Using fuzzy logic for modeling aquifer architecture - Semantic Scholar

J Geograph Syst (2007) 9:289–310 DOI 10.1007/s10109-007-0046-0 ORIGINAL ARTICLE

Using fuzzy logic for modeling aquifer architecture D. M. Allen Æ N. Schuurman Æ Q. Zhang

Received: 27 November 2006 / Accepted: 9 March 2007 / Published online: 31 March 2007 ! Springer-Verlag 2007

Abstract Modeling the geologic architecture of an aquifer and visualizing its three-dimensional structure require lithologic data recorded during well drilling. Uncertainties in layer boundaries arise due to questionable quality of drilling records, mixing during the drilling process, which results in blurred contacts, and natural heterogeneity of the geologic materials. An approach for modeling and visualizing the spatial distribution of aquifer units three-dimensionally based on fuzzy set theory is developed. An indicator is defined for evaluating the possibility of aquifer existence based on fuzzy set theory and probability principles. A specific interpolation method for aquifer 3D spatial distribution requiring only very basic borehole log data is proposed. A 3D modeling and visualization system for aquifers is also developed, which can implement basic GIS functions, like borehole identification and cross-section creation. The methodology developed is tested using real borehole lithology data available for an aquifer in British Columbia, Canada.

1 Introduction Developing geologic models of aquifers is required for most hydrogeological studies. A traditional approach requires a source of subsurface information, typically in the form of well lithology logs, and the construction of a conceptual model of the aquifer architecture that can be used, for example, to construct a D. M. Allen (&) Department of Earth Sciences, Simon Fraser University, Burnaby, BC, Canada V5A 1S6 e-mail: [email protected] N. Schuurman ! Q. Zhang Department of Geography, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

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numerical groundwater flow model. Current approaches for visualizing and mapping the architecture of aquifers assumes that the contacts between the geologic layers are discrete and identifiable in the individual borehole logs. However, these well logs are recorded based on sediment samples that are typically retrieved at the surface during drilling and, due to mixing during the drilling process, the transitions from one geologic unit to the next may be blurred. In addition, much borehole data are of questionable quality due to a lack of training of well drillers in identifying sediment types. Natural heterogeneity of the sediments also confounds the process of mapping discrete aquifer units, and distinguishing those units from less permeable aquitards. Because of this source of uncertainty, geostatistical techniques have been incorporated into some groundwater modeling software packages. For example, transition probability geostatistics can be performed on borehole data (T-PROGS software is incorporated into GMS; Environmental Modeling Systems Inc. 2006). This method generates a specified number of material sets on a 3"D grid, whereby each material set is conditioned to the borehole data and the materials proportions and transitions between the boreholes follow the trends observed in the borehole data (assuming of course that the borehole data are correct). These material sets can be used for stochastic simulations of groundwater flow based on multiple realizations of the geology. Yet, despite the advances made in respect of using geostatistics for describing and representing the heterogeneity of aquifers, defining aquifer architecture based on a layered paradigm remains a common approach, and brings with it the challenges related to uncertainty in defining layer boundaries. In addition, aquifers are generally three-dimensional, and are often interlayered with less permeable layers, or aquitards. Therefore, some means of representing the data in three dimensions is needed. Unfortunately, existing major commercial GIS software, such as ArcGIS, MapInfo and GeoMedia, cannot manipulate the 3D borehole data, which are the basic data for aquifer analysis and display. Other systems, such as CAD-based ones, have been used successfully in geological sciences to display and contour such data (e.g., Logan et al. 2001), but are restricted by their inability to analyze attribute data. Use of fuzzy logic addresses both the uncertainty in the data as well as problems associated with modeling the subsurface. For example, Hseih et al. (2005) employed a fuzzy logic approach to identify the lithology of aquifers using geophysical logs (gamma ray, resistivity and sonic logs). That study showed that the shortcomings of using borehole logs to distinguish between silts and sands, and determining grain size variation in sands, could be overcome by employing a fuzzy lithology system when interpreting the geophysical logs. Ozbek and Pinder (1998) and Gemitzi et al. (2005) used a fuzzy approach to investigate the relation between aquifer vulnerability, and groundwater pollution and health-related risk, respectively. Within the earth sciences, other uses of fuzzy logic have been tested, for example, for generating mineral prospect maps (D’Ercole et al. 2000). The aim of this paper is to develop an approach for modeling and visualizing the spatial distribution of aquifer units three-dimensionally, based on the fuzzy set theory. Specifically, the objectives are (1) to identify an indicator for evaluating the possibility of aquifer existence based on fuzzy set theory and probability principles;

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(2) to propose a specific interpolation method for aquifer 3D spatial distribution, which only needs very basic borehole log data and, thus, can be adopted for most situations of aquifer research and management; and (3) to demonstrate a 3D modeling and visualization system for aquifers, which can implement basic GIS functions, like borehole identification and cross-section creation. The methodology developed is tested using real borehole lithology data available for an aquifer in British Columbia, Canada that has been the focus of recent published work (Scibek and Allen 2006; Scibek et al. 2006).

