JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B02110, doi: /2002jb002278, 2004

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2002jb002278, 2004 Analytical and numerical solutions for alternative overpressuring processes: Application to the Callovo-Oxfordian sedimentary sequence in the Paris basin, France Julio Gonçalvès 1 Agence Nationale pour la Gestion des Déchets Radioactifs, Châtenay-Malabry, France Sophie Violette UMR Sisyphe, Université Pierre et Marie Curie, Paris, France Jacques Wendling Agence Nationale pour la Gestion des Déchets Radioactifs, Châtenay-Malabry, France Received 30 October 2002; revised 27 October 2003; accepted 20 November 2003; published 27 February [1] Previous studies that made use of basin models have shown that the normal geological evolution of the Paris basin does not generate the observed, albeit weak, excess pressures in some shale layers of the basin. Other processes that may have created the overpressures, currently neglected in such models, are investigated here. Terms accounting for osmotic effects and tectonic stress changes are successively added to the diffusivity equation. The effect of changes in outcrop boundary conditions is also calculated with a pseudo-two-dimensional analytical solution. These solutions are applied to the Callovo- Oxfordian shale formation in the eastern part of the Paris basin, France. It is shown that a long-term transient osmotic effect starting in the Tertiary could explain in part the observed excess pressures in the Callovo-Oxfordian shale assuming effective diffusion coefficients of m 2 s 1 in line with the measurements and a pore radius b around 20 Å for the shales. However, because of the uncertainty on the value of the shale pore radius, additional head measurements and osmotic experiments on samples should be made to fully establish the possibility of an osmotic process. Our study also shows that recent changes in hydrodynamic boundary conditions could also explain excess pressure distribution in this shale layer. It is plausible that a combination of the two processes could best explain the distribution and intensity of the overpressures. Tectonic stress changes do not appear to be important; it is shown that for such processes, to maintain high pressures, strong and recent increase in tectonic compressive stress would be required. INDEX TERMS: 1815 Hydrology: Erosion and sedimentation; 1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 3210 Mathematical Geophysics: Modeling; 3230 Mathematical Geophysics: Numerical solutions; KEYWORDS: hydrogeology, clay formation, osmosis, overpressures Citation: Gonçalvès, J., S. Violette, and J. Wendling (2004), Analytical and numerical solutions for alternative overpressuring processes: Application to the Callovo-Oxfordian sedimentary sequence in the Paris basin, France, J. Geophys. Res., 109,, doi: /2002jb Introduction [2] Sedimentary basins, when at hydrological equilibrium, normally show a near-hydrostatic pressure distribution. Under certain conditions some excess pressure or pressure greater than hydrostatic can develop in low-permeability layers (aquitards) or other hydraulically isolated parts of systems. The processes commonly invoked to explain these 1 Now at UMR Sisyphe, Université Pierre et Marie Curie, Paris, France. Copyright 2004 by the American Geophysical Union /04/2002JB002278$09.00 overpressures are compaction, hydrocarbon migration, diagenesis, tectonic stress or more simply topographic effects [Luo, 1994; Neuzil, 1995; de Marsily et al., 2002]. [3] Pressures slightly exceeding the hydrostatic are present in the Callovo-Oxfordian shales of the Paris basin in the Meuse/Haute Marne districts close to Bure where the French National Agency for Nuclear Waste Management (ANDRA) is studying the feasibility of siting a nuclear waste repository. These excess pressures are on the order of MPa, i.e., m of excess head and were determined by in situ slug tests within boreholes segments isolated between two packers. For each borehole, the pressure buildup over several years has been recorded by 1of14

2 Figure 1. Simplified geological map of the Paris basin showing the location of the Bure site for the study of the feasibility of a nuclear waste repository. The main faults and the position of the cross section AB of Figure 2 are shown. in situ probes and the extrapolated pressures were converted into pseudoheads for an equivalent fluid of reference density (1 g cm 3 ). In this process, experimental problems such as boreholes deformation and conversion difficulties due to the variable water density and temperature suggest a possible overestimation of excess heads. Figure 1 shows the location of Bure on a simplified geological map of the Paris basin. The Callovo-Oxfordian aquitard is marly to shaly and surrounded by two aquifers: the underlying Dogger and the overlying Lusitanian. Extensive measurement and monitoring efforts are presently performed by ANDRA. The main parameters for the layers of interest are summarized in Table 1. The classical uncertainty on these parameters is around 1 order of magnitude. The geological layers are characterized by small dips toward the center of the basin (Figure 2). [4] The Bure location, close to the outcrop, is thus more sensitive to the hydrodynamic effects of erosion and might explain why the pressure patterns are different from those in the center of the basin. Figure 3 shows the available head data on a more precise cross section with absolute excess head (relatively to the topography) at borehole MSE101 or relative excess head (as compared to the surrounding aquifers) at boreholes EST107, EST104, and EST103. Their distribution along a vertical profile is presented in Figure 4b. The first attempt at accounting for the abnormal pressures concerns the possibility of a compaction disequilibrium resulting in undercompaction and overpressuring. It was done through numerical simulations of the Paris basin evolution [Burrus, 1997; Gribaa, 1998; Gonçalvès, 2002]. These models describe the stratigraphic and geodynamic evolution of this intracratonic basin with almost constant sedimentation from the Triassic to the Cretaceous. The last general transgression of the late Cretaceous, marked by thick chalk deposits, was followed by the emersion of the basin. The major uplift and erosion that affected the peripheral parts of the basin at the beginning of the Tertiary Table 1. Measured Hydrodynamic and Transport Parameters a,ms 1 K x,ms 1 S s,m 1 D e,m 2 s 1 b,å Lusitanian Callovo-Oxfordian Dogger a See notation section for descriptions of symbols. D e is equivalent to D/R d. 2of14

