Pageoph, Vol. 114 (1976), Birkh[iuser Verlag, Basel

Pore Structure and Physical Properties of Porous Sedimentary Rocks BY M. RINK and J. R. S c n o e P ~ 1)

Summary - Authors' work of recent times on sedimentary petrophysics is reviewed. A report is given on theoretical work about petrophysical relations, based on a versatile model of the pore structure, using statistics and topology; on means for directly measuring pore structure by computerized image analysis of sections; and on an experimental study about an electrical interface conductivity of porous, water saturated rocks and its relations to structure. Some outlook on a subject of very recent interest: recovery of geothermal heat from porous formations, is added.

Introduction

A profound understanding of the petrophysical laws of sediments, rather than field experience only, is vital for the further advancement of applied geosciences and geotechnology. It is needed for Iinking useful quantities that are hard to determine in situ to more easily measurable ones; or, for a safe extrapolation of laboratory measurements to in-situ conditions; also, for general predictions about subsurface processes, requiring an abstraction from particular geological conditions; or, for a separation of truly physical relations between rock material quantities, from - what might be called - geological relations, apparent dependencies that are just caused by common trends during diagenesis: e.g. the effect of consolidation on porosity and tortuosity leading to Archie's equations. The basic influence on physical properties of, and processes in, sediments however is exerted by their pore structure, and microgeometrical parameters are essentially the links between all the macropetrophysical parameters. But how can such a complex geometry as that of the pore space of rocks be approached mathematically ? The only means for a sufficiently accurate theoretical treatment is statistics in combination with topology, using models that are accessible to such mathematical methods on the one hand, and on the other hand describe reality with sufficient fidelity. The necessary effort naturally depends on the scope of physical properties and processes to be incorporated into the model treatment, and on the wanted accuracy of results. But the danger in using simple models is always that of expanding them 1) Institut fOx Geophysik der TU Clausthal D-3392 Clausthal-Zellerfeld, Federal Republic of Germany.

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M. Rink a n d Z R. S e h o p p e r

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beyond their range of validity, and this fallacy is the reason for so many misunderstandings and unsuccessful theoretical treatments in the past.

Model theory

Consider a few physical phenomena happening in the pore space and their dependence on pore structure: capillary forces - electrical conduction in a conductive pore fluid - viscous fluid flow - or Knudsen gas flow (Table 1). The influence of the pore structure shall be shown iteratively, starting out from the simple model of a cylindrical tube and generalizing it then successively. Thereby the validity range of different stages of models will be recognized. Capillary forces, in the simpIe case of a cylindrical tube, are controlled by the inverse radius r, or the inverse square root of the cross-section a. This can be also expressed by the ratio of circumference c to cross-section area a, and this is really the physically fundamental expression, since it stands for the ratio of surface to volume forces. In the more general case of a prismatic tube with an arbitrarily irregular shaped cross-section, we can introduce an equivalent cross-section a', so as to retain the same formalism as before, a' can be linked to a by a shape factor k', which then reads (1)

k" = 4~ra/c 2

which is the inverse square of the so-called' Heywood Factor.' Electrical conduction is Table 1 Effective geometrical factors Capillary Forces m o s t simply

m o r e generally

Electr. Conduction

2

e

r =

= a

a =

=

a?

Viscous Flow m o s t simply

~rr4 a2 81 = 87rl

m o r e generally

aa" k" a 2 ~ = 8~rl

Knudsen Flow m o s t simply

m o r e generally

l

l

a k" = 41r-~

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Pore Structure and Physical Properties of Porous Sedimentary Rocks

