Matrix gas flow through ‘impermeable’ rocks - shales and tight 1 sandstone 2

. The effective pressure sensitivity of gas flow through two shales (Bowland and Haynesville shales) and 8 a tight gas sandstone (Pennant sandstone) was measured over the typical range of reservoir pressure conditions. 9 These are low permeability rocks such as can be exploited as caprocks above reservoirs that might be developed to 10 store compressed air, methane, hydrogen or to bury waste carbon dioxide, all of which may become important 11 components of the forthcoming major changes in methods of energy generation and storage. Knowledge of the 12 petrophysical properties of such tight rocks will be of great importance in such developments. All three rocks 13 display only a small range in log 10 permeability at low pressures, but these decrease at dramatically different rates 14 with increasing effective pressure, and the rate of decrease itself decreases with pressure, as the rocks stiffen. The 15 pressure sensitivity of the bulk moduli of each of these rocks was also measured, and used to formulate a 16 description of the permeability decrease in terms of the progressive closure of narrow, crack-like pores with 17 increasing pressure. In the case of the shales in particular, only a very small proportion of the total porosity takes 18 part in the flow of gases, particularly along the bedding layering.


23
Shales (laminated mudstones) are of particular importance because their fine grain size and tight pore structure 24 gives them a particularly low matrix permeability and hence makes them excellent cap rocks for the containment 25 of oil, water and gases. This includes their future use as a sealant for the storage containment of fuel gases 26 hydrogen and methane, compressed air storage and for the disposal deep underground of waste liquids and gases, 27 including waste carbon dioxide. Organic shales are source rocks for petroleum and become source, reservoir and 28 seal for unconventional natural gas (shale gas). The enormous economic importance of shales cannot be 29 overstated, and this demands an ever-increasing understanding of their petrophysical properties.

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Although it was intended that experiments would be carried out under hydrostatic confinement conditions, the 139 presence of a contrast in elastic properties of the specimen against the porous end plates and the steel loading 140 pistons induces a shear stress along these interfaces. This in turn causes the stress state in the specimen to deviate 141 from hydrostatic and to reduce the average mean stress. Deviations from hydrostatic loading are most severe when 142 the length of the specimen becomes less than twice the diameter. For this reason, mechanical testing of rocks is 143 usually carried out on specimens with a length:diameter ratio of 2.5:1 or more. Finite element analysis (FEA) of 144 the stress state in rocks confined between steel end plates were carried out to assess the expected departures from 145 hydrostatic loading, and the effects predicted must be borne in mind when interpreting the permeability data.   Here, S is cross-sectional area of the sample (normal to flow path), L is specimen length, is downstream 180 volume storativity and is specimen storativity, T is the period of the pore pressure oscillation, k is specimen permeability, and  is viscosity of the pore fluid. Argon gas viscosity as a function of pressure data was reported 182 by Michels et al., (1954). Storativity is the product of the volume of the space occupied by the fluid with the pore 183 fluid (isothermal) gas compressibility. Argon compressibility is non-linear over the pore pressure range used 184 (Gosman et al., 1969) and substantially non-ideal above about 20 MPa. ≈  Vs /Vd where  is specimen effective 185 porosity, Vs is total specimen volume and Vd is downstream reservoir volume. It cannot be assumed that effective 186 (conductive) porosity estimated from permeability measurements will necessarily be equal to total porosity 187 measured independently.

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The apparatus used was the same as used for experiments reported by Rutter and Mecklenburgh (2017;2018).

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Pressure transducers with a resolution of 0.02 MPa were used for pore pressure measurements, and confining 191 pressure was measured to an accuracy better than 0.3 MPa. The minimum pore pressure used was 10.0 MPa. This 192 is sufficiently high to avoid exsorption of gas from mineral surfaces and to avoid slip flow of gas through pore

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Accuracy of reported permeability depends on uncertainties of the parameters in Eq. (1). and can be 197 measured to within about 2% of the true value, and S, T L and  to within 1%. The least certainly known 198 parameter is the downstream volume, which is determined as the difference between the total volume of the pore 199 pressure pipework measured with and without the downstream pipework connected, each measured by the pore 200 pressure change produced by a known volumometer piston displacement. The downstream reservoir volume Vd 201 was measured to be 445 ± 30 mm 3 , including the volume of the downstream porous steel filter. These 202 uncertainties translate to an accuracy of log10 permeability of ± 0.1 log units. This is small, given that permeability 203 varies with pressure by 1 to 3 orders of magnitude.