3 Figure 2. Cross section from interpolated surfaces with the location at depth of the research site at Bure in the Malm deposits and more precisely in the Callovo-Oxfordian shales. The upper part of the Malm is a regional aquifer of the basin: the Lusitanian aquifer. The cross section shows the well logs used. The well La folie de Paris (LFP on the cross section) represents the stratigraphy of the Paris basin. were caused by lithospheric tilting, related to Pyrenean and Alpine orogeny [Guillocheau et al., 2000] are of particular concern for the study area as they resulted in its present-day shape. Since the basin models simulations did not lead to excess pressures, one has to consider alternative processes which were studied at a smaller scale than basin models which currently use meshes of several kilometers. Although osmotic and tectonic effects are less commonly investigated and modeled, we consider them to be potential causes of excess pressures [see, e.g., Marine and Fritz, 1981; Neuzil, 1986; Luo, 1994]. Osmotic effects depend on the salinity gradients that can be found in a shale layer and on its membrane properties. The salinity data for three boreholes are shown in Figure 4b. These salinity profiles are interpreted as the result of a diffusion-dominated process [Remenda et al., 1996; Hendry and Wassenaar, 1999] starting with a homogeneous concentration (seawater salinity) within the shale and subjected at its boundaries to lower and constant salinities. These low salinities in the surrounding aquifers are related to the aquifer recharge and subsequent leaching caused by the uplift and erosion (65 40 Myr). The salinities measured in the surrounding aquifers are higher in the Dogger (5 g L 1 ) than in the Lusitanian (1 g L 1 ). The salinity is otherwise dominated by NaCl. The higher salinity gradients in the upper part of the Callovo-Oxfordian suggest lower diffusion coefficients for the last tens of meters within the shales (A. Vinsot, personal communication, 2002). This area and the entire basin in general have known various changes in the tectonic context particularly during the Tertiary. The principal stress has also rotated during each tectonic period. Variations in the intensity of the compression might have occurred and can be Figure 3. NW-SE cross section showing the head data (m) for five boreholes EST103, EST104, EST107, HTM102, and MSE101. 3of14

4 seen as a possible explanation for the observed overpressures. The role played by these processes since the last major erosion event is investigated through simple onedimensional (1-D) numerical and analytical solutions. A more recent change in hydraulic head boundary conditions, which might be suspected due to the location close to the outcrop of this area, is also investigated by means of a pseudo-2-d analytical solution. For each of the tested processes an inverse approach is used to identify the parameters and geological events that allow the observed overpressures to be reproduced. Their plausibility or relevance are then discussed in the light of the measurements and the geological reconstructions. [5] In the following, we present first the simulations of the compaction history, then the possibility of an osmotic effect. A hydrodynamic effect by a change in boundary conditions and a variation in compressive tectonic stress intensity are successively tested and discussed. Figure 4. (a) Salinity data for different boreholes and (b) mean relative excess head as a function of depth in the Callovo-Oxfordian formation. 2. Compaction Effect Through Geological History Reconstruction 2.1. Basin Modeling Results [6] Several basin models representing processes such as compaction and gravity-driven fluid transfer, heat transfer, solute transport and major uplifts and erosions influencing the pore pressure evolution, have been applied to the Paris basin, first in two dimensions [Burrus, 1997; Gribaa, 1998], then in three dimensions [Gonçalvès, 2002]. Such basin models have often been used elsewhere to explain excess pressures in sedimentary basins [Luo, 1994; Jiao and Zheng, 1998]. Disequilibrium in the compaction is thought to be the most common source of overpressuring in porous media. Overpressuring due to sedimentation and compaction processes is a common feature in mostly siliciclastic sedimentary basins such as deltaic deposits [Burrus, 1997]. This overpressuring occurs when sedimentation rates are high and hydraulic conductivities are very low. The low permeability slows horizontal and vertical release of fluids maintaining the high pressures [Bredehoeft and Hanshaw, 1968; Luo and Vasseur, 1997]. [7] At least three basin models have been built for the Paris basin: two are two-dimensional and one is threedimensional. The first model, proposed by Burrus [1997] with the TEMISPACK code by the Institut Francais du Pétrole (finite volume basin model), was aimed at investigating the thermal evolution of the basin. The authors used the Paris basin as a an example of a pressure-equilibrated basin. Their model reproduces low pressures in the aquitards on the same order as those created by recharge in the aquifers of the basin with or without fault effects (as drains or barriers). A second two-dimensional model was later built by Gribaa [1998] to examine compaction disequilibrium as an explanation for excess pressures observed in the Callovo-Oxfordian shales in the east of the Paris basin, close to Bure. The values of measured excess pressure range from 0.2 MPa to more than 0.5 MPa representing 20 to 50 m of excess head. The simulations with the NEWBAS finite volume basin model from Ecole des Mines de Paris developed by [Belmouhoub, 1996], showed excess pressures that hardly exceeded 0.1 MPa when measured permeability values for the Callovo-Oxfordian shales (around ms 1 [Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA), 1999]) were used. The calculated pressures can be made comparable to the measured ones only if unrealistically small permeability values are used, i.e., two orders of magnitude lower than measured ones [Gribaa, 1998]. The geological evolution and especially, the compaction history cannot explain the observed excess pressures. These results in two dimensions are consistent with the 3-D simulations carried out more recently with the same NEWBAS code [Gonçalvès, 2002]. Weak overpressures are calculated in the course of the simulation and can dissipate very rapidly on a geological timescale in three-dimensional space. This is explained by the low subsidence and sedimentation rates and the rela- 4of14