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controlled simply by the ratio of the true cross-section to tube length, in any case. No shape factor at all is involved here. Laminar flow through a cylindrical tube is controlled by the fourth power of the radius, respectively the square of the cross-section area, divided by the tube length/, according to the Hagen-Poiseuille Law. In the more general case o f a non-circular cross-section, again a shape factor k " enters. But this is not identical with the one k' above. For the flow of gases in the Knudsen range, when the pore dimensions are in the order of magnitude of the mean free path, or smaller, still another power of the crosssection, V ' ~ , and still another shape factor, k " , are effective. Clearly, one can already see some problems of linking physical rock properties to each other by a model theory of the pore structure. One such problem is posed by the differences in the shape factors. With using the usual approximation (3) (2)(1)

k " ~, k " ~ k ' = 4rra/c 2

one should always be aware of making an unknown error. Another problem is the following: we do not have a single channel only, nor have we identical channels in a porous rock. We have a size distribution of crosssection, length and the various shape factors. Instead of determinate quantities, we thus have to use statistical measures in the corresponding laws (Table 1). By applying a binomial or Taylor development, a mean of any power of a statistical variable x can be expressed by the corresponding power of its arithmetic mean and a generally infinite series of moments about that mean:

k=O

(4)

~'~

with iz,~ = M [ ( x

-

ff)~]

~---~= M

- 1

.

(5) (6)

With a polynomial expression, this could even be extended to other functions: M[f(x)]

= M

a , x '~ = Ln=O

J

a , 2 '~ n=O

tz--5~.

(7)

k=O

This way, commensurate parameters can be gained for relating petrophysical phenomena depending in different ways on the pore structure. The question is where to cut off the series. In view of the complex pore structure of consolidated rocks, a fourth-order statistical theory would seem appropriate. However, the mathematical effort practically limits it to the second order. An iteration for the use of such a simplified theory, even in cases of wide and awkward distributions, has been indicated previously [ScHoPPER, 1972].

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The next problem is: with many phenomena, the function to be averaged statistically depends on the way the individual channels are connected to each other: in parallel, in series, or what else. All flow phenomena e.g. follow Kirchhoff's Laws. Either the arithmetic or the harmonic mean has to be used, depending on whether a series circuit or a parallel circuit, and whether a resistance type or a conductance type property is considered. But in a porous rock one neither has a parallel circuit nor a series circuit; but rather a random maze of channels. Thus a three-dimensional channel network model is the only type model truly representative of the pore space of a porous medium. Here is where topology enters the scene. A topological description of a network can be furnished by stating the numbers of branches joining in each node; by the minimum numbers of branches or nodes surrounding each mesh; or by the faces, edges or corners of polyhedral cells formed by the spatial network structure. But since we never know the complete topological structure of the pore space, we can operate with such numbers again in a statistical way only. Thus, a statistical network theory that is quite different from the conventional determinate network theory of engineering is necessary for relating integral physical properties of the network to the corresponding individual physical properties of the branches. Details of a statistical network theory developed for this purpose have been published before [ScHOPPER, 1964, 1966, 1966a, 1967, 1967a, 1972, 1973; RINK and SCHOPI'ER, 1968]. Just its major results, that have come out with unexpected simplicity, are stated here. One single constant describes the influence of the whole complex network topology in a second-order statistical theory. This network constant e is given by integral topological measures, namely the overall number of nodes K divided by the overall number of branches N. 8 =

1 -

K

.~.

(8)

It is a sort of normalized connectivity. Taking large N, e is zero for a series circuit and one for a parallel circuit. Its range in porous rocks is about 89to 88 This constant always appears as a weight factor in conjunction with variance terms in the statistical theory, as in the following equation for a random resistance network: R = ~R0(1 -

es~R)).

(9)

R is some two-terminal resistance of the network; Ro the mean of the branch resistances, and s(~) their variance divided by the square of the mean. ~ is just a scale factor depending on the network shape and the terminal position. We can recognize here an equivalence between connectivity and variance. As long as one single physical property only is concerned, a network model can always be

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transformed into a simpler model, e.g. a bundle model (~ = 1), with a different size distribution, such that the product 8s2 remains constant. The effect of the network constant and mean and variance of geometrical channel parameters on interrelations between different petrophysical properties can be seen, for example, in an equation for the product of formation factor and permeability:

FK=--ff~-k~a~[1 -

(2 - 3,)s~) - (1 -

e)s~,,, -

2s~,) - s~k,,,)].

