GEOSTATISTICAL MODELING OF TRACER TESTS IN A SINGLE FRACTURE. CALIBRATION OF TRANSPORT PARAMETERS. J. JODAR1, A. ALCOLEA1, A. MEDINA SIERRA2 and J. CARRERA1 1
Technical University of Catalonia (UPC). Dep. of Geotechnical Engng. & Geosciences. Module D-2. c/ Jordi Girona 1-3. 08034 Barcelona, Spain.
[email protected] 2 Technical University of Catalonia (UPC). Dep. of Applied Mathematics. Module C-2. c/ Jordi Girona 1-3. 08034 Barcelona, Spain
Abstract. The main objective of this work is to evaluate the usefulness of a geostatistical inverse approach in the estimation the transmissivity field in a single fracture, and the representative transport parameters at the scale of the tracer test. We apply this methodology to data from the GAM Project (Gas Migration). The aim of this project is to provide a better understanding of gas transport mechanisms in shear zones. With third objective several hydraulic and tracer tests were performed at the GTS (Grimsel Test Site, Switzerland). Several studies have attempted to provide a quantitative description of gas transport in fractured formations. However, the main difficulties still remain. A basic step for describing multiphase flow and transport is a proper characterisation of the hydraulic parameters within the fracture. First, one must characterise and conceptualise the system to be studied. To achieve this, a flow model was performed using geostatistical inversion techniques. The objective was to obtain the spatial distribution of the parameters governing the flow equation. This work presents the results obtained from the application of a geostatistical inversion procedure similar to the previous one, but applied now to tracer tests in a shear zone. In this case, only concentration data is used. This paper is organised as follows. First, an overview of the mathematical procedure (based on the Maximum Likelihood Method) is presented. Then a concise information about the model set-up is provided, which is followed by a detailed description of the modelling procedure and the results for the transport model. 1. Introduction 1.1 MOTIVATION Heterogeneity is a fundamental characteristic of nature, present in most of the variables characterising natural phenomena. It is known that heterogeneity of hydraulic
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA conductivity and porosity have a large impact on solute or gas transport through the geosphere. For instance, the long tailing in breakthrough curves of conservative tracers can be attributed to the heterogeneity of the hydraulic conductivity field. Pumping tests are commonly used to assess the spatial variability of transmissivity. Meier et al. (1998) demonstrated that the transmissivity estimates obtained through the interpretation of cross-hole pressure test responses, using Jacob’s method (Cooper & Jacob, 1946), are very close to the effective value of the transmissivity of heterogeneous formations for parallel flow conditions, whereas the storativity estimates are strongly influenced by the heterogeneity. However, conventional interpretation of cross-hole tests do not provide quantitative information of the spatial distribution of the transmissivity, which is required for a posterior interpretation of tracer tests. In this work, we use a geostatistical inversion procedure to estimate a transmissivity field at the scale of the tracer tests, within an heterogeneous shear zone in granite. This method allows us to use the full-time length of pumping tests drawdown data (at several observation boreholes) to estimate the expected value of the transmissivity at a large number of zones. This paper is organised as follows. First, we outline the migration experiment at the GTS (Grimsel Test Site). Then we briefly describe the mathematical formulation of the inversion procedure. Finally, we present a complete description of the modelling procedure and the results of the flow and transport models. 1.1 OVERVIEW OF THE GAM PROJECT In the post-operational phase of a nuclear waste repository, gas is released from the disposal caverns, due to anaerobic corrosion of metals and chemical and microbial degradation of organic substances. Previous investigations indicated that discrete waterconducting fractures (e.g. shear zones) are the primarily responsible for the gas transport from the caverns through the geosphere. Gas transport in shear zones takes place preferentially along channels with the highest transmissivity and aperture. Thus, the first step for the characterisation of gas transport processes is to determine the spatial distribution of these properties. 2. Structural and hydraulic information 2.1 STRUCTURAL INFORMATION The GAM shear zone is located in the southern part of the Grimsel Test Site (GTS). In this part of the GTS all the detected shear zones exhibit similar structure. The fractures dip steeply and the dominant strike is EN-SW. Macroscopically, the shear zones are characterised by zones of ductile deformation with high damage intensity with mica-rich mylonite bands, and brittle fault breccia horizons with a thickness ranging from a few millimetres to one centimetre, located at the zones of the highest ductile deformation. These zones contain fine-grained, non-cohesive gauge material (figure 1).
