Anisotropy and Heterogeneity

2.5.4 Anisotropy and Heterogeneity

Sediments and other rocks can be homogeneous or heterogeneous within some represen- tative volume of observation. Clean beach sand made of pure quartz grains of similar size is one example of a homogeneous rock (unconsolidated sediment). If, in addition to quartz grains, there are other mineral grains but all uniformly mixed, without groupings of any kind, the sediment is still homogeneous. various possible scales, say centimeter to decameter, it is hardly ever satisfied for rock volumes representative of an aquifer or an aquitard. For simplification purposes, and when different groupings of minerals within the same rock, or sediments of different sizes within one sedimentary deposit behave similarly relative to groundwater flow, one may consider such volume as homogeneous and representative. In reality, however, all aquifers and aquitards are more or less hetero- geneous, and it is only a matter of convention, or agreement between various interested stakeholders, which portion of the subsurface under investigation can be considered ho- mogeneous. At the same time, simplification of an aquifer volume appropriate for general

GroundwaterSystem

Pumping well

Contaminant leak

Water table Plume

F IGURE 2.46 An aquifer consisting of predominantly gravel and sand provides water to a well through the entire screen length. At the same time, dissolved contaminants may enter the well through just a few discrete intervals.

water supply purposes may be completely inadequate for characterizing contaminant fate and transport. Figure 2.46 illustrates this point. Alluvial aquifers almost always con- sist of various proportions of gravel, sand, silt, and clay, deposited as layers and lenses of varying thickness. When gravel and sand dominate, with finer fractions forming thin interbeds, the aquifer may be considered as one continuum providing water to a pump- ing well through its entire screen. However, when the aquifer is contaminated, dissolved contaminants will move faster through more permeable porous media which may form quite convoluted preferential pathways intersecting the well at discrete intervals. De- tecting such pathways, although difficult, is often the key for successful groundwater remediation, whereas it may not be of much importance when quantifying groundwater flow rates for water supply.

One important aspect of heterogeneity is that groundwater flow directions change at boundaries between rocks (sediments) of notably different hydraulic conductivity such as the ones shown in Fig. 2.47. An analogy would be refraction of light rays when they enter a medium with different density, e.g., from air to water. The refraction causes the incoming angle, or angle of incidence, and the outgoing angle, or angle of refraction, to be different (angle of incidence is the angle between the orthogonal line at the boundary and the incoming streamline; angle of refraction is the angle between the orthogonal at the boundary and the outgoing streamline). The only exception is when the streamline is perpendicular to the boundary in which case both angles are the same, i.e. −90 degrees.

The mathematical relationship between the angle of incidence (α 1 ), angle of refraction (α 2 ), and the hydraulic conductivities of two porous media, K 1 and K 2 , is shown in Fig.

2.47. The figure applies to both map and cross-sectional views as long as there is a clearly defined boundary between the two porous media. Heterogeneity of the hydraulic conductivity is the main cause of macrodispersion in groundwater systems, which is of particular importance when analyzing capture zones of extraction wells, and transport of contaminants. Figure 2.48 shows two capture zones for the same well, pumping with the same rate, when the aquifer in question is modeled with a homogeneous hydraulic conductivity (right), and with a heterogeneous (spatially varying) hydraulic conductivity (left). Similarly, the shape of a plume of dissolved con- taminants will be significantly influenced by the porous media heterogeneity.

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Streamlines

(flowlines)

Lines of equal

hydraulic head

K 1 tan α 1

K 2 tan α 2

F IGURE 2.47 Refraction of groundwater flowlines (streamlines) at a boundary of higher hydraulic conductivity (top) and a boundary of lower hydraulic conductivity (bottom).

Hydraulic head contour line

15-year capture zone

0 1 2 3 4 km

F IGURE 2.48 Right: 15-year capture zone, defined by flowlines, of a well pumping from a semiconfined aquifer modeled with uniform average hydraulic conductivity. Left: Capture zone of

the same well when the aquifer is represented by spatially varying (heterogeneous) hydraulic conductivity.

GroundwaterSystem

F IGURE 2.49 Some possible reasons for anisotropy of hydraulic conductivity. (a) Sedimentary layers of varying permeability; (b) orientation of gravel grains in alluvial deposit; (c) two sets of fractures in massive bedrock. (Kresic, 2007a; copyright Taylor & Francis Group, LLC, printed with

permission.)

Anisotropy of porous media is another very important factor influencing directions of groundwater flow and transport of contaminants. It is a result of the so-called geologic fabric of rocks comprising aquifers and aquitards. Geologic fabric refers to spatial and ge- ometric relationships between all elements of which the rock is composed, such as grains of sedimentary rocks, and the component crystals of magmatic and metamorphic rocks. Fabric also refers to discontinuities in rocks, such as fissures, fractures, faults, fault zones, folds, and bedding planes (layering). Without elaborating further on the geologic portion of hydrogeology, it is appropriate to state that groundwater professionals lacking a thor- ough geologic knowledge (i.e., “nongeologists”) would likely have various difficulties in understanding the many important aspects of heterogeneity and anisotropy.