(I0)

F is the formation resistivity factor used in electrical logging, while K is the hydraulic permeability, the rock material constant in Darcy's Law. This product essentially is the ratio of electrical to hydraulic resistance. On the right hand side we find metric and topological pore structure parameters only: the mean shape factor ko, the mean cross-section a0, the network constant e and the normalized variances s~) and s~k,,)of cross-section and shape. This formula also includes the last step of necessary refinement of the model, reflected by the normalized variances s~**)and s~,,,) describing internal variations of cross-section and shape along the individual channels. Such corrugation of the channels, modelling the pore bulbs and pore necks in rocks, must be included to make the model also respond properly to such phenomena as capillary hysteresis and multiphase flow. What can one do now with such a sophisticated theory? Can one get at those microgeometrical parameters in situ? Of course not! In this respect, they are just auxiliaries for theoretically finding new petrophysical laws. Eventually one should have a system of equations from which they can be completely eliminated. Nevertheless, it would be worthwhile to determine those geometrical microparameters directly by laboratory measurements as another source of petrophysical information from rock samples. This idea is pursued now.

Figure I Polished section of a sandstone saturated by Wood's Metal. Pore space, light; matrix, dark.

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M. Rink and J, R. Schopper

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C o m p u t e r i z e d image analysis o f sections

The only way the internal rock structure can be viewed directly is on sections. Figure 1 shows such a view. Since statistical information is needed, microscopic eyeview measurements would be too tedious and automatic image analysis with electronic data processing must be employed. Therefore, a high-contrast, dual-brightness image is needed, wherein just pore space and matrix are discerned as in Fig. 1. The most meaningful image is obtained from polished sections by incident light, since thin sections in transmittant light provide a projection rather than a true section. As the best of several procedures for the necessary preparation of the pore space, high-pressure injection of liquid Wood's metal, and afterwards extreme polishing, has proven. A computer program, RINK [1970, 1973, 1973a, 1974] developed for the automatic image analysis, is able to recognize individual section figures of either the light or the dark component, and to count them and calculate a host of size and shape parameters of each individual figure, so as to describe the geometry of every such figure most completely. A selection of such parameters is listed in Table 2. Table 2 Morphometric parameters determined by automatic image analysis

Symbol

Meaning

a

area

dw = 2 J a

diameter after WADELL

C

circumference hydraulic radius shape factor after HEYWOOD shape factor height "~tangent diameter f $ LO width J after FERET

m = ale k~ = e/2~/~a k" = 4~ra/c2

h w

)t~

pr~176

(L

kc~= P q h ++ w

concavity factor

s .... ~

max. intercept length (KRuMBEIN)

sin, = alq

l dm= all k, = l/d=

mean intercept length +-~ approx, longest extension mean intercept length l l elongation factor

rl)r2 = 88 + C c 2 - 16a) sides of equiv, rectangle (MooaE)

line direction

$ column direction

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Pore Structure and Physical Properties of Porous Sedimentary Rocks

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Individual geometrical analysis of each section figure is very necessary, and the integral picture-frame data of the conventional image analyzers will not give the required information, since pore channel statistics must be done. This statistical analysis of the gained image data is done immediately by the same program, and statistical information about every investigated geometrical parameter printed out: mean, variance, skewness, kyrtosis, complete frequency distribution, etc. Since in a section image (Fig. 1) usually too many pore sections appear connected to each other, an image process can be included in the program before the analysis, that automatically cuts all narrow bridges between larger figures, beyond a pre-set limit. Image print-outs in Fig. 2 show the cutting process. A good separation has been obtained, while the deformation of the remaining figures is kept to a minimum. A fundamental problem in the evaluation of section images lies in the thct that one can get two-dimensional information only. However, since the pore structure is more or less a random structure in space, statistical information from a random section is principally representative of the space, according to the principles of stereology, although the proper transformations for the purposes here are not yet available.

...!I!......i..................!i....i....................... .............

