medium. For example, the bundle of capillary tubes to model mercury intrusion, where its geometry is only a representation compatible with the behaviour to be modelled but does not mean that the medium is in fact a bundle of capillary tubes. The term model will be applied to a representation of the pore space; we can have physical models (which can have simple idealized features), or a mathematical model which is a set of equations that represent the pore space and its behaviour. The most difficult part in modelling a porous medium is to select the correct geometric and pore structure characteristics which control the petrophysical properties such as permeability, electrical resistivity, porosity and multiphase flow. There have been many models suggested over the years in the literature [6] and most have considered the rocks to be macroscopically homogeneous and isotropic, but it is now necessary to re-examine them at the pore scale to understand how the pore structure and its anisotropy contributes to the underlying physics.
2.1. Mathematical networks
The pioneering studies on porous media structure as networks were used to study immiscible fluid displacements [ 7,8 ]. These models considered that the pore system has two kinds of voids, Fig. 1 (b). The larger voids (pores or nodes) are connected to the smaller ones (tubes or bonds), which can have circular, ellipse or square shape and a random size distribution. Commonly, the regular networks have a unique coordination number (Z), which is the number of branches converging to a pore or node. Several types of network have been used as models of porous media and differ mainly in the tube sizes, transverse section shape and connectivity. The network tube radius distribution can be arbitrary specified or qualitatively obtained from experiments such as mercury porosimetry (which already assumes a model) or from thin sections. The tubes can have constant lengths or they can be proportional to their diameter (e.g., the wider tubes can be shorter) and the nodes (pores) can have volume or not [6].
3. Packs of spheres Regular packs of spheres are a 3-D representation of natural porous rocks albeit in simple form. They have the advantage that there is full control over the geometry, since the shape and size of pores and throats are defined by the pack characteristics and the size of the spheres [9]. Also the pore structure and internal dimensions can be accurately visualized by pore casts and the dimensions calculated and, as will be shown, used to construct the equivalent network of the pore space and mathematical model. Packs of spheres have been used as a model of unconsolidated or poorly consolidated porous media previously. Graton and Fraser [ 10] in their pioneering work presented many pore structures to study the effects of packing characteristics on porosity and permeability. Rose [ 11 ] studied the effect of pack structure on the surface area and volume of pendular rings. Smith [ 12-14] presented several experimental and theoretical papers relating to different aspects of packs of spheres, such as structure of random packs, fluid distribution, liquid capillary rise and capillary retention. Sundberg [ 15] calculated the electrical resistivity and porosity of different packs. However, none of these earlier workers attempted the description of the pore space as a mathematical network of pores and throats mainly because of the computational difficulties. Nevertheless they made a very detailed description of the physical pore space and the principles involved in pore scale modelling. Most pore structures, as we shall see, are anisotropic, therefore their properties (permeability, electrical resistivity) will depend on the direction in which they are measured. Thus these properties cannot be in reality, true scalar quantities. However, there are (macroscopic) scalar quantities (porosity, water saturation) which are by definition an average value within the pore structure and will not fully reflect the porous
C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
145
media composition or the directional nature of the internal pore structure.
4. Description of the packs of spheres studied Regular packs of spheres (also called systematic packs) are used in this work to study the relationships between the void space at the pore scale and its vectorial characteristics. Six cases of simple packing are analysed and their void space described by an equivalent network that contains all the relevant information about the pore space (pore and throat dimensions, coordination number and spatial interconnections). The arrangements of spheres within the packs will be described using the layer as the basic element of the pack. The void space within the spheres gives the principal characteristics of the pore volume. The description of packs of spheres used in this work is developed from Ref. [ 10]. Some definitions will be introduced for clarification. A row is an arrangement of solid spheres of uniform diameter, d, with their centres located on a straight line; layers are a group of rows in contact with each other forming an angle between 60 ° and 90 ° as shown in Fig. 2. A systematic pack is an arrangement of spheres formed by layers on top of each other. The layer is the basic building block and is used here to obtain different packs by stacking the layers on top of each other. Several different types of structure can be obtained by displacing adjacent layers. Three main ways of stacking layers one upon another to form a pack will be considered and are shown in Fig. 3: (i) in each layer each sphere centre is directly on top of the one below i.e., the upper layer is on top of the one below; packs 1 and 4. (ii) The upper layer is shifted a distance d/2 in one direction but the rows are directly on top of the ones below; packs 2 and 5. (iii) The upper layer is shifted so the rows and the spheres are not overlying the one below (not directly on top of each other), and the distance between the centres of the layers is minimum; packs 3 and 6. The packs generate a 3-D interconnected porous network in which the shape, size of pores and throats are defined by each pack and for some processes by the size of the spheres. In order to visualise fully the pore morphology the six packs
(a)
(b)
(c)
Fig. 2. Plan structure of a layer. (a) Rows at 90 °, square layer; (b) rows at 750; (c) rows at 60 °, rhombie layer.