GEOSTATISTICAL MODELING OF TRACER TESTS Bossart & Mazureck (1991) investigated the macroscopic arrangement of this flow paths. Estimates porosity of the gauge material ranges from 0.1 to 0.3. On the other hand, recent studies of Marschall et al. (1999) showed that the aperture distributions across the brittle structures is typically described by a log-normal distribution, with mean values of 0.2 mm-0.5 mm and cumulative thickness in the centimetre range (figure 1). Moreover, the porosity of the ductile areas is comparable to the matrix porosity. Four types of pore spaces were identified on a micro-scaling by Bossart & Mazureck (1994): 1) Grain boundary porosity (connected web-like pore space system along the grain boundaries of the minerals). 2) Mica porosity (pore space parallel to the orientation of the cleavage plane of sheet silicates). 3) Transgranular pores (microfractures and fissures). 4) Solution pores (pores with finely branching cavities extending from both sides of the pore channel into the mineral grain).
Figure 1: Structural characterisation of core samples of shear zones material (from Marschall et al. 1999). (a) bitmap of a core section drilled from the GAM fracture and (b) aperture distributions of 10 cores samples.
Results arising from quantitative porosimetry (Brodsky et al, 2000) showed that an average porosity for the matrix is found to range from 0.25% - 0.5% (volume). The tortuosity factor of the rock material is assumed to be relatively low.
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA 2.2 HYDRAULIC INFORMATION The GAM shear zone was explored with eighteen boreholes instrumented with packers to measure static pressures and to conduct hydraulic and tracer tests in a single fracture. Figure 2 shows the intersection of the boreholes with the GAM fracture. The intersection of the gallery and the shear zone was sealed, such that a no flow condition was achieved. Several short term pulse tests were conducted. The results showed a highly heterogeneous system with a spatial variability of the transmissivity of almost five FRI87-03
Gallery
GAM98-08
AU83-34 GAM98-07
TPF95-05 TPF95-04
TPF95-06 TPF95-01
TPF95-03 TPF95-02
TPF95-07
GAM98-03
GAM98-02
GAM98-04
GAM98-02
5m
GAM98-06
GAM98-01
Figure 2: Intersection of the gallery and the boreholes at the GAM site with the shear zone
orders of magnitude. Four constant rate injection tests RI1, RI2, RI3 and RI4 were performed in boreholes TPF95.01, TPF95.02, GAM98.02 and GAM98.04 respectively. The obtained data was analysed using the code EPHEBO, using Jacob’s method. The results showed a small variability of the transmissivity estimates and a broad range of storativity, as discussed in Meier et al, (1998), as shown in The following table. Several short term pulse tests were conducted. The results showed a highly heterogeneous system with a spatial variability of the transmissivity ranging almost five orders of magnitude. Four constant rate injection tests RI1, RI2, RI3 and RI4 were performed in boreholes TPF95.01, TPF95.02, GAM98.02 and GAM98.04 respectively. The obtained data was analysed using Jacob’s method. The results showed a small variability of the transmissivity estimates and a broad range of storativity, as expected from Meier et al, (1998), as presented in the following table.
