Journal of Hydrology 532 (2016) 149–162
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Review papers The karst permeability scale effect of Sete Lagoas, MG, Brazil ⇑ Paulo Galvão a, , Todd Halihan b, Ricardo Hirata a a University of São Paulo, Institute of Geosciences, Rua do Lago 562, São Paulo, SP 05508-080, Brazil b Oklahoma State University, Boone Pickens School of Geology, 105 Noble Research Center, Stillwater, OK 74078, USA article info summary
Article history: Collecting and interpreting permeability data in karst systems is considered complicated due to three dis- Received 25 July 2015 tinct properties of these systems. First, the distribution of high permeability features may be one- Received in revised form 30 October 2015 dimensional features difficult to detect with wells, or may be so high in the wells the upper measurement Accepted 16 November 2015 limit is encountered during aquifer testing. Secondly, turbulent flow may make the application of contin- Available online 21 November 2015 uum hydraulic principles difficult. Finally, permeability in these systems commonly increases with the This manuscript was handled by Corrado Corradini, Editor-in-Chief, with the scale of measurement. The aquifer for Sete Lagoas, Brazil, was used to evaluate a permeability combina- assistance of Barbara Mahler, Associate tion methodology testing the permeability structure across a range of spatial scales in order to develop a Editor quantitative model of hydraulically active features consistent across all scales of measurement, from matrix properties to regional-scale flow. The aquifer in this study has some wells without measurable Keywords: drawdown during pumping due to high permeability. Data indicated an increase in permeability from Karst the small- to the well-scale and a decrease from the well- to regional-scale due to the localized develop- Permeability ment of a karst bedding plane dissolution in one structurally controlled region of the aquifer. The matrix Transmissivity permeability in the region is very low and the secondary porosity is mostly filled by secondary precipi- Hydraulic conductivity tation of calcite. Based on measurement technique, the permeability data vary over many orders of mag- Scale effect nitude, while the physical size of permeable features of the aquifer are consistent across the scales of data Brazil collection. The geometry provides a quantitative understanding of the scale effects of permeability measurements. Ó 2015 Elsevier B.V. All rights reserved.
Contents
1. Introduction ...... 150 2. Site description...... 151 3. Methods ...... 151 3.1. Permeability database ...... 151 3.1.1. Small-scale: matrix data, optical microscopy analysis and permeability measurements ...... 152 3.1.2. Well-scale: aquifer test data and empirical relationship analysis ...... 152 3.1.3. Regional-scale: potentiometric data and capture zone analysis ...... 153 3.2. Permeability component data...... 153 3.2.1. Sub-horizontal fracture hydraulic apertures...... 153 3.2.2. Hydraulic conduit diameters ...... 154 3.3. Permeability combination models ...... 154 4. Results...... 154 4.1. Sete Lagoas permeability database...... 154 4.1.1. Small-scale analysis and measurements ...... 155 4.1.2. Well-scale: aquifer test data and empirical relationship analysis ...... 155 4.1.3. Regional-scale: potentiometric data and capture zone analysis ...... 157
⇑ Corresponding author. Tel.: +55 986 677 505. E-mail address: [email protected] (P. Galvão). http://dx.doi.org/10.1016/j.jhydrol.2015.11.026 0022-1694/Ó 2015 Elsevier B.V. All rights reserved. 150 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162
4.2. Permeability feature sizes from inversion ...... 158 4.2.1. Small-scale: matrix data and feature sizes ...... 158 4.2.2. Well-scale permeable feature sizes ...... 158 4.2.3. Regional-scale: overall bedding planes and karst aquifer thicknesses ...... 158 4.3. Permeability combination models ...... 158 5. Discussion...... 159 6. Conclusion ...... 161 Acknowledgments ...... 161 Appendix A. Notation ...... 161 Appendix B. Supplementary material ...... 161 References ...... 161
1. Introduction karstic conduits. Halihan et al. (2000) tested if the Király (1975) hypothesis has an underlying physical basis using a permeability Formation bulk permeability depends on the shape, amount, database for the Cretaceous carbonates from the San Antonio seg- and interconnectivity of void spaces, or permeability structure. ment of the Edwards aquifer and found the Király hypothesis was This is often quantified with testing in the form of hydraulic con- applicable for the aquifer, with nine orders of magnitude variabil- ductivity or transmissivity values, which incorporate fluid proper- ity in permeability. ties and aquifer dimensions into the measurements. Grain size Clauser (1992) analyzed a compilation of different type of crys- analyses, permeameters, aquifer tests and groundwater models talline rocks and noted the permeability can increase three orders are most commonly used to obtain permeability estimates from of magnitude from small- to well-scale, but from well- to regional- small- to regional-scales. However, the permeability values can scale, the increase would not continue. The Halihan et al. (2000) vary over orders of magnitude, depending on the geological forma- model indicated if larger fractures were present, the permeability tion. Permeability data are the fundamental property of the rock would continue to increase with scale. Many of these approaches body and most easily converted to the physical dimensions of compiled data at a range of scales, but few provide a physical basis pores, fractures, or conduits. This work will assume that fractures for the values to allow predictions for various scales. are two dimensional features and conduits are largely one dimen- Martinez-Landa and Carrera (2005) analyzed the hydraulic con- sional flow features. ductivity scale effects in fractured granite in Grimsel, Switzerland. Karstic carbonate rocks are one of the most heterogeneous and The study used different types of hydraulic tests (pulse, recovery, anisotropic of all formations found in nature, because the perme- cross-hole, and tunnel inflow measurements) for matrix- and ability structure of these formations are created by fluid flow gen- borehole-scale, and cross-hole numerical interpretations for erating a hierarchal rock structure of flow features facilitating large-scale flow. The average hydraulic conductivity increases by circulation in the downgradient direction (Mangin, 1975). Conse- orders of magnitude as the test scale grows. The zones of high quently, karst hydraulic parameters are not an independent, inher- hydraulic conductivities are hydraulically interconnected and con- ited static attribute of the rock, but are dynamic over geological trol the permeability at large scales. Other geological and hydroge- time (Huntoon, 1985). Another consideration is this structure can ological investigations related to scale effect using different type of adjust as the flow regime responds to changing hydraulic boundary permeability tests were developed in fractured rocks (Maréchal conditions. Thus, the hierarchy of permeability pathways and their et al., 2004; Illman, 2007), and sedimentary formations (Landon organization within the formation are dictated by the hydrody- et al., 2001; Kurikami et al., 2008; Chapuis, 2013). The close rela- namic characteristics of the flow system operating within the tionship between hydraulic interconnectivities in the rock hetero- recent geological past and the characteristics of older flow systems geneities and hydraulic conductivities increases average (Ewers, 1982). permeability data across the scales. The difficulties in the estimation of karst permeability are One approach to quantify the physical basis of the scale effect is amplified by flow that may be non-Darcian, which leads to the cau- via permeability combination models (PCM). The models provide tious use of continuum estimations assuming Darcian flow. estimates of the averaged permeability generated by a mixture of Depending on the dissolution feature size, some standard aquifer different permeable features in an aquifer, or the numerical effect tests may obtain no measurable drawdown as the test reaches of small-scale features on the well- or regional-scale data (Halihan the upper measurement limit. This results in lower bound estimate et al., 1998, 1999, 2000). The steady-scale geometric models esti- in the calculation of transmissivity and hydraulic conductivity val- mate the effects of different heterogeneities on formation perme- ues. These factors result in uncertainties in making permeability ability structure, assuming measurements on different scales are estimates are a key factor to understanding groundwater flow all valid for the scale the data were collected. These models are and solute transport in the subsurface (Quinlan and Ewers, 1985; modifications of equations for layered aquifers (Leonards, 1962; Rehfeldt et al., 1989). Fetter, 1994). These formations and their heterogeneity can lead to well- To predict formation values at the well-scale in a karstic forma- known