Pack I
Pack 2
Pack 4
Pack 5
Pack 3
Pack 6
Fig. 3. Regular packs and their void space, Packs 1, 2 and 3 have square layers. Packs 3, 5 and 6 have rhombic layers. The packs have two types of throats: concave-square, pack 1; or concave-triangular, pack 6.
described in the next section were physically constructed using plastic balls of 20 mm of diameter with four balls per basic layer and 2 or 3 layers [ 16]. Plaster casts of the pore space were made and cut up in different directions to understand their spatial relationships and studied in detail to identify the influence of the internal pore structure on anisotropy.
4.1. Packs: arrangement of spheres In this work the packs with square (90 °) and rhombic (60 ° ) layers are considered which when combined with the three ways of stacking produce the six types of packs studied. A schematic representation of the six packs and their unit pore volume (described later) are shown in Fig. 3. Pack 1 (simple cubic). This pack has successive square layers stacked so that each sphere is above the one in the bottom layer. The planes that pass through the centres of the spheres form a cube because the square layer structure is present in the three orthogonal directions. This pack has the maximum porosity (with the spheres having contact with each other) of 47.64%. Pack 2 (orthorhombic). Here, as in pack 1, the basic structure is the square layer but the successive layers are displaced a distance d/2 in the direction of one of the rows. Thus the angle between successive layers is 60°; i.e., the second layer is in a lower position than for pack 1, The porosity is 39.54%. Pack 3 (rhombohedral or face-centered). The square layer is also present in this pack but the successive layers are located over the central point of the diagonal of the lower layer (point x in Fig. 2 ( a ) ) . Thus the vertical distance between layers results in 0.707 d giving the minimum porosity of the packs with a square layer (25.95%). Pack 4 (orthorhombic). The basic structure in this pack is the rhombic layer and the spheres in the successive layers are
C.A, Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
146
(a)
(b)
Pack I
Pack 2
structure is present in all directions. It is important to notice that if a sphere is in position y, Fig. 2(c), the position z will remain empty (a sphere will not fit in it because it will be surrounded by spheres in rhombic arrangement) which creates two types of path; one interrupted by the second layer and another that passes through the layers (see Fig. 3). This composite nature with the minimum porosity, 25.95%, is of special importance because it has the same porosity of pack 3 but with a different pore structure.
(c)
Pack 3
(d)
4. 2. Packs: pore space
Fig. 4, Normalization of the unit cell for pack 2. (a) Original cell; (b) cut sector to make it perpendicular to the base; (c) normalized cell; (d) norrealized cells for the packs with square layers.