2 The case study area The Grand Forks area, located at Longitude 1188 W and Latitude 498 N adjacent to the international border between BC, Canada and Washington State, USA, was chosen as a case study site (Fig. 1). The Grand Forks aquifer (34 km2 in area) is contained within the mountainous valley of the Kettle River, and is situated at the junction with the Granby River near the City of Grand Forks (Scibek and Allen 2006). The Grand Forks Valley fill consists of unconsolidated Quaternary and more recent deposits, which overlie relatively impermeable bedrock. The Quaternary deposits include silty clay, silty sand, sand and gravel; a generally coarsening upward sequence, which was deposited following ice retreat in the area. The meandering rivers within the basin (Kettle and Granby Rivers) are believed to have reworked the sand and gravel layers into sorted channels and bars (Scibek and Allen 2006). These same rivers created many oxbow lakes, which could allow silt and clays to accumulate at the lake bottoms. Therefore, it is expected that the upper

Fig. 1 Location of the study area in south-central British Columbia, Canada

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portions of the aquifer will contain primarily gravel and sand, with silty sand, and possibly clay or silt lenses. The gravel and sand form the Grand Forks aquifer, and the silty clay material likely acts as an aquitard. Geologic logs obtained from the British Columbia (British Columbia Ministry of Environment, 2006) WELLS database were used in this study (http://www.aardvark.gov.bc.ca/apps/wells/). Two datasets are available: one is for the general characteristics of a well, including: construction data, well owner, well depth and UTM coordinates, and the other is for the lithology data including: depth intervals and material type. Sometimes, additional information, such as grain size or color is provided. The well tag number (WTN) serves as the unique identifier in both datasets. The geologic log data are assumed to be complete in lithology data, additional to the depth and thickness. Therefore, the lithology data (a range of rock and sediment material types) are selected for analysis. As part of previous work, the lithology data for water wells in the Grand Forks aquifer (Fig. 2) were standardized and classified in order to construct a 3D groundwater flow model (Scibek and Allen 2006). The standardization scheme (1) corrected spelling and grammatical errors in the database, (2) created a continuous depth sequence (i.e., eliminate missing depth intervals), and (3) extracted meaningful lithologic terms from the well records (using a data dictionary). Material types were then classified for all recorded depth intervals (Fig. 3). The standardized borehole data consist of 114 wells (1,111 layer records). In some cases, up to five other material types were identified (Fig. 3).

Fig. 2 Locations of the wells used in this study

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Fig. 3 Sample of standardized and classified lithology data. Dominant and secondary material types were identified for some layers

3 Application of fuzzy logic Spatial data represent points, lines, areas and volumes, and can be classified as ‘‘naturally occurring’’ or ‘‘ordained’’ (by humans). Examples of naturally occurring data are forests, streams and soils, while ordained data represent such things as building footprints, school districts, and roads. Naturally occurring data (phenomena) usually have fuzzy boundaries that cannot be well represented by common GIS paradigms (Burrough 1996). Geographers and cartographers long ago noticed the fuzzy or uncertain nature of boundaries, and developed approaches, such as ‘‘shadow map certainty’’, to assess map accuracy through its spatial extent (Berry 1996). Various other researchers also supported the idea that geographical entities may not be precisely identified; that boundaries generally exist as gradients and should appear as zones rather than lines (e.g., Grigg 1967; Chisholm 1964; Cox 1972; Gale 1972; Alexander and Gibson 1979; Coleman 1980; Robinson 1980; Leung 1982, 1987). For instance, Leung (1982, 1987) improved the concept of regional cores and edges by employing fuzzy set theory. Ordained data (phenomena) tend to have crisp boundaries that can be represented in arcs, polygons or volumes in a GIS. However, as Campari (1996) pointed out, there are still matters of uncertainty in the boundaries of urban space, e.g., administrative boundaries and boundaries of urban artifacts. So, the uncertainty of spatial data is a very common concern. To deal with spatial data correctly and efficiently, the uncertainty in these data must first be addressed. 3.1 Fuzzy set theory The term ‘‘fuzzy set’’, as a generalized form of set theory, was coined by Zadeh (1965). Unlike traditional Boolean logic, which defines whether or not an element belongs to a crisp set (1 or 0), a fuzzy set defines a degree of belonging by a membership function.