permeability scale effects (Rats, 1967; Király, 1975; tion, standard aquifer tests (slug tests, time-drawdown tests and Maclay and Land, 1988; Hovorka et al., 1995; Halihan et al., step-drawdown tests) and the standard analytical equations (e.g. 1999). Scale effects are generally defined as continual increases Theis, 1935; Cooper and Jacob, 1946), assume Darcian flow and in permeability magnitude from small- to regional-scale estimates often provide reasonable fits to the data (Kresic, 2013). These esti- (Király, 1975; Schulze-Makuch et al., 1999). In the Király (1975) mates should be used with caution, due to anisotropic or potential study, made in Mesozoic fractured and karstic limestone aquifers non-Darcian conditions. The estimates may not be valid on a located in the Jura Mountains, Switzerland, it was hypothesized regional-scale, due to scale dependency. In case of regions with the increase from small- to the well-scale was caused by fractures limited pumping test data, but having specific capacity test data, and the largest permeabilities on the regional-scale were caused by an alternative method to estimate permeability utilizes empirical P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 151 relationships between transmissivity and specific capacity (Razack filled by the precipitation of calcite (Tonietto, 2010). The majority and Huntley, 1991; Mace, 1997). In cases with no measurable of groundwater migrates through dissolution features character- drawdown occurring in a well due to high permeability, a lower ized as tertiary porosity. bound estimate can be obtained using a Thiem (1906) method Results from geologic mapping and optical well logging suggest (Halihan et al., 2000). the urbanized area of Sete Lagoas (Fig. 1) is primarily located in a On the regional-scale, groundwater models can be used to graben (Galvão et al., 2015). The area is filled with limestones from obtain permeability estimates (Scanlon et al., 2001, 2003). Another Sete Lagoas Formation and overlying Cenozoic unconsolidated sed- analytical approach is using a regional capture zone analysis, iments, with portions covered by competent rocks from the Serra defined as the area of an aquifer in which all the water will be de Santa Helena Formation (slate, marble, siltstone and argillite). removed by a pumping well or wells within a certain time period The graben limits results in barrier boundaries for groundwater, (Todd, 1980; Grubb, 1993). In some cases, equations can be super- recharging and accumulating water in the aquifer. The most rele- imposed to calculate the capture zone of multiple well systems vant discontinuities for groundwater circulation are from natural (Javandel and Tsang, 1986). karst processes in a low matrix limestone. This explains the devel- This research conducted an evaluation of multi-scale perme- opment of two dominant solutionally enlarged horizontal bedding ability structure, estimating values of permeability at the small-, planes in the Sete Lagoas Formation with high permeability and well-, and regional-scale, along with hydraulic conductivity and significant storage capacity. The two bedding planes are a thick, transmissivity. These data were compared against observations of shallow continuous dissolution zone with conduits 1–8 m thick, the physical size of permeable features, as well as calculations of near the contact with the overlying unconsolidated sediments; the hydraulic dimension of fractures (hydraulic apertures) and and 10–20 m below, a thinner continuous dissolution zone with conduits (hydraulic diameters). Combining different methodolo- conduits that are 20 cm–1 m thick. The limestone dips and become gies, a quantitative hydrogeologic conceptual model was devel- thicker, reaching about 160 m thickness, and is completely covered oped, consistent across all scales of measurement. The aquifer by rocks from the Serra de Santa Helena Formation to the north- studied was the Sete Lagoas karst aquifer (Pessoa, 1996), located east. The two bedding plane conduits become thinner to the north- within the limits of the municipality of Sete Lagoas, Minas Gerais, east (Galvão et al., 2015). Brazil. This aquifer has common complications for karst systems Based on optical well logs and outcrop field observations, the with meter scale pores and some pumping tests obtaining no mea- hydrogeologic conceptual model for the aquifer includes both surable drawdown. Quantifying these parameters through differ- non-karstic and karstic bedding plane apertures. This conceptual ent scales is important because of the increasing demands on the model is based on previous work conducted for a karst risk assess- aquifer. The demands require users to have a quantitative under- ment (Galvão et al., 2015). The terms ‘‘non-karstic bedding planes” standing of how the aquifer functions so that it can be properly uti- for zones without karstification, and ‘‘solutionally enlarged bed- lized and protected. The PCM approach was utilized, comparing ding planes” for dissolution zones will be used here. small-scale permeability data calculated by a permeameter with well-scale parameters calculated by standard pumping tests and 3. Methods an estimated regional-scale value, via capture zone analysis of a multiple pumping well system. These calculations were made to The evaluation of scale effects in this research first involved test possible correlations between different ranges of scale and building a database of permeability measurements from small- to quantify variations in the permeability scale effect. While values regional-scales for the aquifer. Small-scale permeabilities were of permeability vary over many orders of magnitude, the physical measured by means of a hand-held air permeameter. Well-scale size of flow features controlling permeability measurements would evaluation utilized aquifer tests and a relationship derived remain a consistent quantifiable dataset for the aquifer. between specific capacity and transmissivity. A regional-scale esti- mate utilized potentiometric surface measurements and a capture 2. Site description zone analysis approach. Empirical calculations were utilized to invert the database val- The aquifer studied was the Sete Lagoas karst aquifer (Pessoa, ues to determine what size hydraulic features (i.e. pore diameter, 1996), located in the municipality of Sete Lagoas, Minas Gerais, fracture hydraulic aperture, or conduit hydraulic diameters) would Brazil, 70 km northwest of Belo Horizonte (Fig. 1). The city has a be required to obtain measured field values of formation perme- population greater than 200,000 over an area of 538 km2 (IBGE, abilities. These inverted values were utilized in a forward modeling 2010). The area of highest population density is located in the older PCM approach similar to Halihan et al. (1999) to test the stability of urbanized part of the city. The current water supply is almost permeability features across the scales of data. These methods will entirely groundwater from the aquifer sourced from private and be presented sequentially from the bulk permeability measure- public wells. The public supply wells are managed by the Water ments to the inverted estimate of the size of hydraulic features, Supply and Sewage Service (SAAE) [SAAE – Serviço de Abastecimento and finally the PCM forward modeling. de Água e Esgoto]. The highest pumping rates are located in areas The order of magnitude for each scale adopted, for field obser- with the greatest population density (Fig. 1). vations and analytical analysis, was based on Király (1975). These Geologically, the area is located in the São Francisco Craton, dimensions included small-scale from 10 2 to 100 m, by the size of where carbonate argillo-arenaceous sediments are emplaced giv- the rock matrix permeameter analysis; well-scale between 100 and ing origin to the Bambuí Group (Branco and Costa, 1961; 102 m, by pumping test drawdown radius of influence; and Oliveira, 1997; Schöll and Fogaça, 1973; Dardenne, 1978; regional-scale from 102 to 104 m, according to the size of the cap- Schobbenhaus, 1984; Ribeiro et al., 2003; Tuller et al., 2010). The ture zone analysis of the multiple pumping well system. hydrostratigraphic relationships for the Sete Lagoas karst aquifer and spatial distribution are given in Pessoa (1996) and Galvão et al. (2015). The aquifer, which is approximately 75 m thick, con- 3.1. Permeability database sists of Neoproterozoic limestones, composed of two members: the Pedro Leopoldo Member (at the base) and the Lagoa Santa Member For this work, small-scale measurements utilize hand samples (on the top). The primary porosity and matrix permeability can be and thin sections of aquifer matrix. This measurement scale is considered low (Moore, 1989) and the secondary porosity is mostly occasionally divided by authors into a hand-scale and a micro- 152 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162
Fig. 1. Location map in UTM coordinates of Sete Lagoas showing the municipal boundary and urbanized area, the well locations (private and public) and the geology of the study area. The highest pumping rates are located in the areas with the greatest population density.