The following description of pore space is based on our unit volume cell for each pack. The basic structure (minimum portion) of a systematic pack gives a complete representation of the pack and the void contained in it (Fig. 3). The unit volume cell is bounded by the planes which intersect the centres of the adjacent spheres, forming a paraUelepiped with a square or rhombic base (Fig. 4 ( a ) ) . The main characteristics of the packs studied are presented in Table 1. In these packs of spheres the porosity is proportional to the distance between layers and the voids present are interconnected so producing a succession of throats and pores. The systems with square layers have a larger porosity than the ones with rhombic layers for the same way of stacking (compare packs 2 and 5). Packs 1 to 3 (square layer) have the same throat shape (Fig. 3) but as the distance between layers
stacked above the ones in the lower layer. This pack has the same solid structure as pack 2 but rotated 90°; i.e., interchanging rows/layers. The porosity is 39.54%. Pack 5 (tetragonal). This pack, like pack 4, has rhombic layers and the row in successive layers are displaced a distance d/2. The layers form an angle of 60 ° and generate a rhombic structure in two planes and a square structure in the third. The distance between layers is 0.866 d as in pack 2. The porosity is 30.19%. Pack 6 (rhombohedral or hexagonal). The rhombic layer is also the base of this pack but the second layer is placed so that the spheres are located above position y (Fig. 2 ( c ) ) , being the distance between layers=0.816 d. The rhombic Table 1 Basic characteristics of the systematic pack of spheres Pack 1
Pack 2
Pack 3
Pack 4
Pack 5
Pack 6
Type of layer
square
square
square
rhombic
rhombic
rhombic
Spacing of layers
d
dX/r~
dr/2
d
dv/~
dv~
Volume of unit cell
1.00 d 3
0.87 d 3
0.71 d 3
0.87 d 3
0.75 d 3
0.71 d 3
Porosity
47.64
39.54
25.95
39.54
30.19
25.95
(%) Table 2 Comparison of different parameters for packs of spheres Packs
PAR
Z
Throat
~b
L
A
F
1'-2"
j, c
1" d
s-s
~,
,L
=
1 '-3'
~,
? f
~ = =
,[, L ,I, $
1"
J, 1" 1"
,[, J, = =
=
4'-5' 2'-2" b 3'-Y'
s-s t-t s-t s-t
= J, ~
1" ,[
5'-5"
=
=
t-t
=
=
=
1"
" ' perpendicular to the layer. b . parallel to the layer. c ~ decrease in the value. 1" increase in the value. • = no change in the value. f ? not comparable.
¢
1"
147
C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
and the porosity decreases the pore structure (path) becomes more tortuous because of its increasing complexity. In packs 4 to 6 the porosity and the distance between layers follow the same tendency but the paths are more complicated. For example, pack 6 has two types of paths in parallel, one passing readily through the layers and the other enclosed by spheres. It must be noticed that the unit cell of pack 4 has the same basic structure and internal distribution as pack 2, but rotated 90 °. The throat planes (which separate adjacent pores) also intercept the centres of adjacent spheres and therefore the throats are located on the faces of the unit volume cell. There are two types of throat: (i) concave-square formed by four adjacent spheres and (ii) concave-triangle formed by three adjacent spheres. For example, the rhombic layers form a double concave-triangular throat connected by the corners and the square layer form a concave-square throat, see Fig. 3. The same throat shapes are produced by the overlaying of layers. Thus all the six packs have a combination of concavetriangular and concave-square throat shapes in different directions. For example pack 2 has concave-square throats perpendicular to the layer plane but concave-triangle throats in the other directions; this is an indication of the anisotropy of this pack. A close inspection of the void space shows that packs 2-6 all have anisotropy in different degrees but the symmetry in all directions of pack 1 makes it unusually isotropic. The pack have a systematic variation but the pore spaces themselves do not have such a systematic variation and therefore the pore space topology and morphology is difficult to catalogue and interpret. The underlying porous media structure (morphology and topology) which are necessary ingredients for the description of the internal void space affects the physical macroscopic processes (permeability, electrical resistivity, etc.). With this knowledge we can now reduce the pore structure to a network to model the process.
it easier for the determination of the equivalent network and the construction of the experimental packs (e.g., the unit cell of packs 1, 2 and 3 will have a similar shape, as shown in Fig. 4 ( d ) ) . The networks reported later are based on the normalized unit cells. The algorithm to reduce the porous space to the equivalent network consists of three main steps, as follows. 5.1. Reduction o f pore space
If a sphere is allowed to grow at each point on the pore wall until the surface of the sphere touches other points on the pore wall then the centres of those spheres form surfaces which run through the pore space called 'stripwork', which is a connected assembly of strips. This stripwork has some general properties. (i) Each point has associated with it the radius of the tangent sphere to the solid and the number of contacts that it has made with the pore walls. (ii) Double contact points define the surface of the strips. Triple contact points define continuous lines (intersection of surfaces) that bound the strips. Quadruple contact points are usually isolated and are the intersection of triple contact points. (iii) The corners on the pore walls and edges with angles less than 180° are part of the outer boundary of the stripwork. A simple example is shown in Fig. 5, where two triangular tubes converge to form a square section; when reduced to its stripwork each triangular tube will be reduced to three fiat strips, Fig. 5(b) and the square section to four equal strips.
