EFFECTIVE STRESS LAW FOR THE PERMEABILITY OF CLAY-RICH SANDSTONES

SCA22-5 1/6 EFFECTIVE STRESS LAW FOR THE PERMEABILITY OF CLAY-RICH SANDSTONES Widad Al-Wardy and Robert W. Zimmerman Department of Earth Siene and Engineering Imperial College of Siene, Tehnology and Mediine London, UK INTRODUCTION Petrophysial properties of porous sedimentary roks depend on both the onfining pressure, P, and the pore pressure, P p. For example, if hysteresis is negleted, a property suh as the permeability k an be expressed as some funtio = f ( P, Pp ). If the permeability an be expressed as a funtion of the single parameter P P p, i.e., k = f ( P nk Pp ), we then say that it follows an effetive stress law, with being the effetive stress oeffiient, and P nk Pp being the effetive stress. For a rok whose mineral phase onsists of a single mineral, say quartz, the effetive stress oeffiient is not expeted to exeed unity [1]. However, Zobak and Byerlee [2] showed that the effetive stress oeffiient of some lay-rih sandstones an in fat be as high as 3-4. Walls and Nur [3] found that inreased with lay fration, and reahed values as high as 7 for sandstones with volumetri lay frations of 2%. To explain this behavior, Zobak and Byerlee proposed a model in whih the rok onsists of quartz, permeated with ylindrial pores that are lined with a shell-like layer of lay. As the inner lay layer is more ompliant than the outer quartz layer, suh a rok should be more sensitive to hanges in pore pressure than to hanges in onfining pressure. Although this model has frequently been invoked qualitatively, a quantitative disussion of this model does not seem to have been given. A related model is one in whih the lay is in the form of partiles that are only tangentially attahed to the pore walls. In this paper, we utilize newly-developed solutions of the equations of elastiity for these models, along with previously available solutions for visous fluid flow in these geometries, to find the effetive stress oeffiients. Using reently olleted data on the elasti deformation of lays [4], whih show lays to be about twenty times more ompliant than quartz, we find that both models do indeed yield effetive stress oeffiients that inrease with inreasing lay ontent. The seond model gives higher values of, whih are in somewhat loser agreement with those found in the literature. REVIEW OF RESULTS FOR CLAY-FREE ROCKS Before desribing the two models used in our study, it is worth reviewing the results for a rok onsisting of a single mineral. If the pores are assumed to be ylinders of radius a, flow through eah pore will be governed by Poiseuille's law [5], whih states that the hydrauli ondutane of the tube is proportional to a 4. Other fators will influene the

SCA22-5 2/6 overall permeability, suh as the interonnetedness of the pores, but these are assumed not to vary with stress. Hene, to find the dependene of permeability on stress, we need only find the variation of the pore radius a with stress, as shown more rigorously below. It follows from the expressio = f ( P nk Pp ) that the effetive stress oeffiient for permeability,, an be defined as the ratio of the sensitivity of permeability to hanges in pore pressure, to the sensitivity of the permeability to hanges in onfining pressure: ( dk / dpp ) dp = =. (1) ( dk / dp ) dpp= If k depends on the stresses only through the pore radius a, use of the hain rule gives n k ( dk / dp ) ( dk / da)( da / dp ) ( da / dp ) = = = ( dk / dp ) ( dk / da)( da / dp ) ( da / dp ) p dp= p dp= p dp= dpp= dpp= dpp=. (2) Hene, aording to this model, the effetive stress oeffiient for k is essentially the same as that for a (and, for the pore volume). If the elastiity equations are solved for a ylindrial pore in an elasti body, it is found that the effetive stress oeffiient depends on the porosity φ and the Poisson ratio ν of the medium, but never exeeds unity [5,6]. For simpliity, and with little loss of generality, it is onvenient to assume a typial value of ν, suh as.25, in whih ase nk = (2 + φ) /3. (3) If the pores were assumed to be elliptial rather than ylindrial, the effetive stress oeffiient would inrease, approahing unity in the limit of thin rak-like pores, but never exeeding it [5,6]. Hene, it seems that values of >1 should only be expeted to our in a sandstone if it ontains lay. Indeed, the data olleted from various soures by Kwon et al. [7] shows that inreases almost linearly with lay ontent, reahing values as high as 7 when the lay ontent is 2%. DESCRIPTION OF MODELS FOR CLAY-RICH SANDSTONES To explain the results mentioned above in a quantitative way, two different pore-lay models are examined here. The first model is the Zobak and Byerlee shell model, in whih the rok onsists mainly of a single mineral, say quartz, permeated with ylindrial pores that are lined with shell-like layer of lay (Fig. 1a). In the seond model, the rok again onsists mainly of quartz permeated with ylindrial pores, but with the lay situated as partiles that are touhing, but only weakly oupled to, the rok matrix (Fig. 1b). These two somewhat idealized models may be expeted to represent extreme ases with regards to the extent of oupling between the lay and rok.