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The largest apparent uncertainties in reported permeability data arise from hysteretic changes in the behaviour of 205 the rock itself as effective pressure is cycled and will be discussed when the data are presented.

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Bulk modulus measurements as a function of confining and pore pressures were made as far as possible on 208 physically the same samples that were used for the permeability measurements, to avoid any influence of 209 mineralogical or microstructural differences. Measurements were made over a range of total confining pressures 210 up to 200 MPa, after the permeability measurements were made, with constant pore pressures of argon gas, compressibility of the pore spaces. P-wave acoustic velocity measurements were made at the same time, although 214 these data are not reported here.

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Unlike for permeability measurements, porous steel plates were not used at the ends of the specimens for pore 216 fluid displacement measurements. For the relatively porous and permeable Haynesville shale and Pennant 217 sandstone, a short hole, normally 15 mm long and 1.5 mm diameter, was drilled into the end of the specimen 218 facing the pore pressure inlet pipe, to facilitate flow of gas into and out of the specimen. This was thought to be 219 unlikely to be adequate for the lower porosity and permeability Bowland shale, therefore samples were cut in half 220 parallel to the long axis so that a 2 mm thick, porous steel plate could be inserted, to facilitate gas flow over a wide 221 surface area of the rock, yet without affecting the P-wave velocity along the length of the specimen.

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When considering the results, the procedure for pressure application is of importance. For the tests with pore 223 pressure, the application of a confining pressure slightly greater than the eventual pore pressure was made, 224 followed by application of the pore pressure. Then the total confining pressure was increased stepwise away from 225 the constant pore pressure. Thus tests at high pore pressure have been exposed to much higher effective pressures 226 before application of pore pressure, than when the test pore pressure is to be low.

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When pore pressure was made non-zero, constant pore pressure was maintained using a servo-controlled pore 228 volumometer. Each applied increment of the confining pressure caused a small elastic contraction of the pore 229 volume that attempts to raise the pore pressure. The servo-controller backs off the moveable piston in the pore 230 volumometer in order to keep the pore pressure constant. The distance swept by the volumometer piston at 231 constant pore pressure allows the volume of gas expelled to be measured to a resolution of 0.4 mm 3 . In this way 232 the history of pore volume change at constant pore pressure during progressive loading by the confining pressure 233 can be determined. The compressibility of the pore space Cpc is given by the fractional change in pore volume Vp 234 in response to a change in confining pressure Pc at constant pore pressure Pp (Zimmerman, 1991), and is the 235 reciprocal of the dry pore space bulk modulus K : where Vb is the total sample volume. Kdry is the bulk modulus of the porous aggregate. Its 237 reciprocal, compressibility Cbc , the bulk volume change in response to a change in confining pressure at constant 238 pore pressure, is defined by where Vb is the bulk volume, including the pore space. The zero-porosity bulk modulus of the constituent mineral 242 aggregate is defined as Ko (Table 1)  262 m is also given by

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Sample storativity is related to these stiffness parameters by where Kf is pore fluid bulk modulus (Hasanov et al., 2019).

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In all calculations we assume Ko is negligibly sensitive to effective pressure, compared to porous rock stiffnesses 268 such as Kdry, following data for Ko for minerals such as quartz via ultrasonic measurements (e.g. Calderón et al.,   pressure, as inelastic cracks become progressively and permanently closed. Subsequent pressure cycles up to the tendency to reduce permeability slightly with subsequent pressure cycles. The first stage in a suite of permeability 280 measurements covering a wide range of confining and pore pressures therefore must be to take the sample to the 281 maximum effective pressure to which it is to be exposed, to ensure closure of these inelastic cracks and pores up 282 to that pressure. In the regime of elastic behaviour permeability (as log k) is not usually linear, neither on a k vs Pc plot nor even 285 on a log k vs Pc plot but is concave upwards (Fig. 3). The decrease of permeability with effective pressure is due 286 to elastic closure of conductive cracks and pores, and this is expected to become more difficult as the porous 287 material stiffens at higher pressure. Thus although it is common, and useful for the purpose of modelling reservoir

348
Flow normal to layering in shales is often much slower than flow parallel to layering, but not always. Layer-349 normal flow was therefore measured for these rocks using shorter samples than for flow along the layering, and 350 only at 10 MPa argon pore pressure (Fig. 4). However, for Haynesville shale the direction of flow makes little 351 difference, except that pressure sensitivity is reduced for layer-normal flow, as would be expected if flow parallel faster at low effective pressures, but slower at higher effective pressures. Bowland shale shows a small reduction 355 in permeability for flow normal to layering relative to parallel to layering (post the first pressure cycle), and there 356 is also some indication of a reduced pressure sensitivity, although the dataset is small. if the sample storativity is constant (Fig. 5). Thus the effective (conductive) porosity of the sample during the course 362 of the experiment can be calculated. The conductive porosity of many rocks is smaller than the total porosity.