5
5. Representation of morphology and topology through an equivalent network Networks are a good way of representing different processes within the pore space [ 17-19], but there is not yet an available algorithm to reduce the pore space to a general network. However, as will be shown, each physical process needs to be considered separately and have its special characteristics recognized. An equivalent network is a representation of the pore space that contains the topologic and relevant morphologic characteristics of a determined porous space. In this work all the unit volume cells were normalized before reducing the void space to a network. This normalization consists in passing four planes (perpendicular to the layers) around the base as shown for pack 2 in Fig. 4 ( a ) (c). This rearrangement of planes and volumes does not alter the structure or the size of the unit volume cell due to the special characteristics of the systematic packs, but it makes
2
bond
4
3
x-x
node
(c) Fig. 5. Reduction of a pore space to an equivalent network. (a) Original pore spaceand reduction to a stripworkin sectionx-x. 1= edge, 2 = double contact points, 3 =triple contact point, 4 =traces of the strip, 5 =pore wall; (b) reduced stripwork; (c) equivalentnetwork.
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C.A. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
The strips meet at lines which are formed by the triple contact points and these lines join at a quadruple contact point (equidistant point to the surfaces where the triangular and square tubes join). The external borders of the strip are the boundaries of the stripwork. The same stripwork could be obtained by growing the solid, normal to its surface at a constant rate, and stopping the solid growth at the points where two nonadjacent pore-wall surfaces meet.
5.2. Primitive network The stripwork is shrunk by moving its boundary points along the shortest possible distance on the curved surface (geodesics orthogonal to the strip). Two possibilities can arise. (i) Two boundary points may meet each other. In this case one of the points disappears and the remaining point is not moved any more, remains fixed and forms part of the primitive (basic) network. (ii) Two or more strips can join at a line (higher contact line). In this case the boundary point may merge on the line and disappear as a boundary (no further track of the boundary point is then kept). The primitive network is obtained when the system cannot be reduced any further. Each point of this network has associated with it several quantities to model the process (such as radius, transverse area, etc.). Some definitions for the primitive network are: nodes: the positions of local maxima of the equivalent radius of the network and branches; throats: the positions of local minima of the network. Each point of the network has associated with itself the value of the pore level descriptors needed for the pore level laws. For example, the primitive network of the pore space discussed in Fig. 5 (a) reduces to three straight branches joining in a node, shown in Fig. 5(c).
5.3. Working network This is obtained from the primitive network but incorporates some additional geometrical or process criteria to take into account or eliminate some details, e.g., dead end pores could be eliminated because they do not play any role in the flow. The equivalent network does not assume any pre-determined characteristics for the bonds. Therefore, the lines connecting two nodes or pore bodies is only a representation of the topology and does not mean that these connections are straight or they have any special cross-section shape. Then the algorithm used to obtain the equivalent network does not induce any anisotropy effects. Each branch or bond of the network has associated with it the main quantities (characteristics) needed to model the process, and they will be described in the next section.
6. Pore level laws and descriptions The processes to be modelled by a mathematical network requires the pore space to be treated as an assemblage of bonds or branches. The transport of material, fluid, electricity through these segments are incorporated by using the pore level laws (mechanisms--equations). The pore level laws determine how fluids, tracers and ions behave during the transport processes and are used to calculate the average (macroscopic) transport properties. A variety of pore structure descriptors are needed for each process and have to be considered during the construction of the working network. The pore level laws include the descriptors needed ; for example
6.1. Permeability If it is assumed (as is often done) that each segment or branch follows the Hagen-Poiseuille equation: A pTl'rh4
q=
8/xl
(1)
where q is the volumetric flow rate, AP is the pressure difference across the segment,/~ is the viscosity of the fluid, l is the length of the segment and rh is the hydraulic radius [ 5 ], which is defined as the minimum cross-section of flow (q) divided by the wetted perimeter (Wp). Then for each segment the length and mean hydraulic radius are used as descriptors to model the permeability. The total permeability of the porous medium is obtained by solving the equations from all the bonds and nodes. The individual bond characteristics have an influence on the total behaviour of the medium.
