Fractional flow dimensions and hydraulic properties of a fracture-zone aquifer, Leppävirta, Finland J. Leveinen 7 E. Rönkä 7 J. Tikkanen 7 E. Karro

Abstract In order to characterize the hydraulic properties of an aquifer in Finland comprising two subvertical fracture zones, observation-well responses were matched with generalized radial flow (GRF) type curves. The responses in six wells out of seven are consistent with the GRF model. The fractional flow dimensions (1–1.2 and 1.5) were determined by regression analysis of straight-line slopes and type-curve matching. In each test, the flow dimensions in the neighboring fracture zone range from 2–2.25. Comparisons of the late-time responses with the asymptotic GRF solution and the flow dimensions obtained by reversing the pumping and observation points suggest homogeneous hydraulic properties. Deviations in responses can be explained by flow-path tortuosity. After assessments of the extent of the flow and radial distances along the fracture system, hydraulic conductivities and storativities were determined from the results of the type-curve matching procedure. The obtained hydraulic conductivities are 1.3!10 –5 to 7.9!10 –5 m/s and 5.0!10 –6 to 2.5!10 –5 m/s for the western and the eastern fracture zones, respectively. The results were verified by applying them to analytical solutions for pumping wells. The calculated pumping-well responses are consistent with the observations. The analysis of flow dimension also enhances qualitative interpretations on the hydrogeology of fracture zones. Résumé Afin d’obtenir les caractéristiques hydrauliques d’un aquifère de Finlande compris entre deux zo-

Received, April 1997 Revised, September 1997 Accepted, May 1998 J. Leveinen (Y) 7 E. Karro Department of Geology, P.O. Box 11, FIN-00014 University of Helsinki, Finland Fax: c358-9-1912-3466, e-mail: [email protected] E. Rönkä Finnish Environment Institute, P.O. Box 140, FIN-00251 Helsinki, Finland J. Tikkanen Geological Survey of Finland, P.O. Box 96, FIN-02151 Espoo, Finland Hydrogeology Journal (1998) 6 : 327–340

nes de fractures verticales, les réponses de piézomètres ont été confrontées à des courbes types d’écoulement radial généralisé (ERG). Les réponses de six des sept puits sont compatibles avec le modèle ERG. Les dimensions fractionnaires de l’écoulement (1–1,2 et 1,5) ont été déterminées par analyse de régression des pentes des lignes d’écoulement et par comparaison avec les courbes types. Dans chaque essai, les dimensions de l’écoulement obtenues dans la zone de fracture voisine sont comprises entre 2 et 2,25. La comparaison des réponses de retard avec la solution asymptotique de l’ERG et les dimensions de l’écoulement obtenues par inversion du point de pompage et des points d’observation indique des propriétés hydrauliques homogènes. Les écarts par rapport aux réponses peuvent être expliqués par la tortuosité des cheminements. Après estimation de l’extension de l’écoulement et des distances radiales le long du système de fractures, les conductivités hydrauliques et l’emmagasinement ont été déterminés à partir des résultats de la procédure de comparaison avec les courbes types. Les conductivités hydrauliques obtenues sont comprises pour les zones de fractures ouest et est respectivement entre 1,3!10 –5 et 7,9!10 –5 m/s et 5,0!10 –6 et 2,5!10 –5 m/s. Les résultats ont été vérifiés en les appliquant à des solutions analytiques pour les puits de pompage. Les réponses calculées pour les puits de pompage sont en accord avec les observations. L’analyse de la dimension de l’écoulement met en valeur les interprétations qualitatives sur l’hydrogéologie des zones de fractures. Resumen Para caracterizar las propiedades hidráulicas de un acuífero en Finlandia, con dos zonas de fractura subverticales, se utilizó el método de ajuste de las respuestas en pozos de observación con curvas tipo de flujo radial generalizado (FRG). Las respuestas en seis de los siete pozos son consistentes con el modelo FRG. Las dimensiones fraccionales del flujo (1–1.2 y 1.5) se determinaron mediante regresión lineal y ajuste. En cada ensayo, las dimensiones del flujo en la zona de fractura vecina oscilaban entre 2–2.25. Las comparaciones entre las respuestas a largo plazo y la solución asintótica de FRG, así como las dimensiones de flujo estimadas repitiendo los ensayos e invirtiendo los puntos de bombeo y observación, sugieren homogeneidad en las propiedades hidráulicas. Las desviaciones en las resQ Springer-Verlag

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puestas se pueden explicar por tortuosidad en las líneas de flujo. Tras caracterizar la extensión del flujo y las distancias radiales a lo largo del sistema de fracturas, se determinaron conductividades hidráulicas y coeficientes de almacenamiento mediante ajuste de curvas tipo. Las conductividades hidráulicas de las zonas de fractura oscilan entre 1.3!10 –5–7.9!10 –5 m/s (zona occidental) y 5.0!10 –6–2.5!10 –5 m/s (oriental). Los resultados se verificaron mediante la aplicación de soluciones analíticas. Las respuestas calculadas en el pozo de bombeo son consistentes con las observaciones. El análisis de la dimensión de flujo mejora la interpretación cualitativa de la hidrogeología de zonas fracturadas. Key words Finland 7 fractured rocks 7 flow dimensions 7 hydraulic testing 7 water supply

Notations h r

Hydraulic head Mean length of the flow paths (the product of the tortuosity of flow paths between two wells and the distance between the same wells) t Time after the start of pumping Q Volumetric pumping rate (constant) K Hydraulic conductivity Specific storage Ss b Extent of the flow region (parameter describing the throughflow area near the pumping well) n Flow dimension (fractional) G (x) Gamma function G (a, x) (Complementary) incomplete gamma function

Background The in situ hydraulic properties of an aquifer can be deduced from pumping-test results, either by typecurve matching or by straight-line analysis of drawdown on specialized plots (e.g., Gringarten 1982). However, the assessment of hydraulic parameters is highly sensitive to the spatial dimension of the test analysis (Doe and Geier 1990; Black 1993). Most classical analytical methods assume a horizontal-slab aquifer and radial-flow geometry (dimension 2). If such a method is applied to a long and narrow aquifer with essentially linear flow, a hundredfold error can be made in hydraulic conductivity (Doe and Geier 1990). Fracture zones in crystalline rocks are intensively fractured and brecciated large-scale tabular discontinuities, in which intensive weathering may induce pronounced disintegration and secondary porosity. Such rocks can be considered as hard-rock aquifers. Deviations from horizontal-slab and radial-flow concepts are common even if equivalent porous media can be assumed. Analytical models in which the flow dimension is assumed to be an integer can poorly explain the Hydrogeology Journal (1998) 6 : 327–340