5 tively high permeabilities of aquitards at the regional scale in this intracratonic basin. The simulations show that the small excess pressures developing in these layers are dissipated very rapidly, on the order of 100 kyr to 1 Myr after sediment deposition has stopped Estimating the Rate of Excess Pressure Dissipation [8] The problem of estimating pressure dissipation has been examined by different authors, e.g., as early as Terzaghi [1925] or Biot [1941], sometimes making use of Carslaw and Jaeger s [1959] solutions, e.g., Bredehoeft and Hanshaw [1968], Tóth and Millar [1983], or Neuzil [1985]. In the case of the Callovo-Oxfordian shale, an analytical calculation can be made to estimate the relaxation time for an excess pressure in a way similar to that proposed in the previous studies. Consider a time at which all the geological forcings outlined by Neuzil [1995] (compaction, tectonic stress, diagenesis, etc.) cease. We hypothesize that so-called abnormal pressures will be released by fluid flow in the vertical direction. We also assume that the underlying Dogger aquifer and the overlying Lusitanian aquifer are at hydrostatic pressures. We define the excess head h* as the difference between the hydraulic head h and the topographic level. This topographic level is set constant through time since the last major geological event in this part of the basin was the uplift and erosion at the beginning of the Tertiary (65 50 Myr). In the following, we will take h* as the variable of concern. The one-dimensional diffusivity equation governing the release of the excess pressure 2 ¼ s In equation (1), the term /S s can be seen as a pressure diffusion coefficient and is known as the hydraulic diffusivity D h. Here the origin of the z axis is taken to be at the center of the shale layer characterized by a 2e thickness. With the initial and boundary conditions h* =h* 0 for e < z < e and h* =0forz = e and z = e, a solution by analogy with thermal conduction or chemical diffusion is deduced from [e.g., Carslaw and Jaeger, 1959, p. 96] h* ¼ 4 X 1 h* 0 p " # ð 1Þ n 2n þ 1 exp D hð2n þ 1Þ 2 p 2 t ð2n þ 1Þpz 4e 2 cos 2e Similar solutions can be found in the literature [e.g., Terzaghi, 1925; Albarede, 1995]. By performing sensitivity tests for the values of and S s based on the values measured by ANDRA [1999] on the Callovo-Oxfordian ( =10 13 ms 1 and S s = 10 5 m 1 ) and with 2e = 100 m, we see from Figure 5 that it will take between 10 kyr and a few hundred thousand years to relax an excess pressure by vertical fluid flow once geological forcings no longer affect the system. [9] After the last major event at the beginning of the Tertiary and except for late erosion processes and climate changes, none of the processes described above have affected the system for the last 10 Myr. Therefore one has to consider a more recent hydrological process or other long-duration geological processes to explain the observed weak excess pressures in the Callovo-Oxfordian layer. The processes investigated in this article are osmotic effect, ð1þ ð2þ Figure 5. Excess head decrease in the center of the shale after the end of any geological forcing as a function of time for different hydraulic diffusion coefficients D h (in m 2 s 1 ). changes in the hydraulic head boundary conditions, and tectonic stress effects. 3. Osmotic Effects on Pressures in a Clay Layer 3.1. Theory of Osmotic Flow [10] Osmotic effects are sometimes invoked to explain some abnormal pressures in clay formations [e.g., Marine and Fritz, 1981; Neuzil, 1986, 2000] or in more general coupled-process investigations [Bachu, 1995]). Historically, the osmotic effect was recognized in biology at the beginning of the 18th century with the observation of fluid flow driven by salinity gradients and solute exclusion across biologic membranes by the French physician Dutrochet ( ) [Rich, 1926]. Later, the physical theory was established; a remarkable contribution is that of Van t Hoff at the end of the 19th century, extrapolating P, V, T relationships from gas to solutions [see Van t Hoff, 1887] (relation (5)). [11] This theory has been extended to geological situations with the observation that shales can behave as semipermeable membranes. If the geological membrane is perfect, only water can flow in response to salinity gradients. In most case studies, the semipermeable layer is not an ideal membrane and some solute transport is possible. Hence these osmotic coupled processes include both solute transport and fluid flow. Osmotic flow occurs when there is a concentration gradient within the geological medium which causes solute transport dominated by diffusion in such tight environments [Remenda et al., 1996]. The chemical potential will drive the fluid flow from the low-concentration zones to the high-concentration ones. In his review of overpressuring processes, Neuzil [1995] stated that membrane properties of the shales that cause osmotic effects have been demonstrated at the experimental scale [Olsen, 1972; Fritz, 1982; Fritz and Marine, 1983; Keijzer et al., 1999] but that there is a lack 5of14