~x~; i!~z ~

~iii ii~i!i~ ~ ~ii~

J~

~i~i~ii~

~

~iii!i~~ii~ii~~i~ ~i! ~

..... ~ili

Fi~'iil;,!I~~ ~..~.0~

i iiiii'~'~0~ , ~

~

~

xx : ~

i:i

9...... i~......... ~r!I ~ i~

~

~!ii

il

Figure 2 Parts of image print-outs showing the 'cut process' (a) original image; (b) 'dear cut' image.

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Nevertheless, quite rough approximations have already furnished satisfying results: hydraulic permeability and mean capillarity could be determined by image analysis within the order of magnitude [RrNIr 1973, 1974].

Experiments on interface conductivity

In this section a report is presented on some other work that is a good example of mutual stimulation of field work, laboratory experiments and petrophysical theory of sediments: the observation and investigation of an interface conductivity in electrolytesaturated porous rock, that exists in addition to the electrolytic conductivity inside the pore fluid. Field experience has shown a proportionality between bulk rock resistivity (Po) and pore fluid resistivity (Pw) for many sedimentary rock materials and many natural pore waters. Po = Fpw.

(11)

The constant of proportionality F, commonly called formation factor, is then a pure rock material constant. However, in many other cases, namely in shaley rocks or with fresh pore waters, large deviations from such a proportionality have been observed. Already the early geophysicists, working in geoelectrics, knew empirically of telling fresh-water sands from gravels by their resistivity, although such a grain size dependence later-on has been disputed. For a closer study on the conductivity of porous rocks, the authors investigated about 100 samples from clean and shaley sandstones and some artificial materials; with NaC1, KC1, MgCt2 and CaCI2 solutions at 9 different concentrations each, from 6 ppm to 60,000 ppm, logarithmically subdivided [RINK and SCI-IOPPER, 1973, 1974]. They measured bulk rock conductivity •0 versus fluid conductivity Kw, and found qualitatively the same behavior of all materials, regardless of shaliness. A selection of typical results is given in Fig. 3. While the curves in this double logarithmic graph approach a line of proportionality (dashed) at the high conductivity end, a constant value Kqof the rock conductivity, independent of the fluid conductivity, is approached at the low conductivity end. The results can be described by a linear inhomogeneous conductivity equation 1 K0 = ~ Kw + ,ca

(12)

where F and Kq are material constants. Thus Kq must be a constant conductivity shunting the pore fluid conductivity. But it always disappears in dry samples. Therefore, it is neither a fluid property nor a matrix property. It must be due to an interaction of fluid and matrix. That means a

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Pore Structure and Physical Properties of Porous Sedimentary Rocks

281

10 ~ )~o[eSc,.-V

1 103I

'

o B 49[2

F= 48

14.q= 150 tuScm-~

e B 53]3 = B 66[2

F= 17 F= 33

§ X 1/2

"F= 6

X q = 23 ktScrn-' "K.q = 16 ~Scm-' 34.q = 75 b~Scrtr'~

"!

102

_

10 7 10~

,-

.o

/~// /

/

,,

,,/ ,,,/,,//

/

a

/,,~

2~

.--------~

i 1 /

/

102

103

X., [t~Sc~-11 i

10 ~

10 5

Figure 3 Bulk rock conductivity ~0 vs. fluid conductivity Kw, with constant interface conductivity Kq and constant true formation factor F. Sample B49/2: shaly sandstone, F = 48, Kq = 150 FScm -1. Sample B53/3: clean sandstone, F = 17, ~cq = 23~Scm -1. Sample B66/2: clean sandstone, F = 33, ~cq = 16 t~Scm-1. Sample X 1/2: fritted glass, F = 6, xq = 75 t~Scm-z. double-layer conductivity at their interface, which should be expected p r o p o r t i o n a l to the specific internal pore space surface a n d some negative power of the f o r m a t i o n factor, according to model theory. I n Fig. 4, a correlation plot on this relation is shown, b u t since there were n o reliable direct data o n the specific surface available,

I

IF

FSK >{2= c o n s t

10 2

9

:

..'........