6.2. Microscopic dispersion Here the concentration changes have to be monitored and the mixing of the streams entering in the pore body need to be considered and therefore the velocities in each pore segment (v =q/a) and the volume of the pore body (al) have to be obtained as pore descriptors.
6.3. Electrical resistivity The law used is based on the definition of the resistivity itself: Rb=Rwl/am
(2)
where Rb is the resistance of the segment or branch, Rw is the resistivity of the fluid and am is the mean transverse area of the segment. Then the pore level descriptors needed are the mean transverse area perpendicular to the segment direction and its length [ 16].
CA. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
(a)
(b)
(c)
Fig. 6. Reductionof the unit volumecellforpack 1. (a) Originalporespace; (b) stripwork; (c) equivalentnetworkoftbe cell.
Fig. 7. Representation of the transverse area m different positions along an
elementof pack 1. '-
Pack I
Pack 4
Pack 2
Pack 5
Pack 3
Pack 6
Fig. 8. Equivalentnetworkforthe six packs.The numberin the intersections representthe coordinationnumberof the porebody.
6.4. Other petrophysical parameters Porosity, surface area per unit volume, pore-throats dimensions and distribution can be obtained as a consequence of the three previous parts (permeability, microscopic dispersion and electrical resistivity). The variations of the pore level descriptors (a, am, l, Wp) for a segment of pack 1 can be inferred from Figs. 6 and 7. The descriptors needed for each segment are scalar quanrifles and therefore the anisotropy will not be generated by them but will appear due to the topology of the pore structure (e.g., coordination number and connectivity of the network) and the potential externally applied (for example: pressure, concentration or electrical voltage). Therefore, the anisotropic characteristics of a porous medium will be reflected in different processes, but the magnitude of the effects of the porous anisotropy will depend on the individual process.
7. Packs: equivalent n e t w o r k
The methodology to obtain an equivalent network and the pore level descriptors needed to model the pore level laws have been described and now the equivalent network for our
149
six packs is described and analysed. Pack 1 will be explained in some detail to give the methodology while for the other packs the final results will be shown. As described earlier, pack 1 consists of square layers of overlying spheres and the pore walls are convex surfaces forming six concave-square throats (Fig. 6 ( a ) ) . The stripwork consists of orthogonal planes of width d, as shown in Fig. 6(b), and its edges are the points where a pair of spheres touches each other. When the stripwork is reduced to the primitive network the points (edges of the stripwork) grow forming geodesic circles, Fig. 6(b), and due to the orthogonal position of the spheres the circles meet each other at orthogonal lines (normal to each other at one point, as shown in Fig. 6(c) ). Thus the equivalent network is a repetitive (systematic) structure with coordination number (Z) equal to six. Pores are located where the six branches intersect and the throats (and throat planes) are at half length between pores. The equivalent dimensions of the pores and throats are 0.732 d and 0.414 d respectively, where d is the diameter of the solid sphere. From this the ratio of pore and throat transverse areas (PAR) is 3.12 and the ratio of the maximum and minimum length in the throat plane is 2.41 (Fig. 7). Fig. 8 shows the equivalent networks for the six packs. The topology and structure of the unit void and its equivalent network can show the isotropic or anisotropic nature of the packs and give an indication of the directional behaviour of these packs. The following analysis is based on the void space and the networks of Figs. 3 and 8 respectively. Pack 1 will show an isotropic behaviour because it is symmetric and has only one type of path which is the same in each orthogonal direction. Each node has two symmetric connections with respect to the pore on each orthogonal plane, so producing a coordination number equal to six. Packs 2 and 4 will show similar anisotropy behaviour since they have the same morphology and topology (rotated 90 ° ) having two types of path. The coordination number is equal to seven, with five branches on one plane and the other two perpendicular to it. Therefore, these two packs will be anisotropic in two of the three main directions. Packs 3, 5 and 6 also show anisotropy behaviour. They have very complex topologies with two types of pores in each void cell: one of large size and highly interconnected ( Z = 7 or 8) and another smaller and less interconnected (Z= 3, 6, and 4 respectively). Additionally pack 6 also has two types of path perpendicular to the layer plane, one directly connected to the upper and lower layers and the other restricted within the layer. This suggests that due to the anisotropic structure of the pore space any process considered will be dependent on the direction of measurement. This analysis shows that microscopic anisotropy can occur due to the internal pore structure of the porous media (mainly topology), even though the packs themselves are homogeneous on a bulk scale. For example, different degrees of electrical resistivity anisotropy [4] were experimentally measured in these packs demonstrating that anisotropy can occur even for spherical particles with the same size. Finally,
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CA. Grattoni, R.A. Dawe / Powder Technology 85 (1995) 143-151
the effect of the coordination number on anisotropy is not always clear since there are some parameters that cannot be changed independently such as mean transverse area and length, and their relationship in natural porous media still needs to be identified.