6 of experimental evidence at the geological field scale. However, recent field experiments suggest that osmotic behavior of shales is likely to occur in geological media [Neuzil, 2000]. This osmotic effect has to be taken into account in nuclear waste repository safety calculations as an additional driving force for fluid flow [Gautschi, 2001]. In order to account for this chemical potential driving force, an osmotic pressure term p (Pa) has to be added to the hydraulic potential h to describe fluid flow, assuming linearity of the flow-gradient relations [Horseman et al., 1996]. The osmotic term thus describes the flow of water from an area where its chemical potential is high (i.e., low salinities) to where it is low. Darcy s law in one dimension is then modified as q z þ where s is the dimensionless reflection coefficient, or efficiency coefficient which characterizes the membrane properties (i.e., the difference from ideality) of the medium. It is classically defined as s = P/pj q=0 [Olsen, 1972; Fritz and Marine, 1983], where P is the pressure (Pa), and s is sometimes called the osmotic conductivity. [12] The osmotic pressure is related to the activity of water a w by [Neuzil, 2000] ð3þ p ¼ RT lnða w Þ ð4þ V w where R = Pa L (mol K) 1 is the gas constant, T is the absolute temperature in kelvins, and V w is the partial molar volume of water. For molarities lower than 1 mol L 1, the osmotic pressure can be approximated by [Fritz, 1982] p nrtc where n is the number of types of constituent ions and C is the solute concentration (mol L 1 ). Relation (5) is a linearization of equation (4) for an ideal solution [Robinson and Stockes, 1959]. Rewriting equation (3) using equation (4) for p q z þ ¼ þ with K c = s nrt/rg Analytical and Numerical Solutions of the Coupled Transport Equations Simple Analytical Solution [13] A first approximation of the osmotic effect can be calculated assuming that T, r, s, and are constant through z when in reality they do vary. This is particularly questionable for s which varies with the concentration and the porosity. We also consider that the main solute transport process is diffusion, thus neglecting advection. With these simplifying assumptions, one can determine a first analytical solution which is used to obtain first estimates for relevant parameters. Replacing h by h* in equation (3) and neglecting the topographic variations as compared to head ð5þ ð6þ variations so the mass conservation equation rqþs s ¼ þ s ¼ 0 and coupled with the solute diffusion equation, it yields the following system of partial differential equations to be solved in one ¼ S s C 2 ¼ R d where R d is the retardation factor due to adsorption and ion exclusion processes between the solute and the solid matrix and C is the concentration in mol L 1. By using a Laplace transform, we can calculate an analytical solution to this system (Appendix A). With a z axis origin at the Callovo- Oxfordian center (thickness 2e) and with the initial and boundary conditions h* =0,C =0atz =±e, t > 0 (excess head and solute concentration assumed to be rapidly dissipated in the aquifers), and C = C 0, h* =0for e < z < e at t = 0, the solution for h* and e < z < e is given by X 1 h* ¼ R dk c C 0 4 ð 1 ðr d DS s Þ p 2n þ 1 " # ð2n þ 1Þpz D 2n þ 1 cos exp ð Þ2 p 2 t 2e 4R d e 2 " #! ð2n þ 1Þpz cos exp D hð2n þ 1Þ 2 p 2 t 2e 4e 2 Þ n X1 ð 1 2n þ 1 Þ n ð7þ ð8þ ð9þ ð10þ We applied this solution to the Callovo-Oxfordian in the east of the Paris basin assuming an initial salinity for the Callovo-Oxfordian shale layer of 0.75 mol L 1 (40 g L 1 ). The starting time for the osmotic effect calculations is the end of the major uplift and erosion in the early Tertiary when salt leaching in the surrounding aquifers is expected to have started, establishing the salinity gradient required for this osmotic effect. We used the values of ms 1 and 10 5 m 1 for and S s, respectively, as measured by ANDRA, which yielded a hydraulic diffusion coefficient of 10 8 m 2 s 1. The retardation factor can be calculated from measurements of the sodium adsorption coefficient; it is close to 10. Taking into account that this retardation factor should be close to one for chloride, R d was made variable between 1 and 10. The two parameters likely to have great influence on excess head calculations are the reflection coefficient s and the ratio D/R d. [14] We tested the analytical solution with a simple zero concentration boundary condition for the Lusitanian and the Dogger aquifers. To obtain significant excess heads after 50 Myr of simulation, we had to assume a relatively low ratio D/R d of m 2 s 1 and a reflection coefficient s of 0.8, which is significantly higher than recently measured values, albeit for different shales [Keijzer et al., 1999; 6of14

7 Figure 6. Evolution of the excess head for the osmotic effects calculated with the analytical solution assuming diffusion as the only transport process for salt. (a) Profile across the Callovo-Oxfordian at different times. (b) Excess head evolution over time in the middle of the layer showing the influence of D/R d in m 2 s 1, where D is the diffusion coefficient in porous media and R d is the salt retardation coefficient. Gautschi, 2001], as no s measurements are available at Bure. With these values, we can simulate a transient osmotic effect starting 50 Myr ago and associated with the initial salinity gradient which results in a present-day residual excess head on the order of the observed values, i.e., several tens of meters at the center of the shale layer. [15] Figure 6 shows the results calculated with the analytical solution. The initial steep salinity gradient between the aquifers and the aquitard causes a rapid increase in excess head by a brief fluid influx toward the center of the Callovo-Oxfordian. Then, as the total salinity difference decreases by diffusion, this excess head decreases. It is also evident, from Figure 6 and from equation (10) (with DS R d ), that the intensity of the initial excess head is governed by the ratio (K c / )C 0, and therefore by the value of s. [16] The initial excess head decays by fluid flow toward the aquifers, after a short initial stage of converging flow toward the center of the shale layer. The velocity field then rapidly diverges from the center. The diverging velocities are extremely small, the first velocity term in equation (6) is almost completely balanced by the second term accounting for the osmotic effect. This illustrates a quasi-equilibrium situation between hydraulic pressures and salinity gradients and justifies the neglecting of advection. Thus the relaxation of the excess head is extremely slow and the time constant of the process is that of the solute diffusion process. The excess head is then controlled by the diffusion process and evolves with the salinity profile. Figure 6a illustrates this dependence of h* ond/r d. If we set D/R d to higher values, e.g., m 2 s 1 (higher values of sample measurements), with the boundary conditions defined above, the simulated transient flow and transport result in a present-day quasi-steady state with very low concentration and excess head profiles Numerical Solution of the Coupled System [17] The