scale, but they are combined here to represent measurements of Permeability measurements were performed on dry hand sam- carbonate matrix pore properties. Well-scale data are derived from ples of limestones sourced from the two members of the Sete aquifer tests. Wells reaching the upper measurement limit by not Lagoas Formation. The measurements were collected across dry, being able to draw down the water table are included in the anal- fresh and planar rock surface, free of dust and without any weath- ysis. While many authors utilize groundwater model data for ering effects. Measurements (24 total) were taken to determine the regional estimates (Thorkildsen and McElhaney, 1992; Halihan mean small-scale permeability, ksm, in Darcys. These measure- et al., 2000; Scanlon et al., 2001, 2003), this work utilizes an analyt- ments were converted to square meters, and then estimates of ical capture zone solution. the small-scale hydraulic conductivity, Ksm, and transmissivity, Tsm, of the entire aquifer thickness were calculated. The advantage of this non-destructive method is the rapid and reproducible 3.1.1. Small-scale: matrix data, optical microscopy analysis and results (Fossen et al., 2011). permeability measurements Field observations and optical microscopy using thin section for 3.1.2. Well-scale: aquifer test data and empirical relationship analysis the two members of the Neoproterozoic limestone rocks from the Seven long duration (48 h) transient pumping tests were con- 2 Sete Lagoas karst aquifer were made in order to describe the fea- ducted to measure the well-scale transmissivity, Tw (L /t), using tures of the matrix, such as rock mineralogy, grain size and the Theis (1935) method. Pumping rates ranged from 47 to micro-fractures. A TinyPerm II portable hand-held air permeame- 194 m3/h. For the case of a zero drawdown well test in well PT- ter, manufactured by New England Research (NER), was utilized 01, the Thiem equation (Thiem, 1906) was used for analysis, to measure the permeabilities in the matrix of these members. assuming 0.01 m (1 cm) of drawdown in the pumping well. This P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 153 well and the observation wells at distances of 25 m and 51 m were (L) is the half width of the maximum capture zone, and i (dimen- located in the central portion of the urbanized area. The test sionless) is the hydraulic gradient. 3 3 pumped PT-01 at a constant rate, Qw (L /t), of 130 m /h (Table S1 The regional discharge rate, Qr, was estimated by summing the in the electronic supplementary material (ESM)). The bedding pumping rate data values from public supply wells (data provided plane dissolution can be very large in some regions of Sete Lagoas. by SAAE and Servmar Environmental & Engineering) and private This affects the permeability values as deduced by the Darcian wells (data acquired from the SIAGAS’s database – SIAGAS, 2006) approach and implied turbulent flow. The effect would be noted extracting groundwater within the capture zone. Transmissivity, in the Theis curve, but the Theis method was applied based on Tr, was determined based on the best fit-empirical capture zone the analysis of measurements in the seven pumping tests. This in comparison to the potentiometric capture zone, using the equa- assumption would result in a potential underestimation of hydrau- tion describing the empirical edge of the capture zone for a con- lic dimensions of the conduits, but forward modeling of the upper fined aquifer: limit of pumping tests (Halihan et al., 2000) indicates these tests ¼ hiy ð Þ cannot be used to evaluate features of this size. By obtaining mea- x p 3 tan 2 Twiy surable drawdown, the hydraulically connected feature size must Qr be similar to the estimated values. where x (L) is the distance parallel to the regional hydraulic gradi- Twenty-seven time-drawdown or step-drawdown tests were ent and y (L) is the distance perpendicular to the regional hydraulic done to estimate specific capacity, S (L2/t), utilizing the SAAE’s C gradient (Todd, 1980; Grubb, 1993). A sensitivity analysis was con- supply wells. In mathematical terms, S is defined as the pumping C ducted to evaluate the variability possible in this approach. rate in the well, Q (L3/t), divided by the observed decline in w The final regional transmissivity was considered the average of hydraulic head in the well, Dh (L), from an aquifer test. In the case w both analytical values calculated. The regional-scale hydraulic con- of the step-drawdown tests, every discharge value was divided by ductivity, K (L/t), was calculated by dividing the transmissivity its respective drawdown, Dh , observed in each step, and then the r w values by the average total thickness, b (L) (estimated at mean S was calculated. The results were analyzed using an empir- total C 75 m), and then converting the values to permeability in square ical relationship derived for this karst aquifer by fitting a power meters. law function between measured specific capacity and transmissiv- All the permeability data sets were entered in a GIS database ity, based on the Razack and Huntley (1991) and Mace (1997) and georeferenced in the ArcGIS 10.1 software. The coordinate sys- methods. The goal was to estimate the most representative well- tem was Universal Transverse Mercator (UTM) projection, Zone 23, scale transmissivity, both in non-karstic and solutionally enlarged datum SAD 69, with units in meters. Data from optical well logs bedding planes, and then estimate the well-scale hydraulic con- and analytical calculation results were considered to estimate the ductivity and permeability. Information about the physical thick- regional sizes features of the dissolution zones, b (L). These values ness of bedding planes, b (L), was established with optical well r w allowed a comparison between observed physical apertures and logging and outcrop field observations (Galvão et al., 2015). inverted hydraulic apertures for the various data. A transmissivity map was constructed from the data to observe the variability at the regional scale from well scale data. T values w 3.2. Permeability component data were utilized assuming the long duration pumping test data were more accurate data than the empirical values from specific capac- The permeability data were inverted to determine the physical ity data. The values that were found in the observation wells for size of hydraulic features capable of producing the various perme- their respective pumping wells were located in the same hydroge- ability values and to test the stability of permeability feature sizes ologic setting (Table S1). across various scales of measurement. The hydraulic size of frac- tured apertures and conduit diameters were calculated under 3.1.3. Regional-scale: potentiometric data and capture zone analysis assumed laminar and turbulent flow conditions. Measurements of the water table were taken only in SAAE’s supply wells in the Sete Lagoas karst aquifer to develop a potentio- 3.2.1. Sub-horizontal fracture hydraulic apertures metric map for regional-scale permeability evaluation. The well The conversion between permeability values and the size of a casing elevation data were acquired using a SRTM image (Shuttle sub-horizontal fracture was calculated using the cubic law
Radar Topography Mission). According to Demétrio et al. (2006), (Lamb, 1932) between the intrinsic permeability of a fracture, kf 2 the accuracy in the use of this data is less than 5 m. (L ), and a hydraulic fracture aperture, bf (L): A capture zone analysis for the central high pumping rate well 2 area was performed and an assessment of the accuracy of the ana- bf k ¼ ð4Þ lytical results was estimated. Some assumptions were made, such f 12 as pumping wells in a confined aquifer with a uniform flow, under This equation was used to convert between measured perme- steady-state conditions. A capture zone area was determined, ability values and physical feature size, but is only an approxima- based on estimated flow lines in the potentiometric surface. Ana- tion. The calculations do not provide accommodation for fracture lytical equations, derived for confined aquifers by Todd (1980) plane roughness and variability in fracture geometry. and Grubb (1993) delineating the edge of the capture zone, were The hydraulic fracture aperture was calculated from transmis- 2 used to estimate the regional-scale transmissivity, Tr (L /t). sivity or conductivity data from aquifer tests using the hydraulic aperture of a single low-angle fracture equation (Halihan et al., Q r Tr;x ¼ ð1Þ 1999), b (L): 2pX i f L 1=3 Q r 12v cosðhÞ ; ¼ ð Þ Tr y 2 bf ¼ ðTw KsmbtotalÞ ð5Þ 2YLi g