SCA22-5 3/6 Model 1 In this model, the lay is equally distributed over all of the pore walls, forming a thin layer (Fig. 1a). As in the lay-free ase, the permeability will depend only on the radius of the pore tube, a, and the effetive stress oeffiient will be given by the ratio shown in equation (2). The dependene of a on the two applied stresses will of ourse be different in this ase. If a omposite elasti medium suh as this is subjeted to a onfining pressure P along its outer boundary r = b, and a pore pressure P p along its inner boundary r = a, the radial displaement u will be a funtion only of the radius, r. The displaement will have the form u = Ar + B/ r, where A and B are onstants [8]. Different values of A and B will apply in the lay region, a < r <, and in the rok region, < r < b. The values of these four onstants are found by applying the following boundary onditions: τ ( r = a) = P, τ ( r = b) = P, (4) rr p rr u and τ rr ontinuous at r =, (5) where τ rr is the radial normal stress. This stress is related to the displaement through 4 du 2 u τ rr = ( K + G) + ( K G), (6) 3 dr 3 r where K and G are the bulk and shear moduli. These four onditions allow the four onstants, { Arok, Brok, Alay, Blay}, to be found in terms of the four elasti moduli, the porosity, the lay fration, and the two applied pressures. The resulting expressions are lengthy, and the algebrai details an be found in [9]. Model 2 In this model the lay exists in the form of partiles that are tangentially onneted to the pore walls. In this onfiguration, the lays will have essentially no influene on the effet that the onfining pressure has on the pore geometry. An inrease in onfining stress will ause the pore hannel to deform in the same manner as if the lay were not present, and furthermore will have essentially no influene on the geometry of the lay partiles. The pore pressure will ause the pore wall at r = a to expand radially, exatly as in the layfree ase. But the pore pressure, whih ats over essentially the entire outer boundary of the lay partile, will also ause a uniform hydrostati ompression of the lay partile. Hene, this model does not require the solution to any new elastiity problems. In this model, the permeability will depend on the geometry of the region of the pore that is not oupied by the lay partiles. Hene, we need a solution for the visous flow problem within the open pore spae. In order to yield a tratable, two-dimensional flow problem, we assume that the lay exists as a solid ylinder of radius, touhing the pore wall. (Alternative models ould inlude, for example, spherial lay partiles attahed to the pore walls at random loations.) The region available for fluid flow is then the region

SCA22-5 4/6 between two eentri ylinders, of radii a and, in the limiting ase in whih the inner ylinder is touhing the outer one. Formally, the derivatives appearing in equation (1) must then be alulated as dk dp p = dk da da dp p + dk d d dp p, (7) and similarly for the onfining pressure. The partial derivatives of a and with respet to pressure are already known, as explained in the previous paragraph. The derivatives of k with respet to a and are found by differentiating the solution to the visous flow problem, whih is given in [1] in the form of a ompliated infinite series. After making several simplifiations and approximations to this solution, the details of whih an be found in [9], we eventually arrive at the following expression for the effetive stress oeffiient for model 2: n k φ + 2 (1 φ) γ = +, = (1 φ) F 3 1(.56 ), (8) where φ is the porosity of the rok, F is the lay fration (defined here as the volume of lay divided by the total volume of solids), and γ = G / G is the stiffness ratio. RESULTS AND DISCUSSION Model 1 The effetive stress oeffiient predited by model 1 is plotted in Figure 2a as a funtion of lay fration. The porosity is taken to be 2%, and, the Poisson's ratio of both the rok and the lay are taken to be.25. The different urves represent different values of the stiffness ratio. At zero lay ontent, all urves begin at the value.733, given by equation (3). In the limiting ase on whih the stiffness ratio is 1, i.e., the lay and rok have the same elasti properties, then the system is equivalent to a uniform rok without lay, and the effetive stress oeffiient is onsequently insensitive to lay fration. For higher stiffness ratios, the effetive stress oeffiient inreases with lay ontent, at a rate that inreases with inreasing stiffness ratio. However, no ombination of parameters seem to be apable of yielding values of that are greater than about, say, 3 or 4. Model 2 Again, a rok with 2% porosity is assumed, and the effetive stress oeffiient is plotted as a funtion of lay fration, F, at different stiffness ratios, γ (Figure 2b). The results are qualitatively similar as those of model 1, in that the oeffiient inreases as lay fration inreases, with the effet more enhaned for higher values of the stiffness ratio. But the numerial values of are larger for model 2 than for model 1. For example, for a stiffness ratio of 2 and a lay fration of.2, model 1 yields an effetive stress oeffiient of 1.93, whereas model 2 predits a value of 4.6. rok lay