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According to the crisp set theory, an abrupt boundary is supposed to exist between geographic classes. A geographical entity fully belongs to one class, and is totally excluded from other classes. The decision of inclusion or exclusion of an entity within classes is usually based on some chosen criteria. Two problems exist when trying to deal with spatial data. One is the difficulty to define an appropriate criterion (Burrough and Frank 1996). For example, in a groundwater system, lithology can be a criterion for identifying an aquifer; for example, sandstone can belong to the aquifer class. However, several factors affect this criterion, such as grain size, grain size fraction, texture, and the structure of the sandstone (Csillag et al. 2000). The other problem is the uncertainty of boundaries, as mentioned in the above section. Fuzzy set theory is a way of expressing the uncertainty or imprecision of spatial data. We define X as a universe of discourse having its generic elements x, or X = {x}. A fuzzy set F in X is characterized by a membership function, lF(x), which maps X to the membership space [0, 1]. lF(x) represents the grade of membership of x in F. For the continuum and discontinuum X = {x}, the fuzzy set can be expressed as Eqs. 1 and 2, respectively. Z F= xl ðxi Þ=xi ð1Þ F

F = flF ðxi Þ=xi g

ð2Þ

In essence, fuzzy set theory deals with sources of the uncertainty or imprecision that are vague and non-statistical in nature (Zeng and Zhou 2001). For example, a traditional set might comprise the set of all tall people in the classroom. This would result in a binary where all people over 180 cm are classified as tall, and persons under are not. Fuzzy set theory would instead assign degrees of membership in the tall set. A person who is 173 cm might be assigned a 0.96 degree of membership. In this example, fuzzy set theory allows more meaningful membership in a set (McNeil and Freiberger 1993). Fuzzy set theory has the advantage of being closely linked to classifical logic, but in many instances, it is difficult to determine how to assign membership (Duckham et al. 2001). Since the beginning of the 1970s, fuzzy set theory has been introduced and applied to classification of geographic entities due to the ambiguity of class definition (Cheng 2002). Applications include regionalization (Leung 1982, 1987), soil classification and definition of boundaries between soil classes (Burrough 1989; Kollias and Voliotis 1991; Banai 1993; Davidson et al. 1994), and boundaries in geographic space (Mark and Csillag 1989; Wang and Hall 1996). Recently, fuzzy set theory has been used to model the uncertainty of the geometric aspects of mapping units (Lowell 1994; Edwards and Lowell 1996; Usery 1996; Brown 1998), and topology (Dijkmeijer and De Hoop 1996; Clementini and Felice 1996). When ‘object’ became a buzz in the world of computer science, the concept of fuzzy objects also appeared in GIS literature to represent objects with indeterminate boundaries (Burrough and Frank 1996; Shi et al. 1999; Cheng 2002).

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3.2 Application of fuzzy set theory to aquifer visualization An aquifer is an inherently ‘‘fuzzy’’ 3D geological object, with uncertain properties and boundaries. The uncertainty of an aquifer appears in two aspects. First, is that the components of an aquifer possess uncertainty. An aquifer can consist of unconsolidated deposits, such as gravel and sand, or rocks, such as sandstone, limestone and/or fractured igneous rocks. For each rock/sediment type (lithology) there is a possibility of being a member of ‘‘aquifer’’, and its probability is different in various conditions and locations. The other aspect of uncertainty appears in the boundary of aquifer, because the lithology itself sometimes changes gradually and continuously. Most applications of fuzzy set theory in groundwater are related to groundwater quality (Lee et al. 1995; Woldt et al. 1996; Cameron and Peloso 2001; Marczinek and Piotrowski 2002), groundwater level (Hong et al. 2002), and groundwater flow (Dou et al. 1995). Some of these works cover aquifer data management and visualization, such as aquifer spatial properties and regionalization (Piotrowski et al. 1996; Passarella et al. 1997). Piotrowski et al. (1996) demonstrated an application of the fuzzy kriging method in regionalization of hydrogeological data, in which the set of conventional and crisp values was supplemented by imprecise information subjectively estimated by an expert. The approach was based on the crisp data, with the fuzzy data used as a supplement. The study found that reliability of regionalization was higher in comparison with the regionalization performed only with the crisp dataset. In that particular case, 172 imprecise (fuzzy) values were used for the spatial interpolation of a major aquitard thickness in northwestern Germany, while 329 crisp values from boreholes were used. It might be anticipated that the result would be more reliable if the approach were fully fuzzy object-based. 3.3 Fuzzy aquifer indicator Porosity and permeability constitute the primary hydraulic properties of aquifers (see Brassington 1998 for relations to specific storage and hydraulic conductivity). Therefore, an aquifer (A) can be defined a function of the hydraulic properties of the rocks or sediments: porosity (P), permeability (k), i.e., A = f ðP; kÞ:

ð3Þ

However, porosity and permeability data are difficult, if not impossible, to obtain for an aquifer because borehole data are mainly recorded by well drillers with limited geologic background. The only data set available is lithology, from which the hydraulic properties of aquifer can only be inferred. According to hydrogeological principles, an unconsolidated gravel with good sorting would be an ideal aquifer, and clay would be an ideal aquitard. If the crisp set theory is used, we can define the possibility of being an aquifer in the former case as 1, and in the latter case as 0. It is difficult to express the in-betweens, such as a poorly sorted sand, or a silty sand. A similar relation exists for consolidated