GEOSTATISTICAL MODELING OF TRACER TESTS Table: Values of Transmissivity and storativity estimates obtained from the analysis of cross-hole pressure responses (in RI1, RI2, RI3 and RI4) by Jacob's method. Small values in storativity estimates indicate good hydraulic connection between observation borehole and pumping borehole. (I.B. stands for injection borehole) Constant Rate Injection Tests RI1 T*10-10 S*10-6
Observ. Boreholes
RI2 T*10-10 S*10-6
TPF 95.01 TPF 95.02 TPF 95.03
I.B. 2.72 3.27 2.40 12.30
4.06 I.B. 4.66
TPF 95.04
3.85
18.10
TPF 95.05
0.86
2.24
TPF 95.07
2.84 ---------
GAM 98.01 GAM 98.02 GAM 98.03 GAM 98.04 GAM 98.05 GAM 98.06 GAM 98.07 GAM 98.08
-------
1.35
RI3 T*10-10 S*10-6
RI4 T*10-10 S*10-6
7.13
3.85 3.36 ---
2.05 4.70 ---
6.27 5.35 ---
2.75 1.68 ---
4.27
4.25
3.93
2.56
6.29
2.50
0.88
1.41
2.24
2.34
1.98
4.17
5.20
5.06
0.70
3.66
1.05
5.82
2.44
-----------------
-----------------
-----------------
1.83 I.B. 2.27 3.63 3.40 11.00 9.19 5.68
4.40 397.0 2.19 6.60 20.60 66.70 10.00
1.67 8.26 5.81 0.98 2.85 9.33 I.B. 5.48 3.33 10.90 2.01 16.10 28.30 10.10 11.20
With respect to transport data, three dipole tracer tests were conducted: PT1, between boreholes GAM98.02 (injection) and GAM98.04 (pumping) and PT2 between boreholes TPF95.01 (injection) and GAM98.02 (pumping), and PT3 between boreholes TPF95.01 (injection) and GAM98.04 (pumping). Conservative tracers (Uranine, Napthionate and Sulphurodamine) were used in PT1, PT2 and PT3 respectively. 3. Mathematical framework 3.1 DIRECT PROBLEM EQUATIONS Once the motivation and the available information are presented, we will now briefly describe the flow equation and the solute transport equation. We assume these equations are well known and thus skip a complete development. The flow equation can be written as (Bear, 1972):
∇ ( T∇ h ) + q = S
∂h ∂t
in Ω
(1)
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA while the transport equation is (without radioactive decay, matrix diffusion and sink/source terms):
bφR
∂c = ∇(D∇c) − q∇c ∂t
in Ω
(2)
where h is piezometric head, T is the transmissivity tensor, S is the storage coefficient, q is arial recharge, and Ω is the definition domain, D is the dispersion tensor, including dispersion and molecular diffusion, q is Darcy’s velocity, φ is porosity, b is the aquifer thickness and R denotes the retardation factor due to adsorption phenomena. Equations (1) and (2) are solved with appropriate boundary and initial conditions. The latter can be arbitrary or the solution of a previous steady-state situation. 3.1 INVERSE PROBLEM EQUATIONS The ideal statement for any kind of inversion procedure would be to find the set of parameters that make the solution of the model equal to the true heads and/or concentrations. This statement can never be achieved, due to some errors that can not be known separately (wrong conceptualisation of the system, numerical and measurement errors). The usual way to obtain a good agreement between measured (z*) and computed values (z) consists in defining an appropriate distance between these vectors. There are many ways, including Least Squares, Generalised Least Squares, etc. We have selected the Maximum Likelihood Method, as it can account prior information of the parameters to be estimated in a natural way. With the hypothesis that residuals (z-z*) and prior estimates (p*) have a multigaussian distribution, and assuming the independence of errors of heads, concentrations and model parameters, the likelihood function of a hypothesis p on the parameters to be estimated can be written as (Medina and Carrera, 1996): n
(
− L = ( 2π ) 2 Ch Cc Cp
(
) ( t
* * −1 + p − p Cp p − p
)
)
−12
}
{
(
) ( t
)( ) ( ) t
* * * * −1 −1 exp − 1 h − h Ch h − h + c − c Cc c − c 2
(3)
where h and c are computed head and concentration vectors, respectively, h* and c* are measured head and concentration vectors, p is the array containing the estimates of uncertain parameters, p* is the prior information array, Ch, Cc and Cp are the heads, concentrations and parameters covariance matrices, respectively, and n is the total number of data. It is also assumed there exist some unknown scalars τh, τc, τi, (subscript i ranges from 1 to np, number of parameters to be estimated), such that
GEOSTATISTICAL MODELING OF TRACER TESTS Ch=τhVh ; Cc=τcVc ; Ci=τiVi
(4)
where Vh , Vc and Vi are known positive definite matrices. These matrices are based on prior information on the covariance structure of heads, concentrations and parameter prior information, respectively. If τh, τc, τi are assumed to be known, the maximisation of (3), taking (4) into account, can be reduced to the minimisation of: np
S = S h + λc S c + ∑ λi Si
(5)
i =1
(h − h* )t Vh−1 (h − h* ) t S c = (c − c * ) Vc−1 (c − c * ) Sh
=
(p i − pi* ) V (p i − p i* ) t
S
i