these techniques should bring a better understanding of the influence of pore scale structure upon on different processes; i.e., fluid dispersion, permeabilities, capillary pressures and electrical resistivity.
9. List of symbols 8. Discussion a
The pore spaces of six systematic packs of spheres have been analysed to assess the effects of pore structure on anisotropy of porous media. Their pore space have been reduced to an equivalent network that considers all the relevant parameters needed to model their behaviour: coordination number, morphology, topology, length of connections and mean equivalent area (which includes pore-to-throat aspect ratio, pore and throat shape variations and throat dimensions). The systematic pack of spheres are simple porous media but even they give complex pore spaces. The relationships between pore space structure and the processes occurring in natural or artificial porous media can be modelled by using equivalent networks if they include the main internal dimensions of the porous medium. This can help in the understanding of the basic process physics. The packs analysed have shown the existence of microscopic anisotropy (due to the internal pore structure) even if the medium is macroscopically homogeneous and is composed of uniform size spherical particles. It has been shown that reducing the pore space (at a microscopic scale) to an equivalent network is a valid and useful technique for modelling the porous system behaviour. Irregular packs of spheres of the same diameter are a combination of the packs described in this paper and the macroscopic properties could be obtained from the networks of Fig. 8. For example, using the unit volume cells presented in Fig. 4 an irregular pack can be built as follows: rotating pack 3 ( 180° in the vertical direction) stacking pack 1 over it and pack 2 over pack 1. Therefore the macroscopic properties of this compound pack can be easily calculated from the networks, pore level laws and descriptors presented in this paper. The methodology presented can also be applied to more complex porous media (random packs with grains of different sizes) providing that the shape of the grains is known and the positions of the grains can be inferred. For example, by using the grain consolidation model [6] where spheres of different diameters are randomly packed, then if the spheres are allowed to grow evenly in all directions, or preferentially in one direction, and the growing process is stopped when a desired porosity is reached, depending on the conditions imposed, different natural and artificial porous media can be simulated. In a similar way other grain shapes can be used to represent other pore spaces. Once the matrix structure is defined the methodology presented in this paper can be readily applied. Such applications to natural porous medium (e.g. rocks) will be feasible in the near future when 3-D pore space reconstruction, see Ref. [20], using computer image analysis and thin sections, becomes more available. The merging of
am
A d F l L AP
PAR q rh
Rb Ro Rw s t u
% Z
minimum cross section of flow mean transverse area of the segment or branch total transverse area of the unit cell diameter of the sphere formation factor, Ro/Rw length of the segment or branch total length of the unit cell pressure difference across the segment or branch ratio of the pore and throat transverse areas volumetric flow rate hydraulic radius, a/Wp resistance of a segment or branch resistivity of the network resistivity of the fluid square throat triangular throat velocity in a pore segment or branch, q/a wetted perimeter coordination number, number of branches converging to a pore
Greek letters /x ~b
viscosity of the fluid porosity
Acknowledgements We would like to thank Deminex UK Oil and Gas Ltd., the EU under the Joule II programme and PSTI for financial support.
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