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Table 1 Fuzzy indicator values of lithologies in Grand Forks aquifer Lithology

Porosity (%)a,b

Hydraulic conductivity (cm/s)c,d

Fuzzy value (AF)

Gravel (well sorted) Cobbles Pebbles Boulders

20–40

10"1 to 102

0.9

Sand and Gravel Sand (well sorted)

25–50

10"2 to 101

0.8

(Silty) Sand

35–50

10"4 to 10"3

0.6

(Silty) Clay

5–21

10"7 to 10"5

0.3

Bedrock (undifferentiated)

0–2

10"9

0.1

a

b

c

d

Monroe and Wicander (1998), Hudak (2000), Bear (1972), Fetter (1994)

lithologies, such as sandstone (aquifer) or shale (aquitard), although due to fracturing of consolidated units, which tends to increase the permeability, there are additional complexities. As suggested earlier, an aquifer can be represented as a fuzzy object. An aquifer usually consists of unconsolidated deposits, such as gravel and sand, or rocks, such as sandstone, limestone and/or fractured igneous rocks. For each lithology there is a possibility of being a component of ‘‘aquifer’’, depending on its hydraulic properties. So, for every rock/sediment type, a certain fuzzy value can be given for its possibility of being a component of ‘‘aquifer’’. This is called a fuzzy aquifer indicator (AF). The fuzzy indicator value for specific lithology should be between 0 and 1. 0 % AF % 1

ð4Þ

Fuzzy indicator values were assigned to all standardized sediment types found in the well logs for Grand Forks (Table 1). AF values were assigned based on the logarithms of the expected saturated hydraulic conductivity.1 Hydraulic conductivity is directly related to permeability. The logarithm of lowest value in the range specified in Table 1 was used. Our rationale for picking the lowest value was somewhat arbitrary, but is expected to result in conservative estimates of the AF values. For example, silty sand has a range of hydraulic conductivity of 10"4 to 10"3. The logarithm of the low end of this range is -4. If this value is divided by 10 (=0.4) and then subtracted from one, a value of 0.6 is obtained. Similarly, a wellsorted gravel is assigned 0.9, representing the highest likelihood that this material type acts as an aquifer. Other material types fall in between. Undifferentiated bedrock is assigned a mid-range hydraulic conductivity value for unfractured rock. This material type forms the valley sides and underlies the surficial deposits at depth. It has a corresponding AF value of 0.1, based on its expected low permeability.

1

Comparison of hydraulic conductivity values is commonly done using logarithm values due to ranges of this property that span several orders of magnitude.

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In this particular study area, the values were determined based on literature estimates of hydraulic conductivity, but are reasonably representative of the actual hydraulic conductivities. The average hydraulic conductivity for the sand and gravel is 1.56 · 10"1cm/s, with a range of 7.1 · 10"1 to 1.6 · 10"2 cm/s, based on pumping test results in 16 wells (Allen et al. 2004). The calibrated numerical groundwater flow model constructed for the aquifer (Scibek and Allen 2006; Scibek et al. 2006) used values of 9.3 · 10"2 cm/s for the gravel and 2.3 · 10"2 cm/s for the sand, values consistent with those in Table 1. The values used in the model for the silty sand and silty clay are slightly higher at 3.5 · 10"1 cm/s and 5.8 · 10"5, respectively. There were no data to confirm these values (no pumping tests in such materials), but because most of the groundwater flows in the gravel and sand, the values used for silt and clay had little effect on model calibration. The groundwater flow model was well calibrated, achieving a normalized root mean square error of less than 8% when simulated water levels were compared with observed water levels in the aquifer. This high degree of calibration lends support to the conceptual model in respect of aquifer geometry and hydraulic properties. Thus, the values in Table 1 appear to be reasonable estimates of the hydraulic conductivities based on field and model data. In other environments, with different sediment or rock types, or other mixtures, fuzzy values can be set accordingly based on either literature values (as done here) or field measurements of the permeability for the various material types. 3.4 Fuzzification of borehole data Borehole lithology logs contain a record of the material type(s) (e.g., gravel, sand, silt, etc.) present over specified depth intervals (layers) from the ground surface to the well completion depth. A specific fuzzy indicator value can be assigned for each lithology based on the relative permeability of each material type (Table 1). In order to manipulate these data, all layers need to be assigned an associated indicator value. The term fuzzification is used to describe this process. Layers are divided into two groups; one contains only one material (simple record) and the other contains multiple (2–5) materials (complicated record) (see Fig. 3). For a simple record, it is straightforward to assign a fuzzy indicator value (Table 1) to the layer, as there is only one material type. However, most of the borehole records in this particular study contain layers with multiple material types, demanding a methodology for integrating the multiple material types. One solution is to assign each material an estimated proportion of the total, according to the general rule of describing lithology in geology, namely that the first recorded lithology is dominant and the following recorded lithologies are less dominant. For example, in the 2-material case, Material 1 could be assigned a value 0.7 and Material 2 a value 0.3. This means that 70% of the material present at that depth interval is comprised of material 1, and 30% is comprised of material 2. Alternatively, a 50% indicator value could be assigned, giving equal weighting to the two material types. The decision on indicator values could be based on measured grain size distributions in selected wells, or perhaps based on discussion with the driller. Key words in the database, such as ‘‘with’’ or ‘‘and’’ can also be used to