2 2 where Tr,x and Tr,y (L /t) are the regional transmissivities in their where ʋ is the kinematic viscosity of the water (L /t), Tw is the well- 3 2 respective directions, Qr (L /t) is the overall pumping rate or regio- scale transmissivity (L /t) calculated by pumping test, Ksm is the nal discharge rate, XL (L) is the distance from the pumping well to small-scale hydraulic conductivity of the matrix (L/t), btotal is the 2 the downgradient edge of the capture zone (stagnation point), YL total thickness of aquifer (L), g is gravitational constant (L/t ), and 154 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 h is the angle between the fracture and the horizontal plane. If the estimated from optical logging or outcrop evaluation, as they are fracture is horizontal, the angle is zero. idealized hydraulic dimensions, not physical ones dealing with roughness and connectivity. This set of equations allowed inver- 3.2.2. Hydraulic conduit diameters sion of the permeability database. The same concept can be used
The hydraulic conductivity of a conduit, Kc (L/t), can be related to generate forward models for evaluation of scale effects. to a conduit hydraulic diameter assuming both laminar and turbulent flow (Turcotte and Schubert, 1982; Halihan et al., 3.3. Permeability combination models 1998). The hydraulic conductivity equations can also be converted to permeability; however, hydraulic conductivity is simpler to For the permeability combination forward model approach illustrate the turbulent case. The laminar hydraulic conductivity, (PCM) to evaluate the scale effects in the karst aquifer, the analyt- K (L/t): c,lam ical models developed for small- to regional-scale were combined 2 and compared. These models are steady-state and modifications of gdc K ; ffi ð6Þ c lam 32v equations for layered aquifers. Other assumptions were used to simplify the models. The hydraulic conductivity of the matrix is For turbulent hydraulic conductivity K (L/t): c,turb much less than the hydraulic conductivity of the fracture/conduit. 4 5 7 This can be evaluated by utilizing the permeability database. The ffi : g7 dc ð Þ Kc;turb 4 706 1 7 solutionally enlarged bedding planes are continuous permeable v 7 2 features controlling the hydraulic system, where sub-vertical frac- The single horizontal conduit equation was utilized (Halihan tures do not have significant influence on permeability structure et al., 1999) to estimate the hydraulic conduit diameter, dc (L), from across the scales of measurement (Galvão et al., 2015). aquifer tests. Assuming the hydraulic conductivity of the matrix is much less than the hydraulic conductivity of the conduit, Kc: 4. Results Kcdc Kbtotal Tw K ffi ) dc ffi ) dc ffi ð8Þ btotal Kc Kc The results will be presented sequentially from the bulk perme- ability measurements, to the inverted estimate of the size of Substituting dc in Eq. (6), to estimate the laminar hydraulic con- hydraulic features, and finally the PCM results. duit diameter, dc,lam (L):
1 v 3 d ; ffi 32T ð9Þ 4.1. Sete Lagoas permeability database c lam w g
Substituting dc in Eq. (7), to estimate the turbulent hydraulic The permeability database for Sete Lagoas illustrates scale conduit diameter, dc,turb (L): effects do not always increase with scale. A figure illustrating the Sete Lagoas permeability scale effect through scales, compared !7 1 12 v 7 against the Király (1975), Clauser (1992) and Halihan et al. d ; ffi T 0:349 ð10Þ c turb w 4 (2000) studies, as well as with igneous rock, limestone, karst, and g7 gravel permeabilities (Heath, 1983) was generated to illustrate The use of these equations allows permeability data from the the effect in comparison to other carbonate aquifers (Fig. 2). These range of scales to be inverted into a physical feature size. These studies were chosen for comparison with the Sete Lagoas scale estimates should be smaller than the physical size of the features effect data because they have similar approaches and illustrate
Fig. 2. Permeability scale effects in the Sete Lagoas karst aquifer (red line) in comparison with other studies and with permeability ranges for igneous rock, limestone, karst, and gravel (Heath, 1983). The local scale effect presents an increase of hydraulic conductivities from small- to well-scale and a decrease of these values from well- to regional- scale, having similar behavior with the compilation of crystalline rocks scale effect line (Clauser, 1992 – gray line), with a lower order of magnitude. In the case of Edwards aquifer scale effect line (Cretaceous limestones and dolomites from San Antonio segment, Halihan et al., 2000 – black line) and the Jura Mountains scale effect line (Mesozoic fractured and karstic limestone aquifers, Király, 1975 – dashed gray line) significant increases in permeabilities across scales are common. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 155
Table 1 Values of sub-horizontal fracture aperture, hydraulic conduit diameter in laminar and turbulent flow and its respective K and T values. Zero drawdown pumping test not included.
Measurement Well-scale transmissivity Well-scale hydraulic Estimated fracture Estimated conduit Estimated conduit adopted (m2/d) conductivity (m/d) aperture (m) diameter assuming diameter assuming laminar flow (m) turbulent flow (m) Method/equations
1/3 1/7 4/7 7/12 Well ID Most representative K = Tw/bm bf =[(12t cos(h)/g) Á dc,lam =(32 Á Tw Á t/g) dc,turb =(Tw Á t /g 0.349) 1/3 (Tw Km Á bm)] PT-12 1550 19 2.8 3.9 7.7 PT-13 1550 19 2.8 3.9 7.7 PT-18 1620 20 2.8 3.9 7.9 PT-19 980 12 2.4 3.3 5.9 PT-21 980 12 2.4 3.3 5.9 PT-22 330 3.7 1.7 2.3 3.1 PT-24 1590 18 2.8 3.9 7.8 PT-25 330 3.7 1.7 2.3 3.1 PT-28 1100 18 2.5 3.5 6.3 PT-29 3190 32 3.6 4.9 12 PT-30 3590 36 3.7 5.1 13 PT-36 1320 17 2.7 3.7 7.0 PT-39 940 16 2.4 3.3 5.7 PT-40 940 16 2.4 3.3 5.7 PT-41 980 16 2.4 3.3 5.9 PT-45 440 7.3 1.8 2.6 3.7 PT-46 1020 26 2.4 3.4 6.0 PT-47 1210 30 2.6 3.6 6.6 PT-48 1490 25 2.8 3.8 7.5 PT-51 930 16 2.4 3.3 5.7 PT-52 1025 17 2.4 3.4 6.0 PT-57 450 7.5 1.9 2.6 3.7 PT-63 580 8.3 2.0 2.8 4.3 PT-64 580 8.3 2.0 2.8 4.3 PT-66 650 9.3 2.1 2.9 4.6 PT-73 545 5.5 2.0 2.7 4.2 PT-74 1820 18 3.0 4.1 8.4 PT-77 840 8.4 2.3 3.2 5.4 PT-92 130 1.0 1.2 1.7 1.8 PT-93 90 0.69 1.1 1.5 1.5 PT-99 855 21 2.3 3.2 5.4 PT-105 3190 32 3.6 4.9 12
8 Hydraulic conductivity of the matrix (Km)=10 m/s. cos(h)=1orh = 0°, horizontal fracture. Kinematic viscosity (t)=10 6 m2/s. Gravitational constant (g) = 9.81 m/s2.
Assuming Kmatrix Kconduit.
the differences in the scale effect (Király, 1975; Halihan et al., 1.2 Â 10 2 and 2.6 Â 10 1 m2/d (1.4 Â 10 7–3.1 Â 10 6 m2/s) 2000). (Table S2 of the ESM). The measurements are in accordance with values for limestone in other settings (Freeze and Cherry, 1979), 4.1.1. Small-scale analysis and measurements where permeability values of the matrix for limestone are approx- 13 16 2 Optical microscopy analysis indicates the Neoproteozoic lime- imately 10 –10 m and the hydraulic conductivity is between 6 9 stones from the Pedro Leopoldo Member (at the base) and the 10 and 10 m/s. The results are on the lower end of the range, Lagoa Santa Member (on the top) are low permeability competent but the values are consistent for limestone with secondary calcite bedrock, with the secondary porosity (i.e. micro-fractures) gener- precipitation. ally filled by calcite. The rocks from Pedro Leopoldo Member are composed of 85% micrite, 8% and 7% quartz, classified as micrite 4.1.2. Well-scale: aquifer test data and empirical relationship analysis (Folk, 1959) or mudstone (Dunham, 1962), while the rocks from The seven long duration transient pumping tests were con- Lagoa Santa Member are 55% sparite, 30% ooids, and 15% micrite, ducted in wells with karst bedding planes indicated a range of 2 classified as an oobiosparite (Folk, 1959) or grainstone (Dunham, well-scale transmissivities (Tw) from 90 m /d, in the northeast 1962). These low porosity limestone features are in accordance region, to around 3600 m2/d, close to the Santa Helena Ridge foot- with Moore (1989) and Tonietto (2010). hills. Wells tested in the central urbanized area also showed high 2 Hand sample TinyPerm measurements indicate no significant Tw values, above 1500 m /d, while wells located in the northeast differences in permeability between the Pedro Leopoldo and Lagoa region, indicated low values, below 500 m2/d. The well-scale
Santa members. The matrix permeabilities in the Sete Lagoas For- hydraulic conductivity values found (Kw) are in accordance with 16 mation, combining both members, range from 1.9 Â 10 to the Tw results, showing high values in the central area and low 15 2 4.2 Â 10 m . This results in hydraulic conductivity values, Ksm, numbers in the northeast region (Table 1). between 1.6 Â 10 4 and 3.5 Â 10 3 m/d (1.9 Â 10 9– The results of the specific capacity test analysis indicated 8 4.1 Â 10 m/s). The matrix transmissivities, Tsm, based on the results similar to previous research (Razack and Huntley, 1991; mean total thickness of aquifer, btotal = 75 m, are between Mace, 1997). A plot of the logarithms of transmissivity and specific 156 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162
showed results of the same order of magnitude, which was expected due to some inherent uncertainties of this method (Razack and Huntley, 1991; Mace, 1997).