approach described above is probably oversimplified although it allows an initial analysis of the processes. In the following approach, some of the terms describing K c will be treated as depending on z. This is particularly crucial for s. One of the most critical simplifications in the analytical calculation is indeed to consider a constant reflection coefficient s. Bresler p [1973] established a theoretical relation between s and b ffiffi c, where 2b is the space between two clay platelets and c is the normality in anions (equal to the concentration C in mol L 1 for monovalent ions). As discussed in section 1, the diffusion coefficient is also made variable with a lower value (D 1 ) at the top of the layer (20 m thick zone) and higher values (D 2 ) elsewhere. The full coupling involves an advective term in the transport equation. The system then 2 ð ¼ S R d ð11þ In the transport equation, u = q z /f, where f is the porosity. As previously discussed, the advective term is very low compared to the diffusive one. This term is thus neglected again in the following calculations. We used again h* = 0 for the hydrodynamic boundary conditions, but we can now consider more realistic boundary conditions for the salt concentration, close to present-day measurements, i.e., 5 gl 1 for the Dogger and 1 gl 1 for the Lusitanian. We assumed these concentrations to be constant over time and solved the system (11) numerically with a classical finite difference scheme similar to that used by Greenberg et al. [1973]. The evolution of the calculated salinities and excess heads across the shale layer is shown in Figures 7a and 7b. In this approach, the crucial point is to reproduce the salinity data that determine the possible excess head profile which also depends on the b value. For a leaching time of 50 Myr, the best fit for both the salinity and the excess head data of borehole EST104 is obtained with D 1 /R d =10 12 m 2 s 1, D 2 /R d = m 2 s 1, and b = 7of14

8 Figure 7. Evolution of the excess head for the osmotic effect calculated with the numerical solution assuming diffusion as the only transport process for salt. (a) Salinity profile across the Callovo-Oxfordian at different times. (b) Excess head evolution. 20 Å (a reasonable value, see discussion below). Although lower than the measured values, the calculated excess heads are on the order of the measurements. The sensitivity analysis of the parameters shows that the maximum excess head values can be attained by decreasing D 1 and D 2, which yields higher calculated salinities (Figure 8). The initial reflection coefficient is very low (0.03) due to high initial salinities (40 g L 1 ) before rising to higher values in the course of the diffusion process (0.5 < s < 0.8). Figure 7a shows the excess head distribution calculated with this salinity asymmetry Discussion [18] Although central to the osmotic effect analysis, estimating the b value is problematic due to experimental difficulties and a conceptual model of the porous medium geometry based on mercury porosimetry. These measurements cannot explore all the pore classes. Some measurements in the Callovo-Oxfordian give a pore size distribution dominated by two classes: 2 mm and 100 Å. The larger pore class is attributed to quartz and calcite grains unbedded in a clay matrix characterized by the smaller pores. Because of experimental difficulties, the values of 100 Å for b is determined with an estimated 50% uncertainty yielding a Å range. A second indirect method proposed by Neuzil [2000] is based on the relation between A s the specific surface (m 2 g 1 ), the porosity f, the clay density r s, all measured by ANDRA [1999], and the unknown b. This relation is A s b = f/(1 f)r s which represents the volumetric water content per unit mass of clay. A b value of 20 Å is obtained from this relation with f = 0.15, A s = 30 m 2 g 1, and r s = 2.8 g cm 3 as estimated by ANDRA [1999]. These two approaches illustrate the uncertainties on the value of the pore radius. The smaller pores are of particular concern in the osmotic effect since the salt exclusion of a clay that impacts the transport processes and explains its semipermeable membrane properties is Figure 8. Sensitivity on D 1 /R d and D 2 /R d (in m 2 s 1 ) and b for the excess head and on the effective diffusion coefficients for the salinity. Comparison is made with the measurements. 8of14

9 Figure 9. Conceptual model and boundary conditions for the calculation. mainly attributable to the narrow spaces between platelets [Horseman et al., 1996]. [19] The diffusion coefficients inferred by trial and error to fit the salinity data are low compared to measured ones. In the present analysis they are, however, dependent on the time of leaching which can be treated as an unknown. For a time of leaching of 40 Myr instead of 50 Myr, the same process needs 33% higher values of D/R d to obtain the same results as those in Figures 7 and 8. The best fit obtained for the salinity profile which determines, with the b value (taken here to be 20 Å), the excess head profile, provides lower calculated excess head values than the measured ones. 4. Changes in Hydraulic Boundary Conditions 4.1. Conceptual and Mathematical Model [20] A quasi-2-d model was used to investigate recent changes in boundary conditions at the outcrop as a possible cause of excess heads. Such changes in boundary conditions are known to have occurred in the last million years due to fluvial erosion, which considerably deepened the river beds. We tried to determine if the highly transient shale system has kept the memory of higher head boundary conditions from the past. We assumed an initial equilibrium between the Callovo-Oxfordian, the Lusitanian and the Dogger with a homogeneous hydrostatic higher head h* 0 as compared to the present-day one. Assuming a past rapid decrease in head at the outcrop (instantaneous erosion), the perturbation propagates horizontally in the underlying Dogger and overlying Lusitanian aquifers. As a consequence, this horizontal propagation of the perturbation in the aquifers changes the upper and lower boundary conditions for the flow in the Callovo-Oxfordian. This conceptual model is represented in Figure 9. We assumed that in response to this change in boundary conditions, the flow in the aquitard becomes mainly vertical by leakage to equilibrate the new boundary