drawdown responses in fractured-rock aquifers in general. More realistic predictions of pumping-test responses commonly require allowing the flow dimension to take fractional values. Faults and fractures of crystalline rocks commonly appear scale-invariant and resemble fractals. Fractal objects show a power-law relation between their number and a characteristic linear dimension (Mandelbrot 1983; Turcotte 1992). The power describing such a relation is referred to as a fractal dimension. Attempts to characterize natural fracture patterns by a fractal dimension are commonly done by superimposing a grid with square elements (boxes) of given length over a map of fracture traces. The number of boxes containing fractures is counted and the procedure is repeated for boxes of different sizes. Plotting the number of occupied boxes against the length of their side on a double logarithmic scale yields a straight line from which the fractal dimension can be calculated (Mandelbrot 1983; Walsh and Watterson 1993). The fractional flow dimension has been equated to the box-counting fractal dimension (Bangoy et al. 1992). Analytical models relying on fractional values of flow dimension (Barker 1988; Hamm and Bidaux 1996) suggest that long-term responses of wells to constantrate pumping show power-law relationships between time and drawdown in double logarithmic plots (Barker 1988; Acuna et al. 1995). This behavior is also a characteristic of fracture networks with fractal geometry. However, analytical solutions and numerical models for fractal systems indicate that the box-counting dimension is not solely adequate to describe the power-law relationships of head changes in simulated fracture networks with realistic fracture-length distributions (Chang and Yortsos 1993; Acuna and Yortsos 1995). Instead, the slope of the straight-line on double logarithmic plots is a function of two components. However, the components describing the geometry and conductance of the fractal fracture system cannot yet be separated out on the basis of well-test responses (Acuna et al. 1995). Furthermore, the applicability of the boxcounting procedure on maps of fracture traverses has been a subject of controversy (Walsh and Watterson 1993). Real fracture systems can probably be modeled as fractals only over a finite range between upper and lower scale limits. The flow dimension deduced from the slope of the log-log straight line is referred to in this paper as fractional rather than fractal, to emphasize that it can be different from the box-counting dimension of the fracture network. Fractional flow-dimension models have been used for the characterization of fracture-zone hydraulics in nuclear-waste repository sites, oil reservoirs, and geothermal fields (Bangoy et al. 1992; Olsson 1992; Acuna et al. 1995). Possibly because the resulting hydraulic parameters are expressed in terms of fractional dimension, the approaches are not widely adopted in practical hydrogeological exploration investigations. The objective of this study is to estimate the hydraulic properties Q Springer-Verlag

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of a fracture-zone aquifer by fractional flow-dimension models. The following analysis of pumping tests is based mainly on the generalized radial flow (GRF) model of Barker (1988). The pumping-test responses are also compared to a recent dual-porosity model with transient block-to-fissure flow and fracture skin effect (Hamm and Bidaux 1996). The example area is in Leppävirta, south-central Finland, where the Finnish Environment Institute, North Savo Regional Environment Centre, and Geological Survey of Finland have drilled several boreholes in fracture-zone aquifers.

Concepts of the GRF Model In the GRF model (Barker 1988), the partial differential equations describing the boundary condition at the well and the flow in a homogeneous aquifer are expressed in terms of hydraulic parameters, distance, and a fractional spatial dimension. The drawdown during constant-rate pumping in an infinite flow region with zero wellbore storage and skin coefficient at the source can be expressed as h (r, t) p

Q r 2Pn n 2

4p Kb

n

G ( 2 P 1, u),

(1)

3Pn

where up

Ss r 2 4Kt

and G (n/2–1, u) is the incomplete gamma function given in the Appendix; n is the flow dimension; K and S s are the hydraulic conductivity and specific storage of the fracture system, respectively; r is the distance along the fracture system, i.e., the mean length of the flow paths; Q is the pumping rate; and t is time after pumping started. The parameter b in Eq. (1) is a measure of “the extent of the flow region” (Barker 1988). When the flow dimension n is equal to 2, Eq. (1) reduces to the well equation introduced by Theis (1935). In the GRF model, the pumping (or injection) well is conceptualized as “a projection of a sphere over n–dimensional space” (Barker 1988). For integral dimensions the mathematical formulation reproduces source terms commonly used in analytical well-test models. For perfectly linear flow (np1), the well is mathematically implemented as a plane, and b is equal to the square root of the throughflow area at the source. For an ideally two-dimensional case (np2), the parameter b is the thickness of the aquifer perpendicular to the flow. For perfectly spherical flow (np3), the term b 3– n is reduced to unity, and hence the value of b is irrelevant. Owing to the generalizations of the boundary conditions at the pumping well and the flow dimension, the parameter b has concrete meaning only when the flow dimension is an integer. The model may not successfully estimate head values in the pumping well; however, Hydrogeology Journal (1998) 6 : 327–340

it should be able to predict head changes in the observation wells. As the distance to the pumping well increases, the influence of skin and wellbore storage to the response become insignificant, and the head in an observation well is given by Eq. (1). The apparent throughflow area is assumed to increase proportionally to r n –1. The idea of the fractional dimension becomes more comprehensible if it is not considered as a parameter describing the geometrical structure of an aquifer. Instead, the flow dimension can be related to the connectivity of the fracture system (Billaux 1989; Chiles and de Marsily 1993; Acuna et al. 1995). Therefore, the dimension reflects the deficit or excess of connections compared to fracture networks that provide perfect connections in 1, 2, or 3 dimensions. If the distance r were known, two parameters could be estimated by applying the type-curve matching procedure. These are K b (3–n) and the hydraulic diffusivity (i.e., the ratio of fracture-system conductivity and specific storage). In the GRF model, hydraulic properties of the aquifer are assumed to be homogeneous. However, fractured rocks are probably exclusively structurally heterogeneous in a manner that produces tortuosity of flow paths. The tortuosity can be defined as the ratio of the mean length of the flow paths between two points (r) to the distance between the same points. In practice, errors in hydraulic parameters result from a failure to assess the tortuosity. The magnitude of the induced errors in the estimated hydraulic parameters depends on the flow dimension, but in general, they are of same order as tortuosity (less than an order of magnitude). However, for the 2-dimensional case, the term r n–2 reduces to unity and these errors should have no effect on the determination of the hydraulic transmissivity (K b). Type curves similar to those shown in Figure 1 could be obtained from a linear aquifer with fixed

Fig. 1 GRF type curve (incomplete gamma function) Q Springer-Verlag

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throughflow area and hydraulic properties changing as a power function of distance (Doe and Geier 1990). The distinction between the models can be seen by performing pumping tests in several wells and comparing the flow dimensions obtained by reversing observation and pumping points. In an aquifer with heterogeneous hydraulic properties, the log-log slope would give different apparent flow dimensions if pumping and observation wells were reversed (Doe and Geier 1990). An approximation of Eq. (1) can be obtained by replacing G (n/2–1, u) with the terms of its series expansion that are significant at large values of t. This produces the asymptotic solution given by Barker (1988) as h (r, t) p