Utilizing the best Tw data over the various portions of the aqui- fer (Tables 1 and S2), a transmissivity map was made (Fig. 4). The 2 highest well-scale transmissivities, Tw = 1500–3600 m /d (1.7 Â 10 2–4.1 Â 10 2 m2/s) and the zero drawdown well test (located in the area of the solutionally enlarged bedding plane) are concentrated in the central area, within the graben area and close to the Santa Helena Ridge foothills. A decrease in these values 2 3 occurs in the northeast direction, Tw = 90–250 m /d (1.1 Â 10 – 2.9 Â 10 3 m2/s). These values coincide with the optically observed fracture apertures of the solutionally enlarged bedding planes described by Galvão et al. (2015). The thicker apertures are located in the central graben area and close to the Santa Helena Ridge foothills. The data suggest a direct relationship between karstified frac- ture apertures and high well transmissivities. Dividing the results Fig. 3. Relationship between specific capacity and transmissivity values for the Sete of the central and the north portions by the respective thickness Lagoas karst aquifer, showing the best-fit line. of the aquifer, the hydraulic conductivity values range from 1 to 36 m/d (1.2 Â 10 5–4.2 Â 10 4 m/s) (Tables 1 and S2 of the ESM).
This would result in well-scale permeability, kw, values of capacity values for the Sete Lagoas karst aquifer appears linear in 1.2  10 12 and 4.2  10 11 m2, respectively. These results repre- log–log space (Fig. 3). The best-fit line to these data was: sent an increase of 2–4 orders of magnitude above the highest : small-scale permeability values. T ¼ 330ðS Þ0 21 w c The Thiem equation was used to estimate a transmissivity value where T and S are in m2/d. The coefficient of determination, for the zero drawdown well, pumping well PT-01. A transmissivity w c  4 2  1 2 R2, was 0.55. of 6 10 m /d (6 10 m /s) was determined using a pumping rate of 130 m3/h from the test, and assuming a drawdown of To evaluate if this empirical equation was reliable, a Tw recalcu- 0.01 m (1 cm), and the respective distances of 25 and 51 m of the lation was made and compared to the Tw values obtained in the seven pumping tests (Table S1 of the ESM). This comparison piezometers PT-02 and PT-03 from the pumping well PT-01. Based
Fig. 4. Well-scale map showing the variation of values of transmissivity. The highest values are concentrated in the central area, close to the Santa Helena Ridge foothills, and in the eastern portion of the city. The majority of these high values are located within a central graben area, as well as the ‘‘zero drawdown wells” controlled by the solutionally enlarged bedding planes. The dashed lines indicate uncertainties. The graben area boundary was based on the Galvão et al. (2015) study. P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 157
Fig. 5. Regional-scale permeability evaluation of Sete Lagoas. (a) Potentiometric surface map, flow lines, and capture zone. (b) Description of the capture zone model shape, its parameters and respective variables. (c) Comparison between the best-fit empirical capture zone curves (black continuous and dashed lines) and the potentiometric surface capture zone (gray area) assuming variations in the regional-scale transmissivity. (d) Sensitivity analyzes using different discharge values for a regional-scale transmissivity = 900 m2/d. Note that the x-axis has been aligned with the regional groundwater flow direction. on an aquifer thickness of 75 m, the result would be a local hydrau- = 0.008 (Fig. 5a and b), the analytical regional-scale transmissivi- 3 2 lic conductivity of 700 m/d (8  10 m/s) and a permeability of ties, Tr (L /t), were estimated: 8  10 10 m2. The uncertainty in these calculations may limit them 75;000 as permeability data, but ignoring them adds bias. These data rep- ¼ ffi 2= ð :  3 2= Þ Tr;x à : Ãð Þà : 750 m d 8 7 10 m s resent the highest well permeabilities in the aquifer and are 5 2 3 1416 2000 0 008 ; orders of magnitude above the highest small-scale measurements. 75 000 2 2 2 T ; ¼ ffi 1170 m =d ð1:3  10 m =sÞ r y 2 à 4000 à 0:008
The results area analytical regional-scale hydraulic conductivi- 4.1.3. Regional-scale: potentiometric data and capture zone analysis ties from 10 to 16 m/d (1.2 Â 10 4–1.8 Â 10 4 m/s) and regional- A capture zone method in a multiple well system was utilized to scale permeabilities of 1.2 Â 10 11–1.8 Â 10 11 m2, with transmis- estimate the regional-scale permeability. According to the poten- sivities between 750 and 1170 m2/d. This represents a three order tiometric data and approximate flow lines (Fig. 5), the Santa of magnitude increase over the small-scale measurements, but a Helena Ridge is a watershed boundary. The eastern portion of the decrease relative to many of the well-scale data. map has the groundwater flowing to the northeast, while the west- To evaluate the possible variability using this analytical ern portion has the groundwater flowing to the northwest. There is methodology, a local sensitivity analysis was conducted adjusting a flow convergence generated by a considerable cone of depression Tw values within this capture zone (Fig. 5) to assess the relative in the central area. This is explained by the high pumping rate in accuracy (Fig. 5c) (Pianosi and Wagener, 2015). The best-fit empir- the central wells. A capture zone was recognized in this area ical capture zone was found with a transmissivity = 900 m2/d (Fig. 5a). (1.1 Â 10 2 m2/s – see black continuous and dashed lines in
Utilizing Eqs. (1) and (2), with a stagnation point (XL) Fig. 6c), which would result in a hydraulic conductivity of 12 m/d 4 11 2 = 2000 m; half width of capture zone (YL) = ±4000 m; regional (1.4 Â 10 m/s) and permeability of 1.4 Â 10 m . The calcula- 3 pumping rate (Qr) = 75,000 m /d; and hydraulic gradient (i) tion of the best-fit empirical capture zone can be seen below: 158 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162
Fig. 6. Schematic of the hydrogeologic conceptual model of Sete Lagoas showing the non-karstic and the central solutionally enlarged bedding plane apertures, based on optical well logs and field observations. The cross-sectional view (left) illustrates the relative location of the bedding planes and the map view (right) illustrates the karst zone in the central graben area, located in the central urbanized area seen in Fig. 1.