conditions. Thus, for each location x we can solve a one-dimensional diffusivity equation with time-dependent boundary conditions. The parameters K x =10 8 ms 1 and l S s = 10 5 to 10 4 m 1 [ANDRA, 1999], the horizontal permeability and specific storage of the aquifers, respectively, are almost identical for both aquifers and produce the same values for the upper and lower boundary conditions of the aquitard, denoted h* l (x, t). Given the initial and boundary conditions h* =h* 0 everywhere at t = 0 and h*(x, z, t) = h* l (x, t) atz = e (base of the Callovo-Oxfordian) and at z = +e (top) with 2e = 100 m for t > 0, an analytical solution is derived from Carslaw and Jaeger [1959, pp ] for the head h*(x, z, t) in the aquitard, at any abscissa x: h* ðx; z; tþ ¼ 1 X 1 exp D hn 2 p 2 t h* 0 e 4e n¼1 2 cos npz sin np 2e 2 2e ½ np ð 1Þn 1Šþ nd Z t hp exp D hn 2 p 2 t 2e 0 4e 2 h* l ðx; tþ ½1 ð 1Þ n ŠdtŠ ð12þ h* 0 However, we must first calculate the evolution of the excess head h* l in the aquifers. This excess head drops from h* 0 (initial value) to eventually zero according to the onedimensional flow equation: l ðx; K 2 ¼ Ss l l ðx; tþ þ q l ð13þ where q l (negative for inflow) is the leakage (volumetric flow rate exchanged with the aquitard [de Marsily, 1986]). Because of the symmetry of the boundary conditions, this leakage can be approximated using the flow rate K*(h*/e), z where h* is the difference in excess head between the center of the aquitard and the boundaries and e is the half thickness. Equations (12) and (13) are solved for h*/h* 0 simultaneously, numerically for equation (13) and on a regular grid for equation (12) Results and Discussion [21] Figure 10 shows the excess head distribution in the layer as a function of the distance x from the outcrop. From this we can see that the calculated vertical excess head gradients are weak. The time constant for the horizontal excess head dissipation over tens of kilometers is comparable to that of the aquitard for the dissipation over tens of meters. Thus the aquitard equilibrates rapidly with the boundary conditions imposed by the aquifers. Although the vertical excess head distribution of Figure 4b is not reproduced, the distribution along the x direction is in better agreement with the observation of a relative excess head that increases toward the center of the basin (Figure 3). Figure 11 shows that the horizontal excess head profile in the center of the layer is in good agreement with the 9of14

10 Figure 10. Distribution of excess head within the Callovo-Oxfordian as a result of a change in the outcrop boundary conditions at (a) 50 kyr, (b) 100 kyr, and (c) 150 kyr. The boundary with the Dogger aquifer is at z = e. observed excess head distribution if we assume, e.g., a drop h* 0 of m occurring 100 kyr ago. We can see the l influence of the choice of the hydraulic diffusivity D h = K x /S l s for the aquifers as boundary conditions in Figure 11, where it has been varied by 1 order of magnitude. For a given time, high excess head values are found at greater distances from the outcrop when the hydraulic diffusivity in the aquifers is increased. [22] The significance of this process must now be examined. The Pliocene is characterized by many climate changes with glaciations and eustatic variations. These variations caused river incisions. An age around 150 kyr is proposed for the beginning of the Meuse river incision 15 to 20 km to the east of the Bure site. This incision can represent a drop of several tens of meters [Le Roux and Harmand, 1998]. Although our calculation involves simplifying assumptions such as symmetry conditions, it is consistent with this erosion reconstruction. This mechanism may therefore be a process contributing to explain the observed excess heads. 5. Effect of Tectonic Stress [23] The last process that we investigated is the effect of the tectonic compressive stress on the pore pressure evolution. Tectonic stress variations are thought to cause transient fluid motion and generate abnormal pressures in clay layers [Neuzil, 1995]. This effect is rarely included in models except in those proposed by Luo [1994] [Shi and Wang, 1988; Screaton et al., 1990; Ge and Garven, 1992] or McPherson and Garven [1999]. The evolution of the tectonic stress regime in France has been studied for Tertiary times, e.g., by Bergerat [1987] and for the present-day by Cornet and Burlet [1995]. Tectonically, the Oligocene was an extension phase. In the study area, the Gondrecourt graben results from this event (Figure 3). The Oligocene extension was followed by a compression which was first NE to SW in the early Miocene, then this principal stress rotated during the late Miocene to a NW to SE direction related to the Alpine orogeny [Bergerat, 1987]. This direction is still observed at present. From the beginning of the Miocene, the paleostress magnitudes are governed by the Alpine orogeny, the maximum values are reported from the Jura region in southwestern Germany Figure 11. Excess head in the center of the layer along the x direction (toward the center of the basin) and data for different values of the hydraulic diffusivity coefficient of the aquifers (D l h in m 2 s 1 ) at 100 kyr and for h* 0 =70m. 10 of 14

11 in the early Miocene with magnitudes between 50 and 130 MPa [Bergerat et al., 1985]. For the late Miocene, Lacombe and Laurent [1992] proposed a 50 MPa compression stress based on a calcite twin study of the Burgundy platform in the southeast of the Paris basin. Finally, a present-day MPa horizontal principal stress has been measured by Cornet and Burlet [1995]. [24] We examine, in a simplified manner, the effect of a tectonic stress pulse on pore pressure and the relaxation time needed to obtain excess heads of the same order of magnitude as the measured ones. Hence we consider a single step function when the general trend is a linear decrease from 100 to 20 MPa in compression stress. The evolution in compression stress intensity is known discontinuously but shows large variations in a changing tectonic context, so that a sudden increase in the stress intensity is possible. If we consider a simple vertical column of unit surface area, the diffusivity equation in excess head, accounting for the stress effect on strain and pressure, is written according to ¼ S ða a s t ð14þ with s t =(s v + s T )/2 the mean stress, s v is the