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Table 2 Estimated proportion values for a particular layer in a borehole based on the number of material types that are used to describe the lithology of the layer Number of materials

Material 1

Material 2

Material 3

Material 4

1

1.0

2

0.7

0.3

3

0.5

0.3

0.2

4

0.5

0.3

0.1

0.1

5

0.4

0.2

0.2

0.1

Material 5

0.1

distinguish between a lower proportion of the secondary material type and an equal proportion of the material type, respectively. In this particular study, the proportion of each material type present was based on a scheme as shown in Table 2. Thus, for each layer in the borehole, the fuzzy aquifer indicator is calculated based on its material(s).

4 Three-dimensional interpolation of borehole data Defining the boundaries of an aquifer (or aquifer units within a system) typically involves using a variety of data types, such as geologic logs, borehole geophysical logs (occasionally), and rarely numeric point data, including porosity, permeability, grain size, material density, and water content. All data are usually limited in number, contain some degree of uncertainty, and are irregular in distribution both in the horizontal and vertical directions. Therefore, an interpolation method that models the spatial distribution of regionalized sample data is needed. Geostatistical approaches have long been used for modeling spatial data (e.g., Delhomme 1989; Abourifassi and Marino 1984; Hoeksema and Kitanidis 1989; Philip and Kitanidis 1989; Neuman and Jacobson 1984; Weissmann et al. 1999; Yates and Warrick 1987; Painter 1996; Piotrowski et al. 1996; Gumbricht and Thunvik 1997; Martin and Frind 1998; Desbarats and Srivastava 1991; Whittaker and Teutsch 1999; Koike et al. 2001). As most data for aquifers are obtained from borehole (or water well) logs, interpolation methods used for aquifer modeling and visualization necessarily require point-based 3D interpolation (Tobler 1999). However, the spatial correlation structure of borehole data changes locally and it is sometimes difficult to approximate by one of the theoretical semivariogram models over a 3D space. Geologic discontinuities, such as faults and unconformities, can make the semivariogram-based approaches less tangible. As well, despite their advantages in numerous situations, variograms provide only a crude characterization of complex ‘‘multi-point’’ correlation structures implicit in many sedimentary features such as meandering channels. Koike et al. (2001) developed the 3D OPM (optimization principle method), which relates a certain estimated grid value to only 25 values of the neighboring grid points. The method is able to construct directly through interpolation a complicated distribution model from sample data with obscure spatial correlation

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and from sample data whose spatial correlation changes locally. Their methodology consists of two steps. The first involves the transformation of screen locations, locations of sand and gravel layers, and resistivities obtained by electrical logging into indicator values. The second is 3D interpolation using OPM to produce three kinds of distribution models. The surfaces produced, however, are too smooth compared to the real geologic surfaces, even though an attempt was made to reduce this unrealism by adopting the concept of stochastic simulation. The problem lies in the stratification of geologic entities in most sedimentary basin environments. Another pertinent problem is that borehole geologic logs are actually not in points or voxels, but in cylinders with a certain thickness (cylinder height). Furthermore, the cylinder heights are not same for different records. That means each layer cannot be treated as a point or voxel. So, before any interpolation can be applied, the true point or voxel data must be extracted from the original cylinder data. In order to solve the problems above, a 3D interpolation method, specialized for dealing with the stratiform geologic entities and cylinder data is proposed. This method is called horizontal inverse distance weighted (HIDW). Inverse distance weighted (IDW) interpolation is one of the most commonly used techniques for interpolation of scatter points, and while not as rigorous as geostatistically-based methods, such as kriging, it is still commonly used. Inverse distance weighted methods are based on the assumption that the interpolating point should be influenced most by the nearby points and less by the more distant points. The interpolating surface is a weighted average of the scatter points and the weight assigned to each scatter point diminishes as the distance between the interpolation point and the scatter point increases. The simplest form of inverse distance weighted interpolation is sometimes called ‘‘Shepard’s method’’. For a three-dimensional point interpolation, the equation used is as follows: P Pi where: P(x,y,z) Pi e Di

Pðx,y,zÞ ¼ P

is is is is

Dei 1 Dei

ð5Þ

the point whose value needs to be estimated; the prescribed function values at the scatter points; a positive real number called the weighting exponent; the distance from the scatter point to the interpolation point.