Q n 2

n

4 p Kb 3Pn (1P 2 ) 4 Kt Ss

31 2

1P

n 2

n

4

PG ( 2 ) r 2Pn , n(2

(2)

For flow dimensions less than 2, the time-dependent term dominates in Eq. (2) over the distance-dependent term in long-term responses (Barker 1988; Hamm and Bidaux 1996). Distance-independent behavior at long times is also suggested by the dimensionless pressure solution for a well in a fractal fracture network (Chang and Yortsos 1990). The responses approach a straight line in double-logarithmic plots (Figure 1). The slope of the log-log straight line v is given by Eq. (3), as follows n np1 P , n~2 2

(3)

In addition to the development of the log-log straight line, as illustrated in Figure 2, drawdowns approach similar values in a homogeneous aquifer and plot as swarms of asymptotically conjoining log-log curves at

Fig. 2 A theoretical example of asymptotic drawdown (h) as a function of pumping time for wells locating at 50, 150, and 200 m from the pumping well. Specific storage is 10 –5 m –1, the extent of flow region is 30 m, and the fractional flow dimension is 1.5. The three upper curves are calculated by using hydraulic conductivity of 10 –5 m/s (0.036 m/h), and the three lower curves by using 10 –4 m/s (0.36 m/h) Hydrogeology Journal (1998) 6 : 327–340

long times, even if the observation wells are far apart. Homogeneity of hydraulic properties can also be assessed qualitatively by comparing the long-term responses of two or more observation wells. For flow dimensions greater than 2, the time-dependent term in Eq. (2) approaches zero at large values of time, and the drawdown is stabilized. For a flow dimension less than 2, the stabilization of water levels is possible only if a constant-head boundary exists at a finite distance from the well. The readings made before the stabilization represent a period before the cone of depression has reached the outer (constant-head) boundaries and the aquifer responds as if it were infinite. Analysis of the drawdown before stabilization can suggest whether the stabilization results from a flow dimension greater than 2 or a constant-head boundary in a finite aquifer with flow dimension less than 2.

Regional Geology and Fracture-Zone Tectonics In Finland, mosaic-like networks of fracture zones occupying topographical depressions transect the bedrock throughout the country. Such lineaments are commonly surface expressions of Precambrian shear zones and tension faults (Talvitie 1971; Tuominen et al. 1973; Kuivamäki and Tuominen 1985; Vuorela 1982). Topographical lineaments in Finland and in the study region are illustrated in Figure 3a, b. Horizontal strain and microseismic activity suggest that some of the fracture zones are still active zones of weakness in the prevailing tectonic stress fields (Slunga 1991; Kakkuri and Chen 1992; Saari 1992). The fracture zones in the investigation area are shown in Figure 3c. They were identified tentatively as the traces of topographical and aerogeophysical lineaments; this identity was subsequently confirmed by seismic-refraction sounding and VLF measurements along roughly perpendicular profiles. The soil layers were investigated by ground-penetrating radar (Tikkanen et al. 1994). Within the investigation area, the main rock type is migmatitic gneiss with a few gabbro exposures along the eastern (N-striking) fracture zone. In accordance with the regional tectonic models (Talvitie 1971; Ekdahl 1993), the roughly N-striking fracture zones in the investigation area trace subvertical shear zones that functioned as feeder channels for mafic dikes and intrusions and probably resulted from a dextral movement along the NW-striking shear zones. The basement is overlain by Pleistocene glacial deposits that consist mostly of tills and clays. The bedrock on topographical highs is either exposed or overlain by soil that is a few meters thick. Glacial abrasion has removed any weathered zones from outcropping areas. In topographical depressions associated with fracture zones, till deposits are 9–15 m thick. Q Springer-Verlag

331 Fig. 3 Topographical lineaments in a Finland, b the Kuopio region, and c the investigation area. Lineaments labeled as W, E, and NW are surface expressions of the western, eastern, and NW-striking fracture zones, respectively. The thick black lines correspond to lineaments that were not investigated by geophysical methods

The Pumping Tests The wells bored in the investigation area are 40–50 m deep. In order to ensure that the well yield and the water quality remain within acceptable limits, multiplerate pumping tests were conducted by the North Savo Regional Environment Centre. Three wells were pumped (P1, P2, and P3; Figure 3c) and seven additional wells were used for observations. The pumping rates Hydrogeology Journal (1998) 6 : 327–340

and the drawdown observations are shown in Figures 4, 5, and 6. Three phases of test pumping were carried out: 1. Well P1 was pumped for about 60 d. The pumping rate was 351 m 3/d for about 24 h. During the next 13 d, the pumping rate was increased in several steps until it reached 550 m 3/d. Subsequently, the production rate was lowered stepwise to about 440 m 3/d. The mean production rate was 488 m 3/d. Q Springer-Verlag

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Fig. 4 Pumping rates and drawdowns during the P3 test. Constant pumping rate of 704 m 3/d for first 660 min. A linear regression line is fitted for the observations 40 min after the start of the test. The slope of the line is 0.26 (B0.0024) with a confidence level of 99.9%

iance were tested against F– and t-statistics. Derivative (i.e., rate of change of drawdown vs time) curves provide an alternative method of analysis, but these are sensitive not only to changes of flow geometry but also to small errors during water-level measurements and to discrete sampling frequencies. In the case of the tests at Leppävirta, such curves are disturbingly ’noisy’. Comparison of the double logarithmic plots (Figures 4, 5, and 6) indicates that during each test the response at well P2.2 is characterized by sudden changes in water level and a lack of similarity to the GRF type curves. The drawdown at long times is distinctively smaller than in the other wells in the same fracture zone (P2 and P2.1). Changes and transient damping of drawdown commonly result from the dual-porosity aspects of fractured-porous media (e.g., Moench 1984). Fluid flow in fracture networks is possible only if fractures are interconnected sufficiently to enable percolation, at least through part of the fractures. Permeable fracture networks probably include dead-end clusters of fractures, which are connected to the main network but which do not provide active flow paths. Such subnetworks may act as porous-matrix blocks and induce dual-porosity behavior in hard rocks with insignificant intergranular porosity (Hamm and Bidaux 1996). Therefore, well P2.2 could penetrate a branch of a fracture zone or subnetwork of fractures that interact with the main network in dual-porosity-like manner. How-

2. At the beginning of the pumping test on P2, technical problems occurred that impeded the interpretation of the initial pumping steps. During the first 122 min, the pumping rate was 144–322 m 3/d with a mean rate of 316 m 3/d. The pumping was continued for 1 month with a mean production rate of 350 m 3/d. Subsequently, pumping was also started at well P1 such that both wells were pumped at the rate of about 350 m 3/d for 3 weeks. The water levels were lowered in P1 and P2 by about 16 and 24 m, respectively. Complete recovery required 1 week following the stoppage of the pump. 3. Well P3 and the observation wells H1 and H2 were installed after the tests at P1 and P2 were completed. At the beginning of the P3 test, the discharge rate was kept at 704 m 3/d for 5 h. Pumping was continued for more than 2 months with variable flow rates.