10

FK ){2 10~

105

tO6

107

10 8

Figure 4 Cross plot of the product FKK~,or the squared ratio of interface conductivity by specific pore space surface, vs. the true formation factor F.

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M. Rink and J. R. Schopper

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it had to be replaced by other measured quantities via model theory, namely the product of formation factor F and permeability K. If in equation (10), the statistical correction in the brackets is neglected and also approximate equality between the mean shape factor k~ and the one valid for the capillary forces, ko is assumed, then one ends up with about half the inverse square of the capillarity C, which in turn approximately equals the specific pore space surface S, according to model theory: k~ao k'oao 4~ra~ FK ,~ ~ ,~ 8~r -- 8~r C~o"~ 89 2 ,.~ 89

(13) (14) (15) (16) (17)

with k~) = 4rr-~

Co

and

Co ~ co z Spot. a0

(18) (19) (20)

In good agreement with theory, Fig. 4 yields F~K~c~ = const

(21)

despite the approximations introduced into the theory and despite the scattering in the graph, resulting from it. Equation (21) has found considerabIe interest in well logging, for avoiding evaluation errors and possibly reading permeability from electric logs.

Outlook on the recovery of geothermal heat from porous formations At the closing of this paper a few words might be added on a subject of very recent interest. In the wake of the energy crisis, the vast amount of heat contained in porous formations, even at normal geothermal gradients, and its recovery by water circulations has come into discussion. But a lot of counteracting effects of the pore structure must be considered here with respect to heat exchange, water flow and heat transport by convection. Some first calculations have shown promising results for formations of intergranular porosity. The heat AQ that can be drawn from a volume V of porous water saturated rock is given for a usable temperature difference AT by AQ = V[r

+ (1 - r

(22)

with r being the porosity, Cw and cmtx the specific heat of water and rock matrix respectively, and Sw and 8mt~ the respective densities. A common sandstone of 20~ porosity thus would yield about 150 MWs/m z of thermal energy for a usable temperatrue drop of 50~ This means a heat production of about 150 MW for 30 years from one cubic kilometer; say, a sheet of 10 km • l0 k m • 10 m, or a zone of 2 km radius and 100 m pay thickness.

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Pore Structure and PhysicalProperties of Porous SedimentaryRocks

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In the latter case, a water flow of 0.75 m3/s from a central production well would be required. Assuming flow toward the well under natural hydrostatic head only and applying Darcy's Law, a required formation permeability of about 600 md in a depth of 2 km, 400 md at 3 km or 300 md at 4 km would result (that could be met under favorable conditions), and the respective lifting energy at the well would be 10~, 15~ or 20~o of the produced heat energy (again 50~ usable drop assumed). The heat exchange between rock matrix and percolating water must be fast, compared with the removal of heat by the streaming water, to ensure a steep sweep front and thus a good sweep efficiency. From a calculation of heat flow from heated balls toward a constant temperature environment [CARLSLAWand JAEGER,1973], approximately follows:

~/2~ ,.~ S/(1 - 4)

, = ~

(23) (24)

with a being the heat exchange constant, ,~ the heat conductivity of the matrix, S the bulk specific surface and r the time constant for the heat exchange between matrix and pore water. An estimate shows that heat exchange is sufficiently fast in rocks of intergranular permeability in the above mentioned range. However, it might not be so in many fissured and fractured, or cavernous rocks. Thus the thorough study of the dependencies of physical properties of porous sedimentary rocks on their pore structure is vital also for the solution of such new problems of technical interest, and a basic prerequisite for decisions on the feasibility of projects of geothermal heat recovery from porous rocks under given conditions of particular formations.