According to general geological principles, lithology in sedimentary basins is more stable in horizontal direction than in vertical, i.e., there is stratification. For this specific situation, points close together at the same level (or the same direction) are more likely to have similar values than points above and below. Based on the sedimentary basin stratification, a paradigm called horizontal inverse distance weighted (HIDW) is proposed for the aquifer data interpolation in sedimentary basins (Fig. 4).

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Fig. 4 Local IDW interpolation with a search radius of r

HIDW emphasizes the horizontal homogeneity and vertical heterogeneity in sedimentary basins. When interpolating point values, the algorithm either uses only the data within the same level, or varies the exponent of the vertical component of distance to change the vertical search effort. In order to interpolate the aquifer indicator (fuzzy values) for every point (voxel), HIDW uses the voxel concept to handle the borehole cylinder data. In the study, a voxel is used as a key tool to get the fuzzy indicator values from the layer fuzzy values, which are stored as a cylinder format. The algorithm for this process is to allocate the voxel position in a borehole cylinder and get the corresponding value as the voxel value, which can be used in the interpolation. The equation of HIDW interpolation is as follows

Aðx,y,zÞ ¼

n P Ai

i¼1 n P i¼1

Dei 1 Dei

ð6Þ

where: A(x,y,z) is the voxel whose fuzzy indicator value needs to be estimated; n is the number of scatter voxels within a search range, and there are two kinds of search ranges for HIDW as discussed below; Ai is the known fuzzy indicator values at the voxels within a certain search range; e is the weighting exponent;

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Di

301

is the distance from a known-value voxel#to the interpolation voxel " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di ¼ ðX " XiÞ2 þ ðY " YiÞ2 þ ðZ " ZiÞ2 ; but there are some changes for distance calculation as discussed below.

According to the search ranges, HIDW has two alternatives: disk and spherical HIDWs. 4.1 Disk HIDW The disk HIDW uses a flat cylinder (disk) to determine the data set used for interpolation. As shown in the Fig. 5, the variables h and r are used to control the data set (i.e., only the data points or voxels within the disk are calculated for the interpolation of the estimated voxel). With this alternative, the distance (Di) calculation is normal. When the disk height is equal to the height of voxel, the distance calculation becomes 2D one, i.e., qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di ¼ ðX " XiÞ2 þ ðY " YiÞ2 Because the borehole data are in a cylinder format, an essential procedure is to get a value presenting only a voxel (not the whole borehole cylinder) at the same level. It is obvious that one voxel datum can be obtained for each well as long as it is drilled as deep as the estimated voxel.

Fig. 5 Disk HIDW with a disk search range. Note: the pink cube is a voxel and a red ball represents the center of a voxel

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4.2 Spherical HIDW Another alternative, spherical HIDW (Fig. 6), uses a sphere as the search range (just as the general IDW method), but gives the vertical distance less weight by lowering the exponent of the Z component. So the ‘‘distance’’ equation is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Di ¼ ðX " XiÞ2 þ ðY " YiÞ2 þ ðZ " ZiÞv where v is the exponent of the Z component of distance. 4.3 Parameters in HIDW There are several parameters that need to be determined for HIDW: 4.3.1 Voxel size Selecting a good voxel size is the first step. The Nyquist rule can be the first reference. The Nyquist rule states that there should be 2–3 pixels (here, voxel) between average spacing of data points (Viljoen 2007). The height of the voxel (Z) can be less than the horizontal components (X, Y), because aquifers are usually lie flat. In this study, the horizontal dimensions (X and Y) of a voxel are set to 50 m, while the vertical dimension (Z) of a voxel is only 10 m.

Fig. 6 Spherical HIDW with a sphere search range. Note: the pink cube is a voxel and a red ball represents the center of a voxel

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4.3.2 Weighting exponent (e) The weighting exponent for distance allows the user to adjust the emphasis placed on observed points according to their distance from the interpolation point. The weight assigned to each scatter point diminishes as the distance from the interpolation point to the scatter point increases. The value is usually set to 1 or 2, according to geologic setting, such as stability of lithology. The higher the weighting exponent (e.g., 2), the more emphasis is placed on closer points. In most cases, just as in this study, the value is set to 1. However, it can be adjusted by the user if more emphasis is desired on closer points. 4.3.3 Search radius (r) The search radius defines the maximum size, in map units, of a circular zone centered around each grid node within which point values from the original data set are averaged and weighted according to their distance from the node. A large search radius incorporates observed values that are very distant from the point of interest and can lead to smoother interpolated surfaces. However, a large search radius may also use a sample that is so distant that its relationship to the values at the location of interest is tenuous. Conversely, a small search radius can fail to generate interpolated values for a large subset of locations, but generally provides more conservative estimates of values. Little guidance exists for establishing an optimal value for search radius, and generally this is done be experimentation with the dataset. In this study, we employed a search radius of 200 m, and experimentation with other search radii did not exert a strong influence on the results. 4.3.4 Search range There are two choices—disk or spherical. A different weight calculation equation is used for each search range as discussed above. 4.3.5 Exponent of Z component of distance (v) The value of v in the spherical HIDW is determined from the geologic background of an aquifer. For example, v would be lower in the sedimentary basins filled with thin-bedded sediments than those filled with thick-bedded ones.