Evaluation of Flow Dimensions Analysis of Test Results The fractional flow dimension at each individual well (excluding well P2.2, which had anomalous results) was estimated by linear regression analysis of log-log curves of drawdown vs time or by manual type-curve matching. The significance of the regression slope and its varHydrogeology Journal (1998) 6 : 327–340

Fig. 5 Pumping rates and drawdown during the first 10 d of the P1 test. Constant pumping rate of 351 m 3/d for the first 24 h. The regression line fitted to the curve slopes 0.255B0.09 at 95% confidence level Q Springer-Verlag

Hydrogeology Journal (1998) 6 : 327–340

2 2.25 2 1.2 2 P4.23 P5.52 P4.28 P5.91 P4.77 P4.66 P4.94 5.86EP05 2.99EP06 5.20EP05 1.28EP06 1.71EP05 2.20EP05 1.14EP05 2.05EP03 4.30EP05 1.82EP03 7.44EP04 5.98EP04 1.05EP03 5.90EP04 P4.64 P5.28 P4.73 P4.95 P4.94 P4.86 P4.91 2.31EP05 5.24EP06 1.85EP05 1.14EP05 1.15EP05 1.39EP05 1.24EP05 P4.05EP01 2.44EP01 P4.50EP01 9.63EP01 P1.71EP01 3.62EP02 3.63EP02 3.94EP01 1.75Ec00 3.55EP01 9.18Ec00 6.75EP01 2.47Ec00 1.09Ec00 8.08EP04 7.54EP05 6.46EP04 6.83EP03 4.04EP04 1.75EP03 6.41EP04 Eastern fracture zone P1 P2 P3 P2 P1 P2.1 P2 P2.1 P3 P2.1 Arithmetic mean Geometric mean

P3.09 P4.12 P3.19 P2.16 P3.39 P2.76 P3.19

2.25 1.5 2.25 1.5 1.5 2 1.5 P4.36 P5.75 P4.59 P5.80 P5.77 P4.01 P6.29 P4.61 P5.23 4.31EP05 1.79EP06 2.56EP05 1.58EP06 1.71EP06 9.72EP05 5.15EP07 2.45EP05 5.96EP06 6.21EP04 3.70EP04 3.68EP04 3.28EP04 3.54EP04 3.40EP03 1.07EP04 7.93EP04 4.47EP04 P4.83 P4.04 P4.85 P4.43 P4.59 P4.79 P4.17 P4.42 P4.53 1.49EP05 9.13EP05 1.40EP05 3.75EP05 2.54EP05 1.62EP05 6.70EP05 3.81EP05 2.96EP05 P4.63EP01 1.71Ec00 P2.62EP01 1.38Ec00 1.17Ec00 P7.78EP01 2.11Ec00 1.50Ec00 6.95EP01 3.44EP01 5.11Ec01 5.47EP01 2.37Ec01 1.49Ec01 1.67EP01 1.30Ec02 3.16Ec01 4.96Ec00

Log (K) Log (K/Ss) Log (Kb 3Pn)

P3.67 P1.72 P3.69 P2.11 P2.28 P3.25 P1.86 P2.18 P2.65 2.14EP04 1.90EP02 2.02EP04 7.77EP03 5.27EP03 5.67EP04 1.39EP02 6.69EP03 2.22EP03 Western fracture zone P2 P1 P3 P1 P2 P1.1 P3 P1.1 P1 P1.2 P2 P1.2 P3 P1.2 Arithmetic mean Geometric mean

Log (Ss)

Flow dimension n Specific storage Ss Generalized storativity Ssb 3Pn Hydraulic conductivity K Hydraulic diffusivity K/Ss Generalized transmissivity Kb 3Pn Obs. well

ever, the full analysis of the response by a dual-porosity model suffers from variable flow rates and scarce drawdown measurements. Alternatively, the damped response could be attributed to well damage or to poor development of the well, which was intended to be used for drawdown observations only. During the P3 pumping test, the drawdown at the pumping well forms a distinctive log-log straight line 30 min after the start of the test (Figure 4). The regression slope is 0.26 (B0.0024 with 99.9% confidence level), corresponding, according to Eq. (3), to a flow dimension of 1.48 for the readings 40 minutes after the start of the test and before the end of the initial period of constant pumping rate. Subsequent changes in the pumping rate induced temporary deviations from the log-log straight-line behavior. As predicted by the GRF model, at long times the observations plot a log-log straight line with a slope that is independent of pumping rate. Because the slope is not controlled by pumping rate, the multiple-rate observations plot fairly similar curves to the constant-rate type curves in Figure 1. Type curve matching of the observation-well readings in the pumped fracture zone suggests a flow dimension of about 1.5. The results of the type-curve matching are listed in Table 1.

Test

Fig. 6 Pumping rates and drawdowns (for 30 d) during the P2 test. During the first 122 min, the pumping rate was 144–322 m 3/d with mean rate of about 316 m 3/d. Linear regression slope of a log-log straight line between 20 and 45 min is 0.50 (B0.021). The observations between 65 and 110 min have a regression slope of 0.389 (B0.0062) with confidence level of 99.9%

Table 1 Estimated hydraulic parameters and flow dimensions at observation wells in the western (wells P1, P1.1, P1.2 and P1.3) and the eastern fracture zone (wells P2 and P2.1)

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At the end of the P3 test, a rapid increase of drawdown is observed at pumping well P3, which is also the northernmost well in the western fracture zone (Figure 3c). This increase suggests a no-flow boundary north of well P3, which is supported by the fact that north of P3 the fracture zone splays and grades into a wider and moderately fractured zone with lower seismic velocities. For test P1, a rapid stabilization of drawdown in the pumped fracture zone follows each step-like increase of pumping rate. A tendency toward a steady state is observed also in wells P2 and P2.1 during the P2 pumping test (observations around 10 3 min in Figure 6). These observations can be attributed to a flow dimension greater than 2, or to a finite aquifer with a constanthead boundary. In order to make the distinction for the P1 and P2 test responses, assessment of flow dimension is based on measurements taken before the stabilization during the initial constant-flow period. During the P1 test, measurements were taken at relatively widely spaced time intervals (Figure 5). The type-curve match procedure with the data from well P1.2 suggests a flow-dimension value around 1.5–1.6. The scarce data measured before the stabilization from the pumping well suggest a log-log straight line with regression slope of 0.255B0.09 at 95% confidence level (i.e., np1.49). The readings made before the start of stabilization at the observation well P2.1 during the P2 test (Figure 6) match with the type curve corresponding to flow dimensions of 1.2. The readings recorded by the minute suggest a log-log straight line between 20 and 45 min with 0.500 regression slope (B0.021 with 99.9% confidence level). The subsequent increase of pumping rate from 295 to 322 m 3/d is not reflected as a distinctive “jump” in the log-log curve. However, during this pumping period, the slope decreases and the observations between 65 and 110 minutes have a regression slope of 0.389 (B0.0062 with 99% confidence level).