Part A: half width of the capture zone, XL (L): (Table 1). The mechanical apertures seen in the optical well logs showed local values between 0.2 m and 8 m. This illustrates a large 75; 000 Y ¼ 5210 m difference between hydraulic and physical feature size, with L 2 Ã 900 Ã 0:008 hydraulic sizes being orders of magnitude smaller. The empirical Part B: distance to the stagnation point, XL (L): hydraulic values are a combination of both the flow from the upper 75; 000 and lower bedding planes. The values of Tw used in Eqs. (5), (9) and XL ¼ ffi 1660 m (10) were from pumping tests extracting water from the two con- 2 Ã 3:1416 Ã 900 Ã 0:008 duits simultaneously. Part C: shape of the curve describing the capture zone, x (L). The following values are for one-half of the capture zone, which is symmetrical about the x-axis: 4.2.3. Regional-scale: overall bedding planes and karst aquifer thicknesses y x ¼ hi¼ y=tanð0:000603yÞ Using values of Tr found via regional capture zone method to 2Ã3:1416Ã900Ã0:008y tan 75;000 estimate the values of regional hydraulic fracture aperture, laminar and turbulent conduit in that range, the respective Eqs. (4), (9), and A possible lower estimate of pumping rate value from private (10) were assumed. The combined results ranged between 2 and wells was not included in the analysis (no data or not being regis- 6 mm (fracture aperture: 2–3 mm, laminar flow conduit: 3– tered) and was not added to the final Qr value used in the calcula- 4 mm, and turbulent flow conduit: 5–6 mm), slightly lower than tions above. A local sensitivity analysis was performed to test the the well-scale hydraulic size. effects of pumping rate (Fig. 5d). The transmissivity was fixed at 900 m2/d (best-fit) adjusting only the pumping rate. The result 3 4.3. Permeability combination models showed that, using Qr = 100,000 m /d, the value of XL had no signif- icant change, while YL had a small change. Using Qr = 50,000 and 3 It was possible to accommodate the small-, well-, and regional- 150,000 m /d, there was a significant change, both in XL and YL. While some private wells data have not been included, the major- scales of measurement and compare the values against a single ity were residential wells with lower pumping rates. The well data conceptual model of hydraulic features by combining the approaches for different scales. The forward models were used to are not significant in the total sum of Qr. The public wells located within the captured zone that were included in the analysis repre- test whether the hydraulic features sizes of the inverse conceptual sent the highest pumping rates and have the greatest impact on model for the aquifer. The only significant permeable features of the discharge value used in the equations. the Sete Lagoas permeability combination model were the two large bedding planes in the aquifer (Fig. 6). The model assumes the measurements at the various scales are valid estimated of 4.2. Permeability feature sizes from inversion permeability. In small-scale, the mean permeability of the matrix is very low 4.2.1. Small-scale: matrix data and feature sizes (from 10 16 to 10 15 m2)(Fig. 2), because the matrix is composed The permeability of the matrix is between 10 16 and 10 15 m2, of a Neoproterozoic fine-grained (Pedro Leopoldo Member, at the according to permeameter and microscopy analysis for these types base) and a medium-grained (Lagoa Santa Member, on the top) of lithologies. This is classified as low permeability competent bed- limestone, with the secondary porosity (micro-fractures) filled by rock, with the secondary porosity generally filled by calcite. The calcite. The values can be approximated on time scales for pump- hydraulic fracture aperture, bsm, required to generate this level of ing or regional flow estimates as zero relative to the bedding permeability would be considered between 10 8 and 10 7 m (or planes features. A value of 10 15 m2 was utilized to represent the 0.01 and 0.1 lm). These values are in accordance with values for matrix permeability for the simple conceptual model across the limestone pore diameters of the matrix, ɸsm, around 0.1 lm scales. (Zinszer and Pellerin, 2007). An increase of between four and six orders of magnitude of well-scale permeability features in comparison to the matrix 4.2.2. Well-scale permeable feature sizes (10 12–10 10 m2) is noted (Fig. 2), due to the solutionally enlarged Using empirical hydraulic equations for estimating sub- bedding planes. Regarding the zero drawdown wells, the result horizontal fracture apertures, the results analyzed for each well showed permeability around 10 9 m2. The inversion indicates showed that the hydraulic fracture aperture, bf, has a range of 1– these can be simulated as a single hydraulic fracture that is 1– 4 mm. The estimated hydraulic conduit diameter, assuming lami- 2 mm in size in the non-karstified bedding plane regions (values nar flow, dc,lam, is between 2 and 5 mm. Assuming turbulent flow, from fracture apertures) and 2–13 mm in size in the karstified the hydraulic conduit diameter, dc,turb, ranges from 2 to 13 mm region depending (values from laminar, 2–5 mm, and turbulent P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 159
Fig. 7. Schematic scale effect conceptual model. In a Neoproterozoic karst aquifer governed by solutionally enlarged bedding planes, with negligible permeability of the matrix, the chance to have a considerable increase in the scale effect from the well- to the regional-scale is low.
conduit diameters, 2–13 mm) on whether the model allowed for non-karstic zones, the results are a regional permeability of turbulent flow (Table 1). A non-karstic fracture with a 1 mm aper- 10 11 m2. ture and a karstic fracture with a 5 mm aperture were used for the simple conceptual model of the system. The fractures yield perme- abilities of 10 12 m2 in non-karstic areas and 10 10 m2 in karstic 5. Discussion areas at the well-scale in a 75 m thick aquifer. The regional-scale permeability was approximately 10 11 m2 A schematic scale effect conceptual model block diagram for Sete (Fig. 2). The values of permeability have a slightly smaller value Lagoas karst aquifer was made (Fig. 7) to show how a simple model in comparison to the largest well-scale values. The inverted perme- of hydraulic feature size affects each scale. The diagram considers ability sizes were 2–6 mm. This is consistent with the non-karstic permeabilities measured and the inverted hydraulic feature size for- and karstified bedding plane estimate at the well-scale. The value ward model. The changes in permeability (k) or associated hydraulic of permeability was 10 11 m2 for the area encompassed by the cap- values (K, and T), the respective size permeable features from small- ture zone analysis for the simple conceptual model of the regional- to well-scale, and a decrease of these values in some areas from well- scale. The estimate can be reached by assuming a 2 mm fracture to regional-scale were quantitatively observed for the aquifer. The existing over the regional-scale, or by constructing the regional features are noted through the inversion of field measurements permeability from the well-scale permeabilities. Using the two and the forward modeling of a simple conceptual model with a sin- well-scale permeabilities from the simple model of 1 and 5 mm gle fracture of both karstic and non-karstic settings. fractures, the karstic zone in the center of the pumping area can The porosity and permeability, as well as the aperture of the be modeled as a layered aquifer averaging the flow with the result fractures in the matrix are very low. The hydraulic conductivity being similar to the larger value. When the lower value is utilized of the matrix is much lower than the hydraulic conductivity of near the southwest portion of the aquifer, this would be flow the bedding planes intercepted at the well-scale. Increases across the layers. With reasonable approximations for karstic and between four and six orders of magnitude in the parameters from 160 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 small- to well-scale were observed. However, well- to regional- (Rovey, 1994; Halihan et al., 2000). The limitation of aquifer tests scale had a small decrease in the values (Figs. 2 and 7). The same to predict permeabilities was examined using the Thiem equation, behavior in the empirical size permeable features was observed, for steady-state flow in a confined aquifer. Estimating the highest where values of hydraulic aperture of 0.01–0.1 lm (small-scale) local permeabilities values caused by the solutionally enlarged are increasing to 2–13 mm (well-scale) and decreasing to 2– bedding planes was unworkable for well testing. 