overburden and s T is the tectonic compressive stress (Pa). In equation (14), we neglect processes such as the thermal effect [Luo and Vasseur, 1995] and pressure solution. For the sake of simplicity, we assume constant values for the parameters, S s, and a, whereas they depend on porosity [Luo, 1994]. Assuming that sedimentation and erosion processes have been very weak since the end of the major uplift and erosion of the early Tertiary, we can neglect the variations in s v. Thus the t / accounting for the fluid flow caused by porosity changes [Neuzil, 1986; Luo and Vasseur, 1995], reduces to 1 2 (@s T/). Here the porosity changes are caused by variations in tectonic stress. We assume zero initial stress as a reference and an instantaneous stress increase s 0 at t = t 0 then s T ðþ¼ht t ð t 0 Þs 0 ð15þ where H(t t 0 ) is the step function. One can then solve equations (14) and (15) by a Laplace transform as used in previous studies for the more general compaction equation [Gibson, 1958; Bredehoeft and Hanshaw, 1968]. The initial and boundary conditions are h* =0forz =±e and t >0 which corresponds to a hydrostatic head distribution in the aquifers and h* =0for e < z < e at t = 0. The solution is (see Appendix B) ( ð h* ¼ a a sþ s X 1 ð 1Þ n ð2n þ 1Þpz cos 2S s p 2n þ 1 2e "!#) 1 exp D hð2n þ 1Þ 2 p 2 ðt t 0 Þ 4e 2 ð16þ Figure 12 shows this solution for the parameters of the Callovo-Oxfordian and (a a s )=10 8 Pa 1. The pulse of tectonic stress at t = t 0 causes a pulse in pressure. The release of this excess head is then governed by the hydraulic diffusivity coefficient D h. Figure 12 shows that the Figure 12. Tectonic stress effect with an instantaneous increase of head which relaxes briefly after a pulse of tectonic stress. D h is the diffusivity of the clay in m 2 s 1. relaxation time for this abnormal pressure created by the tectonic stress is on the order of kyr similar to that obtained for compaction in the first part of this paper. The observed excess heads can only be obtained from a 1 MPa pulse kyr ago. These values of tectonic stress increase are of the same orders of magnitude as those found in the literature, albeit as a decrease rather than an increase. Thus, if one wants to explain excess pressures by tectonic effects, this requires a stress increment in the last 100 kyr, which appears most unlikely. 6. Discussion and Conclusions [25] As outlined by Neuzil [1995], the hydrodynamic response to geological or chemical forcing is highly transient in aquitards. A pressure disequilibrium will be released by pressure diffusion processes for which the governing equation has been solved in different ways in this article. The normal evolution of the Paris basin does not lead to undercompaction and overpressures. It is shown that the developing weak excess pressures due to undercompaction are rapidly released, i.e., in a few tens of thousand years. Other processes have been tested to explain the abnormal pressures recorded in the Callovo-Oxfordian. Additional terms accounting for plausible processes able to cause some excess heads were successively included in the well-known diffusivity equation in which hydraulic head is replaced by excess head. Whenever possible, simple 1-D or 2-D analytical solutions were developed to provide a first-order interpretation of the influence of such processes in the Callovo-Oxfordian in the east of the Paris basin. The initial and boundary conditions were simple and only a single perturbation was always assumed, whereas it is obvious that the boundary conditions, for instance, have experienced continuous changes. This simple approach could be extended to more complete calculations. Among the processes inves- 11 of 14

12 tigated here, the osmotic effect is the one most likely to explain, at least partly, the observed excess heads, over a long period of time, i.e., for the duration of the existence of salinity gradients. The chemical gradient that might have developed through the clay at the end of the early Tertiary erosion, when salinity reduction occurred in the surrounding aquifers, might be the cause of a sudden brief pulse of fluid migration toward the center of the Callovo-Oxfordian layer, thus increasing pressures. At the same time, however, the concentration gradient started to decrease through solute diffusion, and the pressure increase was slowly dissipated. Fluid migration goes back from the center of the clay layer toward the aquifers, but is so slow that it is virtually zero. [26] We have shown that to explain the present-day observed excess heads by an osmotic effect with transient flow and transport starting with the Tertiary uplift and erosion starting 50 or 40 Myr ago, we had to assume low values for the diffusion-retardation ratio (D/R d = 1 to m 2 s 1 ). We also had to assume a value of 20 Å for the pore radius b which, together with the concentration C determines the variable value of s, the osmotic reflection coefficient. The present-day salinities within the Lusitanian and Dogger aquifers are introduced as boundary conditions (1 and 5gL 1, respectively). This natural salinity is sustained in the Dogger by an upward salt flux from the Triassic halite toward the overlying layers. With these conditions, the calculated transient flow and transport processes have a salinity profile approaching the measured one and a few tens of meters of residual excess head. However, the measured heads are higher than the calculated ones with the best fit for the salinity and a reasonable value for b given its likely range. [27] The natural upward salt flux controlling the boundary conditions in the adjacent aquifers of the Callovo-Oxfordian may justify an extension to more complete 2-D or 3-D calculations despite the difficult estimation of the equivalent parameters for each mesh required in such modeling. These results depend strongly on the value of the pore radius b, and on the salinity profile. Given the uncertainty on the pore radius b, this study cannot be fully conclusive regarding osmosis. To reinforce our conclusions, additional head data are required, especially at the bottom of the shale layer where no excess head data are available as well as osmosis experiments. However, these experiments on osmosis at the sample scale and especially the osmotic efficiency estimation rise the question of the upscaling at the field scale. [28] We must now