Hydrogeological Interpretation The drawdown measurements suggest flow dimensions less than 2 for the western and the eastern fracture zones. A constant-head boundary allowing the stabilization during tests P1 and P2 is attributed to a small lake at the southern boundary of the investigation site. Flow dimensions close to 1.5 for tests (P1 and P3) involving all four of the wells in the western fracture zone (P1, P3, P1.1, P1.2) suggest geostatistically invariant connectivity of fractures. The invariance is also notable because well P3 was intentionally drilled in the intersection of two major fracture zones in order to obtain a greater yield. However, the flow dimension, and thus the connectivity of the fracture system, is not greater at P3. This suggests similar flow dimensions in both fracture zones or poor conductivity for the crossing fracture zone. The latter is consistent with the abunHydrogeology Journal (1998) 6 : 327–340

dant clay in drilling cuts (suggesting intensive weathering to clay) and the lack of response in observation wells H1 and H2 during the P3 pumping test. In general, the spatially invariant dimensions in the western fracture zone suggest that none of the wells extracts water from a higher dimensional domain with substantial storage (such as a branching fracture zone or soil) that would not contribute as significantly to the flow in other wells in that particular fracture zone. The P2 test responses suggest variable flow dimensions in the eastern fracture zone. The observation well P2.1 is near a lineament that cuts both studied fracture zones (Figure 2c), but the lineament has not been investigated by geophysical methods. If a fracture zone is intersected by another with a higher flow dimension (i.e., higher connectivity of fractures), wells near the intersection (such as the well P2.1) could have higher flow dimensions than wells farther from it. Therefore, the observed increase of flow dimensions at the pumping well (P2) possibly results from an increasing fracture connectivity due to a crossing fracture zone. In all three tests, pumping produced similar flow dimensions of around 2–2.2 in the observations wells in the neighboring fracture zone. The flow dimension (and the connectivity) in the network of fracture zones at the investigation area is higher than in a single fracture zone. Although reversed flow directions produce apparent “radial” flow dimensions for the pair of wells P1 and P2, the results do not necessarily indicate homogeneous hydraulic connections between the points. The reversibility of flow dimension can help to reject the possibility of a heterogeneous, tube-like aquifer with a power-law relationship of hydraulic properties and fixed throughflow area (Doe and Geier 1990). However, the method is probably not applicable to a network of fracture zones. Therefore, its application is limited to wells that penetrate the same fracture zone. The observations at well P1.1 during P1 and P3 tests produce a good match with the type curve corresponding to a flow dimension of 1.5. Because the responses to tests were recorded in the same fracture zone during opposite flow directions, the results suggest homogeneous hydraulic connections between these wells. Homogeneity was also assessed qualitatively by comparing the long-term responses. In all the tests, not only are the shapes of drawdown curves similar in a particular fracture zone but also they plot as asymptotically conjoining curves, suggesting homogeneous properties for both fracture zones in the investigation area.

Assessment of Hydraulic Parameters Because the pumping rates varied during the tests completed in Leppävirta, their full analysis would require the use of multiple-rate well-test methods. However, the existing multiple-rate pumping methods are probably not suitable, because they rely on an assumption of a horizontal slab aquifer with radial (two-dimensional) Q Springer-Verlag

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flow, which probably is not an applicable concept for fracture zones. Here, however, the P1 and P3 tests have an initial constant-flow period that is sufficient for quantitative analysis. Observation-well responses during the P2 test were analyzed using the mean pumping rate for the first 122 min.

Assumptions on Flow Paths In order to calculate hydraulic diffusivity (i.e., the ratio of conductivity to specific storage) from the GRF solution, the distance r must be estimated. Here, the fracture zones are considered to provide the minimum tortuosity of flow paths, and radial distance is estimated as the straight-line distance from a pumping well to an observation well. However, the internal structure of fracture zones commonly consists of subparallel cataclastic shear structures that branch and braid in a wide, less intensively fractured matrix. Such structures would manifest their existence as obscured responses that suggest either heterogeneous hydraulic properties or variable tortuosity of the flow paths at the observation wells. A comparison of the responses at the western fracture zone induced by pumping at P3 indicates that the drawdown was greater in P1 than in P1.1, despite P1.1 being closer to the pumping well than P1. Taking into account the conjoining log-log curves of the long-term responses, which are consistent with asymptotic behavior [Eq. (3)] and suggest homogeneous hydraulic properties near wells P1, P1.1, and P1.2, the response at P1 is attributed to variable tortuosity of flow paths rather than heterogeneous hydraulic properties along the western fracture zone. Assumption on the Extent of Throughflow Area Solutions of the GRF model yield values of Kb 3– n and K/Ss. Therefore, to separate hydraulic conductivity and specific storage from hydraulic diffusivity, the extent of the flow region at the pumping well must be estimated. Unfortunately, the mathematical implementation of the pumping well in the GRF model produces problems in defining the appropriate values in cases of noninteger dimensions (Barker 1988). Furthermore, in hard rocks, the flow near a well is strongly controlled by fractures opening into the well. These may not provide a representative sample of the conduits in the fractured system. Therefore, the flow geometry near the well may be different from that in the aquifer as a whole. The drawdown is also influenced by wellbore storage and skin effect. In well testing, these phenomena are reflected as difficulties in matching the early responses in pumping wells with type curves that successfully predict long-time responses in pumping wells or complete data sets from observation wells. The flow to the pumping well spreads until it occupies the whole extent of the fracture system at a finite distance from the well. The flow around a pumping well may be anaHydrogeology Journal (1998) 6 : 327–340