SCA22-5 5/6 Experimental Data from Literature Figure 3 shows the preditions of our two models, ompared with some experimental data from the literature. Data for different lay-bearing sandstones olleted in [7] from various soures [2,3,11,12,13] are shown in the figure, along with our model results. Note that eah experimental data point orresponds to a rok having a different porosity; in order to ompare the data to the model preditions, we hoose a porosity of 2% for the model alulations. Similarly, we use the same stiffness ratio of 2:1 to generate our theoretial urves, although the atual values for the roks probably varied from this value. Nevertheless, we see that the both models give the same trend as is observed in the data. In partiular, model 2 gives a reasonable fit. This fit ould be greatly improved by assuming a stiffness ratio of 3, whih is not unreasonable. CONCLUSIONS We have disussed the impliations of two oneptual models for lay-rih sandstones. The first model is the shell model proposed by Zobak and Byerlee [2], in whih the lays line all of the pore walls, and the seond is a model in whih the lay exists as partiles tangentially attahed to the pore walls. Both models yield effetive stress oeffiients that inrease with inreasing lay ontent. The rate of inrease depends on the stiffness ratio, γ = G rok / Glay. Using realisti values of the stiffness ratio, model 1 annot yield values of larger than about 3 or 4. Model 2, on the other hand, an give muh higher values, and an roughly fit the data set olleted by [7]. REFERENCES 1. Berryman, J. G., Effetive stress for transport properties of inhomogeneous porous rok, J. Geophys. Res. (1992) 97, 17,49-24. 2. Zobak, M. D., and Byerlee, J. D., Permeability and effetive stress, Amer. Asso. Pet. Geol. Bull. (1975) 59, 154-58. 3. Walls, J., and A. Nur, Pore pressure and onfining pressure dependene of permeability in sandstone, 7 th Form. Eval. Symp. Can. Well Logging So., Calgary (1979), paper O. 4. Farber, D. L., Bonner, B. P., Balooh, M., Viani, B., and Siekhaus, W., Observations of water indued transition from brittle to visoelasti behavior in nanorystalline swelling lay, EOS (21), 82, 1189. 5. Bernabe, Y., Brae, W. F., and Evans, B., Permeability, porosity and pore geometry of hot-pressed alite, Meh. Maters. (1982) 1, 173-83. 6. Zimmerman, R. W., Compressibility of Sandstones, Elsevier, Amsterdam (1991). 7. Kwon, O., Kronenberg, A. K., Gangi, A. F., and Johnson, B., Permeability of Wilox shale and its effetive pressure law, J. Geophys. Res. (21) 16, 19,339-53. 8. Rekah, V. G., Manual of the Theory of Elastiity, Mir, Mosow (1979). 9. Al-Wardy, W., Measurement of the Poroelasti Parameters of Reservoir Sandstones, Ph.D. thesis, Imperial College, London (22). 1. White, F. M., Visous Fluid Flow, MGraw-Hill, New York (1974). 11. Zobak, M. D., High Pressure Deformation and Fluid Flow in Sandstone, Granite, and Granular Materials, Ph.D. thesis, Stanford Univ., Stanford, Calif. (1975). 12. Zobak, M. D., and Byerlee, J. D., Effet of high-pressure deformation on permeability of Ottawa sand, Amer. Asso. Pet. Geol. Bull. (1976) 6, 1531-42. 13. David, C. and Darot, M., Permeability and ondutivity of sandstones, in Rok at Great Depth, V. Maury and D. Fourmaintraux, eds., Balkema (1989), pp. 23-9.

SCA22-5 6/6 Rok Clay a b Rok Clay b a (a) Figure 1. Cross setions of ylindrial pore lay models: (a) pore lined with shell-like layer of lay, (b) lay forming a ylindrial rod sitting along side wall of the pore. (b) 5 4 3 2 γ=1 γ=1 γ=2 γ=3 γ=4 model-1 3 25 2 15 1 γ = 1 γ= 1 γ = 2 γ = 3 γ = 4 model-2 1 5.5.1.15.2.25.3.35 Clay fration, F (a).5.1.15.2.25.3.35 Clay fration, F (b) Figure 2. Effetive stress oeffiient, as a funtion of lay fration F for different stiffness ratios γ; for the two models. 14 12 1 8 6 4 2 5 1 15 2 25 3 35 Clay fration, F Figure 3. Effetive stress oeffiient, as a funtion of lay fration F. The points refer to data from literature and the lines refer to results from the models. Z ZB1 ZB2 WN DD M1 M2