In hydrogeology, a porous medium is considered anisotropic when hydraulic con- ductivity varies in different directions. All aquifer types are more or less anisotropic, with fractured rock and karst aquifers often exhibiting the highest degree of anisotropy; such aquifers may have zones of extremely high hydraulic conductivity with almost any shape imaginable. Figures 2.49 and 2.50 illustrate just some of many possible causes of anisotropy in various types of rocks. It is important to understand that a varying degree of anisotropy can (and usually does) exist in all spatial directions. It is for reasons of simplification and/or computational feasibility that hydrogeologists consider only three main perpendicular directions of anisotropy: two in the horizontal plane and one in the vertical plane; in the Cartesian coordinate system these three directions are represented with the X, Y, and Z axes. Figure 2.51 illustrates the importance of aquifer anisotropy in determining well capture zones.

Again, for reasons of simplicity or feasibility, one may decide that the groundwater system under consideration, or any of its parts, could be represented by a volume includ- ing “all” important aspects of heterogeneity and anisotropy of the porous media present. Such volume is sometimes called representative elementary volume (REV) and is defined by only one value for each of the many quantitative parameters describing groundwater flow, and fate and transport of contaminants. The REV concept is considered by many to be rather theoretical, since it is not independent of the nature of the practical problem

to be solved. For example, less than 1 m 3 (several cubic feet) of rock may be more than enough for quantifying phenomena of contaminant diffusion into rock matrix, whereas this volume would be completely inadequate for calculating groundwater flow rate in a fractured rock aquifer where major transmissive fractures are spaced more than 1 meter

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F IGURE 2.50 Cross-bedded sandstone of the Cutler Formation in southern Utah. The banding within the outcrop represents cross-stratification of river (fluvial) deposits. Note how cross stratification truncates underlying strata. Rock hammer for scale. (Photograph courtesy of Jeff Manuszak.)

apart. Deciding on the representative volume will also depend on the funds and time available for collecting field data and performing laboratory tests. Extrapolations and interpolations based on data from several borings or monitoring wells will by default

be very different than those using data from tens of wells. Another related difficulty, which always presents a major challenge, is upscaling. This term refers to assumptions made when applying parameter values obtained from small volumes of porous media (e.g., laboratory sample) to larger, field-scale problems. Whatever the final choice of each quantitative parameter may be, every attempt should be made to fully describe and quantify the associated uncertainty and sensitivity of that parameter.

The following example illustrates how two different choices of two basic hydro- geologic parameters reflecting heterogeneity can produce very different quantitative answers, even though both selections may seem reasonable. Consider the following sce- nario: point of contaminant release and a potential receptor are 2500 ft apart; the regional hydraulic gradient in the shallow aquifer, which consists of “fine sands,” is estimated from available monitoring well data to be 0.002. How long would it take a dissolved contaminant particle to travel between the two points, assuming that the contaminant does not degrade or adsorb to solid particles (i.e., it is “conservative” and moves at the same velocity as water)?

As shown in Fig. 2.37, fine sand can have hydraulic conductivity anywhere between

a little less than 1 and about 40 ft/d. Effective porosity (specific yield) of “sand” can vary anywhere between 20 and 45 percent. Assuming the lowest values from the two ranges,

GroundwaterSystem

(a)

(b)

s Mile

Isotropic and homogeneous single layer Horizontal anisotropy along rows in a single aquifer. Area of contribution is 12.7 mi 2 layer aquifer. Area of contribution is 12.4 mi 2

0 5 Miles

(c)

(d)

Cells with

Fracture cells

Vertical fractures with 100:1 transmissivity of 2

40 bulk matrix. Area of contribution is 12.1 mi 1 10 Columns

F IGURE 2.51 The influence of aquifer anisotropy and heterogeneity on a modeled capture zone for the Central Swamp region, Cypress Creek well field near Tampa, FL. (a) Isotropic and homogeneous 1-layer aquifer; (b) anisotropic hydraulic conductivity with five times greater value along rows; (c

and d) simulation of vertical fractures with “fracture” cells where transmissivity is 100 times greater than in the surrounding “matrix” cells. (Modified from Knochenmus and Robinson, 1996.)

the linear velocity of a groundwater particle, using Eq. (2.34), is: v L =

K×i =

0.8 (ft/d) × 0.002 =

0.016 [ft/d]

n ef 0.2

Based on this velocity, the time of groundwater (and dissolved contaminant) travel between the two points of interest would be 156,250 days or about 428 years (2500 ft- distance is divided by the velocity of 0.016 ft/d). Using the highest values from the two ranges (40 ft/d and 45 percent), the time of travel is calculated to be about 14,045 days or

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38.5 years, which is a very significant difference, to say the least. This simple quantitative example shows inherent uncertainties in quantifying groundwater flow characteristics, even when assuming that the porous medium is “homogeneous.”