lyzed by developing composite models with different mathematical implementations for the surroundings of the well and the remaining aquifer (Karasaki 1987). The results suggest that for integral dimensional flow, the early-time responses at the pumping well reflect the hydraulic properties of the inner region of a composite model comprising the fractures opening into the well. The late responses represent the properties of the outer region corresponding to the aquifer as a whole (Karasaki 1987). Responses at observation wells located far enough from the boundary between the inner and the outer regions are not affected by the properties of the inner region, i.e., the local fracture characteristics at the pumping well. In the GRF model, the throughflow area is a function of parameters b and r n –1. Even for non-integral values of flow dimension, the value of the b–parameter (which does not have a concrete meaning) should be selected so that it properly describes the growth of throughflow area (proportional to r n –1) in the aquifer rather than in the fractures opening into the well. In future studies, it probably will be possible to relate the b–parameter to a geostatistical property of a fracture system, such as a range of fracture connectivity on a variogram. Here, however, the deduction of geostatistical characteristics of the studied (subvertical) fracture zones probably would be difficult, because boreholes (all vertical) are likely to cut a biased sample of fractures, most of which are probably subvertical. For the investigated aquifers, the extent of the flow was first estimated by assuming radial or linear flow. In the investigation area, the seismic p-wave velocities of the fracture zones range from 3600–3800 m/s, suggesting intensive fracturing across their entire width (Sjögren et al. 1979). With a flow dimension equal to 2, the average hydraulic conductivity over the width of the fracture zone could be estimated by using the widths of the seismic anomalies as an estimate of the b–parameter. For a subvertical fracture zone with ideally linear flow, the throughflow area could be approximated as the width of the fracture zone multiplied by the length of the fractured section in the borehole. For ideally linear flow, the b–parameter would be equal to the square root of the throughflow area. However, the width of the seismic anomaly can be greater than the width of the fracture system contributing to the flow. The values of the b–parameter should be considered as estimates of the maximum value. Because the tests suggest “subradial” flow dimensions (about 1.5) for wells P1 and P3, the widths of the aquifer at the two wells (35 and 37 m, respectively) were taken as estimates of b–parameter values for these wells. The drilling records suggest fracturing along the whole length in the boreholes. Therefore, square roots of the throughflow area estimates would be 36–37 m for these wells. Thus, by assuming integer values for the flow dimension corresponding to linear or radial flow geometry, one will get estimates of b–parameter values that are close to each other and reasonQ Springer-Verlag

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able for the interpretation of the “subradial” flow in wells P1 and P3. The pumping-test results at P2 suggest more linear flow dimensions (1–1.2) for the eastern fracture zone. Therefore, instead of using the estimated fracture-zone width (25 m), the square root of the throughflow area (37 m) was applied for the interpretation of the P2 test. The selection was supported by the values for the western fracture zone, which are more consistent with the results of the P1 and P3 tests. Comparison of the Kb 3– n values should be limited to similar flow-dimension values (Table 1). For cases where n is 1.5, the values of Kb 3– n range from 1.3!10 –2 to 5!10 –3 m 2.5/s (10 –2.3 to 10 –1.9 m 2.5/s). Hydraulic diffusivity (K/Ss) is proportional to the square of radial distance [Eq. (2)], and, therefore, in addition to the inaccuracy of the matching procedure itself, diffusivity estimates are also sensitive to failures in assessing the flow-path tortuosity. These liabilities are probably reflected in the relatively wide distribution of obtained values.

Verification of the Results The analysis of P1 and P3 tests was validated by applying the observation-well parameters to the analytical solutions for pumping wells. Calculated and observed responses are compared in Figures 7a and 7b. The P3 results were compared with the Laplace solutions for a pumping well with wellbore storage and skin in an infinite aquifer (Barker 1988, Equation 21). The average values of hydraulic conductivity and specific storativity for the western fracture zone produce a good prediction of P3 responses at long times, which should be independent of skin factors and wellbore storage. The modeled responses are also close to the measured drawdown at short and intermediate times if the skin factor value is adjusted to about 10 (Figure 7a). Analogies to integer flow cases (Ramey 1970; Nutakki 1982; Karasaki 1987) suggest that the apparently high skin factor results from convergence of flow through relatively few fractures opening into the well. The Laplace solution for finite aquifers (Barker 1988, Equation 39) assumes zero wellbore storage and, therefore, is not likely to give a close match to observed short and intermediate time responses. The average parameter values for the western fracture zone predict drawdown values that are too low compared to P1-test results (Figure 7b). However, good but slightly too high estimates of long-time transient and stabilized drawdowns were obtained by using P1-test results for the western fracture zone. A reasonable match between the observed and predicted onset of stabilization was obtained by adjusting the distance to the constant-head boundary. The values of 550–600 m are in close agreement with the distance to a small lake south of the test area. The P2 analysis is probably affected by the variable pumping rates at the beginning of the test. ThereHydrogeology Journal (1998) 6 : 327–340

Fig. 7a Comparison of the observed and calculated drawdown at the pumping well P3. The modeling was done by numerical inversion of the Laplace solution for a pumping well with constant pumping rate, wellbore storage, and skin factor (s) in an infinite aquifer. Wellbore storage was calculated by using the radius of the steel casing in P3 well. Hydraulic conductivities (K) and storativities (Ss) correspond to the arithmetic mean of the observationwell results. b Comparison of the observed and calculated drawdown for the initial constant-rate period at pumping well P1, based on numerical inversion of the Laplace solution for pumping well with constant-head boundary at finite distance R0 given by Barker (1988). Zero wellbore storage and skin. 1 hydraulic parameters at well P1.2 and R0 650 m; 2 and 3 the arithmetic mean values of the observation-well results for the western fracture zone and R0 650 or 500 m. The data plotting out of the range defined by the curves represent variable pumping-rate steps

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P5.2 P4.9 P4.6 P4.7 P5.8 P5.9 P4.5 P4.9 P4.4 P4.8 P4.6 P4.9 P3.2 P4.1 P4.3 P5.0 9.10Ec02 m 2 1.38Ec03 m 2 P1.2 P2.1 P1 P2

4.5EP02 m 3/s 1.4EP02 m 3/s

5.4EP01 m 1.0EP01 m

Log Ss g-mean Log Ss a-mean Log Ss Log K g-mean Cross-sectional area (A) Channel conductivity (K!A)

Channel storage (Ss!ApS!width)

Log K

Log Ss

Log K

Log K a-mean

GRF-model Leaky model by Gustafson (1976)

Ylinen and Väätäinen (1993) formulated type curves for a tube with a source having a constant flow rate and a fixed-head boundary condition at a finite distance from the source, based on the solutions given in Carslaw and Jaeger (1959). These curves produce a partial match with the observation-well data from the pumped fracture zone during the P1 test and a slightly better match for P2 test responses. The type curves for esker aquifers with leakage from the surrounding media (Gustafson 1976) appear to be applicable to long and narrow aquifers such as a fracture zone and they show a good match with the observation-well data from the pumping tests P1 and P2 (Leveinen 1996). The analysis of observation-well data produces estimates of channel conductivity (conductivity multiplied by cross-sectional area) and channel storativity (storage multiplied by width), presented in Table 2. The constant-rate linear flow solutions (Carslaw and Jaeger 1959) suggest that drawdown at the source forms a straight line with a 1/2 slope in log-log plots, with or without a leakage factor similar to that of Gustafson (1976); this relationship is a characteristic of a linear flow regime in general. However, the early observations at the pumping well P1 and the responses both in the pumping well and in observation wells during the P3 test have distinctively lower slopes. Gently sloping log-log curves can indicate a bilinear flow regime, i.e., linear flow in a formation coupled with linear flow from the surrounding media to the formation (Cinco et al. 1978). A bilinear flow regime has been distinguished in single fractures with finite conductivity and in fractured intrusive dike aquifers surrounded by porous media (Cinco et al. 1978; Boonstra and Boehmer 1986; Boehmer and Boonstra 1986). However, the pumping-well observations are not consistent with these models. As outlined by Barker (1988), the GRF model can be extended to cover the case of dual porosity. In a recent paper, Hamm and Bidaux (1996) completed the procedure by generalizing the theory of transient flow from block to fissure with a fracture-skin effect (Moench 1984). The dual-porosity responses are poorly detectable in log-log plots if the flow dimension is greater than 2. With lower flow-dimension values, the transitional change of the drawdown curve can be masked by wellbore storage and skin effects in pumping wells. Although responses at pumping wells P1 and P3 record a decrease in drawdown rate at early to intermediate times, the dual-porosity behavior is not distinctive at the observation wells, and the observed changes in log-log curves can be attributed to different flow geometry around the pumping well (Acuna and Yort-

Well

Discussion

Test

fore, the aforementioned verification procedure was not attempted. However, the results of the tests also appear to produce a consistent range of values for the eastern fracture zone.