Conclusions The microstructure of rocks has a strong influence on their physical properties, often overriding that of the material properties of matrix and interstitial fluid by far. In basic theoretical work, closely fitting structural models are mandatory, and unknown errors can be caused by improper simplifications. But, once a working general model theory is established, effort often can be reduced in special cases, by adjusting the model to the particular problems and the desired quality of the results. Statistical image analysis of section micrographs can provide valuable geometrical data on the microstructure. An intimate interlacing of field experience, physical laboratory experiments, microgeometrical analysis, and a sound model theory is always recommended for studying rock property interrelations. The reviewed work is expected to have much bearing also in studying the hydraulic effects of earthquake precursory phenomena.

284

M. Rink and J. R. Sehopper Acknowledgement

Substantial s u p p o r t o f this w o r k b y D e u t s c h e Forschungsgemeinschaft a n d friendly help from Oil C o m p a n i e s is gratefully acknowledged. M a n y students cont r i b u t e d to the results.

I~.FEI~NCE$ H. S. CARLSLAWand J. C. JAEGER(1973), Conduction of heat in solids, Clarendon Press, Oxford, 1973. M. RaNK (1970), Automatische morphometrische Bildanalyse mit Hilfe eines elektronisehen Digitalrechners, Diss., Clausthal, 1970. M. RaNK (1973), Porengeometrische Untersuchungen an Sedimentgesteinen mit einem Bildanalyseverfahren fiir Digitalrechner, Z. Geophysik 39 (1973), 989. M. RaNK (1973a), An image analysis procedure for digital computers and its application to investigations on pore structures, Proc. RILEM-IUPAC Internat. Syrup. ' Pore Structure and Properties of Materials,' Prague, 1973. Final Report, Part II, C-497. M. RaNK (1974), Untersuchungen am Porenraum yon Sedimentgestein mit bildanalytischen Methoden im Digitalrechner, Z. Prakt. Metallographie, Sonderband 5 'Quantitative Analysen yon Gef[igen in Medizin, Biologie und Materialentwicklung'; Symp. Leoben 1974. M. RINK and J. R. SCrIOPPER (1968), Computations of network models of porous media, Geophys. Prospect. 16 (1968), 277. M. RaN~ and J. R. SCHOPPER(1973), Interface conductivity of liquid-saturated porous media and its relation to structure, Proc. RILEM-IUPAC Internat. Symp. 'Pore Structure and Properties of Materials,' Prague, 1973. Final Report, Part II, C-311. M. RINK and J. R. SCrIOPPER(1974), Interface conductivity and its implications to electric logging, 15th Annual Logging Syrup. Trans. SPWLA, 1974; and 3rd European Logging Symp. Trans. SPWLA, London, 1974. J. R. Scr~oPPER(1964) Untersuchungen fiberZusammenhdnge zwischen elektrischen und hydraulischen Eigenschaften por6ser Gesteine, Diss., Mainz, 1964. J. R. SCnOPI"ER (1966), A theoretical investigation on the formation factor/permeability/porosity relationship using a network model, Geophys. Prospect. 14 (1966), 301. J. R. SCHOPPER (1966a), Untersuchungen iiber elektrische und hydraulische Eigenschaften por6ser Gesteine, Z. Geophys. 32 (1966), Sonderheft, 525. J. R. SCHOPPER(1967), A theoretical study on the reduction of statistical pore system parameters to measurable quantities, Geophys. Prospect. 15 (1967), 261. J. R. SCHOPPER(1967a), Experimentelle Methoden und eine .4pparatur zur Untersuchung der Beziehungen zwischen hydraulischen und elektrischen Eigenschaften loser und kanstlich verfestigter por6ser Medien, Geophys. Prospect. 15 (1967), 651. J. R. SCnOPPER (1972), Theoretische Untersuehung elektrischer, hydraulischer and anderer physikalischer Eigenschaften poriiser Gesteine mit Hilfe statistischer Netzwerke. Habilitationsschrift, Clausthal, 1972. J. R. SCrtOPPER (1973), A statistical network model and theory of porous media. Proc. RILEMIUPAC Internat. Syrup. 'Pore Structure and Properties of Materials,' Prague, 1973. Preliminary Report, Part I, A-533. Further references ibidem. (Received 8th February 1975)