=
λc =
(6)
−1
i
τh τ ; λi = h τc τi
(7)
The maximum likelihood method can be used in a geostatistical framework, for a non-linear estimation of the expected values of the natural logarithm of the transmissivity for a large number of zones. Prior information on the T zones estimates and their covariance, which are both necessary for solving equations 3, 4 and 5, were obtained by block kriging. The input parameters for block kriging were the point T estimates from pulse tests, the geometry of the T zones and a correlation structure in terms of a variogram model, a sill and a range. 4. Conceptual model The fracture is modelled as a rectangular (2D) domain of 184*184 m2, centred on the access gallery (figure 3). Fault gouge partially fills the fracture, existing open fracture zones. Transmissivity is treated as a regionalized variable. Its heterogeneity was established by hydraulic test results. Therefore, the model domain is divided into 483 T zones, which are more finely discretized in the region where the observation points are located (intersection between drilled boreholes and the fracture plane). The storativity was assumed to be homogeneous within the model domain because of the difficulty of obtaining storativity from conventional hydrotest analysis (Meier et al., 1998). Due to the existence of milonitization zones within the fracture domain, there are parts where it behaves as an equivalent porous medium [Fischer et al (1991)]. For this
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA reason, application of the cubic law to obtain the dependence between transmissivity and porosity is not suitable throughout the domain. We have adopted a mixed model for the definition of this dependence, where the porosity is a function of the transmissivity value over each zone. If the zone is considered a highly-conductive part of the domain, the cubic law can be applied, while if it is considered a low-conductivity zone, Kozeny’s law is applied. The threshold value considered of transmissivity for the application of these laws is 10-8 m2/s). Once the aperture for each zone is defined (the transmissivity value) its porosity is computed as φi = Φbi, where Φ is the regional porosity, taken as constant and bi is the zone thickness. Dispersivity is treated as homogeneous, and no matrix diffusion process was considered.
Figure 3: Shear zone plane. Left: Full model domain with the gallery of the rock laboratory and the 483 transmissivity (T) zones. Right: Zone of tracer test with T zones. Black dots are the observations points.
5. Results and discussion 5.1 GEOSTATISTICAL INVERSION The T field estimated had an anisotropic correlation structure. The exponential variogram had a sill (σ2lnT) of 7.4 , ranges of 3.0 m in the horizontal and 0.3 m in the vertical direction and no nugget effect was considered. The assumption of an anisotropic correlation function can be based on the qualitative information on the preferred orientation of connected high transmissivity zones. We present the kriging (previous information) transmissivity field and the estimated transmissivity field (figure 4). Drawdown data fits for the four constant rate pumping tests were pretty good (Ramajo et al., 1999)
GEOSTATISTICAL MODELING OF TRACER TESTS
Log10 (T)
Figure 4: Kriging transmissivity field (upper part) and estimated transmissivity (lower part) field. Black dots are the observations points.
5.2 ESTIMATION OF TRANSPORT PARAMETERS Three dipole tracer tests have been performed in the fracture. PT1 between boreholes GAM98.02 and GAM98.04, PT2 between boreholes 98.02 and 98.01, and finally PT3 between boreholes TPF95.01 and GAM98.04. Only concentration data sets from PT1 and PT2 were used simultaneously to calibrate the transport parameters.
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA Two independent transport parameters were estimated: bφ (thickness-flow porosity) and αL (longitudinal dispersivity). The transversal dispersivity αT was fixed at 3.0E-2 m, because simulation showed a relative small sensitivity of the model with respect to variations in αT. Using the transport parameters obtained after the joint calibration of PT1 and PT2 a transport simulation was performed, using the same dipole configuration, and injection concentration used in PT3, to compare the simulated and the measured breakthrough curves. In the upper graph of figure 5 the quality of the fits between computed and
201
Concentration PT3 ( ppb)
181 Meas. PT3
161
Simul. PT3
141 121 101 81 61 41 21 1 0.E+00
2.E+05
4.E+05
6.E+05
8.E+05
1.E+06
Time (s)
Figure 5. Upper graph: Fits between measured and calibrated (joint calibration) concentrations in PT1 and PT2. Lower graph: Simulated vs. measured concentrations in PT3 using the calibrated transport parameters from PT1 and PT2.
measured concentration can be observed after simultaneous calibration of PT1 and PT2, and in the lower graph the simulated and measured breakthrough curves are presented. All of the calibrated breakthrough curves fit the breakthrough time quite well, but the fit worsens in the tail of the curves. This may be due to several reasons. Firstly it could be due to the threshold value used to determine the application of the cubic law or the Kozeny´s law. The secondly it could be due to the fact that diffusion has been neglected
GEOSTATISTICAL MODELING OF TRACER TESTS in the model. The peak of maximum concentration is bigger in the simulation than that of the measured concentration, which would support the second hypothesis. 6. Conclusions Transport parameters were calibrated in the framework of the GAM Project. Due to the conceptualisation of the model, a non-linear mixed dependence between transmissivity and porosity was considered, arising from the combination of Kozeny’s and cubic laws. This conceptual model leads to a good agreement between computed and measured concentrations. Matrix diffusion seems to be an important process in order to fit the tails of the measured breakthrough curves. It is planned to investigate the role of this process and also the correlation between porosity and transmissivity in shear zones. 7. Acknowledgements This work was performed at the Technical University of Catalonia, in the framework of the GAM Project, funded by ENRESA (Spanish Nuclear Waste Management Company), through grant number 703276. 8. References Bear, J., Dynamics of fluids in porous media. Donver, Mineola, N.Y., 1972 Bossart, P., Mazurek, M., 1991. Structural geology and water flow paths in the migration shear zone. NAGRA Internal Report 91-12, Wettingen, Switzerland. Brodsky, N., Stormont, J., Fredrich, J., 2000. Laboratory measurements of porosity, gas threshold pressures, and gas, liquid, and relative permeabilities for shear zone material from the Grimsel Test Site, Switzerlans. NAGRA Internal Report 00-15, Wettingen, Switzerland. Carrera, J., 1993. An overview of uncertainties in modelling groundwater solute transport. J. Contam. Hydrol., 13, 23-48, 1993. Cooper, H., Jacob, C., 1946. A generalised graphical method for evaluating formation constants and summarising well-field history. AGU, 27(4), 526-534 Jaquet, O., Thompson, B.,1991. Inverse modelling in the macro-K experiment at the Grimsel Test Site. NAGRA Internal Report 90-21, Wettingen, Switzerland. Jódar, J., Alcolea, A., Medina, A., 2000. Geostatistical inversion of flow and transport parameters. The Grimsel Test Site. Proceedings of GROUNDWATER2000. Bjerg, P., Engesgaard, P., Krom, Th.D., Editors. Marschall, P., Fein, E., Kull, H., Lanyon, W., Liedtke, L., Müller-Lyda, I., Shao, H., 1999. Conclusions of the tunnel near field programme (CTN). NAGRA Internal Report 99-07, Wettingen, Switzerland.
J. JODAR, A. ALCOLEA, A. MEDINA AND J. CARRERA Medina, A., Galarza, G. & Carrera, J., 1996. Transin-II. Fortran code for solving the coupled flow and transport inverse problem in saturated conditions. In El Berrocal Project. Topical Report, 4(16). Medina, A., Carrera, J., 1996. Coupled estimation of flow and solute transport parameters.. Water Resources Research 32(10), 3063-3076. Meier, P., M.,Medina, A., Carrera, J., 1998. Inverse geostatistical modelling of pumping and tracer tests whithin a shear-zone in granite. Proceedings of GeoENV II 1998. Geostatistics for Environmental Applications. Gómez-Hernandez, J., Soares, A., and Froidevaux, R. Editors Meier, P., M., Carrera, J. & Sanchez-Vila, J. 1999. Estimation of representative groundwater flow and solute transport parameters in heterogeneous formations. Ph.D. dissertation, School of Civil Engineering, Barcelona. Ramajo, H., Jódar, J., Carrera, J., & Olivella, 1999. Progress report on modelling of gas migration processes. NAGRA Internal Report 99-61, Wettingen, Switzerland.