Table 2 Comparison of the results obtained by an Euclidean model for leaky channels by Gustafson (1976) for wells P1.2 and P2.1 (in Leveinen 1996) and results of GRM analysis (values for same wells, arithmetric and geometric means)

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sos 1995; Leveinen 1996). An attempt to apply the dual-porosity approach was not considered necessary for the Leppävirta case. In contrast to the experiences of Hamm and Bidaux (1996), the GRF type-curve analysis of the observationwell responses produces values of hydraulic parameters that predict the drawdown at pumping wells. Appearance of the deviation observed by Hamm and Bidaux (1996) can result from heterogeneity of the hydraulic parameters of an aquifer. However, for flow dimensions less than 2, the observation-well values may include a significant error factor induced by the deviation of the estimated distance (length of a straight line between wells) from the mean distance along the tortuous flow paths. The quantitative estimates of hydraulic conductivities obtained by Euclidean analytical models (Leveinen 1996) and the GRF model are similar, assuming that the throughflow area of discharge is given by the width of the fracture zone multiplied by the length of the fractured section in the borehole. The specific storativity values appear to be more dependent on the applied analytical model. However, in addition to the greater number of observations that can be analyzed quantitatively, the fractional flow modeling of a set of pumping tests provides qualitative information on major flow paths in a fracture-zone network. Rather than using the information on flow dimension to select a method that is based on Euclidean geometry (Doe and Geier 1990), the results of this study encourage the assessment of a realistic range of b–parameter values and their subsequent application to the fractional-flow type curves. Water samples taken during the pumping tests P1 and P2 record steadily increasing electrical conductivities and chloride contents. This increase could mean that in the studied fracture zones flow lines splay subradially downward, thereby reaching deeper groundwater with increased ion concentrations and longer residence times. The pumping tests at P1 and P2 suggest that a small lake south of investigation area affects the hydraulic responses by providing a constant-head boundary. Alkalinity behaves as a conservative parameter in hydrological mixing processes (Stumm and Morgan 1981). Therefore, for example, the increasing trend of alkalinity during the pumping test P1 could be explained by direct infiltration of an anomalously alkaline surface-water source. Changes in alkalinity, oxygen, ammonium, nitrate contents, and in chemical oxygen demand during the pumping test suggest that the discharge also had contributions from shallow groundwaters in the surrounding rock mass and soil. Observation tubes in soil deposits did not respond to the pumping. Although dual-porosity characteristics are not distinctive, it is possible that these changes in water chemistry during pumping are induced by a leaky flux from the fractured rock mass around the fracture zones. The investigations at Leppävirta and other localities in Finland show that hydraulic properties can vary considerably even in fracture zones situated close to each Hydrogeology Journal (1998) 6 : 327–340

other, apparently due to different tectonic reactivation histories and/or weathering intensities that control their permeability. Because the flow dimension is probably a function of the connectivity of the fracture system (Billaux 1989; Acuna and Yortsos 1995), the comparison of flow dimensions at wells near or in junctions of major fracture zones with observations at some distance from them helps to detect the fracture zones that contribute to the flow and also to assess the need to include them in a monitoring program.

Conclusions The transient drawdown data from six wells out of seven are consistent with the GRF model in the Leppävirta investigation area. Flow dimension can be estimated by linear regression analysis of the slope of the longterm responses in log-log plots, but consistent results, although less accurate, can be obtained by type-curve matching if log-log straight lines have not developed. The obtained values of flow dimension (~2) strongly support the preliminary concept that the flow is primarily channeled along the two roughly N-trending fracture zones. By making simplifying assumptions concerning the extent of the throughflow area and the radial distance along the fracture system, hydraulic conductivities and storativities were estimated from observation-well data by type-curve matching procedures. The obtained hydraulic conductivities are 1.3!10 –5 to 7.9!10 –5 m/s (10 –4.9 to 10 –4.1 m/s) and 5.0!10 –6 to 2.5!10 –5 m/s (10 –5.3 to 10 –4.6 m/s) for the western and the eastern fracture zones, respectively. These are high values compared to values reported for most igneous and metamorphic rocks. Comparable values for hydraulic properties can be obtained from the same data by matching with other type curves that are based on Euclidean flow dimensions. However, none of these models could be applied as comprehensively as the GRF model for the analysis of all the drawdown data from the investigation area. Fracture zones may braid and branch, causing tortuosity of the flow paths and errors in estimated radial distances. Such errors cause scatter in the hydraulic conductivity and storativity estimates, which depend on the radial distance to the power of 2–n. The obtained values from observation wells predict the responses at the pumping wells, and, therefore, accurately characterize the hydraulic properties. Comparison of observed, long-term responses and the asymptotic behavior of the GRF model suggest that the fracture zones are relatively homogeneous conduits with deviations that can be explained in terms of flow-path tortuosity. Because the fractional flow dimension is related to the connectivity of the fracture system, spatial and temporal variations of flow dimension may provide information on possible interconnections of major fracture zones. The experiences from Leppävirta suggest that Q Springer-Verlag

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fractional flow models provide important quantitative and qualitative information on the hydraulic properties of major flow paths in fracture-zone aquifers. In most hydrogeological exploration investigations of fracture zones and other hard-rock aquifers, information on the fracture network is often extremely limited. Therefore, information on flow geometry and major flow paths provided by pumping-test analysis could enhance hydrogeological and geophysical interpretations of the fracture zones. Acknowledgments This work is based on the results of the groundwater explorations funded by the community of Leppävirta. In Finland, only a few municipal water plants utilize hard-rock aquifers. Therefore, the authors acknowledge the community officials of Leppävirta for their open-mindedness and for their confidence in the geologists of the Geological Survey of Finland and the North Savo Regional Environment Centre, who conducted the explorations. These institutions are thanked for providing the information and the test results. Professor Esko Mälkki has been a valuable source of encouragement. David Banks and Joel Geier made very constructive comments on the manuscript that substantially clarified the techniques.

References Acuna JA, Yortsos YC (1995) Application of fractal geometry to the study of networks of fractures and their pressure transient. Water Resour Res 3 : 572–540 Acuna JA, Ershaghi I, Yortsos YC (1995) Practical application of fractal pressure-transient system in naturally fractured reservoirs. Soc Petr Eng Form Eval 9 : 173–179 Bangoy LM, Bidaux P, Drogue C, Plégat R, Pistre S (1992) A new method of characterizing fissured media by pumping tests with observation wells. J Hydrol 138 : 77–88 Barker JA (1988) A generalized radial flow model for hydraulic tests in fractured rock. Water Resour Res 10 : 1796–1804 Billaux D (1989) Influence de la connectivité d’un réseau de fracture sur sa résponse à un essai de pompage. In: Maury V, Fourmaintraux D (eds) Proc Symp Rock Mechanics and Rock Physics at Great Depth, pp 473–480 Black JH (1993) Hydrogeology of fractured rocks – a question of uncertainty about geometry. In: Banks S, Banks D (eds) Hydrogeology of hard rocks. IAH Mem XXIV, Part 2, pp 783–796 Boehmer WK, Boonstra J (1986) Flow to wells in intrusive dikes. PhD Dissertation, Free University of Amsterdam, The Netherlands Boonstra J, Boehmer WK (1986) Analysis of data from aquifer and well tests in intrusive dikes. J Hydrol 88 : 301–317 Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Oxford University Press, Bristol Chang J, Yortsos Y (1990) Pressure-transient analysis of fractal reservoirs. Soc Petr Eng Form Eval 5 : 31–38 Chang J, Yortsos YC (1993) A note on pressure-transient analysis of fractal reservoirs. Soc Petr Eng Adv Tech Ser 1 : 170–171 Chilès J-P, Marsily G de (1993) Stochastic models of fracture systems and their use in flow and transport modelling. In: Bear J, Tsang C-F, Marsily G de (eds) Flow and contaminant transport in fractured rock. Academic Press, San Diego Cinco H, Samaniego F, Dominquez N (1978) Transient pressure behaviour for a well with a finite conductivity vertical fracture. J Soc Pet Eng 4 : 253 Doe TW, Geier JE (1990) Interpretation of fracture geometry using well test data. Stripa Proj Tech Rep 91-03, Swedish Nuclear Fuel and Waste Management Co, Stockholm Hydrogeology Journal (1998) 6 : 327–340

Ekdahl E (1993) Early Proterozoic Karelian and Svecofennian formations and the evolution of the Raahe–Ladoga Ore Zone, based on the Pielavesi area, central Finland. Bull Geol Surv Finl 373 Gringarten AC (1982) Flow-test evaluation of fractured reservoirs. In: Narasimhan TN (ed) Recent trends in hydrogeology. Geol Soc Am Spec Pap 189 : 237–263 Gustafson G (1976) A method of calculating the hydraulic properties of leaky esker aquifer system. Nordic Hydrol Conf. Preprints of papers, Session IV, pp 47–64 Hamm S-Y, Bidaux P (1996) Dual-porosity fractal models for transient flow analysis in fissured rocks. Water Resour Res 9 : 2733–2745 Kakkuri J, Chen R (1992) On horizontal crustal strain in Finland. Bull Geodes 66 : 12–20 Karasaki K (1987) Well test analysis in fractured media. PhD Dissertation, University of California, Lawrence Berkeley Laboratory, USA Kuivamäki A, Tuominen HV (1985) Seismotectonic interpretation of the Loviisa region. Electric Power Company Imatran Voima Ltd, Report 85–1 (in Finnish with English Abstr) Leveinen J (1996) Approach to the analysis of drawdown data form a fracture zone aquifer at Leppävirta, East-Central Finland. Acta Univ Carol Geol 40 : 135–150 Mandelbrot BB (1983) The fractal geometry of nature. Freeman, San Francisco Moench AF (1984) Double-porosity models for a fissured groundwater reservoir with fracture skin. Water Resour Res 7 : 831–846 Nutakki R (1982) Pressure transient analysis of wells in very long reservoirs. SPE-AIME 57th Ann Tech Conf, Soc Pet Eng of AIME, SPE 11221 Olsson O (ed) (1992) Site characterization and validation – final report. Stripa Proj Tech Rep 92–22, Swedish Nuclear Fuel and Waste Management Co, Stockholm Ramey HJ Jr (1970) Short-time well test data interpretation in the presence of skin effect and wellbore storage. J Petr Tech, Trans AIME 249 : 97–104 Saari J (1992) A review of the seismotectonics of Finland. Rep YJT-92-29, Nuclear Waste Commission of Finnish Power Companies, Espoo Sjögren B, Ofsthus A, Sandberg J (1979) Seismic classification of rock mass qualities. Geophys Prosp 2 : 409–443 Slunga R (1991) The Baltic Shield earthquakes. Tectonophys 189 : 323–331 Stumm W, Morgan JJ (1981) Aquatic chemistry, an introduction emphasizing chemical equilibria in natural waters, 2nd edn. Wiley, New York Talvitie J (1971) Seismotectonics of the Kuopio region, Finland. Bull Comm Géol Finl 248 Theis CV (1935) The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Trans Am Geophys Union 2 : 519–524 Tikkanen J, Forss H, Kesola R (1994) Bedrock groundwater – an unexploited groundwater resource in Finland, poster paper. In: Suokko T, Soveri J (eds) Future Groundwater Resources at Risk Conf, Public Acad Finland 4 : 285–291 Tuominen HV, Aarnisalo J, Söderholm B (1973) Tectonic patterns in the central Baltic Shield. Bull Geol Soc Finl 45 : 205–217 Turcotte DL (1992) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge Vuorela P (1982) Crustal fractures indicated by lineament density. Photogramm J Finl 1 : 21–37 Walsh JJ, Watterson J (1993) Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies. J Struct Geol 12 : 1509–1512 Ylinen A, Väätäinen A (1993) Interpretation of the 1989–1992 hydraulic interference tests at the five TVO sites in Finland. Rep YJT-93–10, Nuclear Waste Commission of Finnish Power Companies, Espoo Q Springer-Verlag

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Appendix: Gamma Function and Incomplete Gamma Function

as the complementary incomplete gamma function and can be defined as

The gamma function is defined as

G (a, x) p # e Pt t aP1 dt

e

G (a) p # e Pt t aP1 dt

(A1)

0

In mathematics, the term incomplete gamma function is associated with three closely related functions. The function denoted in this paper by G (a, x) is also known

Hydrogeology Journal (1998) 6 : 327–340

(A2)

x

e

The function can be expressed as a series expansion (P1) m x aPm a(0, P1, P2, . . . (A3). mp0 M! (acm) e

G (a, x)pG (a)P A

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