5 AMV3D and its application in the Grand Forks aquifer Using the fuzzy indicator of aquifer materials and HIDW, a 3D aquifer modeling and visualization system (AMV3D) was developed under a combination development environment of Visual C++ (MFC) and OpenGL (Zhang 2003). Because of the study scope, AMV3D has only basic functions, such as importing data from different sources, borehole data display and query, cross-section creation, and

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Fig. 7 Functional diagram of AMV3D

interpolation and display of an aquifer. Figure 7 is an overview of the system. Details on the use of AMV3D can be found in Zhang (2003). The dimensions of the voxel are 50 m · 50 m · H. Here H is the height of the voxel, with a changeable value according to the vertical exaggeration chosen by the user. Because the deposits in a sedimentary basin change more rapidly in the vertical direction than in the horizontal one, H is much smaller than the horizontal dimensions (50 m). In this study, H is 10 m. Based on the fuzzification of borehole log data, different rock/sediment types are displayed in different colours (Fig. 8). From red to green, the fuzzy indicator value decreases from 0.9 to 0.1, corresponding to material types in Table 1. Lithology cross-sections were then produced to display the sediment type information based on the lithology group. In Fig. 9, red represents the gravel (-cobbles-pebblesboulders); green represents the sand, blue the silt, and grey the clay. Bedrock (not shown) underlies the clay. Recent aquifer architecture modeling (Scibek and Allen 2004) and accompanying numerical flow modeling (Scibek and Allen 2006; Scibek et al. 2006) involved generating a geologic model for the Grand Forks valley using the same database and employing a layered paradigm (Fig. 10). The software GMS (Environmental Modeling Systems Inc. 2006) was used. Because of restrictions with this software for generating continuous layers, the geology is simplified into discrete units; it was not possible to represent lenses of material embedded in other material types. Interpolation of the material types is done by visual inspection of the well logs rather then by employing an interpolation algorithm. The aquifer materials (gravelumber, sand-yellow) are present near the surface. Combined, they are thickest to the valley (top left), and thin to the east (bottom right). Silty sand—green is present across the section, but is thinner to the east. The silty clay—teal is generally only present in the western portion of the valley. A lower sand aquifer (gold) is present on the bottom right. The approximate location of the cross section shown in Fig. 9 is indicated by the dashed line in Fig. 10. There is a good degree of similarity between the two, even to

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Fig. 8 Borehole data displayed in the fuzzy indicator value (from red to green, the fuzzy indicator value decreases from 0.9 to 0.1)

Fig. 9 Lithology cross-section at Y = 5,432,900 (UTM). Red represents the gravel (-cobbles-pebblesboulders) group; green the sand; blue the silt; grey the clay. Note, data are not corrected for elevation in this example

the extent that the lower sand unit is present (although slightly thinner in the fuzzy model). The consistency of the results, and those of other cross-sections (not shown) across the valley supports the fuzzy approach in respect of identifying the overall geometry of the main material types. Of course, to truly validate the fuzzy approach, high resolution well logs would have to be collected throughout the valley bottom; an impossible task. The advantage of the fuzzy model, however, is that it highlights the spatial variability of the geology, particularly in respect of the presence of lenses of less permeable material interbedded with more permeable material. These transitional material types are expected for this particular aquifer, and can be significant for groundwater flow. It is also convenient to illustrate the possible aquifer in the cross-section. In this case, using HIDW, the fuzzy aquifer indicator values are calculated for every voxel. Because the user generally wants to show only the aquifer, and not the full range of fuzzy indicator values, it is necessary to specify a boundary (or cut-off) value. This boundary value determines whether or not a voxel belongs to the aquifer set. One of two methods can be used to determine the boundary value: one is statistically based and the other is expert-based.

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Fig. 10 Three-dimensional geologic fence diagram showing the distribution of geological units in the Grand Forks valley. Umber is the gravel (-cobbles-pebbles-boulders); yellow is the sand; green is the silt; blue is the clay. This geologic model was constructed in GMS by interpreting the borehole lithology logs at coarse scale

The statistically based method depends on the distribution of the aquifer indicators. For example, Koike et al. (2001) stated that the frequency distribution of aquifer coefficients (an indicator of the aquifer they studied) showed approximately a normal distribution. So, they assumed that the aquifer consists of the values larger than the mean plus a standard deviation (m + r). In most cases, the statistical method is a good choice because there is generally at least one aquifer existing in the study area. But in some cases, such as the case where no aquifers are present, the method still can yield a mean (m) and a standard deviation (r). The expert method is based on geological and hydrogeological knowledge of the study area. A boundary value that separates the aquifer data set and the non-aquifer data set can be determined. This approach is used in AMV3D. Figure 11 is an example of an aquifer cross-section. Here, a value of 0.75 (aquifer fuzzy indicator value) is chosen as the boundary value of the aquifer, because sand, gravel and cobbles are components of the Grand Forks aquifer; a value 0.8 or 0.9 is given for these materials. From red to pink, the fuzzy indicator values decrease from