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The total porosity also corresponds to a particular value of . If all of the porosity were to be involved in the flow,

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The log gain vs phase angle data was non-linear least-squares fitted to obtain an average value for for each rock    Kdry can be calculated using Eq. (4) (Fig. 6a). Ko is the mineral bulk compressibility estimated as the VRH 396 average at zero porosity (given for these rocks in Table 1).

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K / is the value of the pore bulk modulus referred to the total volume of the rock, rather than to the pore space 398 volume. K / and Kdry versus Terzaghi effective confining pressure are shown in Fig. 6 for Pennant Sandstone.
The pore pressure coefficient m, describing the effects of pore pressures on elastic distortions of a porous rock,

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and defined in Peff = Pcm Pp is given in terms of the bulk moduli Kdry and Ko in Eq. (7). In Fig. 7   Pore volumometry by the expelled gas volume method during progressive increase in confining pressure was 424 carried out on the two shale samples used (Fig 8). The resolution of the pore volume change data is poor because 425 the specimen size was rather small (1.9 cm long). The rapid increase in slope translates to a rapid rise of calculated

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A large specimen (25 mm diameter and 50 mm long) was used for these measurements on Bowland shale, cored 458 parallel to the layering. Because this is a low permeability rock, a 2 mm thick longitudinal slab of porous sintered 459 stainless steel was deployed as described earlier, to facilitate gas flow between the rock pores and the pore 460 pressure system. During pressure cycling it was necessary to correct data for the storativity of this plate.

466
The poroelastic coefficient m calculated from the volumometry data is shown in Fig. 9b

480
For a gas saturated rock Cf > Cpp, hence B → 0, and a gas-saturated rock will therefore never develop appreciable 481 pore pressures, especially at high porosities and from low initial gas pressures even when undrained, hence was 482 not considered to be an issue in the present experiments.

483
For a liquid-saturated rock however, this will not be true. B will approach 1 when Cpp >> Cf.  491 t is the time required for pressure to decay by factor 1/e at distance L. The ratio k /  (Cf + Cpp) is the hydraulic 492 diffusivity  (dimensions m 2 s -1 ) (Zimmerman, 1991). For water, viscosity  is 0.001 Pa s. Taking the bulk 493 modulus Kf (= 1/fluid compressibility, Cf) to be 2 GPa, and the permeability to be 10 -18.5 m 2 for Haynesville shale 494 at about 5 MPa effective pressure (this is the highest permeability measured, which would apply after an excess 495 fluid pressure had been generated by compaction),  ~ 6 × 10 -6 m 2 s -1 . This leads to t ~ 60 s for L = 2 cm. This 496 assumes water and gas permeabilities are the same at the same pressure conditions, but permeability to water may 497 be about one order of magnitude lower (Faulkner and Rutter, 2001) in foliated clay-bearing rocks. Time t is 498 shorter by a factor 1/30 when the pore fluid is gas owing to its lower viscosity (Gosman et al., 1969). This 499 equation is for constant k, but when k is a strong function of Peff, decreasing perhaps 300-fold at high effective 500 pressures, up to 5 minutes may be required for small pore pressure transients to decay. The simplest approach to describing the influence of pore space geometry and connectivity on permeability is to 503 regard the pores as a bundle of circular capillary tubes, so that the equation for viscous Poiseuille flow can be 504 applied and permeability calculated as a function of capillary tube radius. The circular capillary tube is a special 505 case of flow through tubes of elliptical cross section. In this case the flow rate then becomes acutely sensitive to the short radial dimension of the tube, and the more eccentric the tube cross-section the greater will be the 507 sensitivity of its shape to externally applied effective pressure (Seeburger and Nur, 1984). Ma   permeability. However, Mavko and Nur (1978) and Seeburger and Nur (1984) showed that the bulk modulus of a 545 porous solid of given porosity is not affected by the shape (eccentricity) of the pores. All pores change volume by 546 the same fractional amount. Only the distortion under pressure of the more eccentric ones is likely to affect the 547 permeability, although all pores will affect the storativity, according to how well connected they are. The

548
'connected' porosity estimated from the log gain versus phase shift plot, that is much smaller than the total 549 porosity, is used in Eq. (18). Its small value implies that most of the porosity is not being inflated during the 550 passage of the pore pressure wave, hence during the time-scale of the pressure oscillation the greater part of the 551 porosity is closed off by the action of the effective pressure.

552
Eq. (18) can be fitted to the permeability data log k = f(Peff) measured for rock types studied using the non-linear 553 least-squares fitting routine Solver in MS Excel®, to estimate the parameters N,  and . Via the inferred 554 effective porosity the conductive pore width can also be estimated. The results of the fitting exercise provide the 555 parameters for a bundle of capillary tubes that behaves in the same way as the measured rocks. This is not to say 556 that the geometric arrangement of a simple capillary tube bundle corresponds to the pore space configurations in 557 these rocks, nor that a solution can be found for all rocks. The pressure sensitivity lies in the function that 558 describes m as a function of pressure, obtained from pore volumometry, and incorporating the effective pressure 559 coefficient n. Figure 10 shows the fit to the data for the Pennant sandstone; fit parameters are in Table 3. grains. The 4.6% porosity is contained mostly in the spaces originally between these grains that are now largely 574 filled with phyllosilicate and oxide phases, i.e. about 26% of the total rock volume, and is microstructurally in 575 some ways comparable to a shale. Therefore in Table 3 the estimated conductive channel dimensions are based on 576 flow through this reduced volume fraction. 577 Figure 11 shows the fits to the permeability data for Haynesville shale. The cross-section shape of the elliptical 578 tubes is extremely eccentric and the shorter width of the tubes is measured in nanometres. This is consistent with    and Pennant sandstone at low effective pressures, when the permeabilities are not strikingly different. n is the pore 634 pressure multiplier for the permeability data, N is the number of pores intersecting a 1 m 2 area normal to the flow path, 635 a is the pore shape aspect ratio and  is the Poisson ratio. 2b is the mean short dimension (nm) of the elliptical cross 636 section and s is the average pore spacing (microns).

656
The simple model of a set of similarly-sized and shaped channels that can behave in a comparable way to a real 657 pore network is clearly inapplicable to this rock.

679
As was pointed out earlier, pressure sensitivity of permeability according to the simple capillary bundle model 680 cannot behave in the same way as was observed experimentally for Bowland shale. Also, a single value of n 681 cannot reconcile permeabilities at different pore pressures for this rock. Figure 12 shows the permeability data for

682
Bowland shale separated into measurements at different pore pressures. By extending the collective fit between 683 log permeability and effective pressure shown in Fig. 12 to the data at each pore pressure, the downward 684 divergence of the curves becomes apparent. This can be described empirically by fitting a linear variation of n 685 with Terzaghi effective pressure, such that n = 1 at low effective pressures, rising to n =1.6 at the upper end of the 686 pressure range used. This is interpreted as a further manifestation of the pore structure complexities that mean that 687 this Bowland shale cannot be described by a simple capillary tube bundle model.

722
their aspect ratios are extremely small and the narrow dimension is expected to be in the nanometric range (Table   723 3).

724
• For flow normal to layering, at least in Haynesville shale, storativity is much greater than for flow across the 725 layering, but still implies that over half of the pore space is not participating in the flow.

726
• Permeability in both shales is very low under elevated effective pressures compared to Pennant sandstone,

727
which is of similar overall porosity, implying that connected pore spaces are narrow and/or poorly 728 connected/tortuous.

729
The above observations suggest that the effective configuration of pores spaces corresponds to the sketch shown 730 in Fig. 13b, with a population of highly oriented, crack-like pores parallel to layering that account for only a small

752
High storativity of the sandstone implied that most of the available pore space was involved in the gas flow, but 753 in the shales, for flow parallel to the layering, less than 10% of the available pore space was involved in the flow.

754
For flow in the Haynesville shale across the layering a larger pore space fraction was involved, but still much less 755 than all the available pore space. Thus only a small fraction of the total pore space can be inferred to be well 756 connected in the shales. This implies that whilst the permeability we measure in the oscillating pore pressure 757 experiment is that associated with gas transport through the rock mass, a lower effective permeability applies to 758 the ability of the gas to flow into and out of the storage pores.

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The Authors declare that they have no conflict of interests. deformation at high temperature and pressure. In Fault Mechanics and Transport Properties of Rocks, edited by B.