examine the chemical and physical approximations of our calculations. The linearization of equation (4) is based on the strong assumption of free water and dilute solution. The electrical and chemical properties of the clay matrix and the pore solution are not explicitly modeled although they are taken into account, at least partly, in the relationship between s, b, and C [Bresler, 1973]. The porous medium in clays is characterized by the presence of a so-called double layer: to preserve electroneutrality, cations in solution are attracted to the negatively charged clay surfaces. Only the cations outside this double layer can diffuse in the free solution. The anions are repelled from the clay surface. Moreover, the presence of structured water, which means water organized around ions, increases the viscosity. As a consequence, a modified expression of the concentration in equation (5) should be used as discussed by Horseman et al. [1996]. The calculations could be made more accurate by introducing such physical and chemical properties. [29] The excess head within the Callovo-Oxfordian seems to increase when one moves down dip toward the center of the basin; this would be consistent with the changes of the outcrop elevation scenario. If osmotic effects and outcrop elevation changes can be considered to occur together, and if the excess heads they generate are additive, then this would shift the excess head profile of Figure 8 toward higher values, closer to the observed heads. It would thus constitute a plausible explanation for the excess head in the Callovo-Oxfordian. Finally, the tectonic stress effect, creating pressure pulses, requires strong and relatively recent compression stress variations. So far, we do not have any evidence to clearly establish that such strong stress variations have occurred in the Paris basin. [30] In the current research on the confinement properties of the Callovo-Oxfordian [Marty et al., 2003], the understanding of processes which could explain the observed overpressures attracts particular attention. As noted by Neuzil [2000] and illustrated in this paper, the time constant for hydrodynamic excess head dissipation is one to several orders of magnitude lower than the time constant when an osmotic effect is involved. This fact and the results presented here could justify experimental investigations on osmosis. Appendix A: Analytical Solution of the Osmotic Coupled Transport [31] The diffusivity equation including the chemical driving forces is 2 2 C 2 ¼ s ða1þ If diffusion is the only transport process, the well-known transport equation is 2 ¼ d ða2þ The boundary and initial conditions are h*=0andc = 0 for z = e and z = e and for t > 0 and h*=h* 0 and C = C 0 for e < z < e for t = 0. The Laplace transform of equation (A2) is 2 C R 2 ¼ pc C 0 ða3þ Where C stands for the Laplace transform of C. The complete solution of equation (A3) is C ¼ Aexpðq 1 zþþbexpð q 1 zþþ C 0 p ða4þ with q 2 1 = pr d /D. Applying the boundary condition (C = 0 so C =0atz =±e) yields for symmetry reasons: A ¼ B ¼ C 0 1 ða5þ p coshðq 1 eþ Hence the solution to the Laplace transform equation is given by C ¼ C 0 1 cosh q ð 1zÞ ða6þ p coshðq 1 eþ The inverse Laplace transform of this function is given by Carslaw and Jaeger [1959]. 12 of 14

13 [32] We can now solve equation (A1) by the Laplace transform. If we replace the 2 C/@z 2 by (R d /D)(@C/) and apply the Laplace transform, we 2 ¼ Ss ph* h* ð0þ R dk c D pc C 0 Introducing equation (A6) into equation 2 q2 h* ¼ R dk c C 0 D coshðq 1 zþ coshðq 1 eþ ða7þ ða8þ with q 2 = p/d h and D h = /S s. The complete solution of this equation is K c C 0 h* ¼ A expðqzþþb expð qzþ Dq 2 1 q2 cosh ð q 1zÞ coshðq 1 eþ ða9þ Applying the boundary conditions h* = 0 at z = ±e leads by symmetry to A ¼ B ¼ R dk c C 0 Dq 2 1 q2 1 coshðqeþ Finally, we obtain the solution in the Laplace domain: R d K c C 0 h* ¼ D R d D S 1 coshðqzþ s p coshðqeþ 1 coshðq 1 zþ p coshðq 1 eþ ða10þ The solution of equation (A10) is found by applying an inverse Laplace transform given by Carslaw and Jaeger [1959] for large times: " R d K c C 0 4 X 1 ð 1Þ n ð2n þ 1Þpz h* ¼ cos ðr d DS s Þ p 2n þ 1 2e! D 2n þ 1 exp ð Þ2 p 2 t X1 ð 1Þ n 4R d e 2 2n þ 1 cos ð2n þ 1Þpz 2e!# exp D hð2n þ 1Þ 2 p 2 t 4e 2 ða11þ Appendix B: Analytical Solution of the Diffusivity Equation Including the Stress Effect [33] The diffusivity equation accounting for the stress effect 2 ¼ s ð a a sþ T ðb1þ where s T is the tectonic stress depending only on time. By performing a Laplace transform, we 2 ¼½S ð sph* h* ð0þš a a sþ ½ps T s T ðþ 0 Š 2 ¼ ð q2 h* a a sþ ½ps T s T ð0þš ðb2þ 2 The solution of equation (B2) is h* ¼ A expðqzþþb expð qzþþ a a ð sþ s T s T 2S s p ðb3þ Applying the boundary conditions (h* = 0 so h* = 0 for z = ±e), for symmetry reasons we obtain A ¼ B ¼ a a ð sþ s T s T ðþ 0 1 2S s p coshðqeþ h* ¼ a a ð sþ s T s TðÞ 0 cosh 1 ð qz Þ 2S s p coshðqeþ ðb4þ As we know the inverse Laplace Transform for cosh(qz)/ cosh(qe) and (s T (s T (0)/p)), we apply Duhamel s theorem R t 0 f 1ðt tþf 2 ðtþdt ¼ f 1 f 2 and obtain the solution ( " # ð h* ¼ a a sþ s T s T ðþ 0 D hp X 1 2S s e 2 exp D hð2n þ 1Þ 2 p 2 t 4e 2 ð 1Þ n ð2n þ 1Þpz ð2n þ 1Þcos 2e Z " # ) t exp D hð2n þ 1Þ 2 p 2 t 4e 2 ½s T ðtþ s T ð0þšdt ðb5þ 0 If s T (t) =H(t t 0 )s 0 where H(t t 0 ) is the step function, and if s T (0) = 0, then ( ð h* ¼ a a sþ s X 1 ð 1Þ n ð2n þ 1Þpz cos 2S s p 2n þ 1 2e "!#) 1 exp D hð2n þ 1Þ 2 p 2 ðt t 0 Þ 4e 2 ðb6þ Notation A s specific surface, m 2 g 1. a w activity of water, dimensionless. b pore radius, Å. C, c concentration, mol L 1, and normality. D diffusion coefficient, m 2 s 1. D h hydraulic diffusion coefficient, m 2 s 1. D e effective diffusion coefficient, equal to D/R d,m 2 s 1. e half thickness of Callovo-Oxfordian clay, m. h, h* hydraulic and excess head, m. K x horizontal hydraulic conductivity, m s 1. vertical hydraulic conductivity, m s 1. P pressure, Pa. q l volumetric flow rate, s 1. q z Darcy s velocity, m s 1. S s l S s specific storage coefficient for clay, m 1. specific storage coefficient for aquifers, m 1. R gas constant, Pa m 3 (mol K) 1. R d retardation factor, dimensionless. T temperature, K. V s sedimentation rate, m s 1. V w partial molar volume of water, m 3 mol 1. a bulk compressibility of the porous medium, Pa 1. a s bulk compressibility of the solid, Pa 1. f porosity dimensionless. 13 of 14