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Fig. 11 Aquifer cross-section along the center line of the valley (UTM Y = 5,431,700) determined using the fuzzy approach. From red to pink, the fuzzy indicator values decrease from 0.9 to 0.75; grey denotes aquitard units. Note, data are not corrected for elevation in this example

Fig. 12 Aquifer cross-section along the center line of the valley (UTM Y = 5,431,700) as viewed in the calibrated groundwater flow model. Gravel is shown in white, sand in blue, silt in green, clay in teal, and a lower sand in red

0.9 to 0.75. A corresponding cross section from the groundwater flow model is shown in Fig. 12; here the aquifer materials are in white-gravel, and dark blue-sand. In both sections, the aquifer is thickest to the west (left side of Figs. 11, 12), and thins to the east. The aquifer pinches out in the northeast corner of the valley. In the fuzzy model, the possible aquifers display clearly regardless of whether they exist as a layer or as a lens. This is very important because most existing GISs cannot deal with the lenses for interpolation and display.

6 Conclusions A fuzzy set-based approach for the aquifer modeling and visualization is proposed and a case study used to demonstrate its theoretical and practical significance. The

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fuzzy aquifer indicator is the principal concept of the fuzzy set approach for modeling and visualizing aquifer spatial distribution. It provides a method to convert the lithology descriptions in literal words into numeral; it gives the numeral specific meaning, i.e., the closer to 1, the larger possibility or probability of being the component of an aquifer; and it enables the borehole log data to be interpolated because they are expressed in meaningful numerals. As shown in the case study, HIDW is an applicable 3D interpolation method for stratiform objects, such as aquifers. It allows handling the subsurface lenses, which most existing GISs fail to correctly interpolate and display; and it manipulates the stratiform 3D objects more reasonably than other methods, such as OPM. AMV3D was developed for aquifer modeling and visualization. Although it is preliminary, it has good rendering abilities—AMV3D deploys OpenGL that provides a tremendous 3D object display environment; it has basic GIS functions, such as identification and selection. It also has the ability to cope with lenses and stratiform objects based on the fuzzy indicator and HIDW. From the study results, it is clear that HIDW is applicable for aquifers or other stratiform 3D objects, but it is still necessary to develop a statistical method to evaluate the interpolation quality. Boundary control is another issue related to 3D interpolation. If the bedrock surface and basin boundary data could be used to control the interpolation boundary, the results from HIDW will improve significantly. References Abourifassi M, Marino MA (1984) Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity. Math Geol 16:19–35 Alexander JW, Gibson LJ (1979) Economic geography. Prentice-Hall, Englewood Cliffs Allen DM, Scibek J, Whitfield P, Wei M (2004) Climate change and groundwater: summary report. Final report prepared for natural resources Canada, Climate Change Action Fund, March 2004, 404 pp, http://www.adaptation.rncan.gc.ca/projdb/index_e.php?class=118Grand Forks Banai R (1993) Fuzziness in geographical information systems: contributions from the analytic hierarchy process. Int J Geograph Inform Syst 7(4):315–329 Bear J (1972) Dynamics of fluids in porous media. Dover, New York Berry JK (1996) Beyond mapping: Concepts, algorithms, issues in GIS. John Wiley and Sons Ltd., 246 pp Brassington R (1998) Field hydrogeology. Wiley, New York British Columbia Ministry of Environment (2006) WELLS database http://www aardvark.gov.bc.ca/apps/ wells/ Brown DG (1998) Classification and boundary vagueness in mapping presettlement forest types. Int J Geograph Inform Sci 12:105–129 Burrough PA (1996) Natural objects with indeterminate boundaries. In: Burrough PA, Frank AU (eds) Geographic objects with indeterminate boundaries, Taylor & Francis Inc., Bristol, pp 3–28 Burrough PA (1989) Fuzzy mathematical methods for soil survey and land evaluation. J Soil Sci 40(3):477–492 Burrough PA, Frank AU (eds) (1996) Geographic objects with indeterminate boundaries. Taylor & Francis, London, pp 345 Cameron E, Peloso GF (2001) An application of fuzzy logic to the assessment of aquifers’ pollution potential. Environ Geol 40(11–12):1305–1315 Campari I (1996) Uncertain boundaries in urban space. In: Burrough PA, Frank AU (eds) Geographic objects with indeterminate boundaries, Taylor & Francis Inc., Bristol, pp 57–70 Cheng T (2002) Fuzzy objects: their changes and uncertainties. Photogrammetric Eng Remote Sensing 68(1):41–49 Chisholm M (1964) Rural settlement and land use. Wiley, New York

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