6 mm (regional-scale). These inverted hydraulic sizes are signifi- A capture zone with dimensions of 8 km by 8.5 km (68 km2) cantly smaller than outcrop fracture or conduit sizes consistent was present at the regional-scale, caused by the high pumping rate with previous investigations, based on connectivity and fracture in wells estimated in 75,000 m3/d (Fig. 5). Regional-scale has a lit- variability (Halihan et al., 2000). tle decrease of the permeability values compared to well-scale Other karst areas, such as Edwards Aquifer located in the United (Figs. 2 and 7) in that range of measurement. The groundwater is States (Halihan et al., 2000) and Juras Mountain in Switzerland also governed by the same karstic conduits observed at the well- (Király, 1975), have increases in permeability in scale effects from scale. The value is decreased by incorporating non-karstic areas small- to regional-scale. The quantitative relationships between in the regionally averaged value. Higher permeabilities than gener- feature size and hydraulic properties showed this behavior does ated using the capture zone analysis may be required to simulate not happen at this range of measurement in the Sete Lagoas karst the system with lower adjacent values to represent non-karstic aquifer, in Brazil (Fig. 2). The relationships are in accordance with areas in a groundwater model at the regional-scale with grid Clauser (1992) results, where permeabilities increase from small- dimensions below a kilometer. The permeability scale effect data- to well-scale, but from well- to regional-scale they do not continue. base for such a model indicate a leveling off of values between The difference in the Clauser (1992) study is the effects happened well- and regional-scales instead of a decrease in permeability. In in crystalline rocks (Fig. 2). According to Király (1975) and Halihan an aquifer modeled with a karst model including numerical frac- et al. (2000), increases from small- to the well-scale are caused by tures or conduits, the physical basis for this model suggests evalu- fractures being incorporated into the well-scale measurements of ating feature sizes in a grid block can be applied to models across permeability; and the largest permeabilities on the regional-scale various scales. As a great deal of uncertainty exists on conduit loca- were caused by karstic conduits being incorporated. The increase tion and connectivity, the results suggest any permeability data can obtain nine orders of magnitude variability (Halihan et al., obtained at any scale can be integrated across scales to obtain 2000) (see Edwards aquifer scale effect line in Fig. 2). This can be greater certainty of permeability estimates. interpreted as the scale effect and dependency in karst aquifers An increase of measurement scale does not always imply an is more related to fracture and conduit connectivity. increase of permeability in these settings; it will depend of the Based on the geologic mapping and optical well logs data type of lithology and the connectivity of high permeability fea- (Galvão et al., 2015), the sub-horizontal solutionally enlarged bed- tures. The scale effect had a reduction of the permeability values ding planes have the primary role in providing aquifer transmissiv- in the study area, due to the localized development of karst bed- ity. All the wells are intercepting sub-horizontal bedding plane ding plane dissolution in one structurally controlled region of the discontinuities, extracting groundwater from the Neoproterozoic aquifer. Measurements can be applied at different scales if the con- Sete Lagoas karst aquifer from the main two continuous bedding nectivity and structure is well understood to develop a robust planes. The bedding planes are more solutionally enlarged in the quantitative model of the hydraulic structure of the aquifer. Trans- central area and less karstic to the northeast (Figs. 4, 6 and 7). port should not be calculated at a given scale using continuum Low matrix permeability in a karst aquifer governed by bedding approximations, as they will underestimate transport times by planes and low sub-vertical fractures are not likely to have a con- averaging low permeabilities with high permeability values. Tur- siderable increase in the scale effect from well- to regional-scale. bulent effects need to be considered and determine whether they The largest permeabilities occur due to the same karstic solution- are important for estimating permeability or the hydraulic size of ally enlarged bedding planes in both well- and regional-scale. This a feature supplying flow. interpretation is in accordance with Singhal and Gupta (2010), Compared to other regions of the world, a single quantitative where carbonate rocks of Paleozoic and Mesozoic ages often have model of permeability scale effects can be elucidated from this water flow primarily through conduits and fractures. and other associated studies. Aquifers are composed of a range of There is variation between low and high values of transmissiv- hydraulically connected permeable features. As these systems are ity and hydraulic conductivity at the well-scale. The central area sampled, the features sizes and connections vary depending on has the greatest permeabilities values, including the zero draw- the choice of measurement method. Determining a strong down well pumping test. This area is located within a graben area evidence-based conceptual model of what features control perme- and with more karstification. The solutionally enlarged bedding ability allows us to make prediction about quantitative values of planes are the most important structural features in the Neopro- permeability expected at various scales. Matrix permeability is terozoic limestones of the Sete Lagoas karst aquifer, providing often small in these systems, but should not be assumed to be pathways for groundwater flow and storage capacity. Smaller val- neglected. A number of young carbonate systems have significant ues of transmissivity are more common in the northeast portion of permeability in the matrix of the system. Well-scale permeability the area, suggesting less karstic bedding planes (confirmed by is expected to vary significantly in most cases as fracture aperture lithologic well logs, by Galvão et al., 2015). The depth of these lime- distribution may be strongly variable, as in the Edwards aquifer of stones and its lithologic contacts could explain these different Texas (Halihan et al., 2000). The aperture of single horizontal frac- degrees of karstification. Transmissivity values, and consequently tures can also be highly variable as they are sampled spatially (this hydraulic conductivity and permeability, can be expected to vary study; Muldoon et al., 2001). Understanding connected conduits is according to vertical position in karst rock (Ford and Williams, important for many karst aquifers on a regional-scale. 2007). It will be highest in the most weathered zone of limestone This study illustrates conduit permeability may not be a strong near the surface (epikarst) and usually diminishes exponentially regional effect if large conduits are not well connected on the scale with depth. of measurement. The regional-scale findings could vary depending Regarding the zero drawdown aquifer test, located in the cen- on the measurement method, data availability, and the size of scale tral urbanized area, some researchers believe these tests are unus- studied. Conclusions should be made with caution for regional able or useless. Others suggest permeability may be at ‘‘practical areas which could incorporate conduits connected to a poorly sam- infinity” or at the testing limit for aquifer testing methodologies pled domain. A smaller grid size regional numerical model would P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162 161 likely observe some high permeability grid blocks in the central Appendix A. Notation portion of this aquifer, but these values would be well supported by the conceptual model built from aquifer testing and optical log- ging of the system. The primary lesson obtained from these data sets is under- bf hydraulic aperture of a single low-angle fracture (L) standing the hydraulic size and geometry is critical to predictions btotal total thickness of aquifer (L) of flow and transport, and data from all scales can contribute to bw well-scale bedding plane aperture (L) quantifying the hydraulic properties. The size and connectivity of br regional-scale aperture (L) permeable zones is somewhat more important than flow regime, dc hydraulic conduit diameter (L) but when evaluating large karst features, their regional connectiv- d ; laminar hydraulic conduit diameter (L) ity at a particular scale results in hydraulic sizes generally on the c lam d ; turbulent hydraulic conduit diameter (L) millimeter scale, even when these features have meter scale c turb g gravitational constant (L/t2): 9.81 m/s2 dimensions in portions of the flowpath. Models on regional- i hydraulic gradient (dimensionless) scales must accommodate the ends of large features for flow pre- k intrinsic permeability of a fracture (L2) dictions, but maintain an understanding of void sizes along the f 2 pathway for storage and mixing issues. ksm small-scale permeability (L ) 2 kw well-scale permeability (L ) 2 kr regional-scale permeability (L ) 6. Conclusion Kc conduit hydraulic conductivity (L/t) Kc;lam laminar hydraulic conductivity of the conduit (L/t) Unlike other karst areas with increases in permeability in scale Kc;turb turbulent hydraulic conductivity of the conduit (L/t) effects from small- to regional-scale, such as Edwards Aquifer Ksm small-scale hydraulic conductivity of the matrix (L/t) located in the United States (Halihan et al., 2000) and Juras Moun- Kw well-scale hydraulic conductivity (L/t) tain in Switzerland (Király, 1975), this behavior does not happen in Kr regional-scale hydraulic conductivity (L/t) Sete Lagoas karst aquifer. An increase of about four to six orders of 3 Q w pumping rate in the well (L /t) magnitude in the permeabilities from small- to well-scale is com- Q regional multiple well pumping rate (L3/t) mon, but from well- to regional-scale, the value decreases in this r R2 coefficient of determination (dimensionless) area. 2 Sc specific capacity (L /t) The highest permeability features at the well-scale are concen- T transmissivity (L2/t) trated in the central area and close to the Santa Helena Ridge foot- 2 Tsm small-scale transmissivity (L /t) hills, within the graben area, and in the east portion of the study 2 Tw well-scale transmissivity (L /t) area. The data coincide with the thickness of the karst bedding 2 Tr regional-scale transmissivity (L /t) planes, suggesting a direct relationship between solutionally x distance parallel to the regional hydraulic gradient (i) enlarged bedding planes thickness and formation permeability. (L) However, the Sete Lagoas karst aquifer has some zones of high XL distance from pumping well to the downgradient edge karstification. The standard aquifer test used is an insufficient of the capture zone (stagnation point) (L) method for estimation, evidenced by zero drawdown and illustrat- y distance perpendicular to the regional hydraulic ing the limits of aquifer tests. These zones are located in the central gradient (i) (L) portion of the urbanized area. Non-Darcian flow likely occurs in YL half width of the maximum capture zone (L) these wells. There is a potentiometric flow convergence in the cen- Dhw hydraulic head in the well (L) tral portion of the study area at the regional-scale. The flow con- p pi = 3.1416 vergence is setup by a cone of depression and high pumping rate h angle between fracture and horizontal in wells. v kinematic viscosity of the water (L2/t): 10 6 m2/s The results indicate that multi-scale evaluation using different /sm porosity of matrix (lL) approaches is useful to develop a quantitative hydrogeological conceptual model in karst aquifers consistent across scales of mea- surement, from small- to regional-scale flow. While hydraulic con- ductivity or intrinsic permeability estimates vary over 6 orders of magnitude (from 1.9 Â 10 9 m/s to 8.0 Â 10 3 m/s or Appendix B. Supplementary material 1.9 Â 10 16 m2 to 8.0 Â 10 10 m2), the interconnected hydraulic feature sizes controlling the permeability structure remain consis- Supplementary data associated with this article can be found, in tent across the datasets vary less than one order of magnitude (1– the online version, at http://dx.doi.org/10.1016/j.jhydrol.2015.11. 8 mm) and are well constrained by the data. The methodologies 026. obtained results suggesting Neoproterozoic karst aquifers with negligible porosity and permeability of the matrix, governed References mainly by bedding planes discontinuities, would not likely have increases in permeability from well- to regional-scale unless a zone Branco Jr., Costa, M.T., 1961. Belo Horizonte–Brasilia roadmap tour. In: Brazilian Congress of Geology, Brasilia. Radioactive Research Institute, Federal University of more significant karstification occurred on that scale. of Minas Gerais (UFMG), Belo Horizonte, p. 25, Publication 15. Chapuis, R.P., 2013. Permeability scale effect in sandy aquifers: a few case studies. In: Proceedings of the 18th International Conference on Soil Mechanics and Acknowledgments Geotechnical Engineering, Paris, pp. 507–510. Clauser, C., 1992. Permeability of crystalline rocks. EOS, Trans. Am. Geophys. Union 73 (21), 233–238. This work was funded by Servmar Environmental & Engineering Cooper Jr., H.H., Jacob, C.E., 1946. A generalized graphical method for evaluating and by Fundação de Amparo à Pesquisa do Estado de São Paulo formation constants and summarizing well field history. Trans. Am. Geophys. (FAPESP) [São Paulo Research Foundation] (process 2012/12846-9). Union 27, 526–534. Dardenne, M.A., 1978. Synthesis on the stratigraphy of Bambuí Group in Central Special thanks go to Prof. Dr. Renato Paes de Almeida for providing Brazil. Brazilian Congress of Geology, 30, vol. 2. Annals Recife: Brazilian Society the TinyPerm II portable hand-held air permeameter. of Geology, Recife, pp. 597–610. 162 P. Galvão et al. / Journal of Hydrology 532 (2016) 149–162
Demétrio, J.G.A., Correia, L.C., Saraiva, A.L., 2006. Using SRTM images to developing fractured layer of a hard rock aquifer. Water Resour. Res. 40, W11508. http://dx. potentiometric maps. In: Brazilian Groundwater Congress. Abstracts. ABAS, doi.org/10.1029/2004WR003137. Curitiba, 176p. Martinez-Landa, L., Carrera, J., 2005. Na analysis of hydraulic conductivity scale Dunham, R.J., 1962. Classification of carbonate rocks according to depositional effects in granite (Full-scale Engineered Barrier Experiment – FEBEX), Grimsel, texture. In: Ham, W.E. (Ed.), Classification of Carbonate Rocks, Memoir 1. Switzerland. Water Resour. Res. 41 (W03006), 1–13. http://dx.doi.org/10.1029/ American Association of Petroleum Geologists, Tulsa, Oklahoma, pp. 108–121. 2004WR003458. Ewers, R.O., 1982. An analysis of solution cavern development in the dimensions of Moore, C.H., 1989. Carbonate diagenesis and porosity. Development in length and breadth. Ph.D. dissertation, McMaster University (Hamilton, Ontario, Sedimentology, vol. 46. Elsevier, Amsterdam, p. 338. Canada), 398p. Muldoon, M.A., Simo, J.A., Bradbury, K.R., 2001. Correlation of hydraulic Fetter, C.W., 1994. Applied Hydrogeology. Macmillan, New York, p. 691. conductivity with stratigraphy in a fractured-dolomite aquifer, northeastern Folk, R.L., 1959. Practical petrographic classification of limestones. Am. Assoc. Wisconsin, USA. Hydrogeol. J. 9 (6), 570–583. Petrol. Geol. Bull. 43, 1–38. Oliveira, L.M., 1997. The management of urban geological risks in karst areas. Ford, D.C., Williams, P.W., 2007. Karst Geomorphology and Hydrology, second ed. Undergraduate thesis. Pontifícia Universidade Católica do Paraná (PUC-PR), 46p. John Wiley & Sons, Chichester, UK, p. 562. Pessoa, P., 1996. Hydrogeological characterization of the region of Sete Lagoas – Fossen, H., Schultz, R.A., Tobari, A., 2011. Conditions and implications for MG: Potentials and Risks. Master Thesis. Department of Geosciences, University compaction band formation in the Navajo Sandstone, Utah. J. Struct. Geol. 33, of São Paulo, São Paulo. 1477–1490. Pianosi, F., Wagener, T., 2015. A simple and efficient method for global sensitivity Freeze, A.R., Cherry, J.A., 1979. Groundwater: Englewood Cliffs. Prentice-Hall, New analysis based on cumulative distribution functions. Environ. Model. Softw. 67, Jersey, p. 604. 1–11. http://dx.doi.org/10.1016/j.envsoft.2015.01.004. Galvão, P., Halihan, T., Hirata, R., 2015. Evaluating karst geotechnical risk in the Quinlan, J.F., Ewers, R.O., 1985. Groundwater flow in limestone terrenes: strategy, urbanized area of Sete Lagoas, Minas Gerais, Brazil. Hydrogeol. J. http://dx.doi. rationale, and procedures for reliable, efficient monitoring of groundwater org/10.1007/s10040-015-1266-x. quality in karst areas. In: National Symposium and Exposition on Aquifer Grubb, S., 1993. Analytical model for estimation of steady-state capture zones of Restoration and Ground Water Well Assoc, pp. 197–234. pumping wells in confined and unconfined aquifers. Ground Water 31 (1), 27– Rats, M.V., 1967. Neodnorodnost Gornykh Porod i Ikh Fizicheskikh Svojstv. Nauka 32. Moskva. Halihan, T., Wicks, C.M., Engeln, J.F., 1998. Physical response of a karst drainage Razack, M., Huntley, D., 1991. Assessing transmissivity from specific capacity in a basin to flood pulses: example of the Devil’s Icebox cave system (Missouri, large and heterogeneous alluvial aquifer. Ground Water 24, 519–524. USA). J. Hydrol. 204, 24–36. Rehfeldt, K.R., Hufschmied, P., Gelhar, L.W., Schaefer, M.E., 1989. Measuring Halihan, T., Sharp Jr., J.M., Mace, R.E., 1999. Interpreting flow using permeability at hydraulic conductivity with the borehole flowmeter. In: Research Report EPRI multiple scales. In: Palmer, A.R., Palmer, M.V., Sasowsky, I.D. (Eds.), Karst EN-6511, Project 2485-5. Electric Power Research Institute, Palo Alto, California, Modeling. Karst Waters Institute Special Publication No. 5, Charlottesville, pp. 209p. 82–96. Ribeiro, J.H., Tuller, M.P., Danderfer Filho, A., 2003. Geological mapping of the region Halihan, T., Sharp Jr., J.M., Mace, R.E., 2000. Flow in the San Antonio segment of the of Sete Lagoas, Pedro Leopoldo, Matozinhos, Lagoa Santa, Vespasiano, Capim Edwards Aquifer: matrix, fractures, or conduits? In: Wicks, C.M., Sasowsky, I.D. Branco, Prudente de Morais, Confins and Funilândia, Minas Gerais State, Brazil (Eds.), Groundwater Flow and Contaminant Transport in Carbonate Aquifers. (scale 1:50,000), second ed. Belo Horizonte, 54p. Balkema, Rotterdam, The Netherlands, pp. 129–146. Rovey, C.W.I.I., 1994. Assessing flow system in carbonate aquifers using scale effects Heath, R.C., 1983. Basic Ground-Water Hydrology. U.S. Geological Survey Water- in hydraulic conductivity. Environ. Geol. 24, 244–253. Supply Paper 2220, 86p. Scanlon, B.R., Mace, R.E., Smith, B., Hovorka, S., Dutton, A.R., Reedy, R., 2001. Hovorka, S.D., Mace, R.E., Collins, E.W., 1995. Regional distribution of permeability Groundwater availability of the Barton Springs segment of the Edwards aquifer, in the Edwards Aquifer. Draft Contract Report to the Edwards Underground Texas: Numerical simulations through 2050 prepared for Lower Colorado River Water District under Contract No. 93-17-FO. Bureau of Economic Geology, The Authority under contract number UTA99-0, Austin, Texas. University of Texas at Austin, 77p. Scanlon, B.R., Mace, R.E., Barrett, M.E., Smith, B., 2003. Can we stimulate regional Huntoon, P.W., 1985. Gradient controlled caves, Trapper-Medicine Lodge area, groundwater flow in a karst system using equivalent porous media models? Bighorn Basin, Wyomig. Ground Water 23, 443–448. Case study, Barton Springs Edwards aquifer, USA. J. Hydrol. 276, 137–158. IBGE – Brazilian Institute of Geography and Statistics, 2010. Basic Municipal Schobbenhaus, C., 1984. Geology of Brazil. National Department of Mineral Information. Available: