# Comparison of alternative representations of hydraulic-conductivity anisotropy in folded fractured-sedimentary rock: modeling groundwater flow in the Shenandoah Valley (USA)

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DOI: 10.1007/s10040-008-0431-x

- Cite this article as:
- Yager, R.M., Voss, C.I. & Southworth, S. Hydrogeol J (2009) 17: 1111. doi:10.1007/s10040-008-0431-x

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## Abstract

A numerical representation that explicitly represents the generalized three-dimensional anisotropy of folded fractured-sedimentary rocks in a groundwater model best reproduces the salient features of the flow system in the Shenandoah Valley, USA. This conclusion results from a comparison of four alternative representations of anisotropy in which the hydraulic-conductivity tensor represents the bedrock structure as (model A) anisotropic with variable strikes and dips, (model B) horizontally anisotropic with a uniform strike, (model C) horizontally anisotropic with variable strikes, and (model D) isotropic. Simulations using the US Geological Survey groundwater flow and transport model SUTRA are based on a representation of hydraulic conductivity that conforms to bedding planes in a three-dimensional structural model of the valley that duplicates the pattern of folded sedimentary rocks. In the most general representation, (model A), the directions of maximum and medium hydraulic conductivity conform to the strike and dip of bedding, respectively, while the minimum hydraulic-conductivity direction is perpendicular to bedding. Model A produced a physically realistic flow system that reflects the underlying bedrock structure, with a flow field that is significantly different from those produced by the other three models.

### Keywords

USAFractured rocksNumerical modelingBedrock structureAnisotropy# Comparaison de représentations alternatives de l’anisotropie de la conductivité hydraulique dans des roches sédimentaires plissées et fracturées: modélisation de l’écoulement des eaux souterraines dans la vallée de Shenandoah (Etats-Unis)

## Résumé

Une représentation numérique qui traduit de manière explicite l’anisotropie généralisée en trois dimensions de roches sédimentaires plissées et fractures, dans un modèle hydrodynamique, reproduit le mieux les caractéristiques essentielles du système d’écoulement de la Vallée de Shenandoah (Etats-Unis). Cette conclusion repose sur la comparaison de quatre représentations possibles de l’anisotropie dans lesquelles le tenseur de perméabilité représente la structure du bedrock comme (modèle A) anisotrope avec des directions et des pendages variables, (model B) horizontalement anisotrope avec une direction uniforme, (modèle C) horizontalement anisotrope avec des directions variables, et (modèle D) isotrope. Des simulations à l’aide du modèle d’écoulement et de transport SUTRA de l’U.S.G.S. sont basées sur une représentation de la conductivité hydraulique concordante aux plans de stratification dans un modèle structural en trois dimensions de la vallée, qui transcrit l’agencement des roches sédimentaires plissées. Dans la représentation la plus générale, (model A), les directions des conductivités hydrauliques maximales et médianes concordent avec les directions et pendages des couches, respectivement, tandis que la direction des conductivité hydrauliques minimales est perpendiculaire à la stratification. Le Modèle A fournit un système d’écoulement physiquement réaliste qui reflète la structure du substratum sous-jacent, avec un champ d’écoulement significativement différent de ceux produits par les trois autres modèles.

# Comparación de representaciones alternativas de la anisotropía de la conductividad hidráulica en rocas sedimentarias fracturadas plegadas: modelación del flujo subterráneo en el Shenandoah Valley (EEUU)

## Resumen

Una representación numérica que representa explícitamente la anisotropía generalizada tridimensional en rocas sedimentarias fracturadas plegadas en un modelo de aguas subterráneas es la que mejor reproduce las características más destacadas del sistema de flujo en el Shenandoah Valley, EEUU. Esta conclusión resulta de una comparación de 4 representaciones alternativas de anisotropías en las cuales el tensor de la conductividad hidráulica representa la estructura del basamento como (modelo A) anisotrópico con inclinación y rumbo variable, (modelo B) horizontalmente anisotrópico con un rumbo uniforme, (modelo C) horizontalmente anisotrópico con rumbos variables, y (modelo D) isotrópico. Las simulaciones usando el modelo de flujo y transporte del agua subterránea del US Geological Survey SUTRA están basadas en una representación de la conductividad hidráulica que se conforma con planos de estratificación en un modelo estructural tridimensional del valle que duplica el esquema de rocas sedimentarias plegadas. En la representación más general, (modelo A), las direcciones de la conductividad hidráulica máxima y media se adaptan al rumbo e inclinación de la estratificación, respectivamente, mientras que la dirección de conductividad hidráulica mínima es perpendicular a la estratificación. El modelo A produjo un sistema de flujo físicamente realista que refleja la estructura del basamento subyacente, con un campo de flujo que es significativamente diferente de aquellos producidos por los otros 3 modelos.

# 褶皱裂隙沉积岩中水力传导度各向异性数字仿真的比较: 美国Shenandoah流域的地下水流模拟

## 摘要

地下水模型中可正确描述一般褶皱裂隙沉积岩三维各向异性的数值表述能很好的复制美国Shenandoah流域地下水流特征。该结论由下述四种描述各项异性的数值表述得出: 即渗透系数张量所代表的基岩结构为不同走向和倾向的各向异性 (模型A) , 走向一致的水平向各向异性 (模型B) , 走向不同的水平向各向异性 (模型C) , 和各向同性 (模型D) 。利用美国地质调查局地下水流与运移模型SUTRA的模拟是基于代表相应于流域中褶皱沉积岩模式三维结构模型中层面的渗透系数。在最常规的表述 (模型A ) 中, 渗透系数最大值和中间值的方向分别受层面的走向和倾向控制, 而最小值方向则与层面垂直。模型A产出了一个物理上现实的能反映下伏基岩结构的流动系统, 具有与其它三个模型显著不同的流场。

# Comparação entre representações alternativas da anisotropia na condutividade hidráulica em rochas sedimentares fracturadas e dobradas: modelação do fluxo de água subterrânea em Shenandoah Valley (EUA)

## Resumo

A reprodução mais adequada das características relevantes do sistema de fluxo em Shenandoah Valley, EUA, é conseguida, num modelo de fluxo das águas subterrâneas, através de uma representação numérica explícita da anisotropia tri-dimensional generalizada de rochas sedimentares fracturadas e dobradas. Esta conclusão resulta da comparação entre quatro representações alternativas da anisotropia em que o tensor da condutividade hidráulica representa a estrutura do bedrock como (modelo A) anisotrópico com atitudes variáveis, (modelo B) anisotrópico na horizontal com direcção uniforme, (modelo C) anisotrópico na horizontal com direcção variável; (modelo D) isotrópico. As simulações conduzidas com o modelo de transporte e fluxo SUTRA, do USGS, baseiam-se na representação da condutividade hidráulica orientada segundo os planos de estratificação num modelo estrutural tri-dimensional que reproduz o padrão de dobramento das rochas sedimentares. Na representação mais geral (modelo A), as direcções de condutividade hidráulica máxima e média orientam-se, respectivamente, segundo a direcção e a inclinação da estratificação, enquanto a direcção de condutividade hidráulica mínima é perpendicular à estratificação. O Modelo A produziu uma representação fisicamente realista do sistema de fluxo que reflecte as características do bedrock, com uma representação de fluxo significativamente diferente da produzida pelos outros três modelos.

## Introduction

Anisotropic hydraulic behavior has long been observed in inclined fractured-sedimentary rocks (e.g. Vecchioli 1967). The anisotropic behavior is a consequence of preferential flow through the bedding-plane fractures and joints in the rock which form a three-dimensional fracture network. Simulation studies of inclined fractured-rock aquifers represent anisotropy in the fracture network through the three-dimensional, hydraulic-conductivity tensor (herein referred to as the conductivity tensor), which defines the magnitudes and directions of maximum, medium and minimum hydraulic conductivity in the aquifer system. Simulations that utilize classical finite-difference (FD) models are limited to an orthogonal grid oriented parallel to the principal axes of the unidirectional conductivity tensor. As a result, the simulated conductivity tensor is either (1) horizontal and does not account for the dip of the bedding, or (2) if the FD grid is deformed to parallel the dip of the bedding, the axis of minimum hydraulic conductivity is vertical, rather than perpendicular to the dip. Thus, representation of the complicated structural geometry of deformed or folded rocks using FD grids is logistically difficult or impractical.

In contrast, the orientation of the conductivity tensor in finite-element (FE) models is defined independently of the model coordinate axes, so the directions of maximum, medium and minimum hydraulic conductivity can vary throughout the model domain. As a result, it is possible to more accurately represent anisotropy in groundwater flow through complicated structural settings using FE models. In this study, application of both FD and FE methods are compared through simulation of groundwater flow in the Shenandoah Valley, Virginia, which contains a 5-km thick strata of folded sedimentary rock referred to as the Massanutten syncline.

### Anisotropy in fractured-sedimentary rocks

Consolidated sedimentary rocks generally have little primary permeability, so the anisotropic behavior of groundwater flow is a consequence of the orientations of bedding and fractures that provide preferential flow paths within the rock. In both flat-lying and inclined sedimentary rocks, at least two orthogonal sets of fractures typically cut across the bedding to provide hydraulic connection between adjacent bedding-plane fractures (Michalski and Britton 1997; Carleton et al. 1999; Burton et al. 2002). The density and spacing of joints that cut across the bedding determines the degree of anisotropy. In these rocks, the direction of maximum hydraulic conductivity is oriented parallel to bedding and the direction of minimum hydraulic conductivity is normal to bedding. Thus, in flat-lying rocks the conductivity tensor is oriented horizontally and vertically, while in inclined rocks, the conductivity tensor is oriented parallel to the strike and dip of bedding. In folded rocks, the conductivity tensor can be oriented in any direction and the direction can vary from place to place, with the minimum hydraulic-conductivity direction oriented normal to the dip of the bedding.

### Alternative representations of flow in folded fractured-sedimentary rocks

Computer codes used to develop groundwater models are based on the theory of flow through porous media. However, they are commonly applied to fractured-rock aquifer systems, especially at a regional scale, by accounting for anisotropy in the hydraulic properties of fractured rock. Computer codes are available to simulate flow through discrete fractures embedded in porous media (e.g. Therrien and Sudicky 1996), but have typically not been applied to regional scale models where the geometry of discrete fractures is poorly defined. One exception is a field-scale (15 km) model of flow through gold-bearing veins near a shear zone in the Canadian Shield in Ontario where the geologic structure has been extensively mapped (Beaudoin et al. 2006).

*K*

_{max}and

*K*

_{med}) are generally assumed to lie within bedding planes, while the minimum hydraulic conductivity (

*K*

_{min}) is oriented perpendicular to the bedding (i.e. parallel to the joint sets). A perspective diagram of a block of folded sedimentary rock (Fig. 2a) illustrates this approach by showing hydraulic-conductivity directions at selected points on the upper edge of the block.

*K*

_{max}directions are oriented parallel to the strike of the bedding in the horizontal plane. The

*K*

_{med}and

*K*

_{min}directions are parallel and perpendicular to the dip of the bedding, respectively.

Simulation studies of folded or inclined fractured-rock aquifers typically utilize finite-difference (FD) models that are formally limited to an orthogonal grid oriented parallel to the principal axes of the conductivity tensor. These studies have used several approaches to represent anisotropic conductivity. The examples given here were three-dimensional models constructed with the FD MODFLOW code (Harbaugh et al. 2000). Under the first approach (uniform strike), a horizontal FD grid is aligned parallel to the strike of the bedding and horizontal anisotropy is specified to decrease hydraulic conductivity in the dip direction. Vertical hydraulic connections are controlled in this approach by specifying vertical anisotropy to decrease the hydraulic conductivity in the vertical direction. An illustration of the uniform-strike approach (Fig. 2b) indicates *K*_{max} and *K*_{min} directions parallel and perpendicular, respectively, to the strike of the bedding (in the horizontal plane), and the *K*_{med} direction oriented vertically. Because the simulated conductivity tensor is oriented horizontally and vertically, it does not conform to the actual dip of the bedding.

Finite-difference models constructed to simulate groundwater flow through inclined fractured-sedimentary rocks

Anisotropy | |||||
---|---|---|---|---|---|

Model design and geologic setting | State | Dip of bedding, degrees | Horizontal (within layer) | Vertical (between layers) | Reference |

Horizontal grid (uniform strike, Fig. 2b) | |||||

Newark Basin | PA | 11 | 11 | 1 | Senior and Goode (1999) |

Appalachian Valley and Ridge province | TN | 20 | 2–8 | 10 | Haugh (2002) |

Appalachian Valley and Ridge province | PA | 60–80 | 5 | 24 | Lindsey and Koch (2004) |

Inclined grid (Fig. 3a) | |||||

Newark Basin | NJ | 27 | 2 | 1.8 × 10 | Carleton et al. (1999) |

Newark Basin | PA | 8 | 1 | 138–7,500 | Goode and Senior (2000) |

Newark Basin | NJ | 15–70 | 1 | 10 | Lewis-Brown and Rice (2002) |

Newark Basin | PA | 10–30 | 5 | 10 | Risser and Bird (2003) |

Newark Basin | NJ | 8 | 1 | 190 to 660 | Lewis-Brown et al. (2005) |

Stair-step grid (Fig. 3b) | |||||

Newark Basin | PA | 12 | 1 | 3.8 × 10 | Goode and Senior (2000) |

The second approach is a variant of the uniform-strike approach and uses the LVDA package in MODFLOW (Anderman et al. 2002) to represent variable-direction anisotropy within a two-dimensional plane. In this approach (variable strike) the *K*_{max} and *K*_{min} directions (in the horizontal plane) can vary to represent changes in the strike of the bedding throughout the model domain, but the *K*_{med} direction remains vertical (Fig. 2c). Neither the uniform-strike approach nor the variable-strike approach can simulate the anisotropic hydraulic behavior within dipping beds, nor accurately represent the vertical movement of groundwater along bedding.

*K*

_{min}direction is vertical rather than perpendicular to the dip of the bedding. These approaches are best suited to relatively simple geologic structures such as homoclines, although the stair-step approach could be applied to more complex geologic structures if the spacing of the FD grid was sufficiently small.

A number of studies have used these latter two approaches to simulate flow on local and regional scales (Table 1). Hydraulic conductivity of model layers representing fracture zones is typically assumed to be isotropic, although values of anisotropy (ratio of strike-parallel to strike-perpendicular hydraulic conductivity) were estimated by Carleton et al. (1999; 2:1) and specified by Risser and Bird (2003; 5:1). Vertical anisotropy (ratio of strike-parallel to cross-bedding hydraulic conductivity) was generally an insensitive parameter in model calibration, except in models that simulated aquifer tests (Carleton et al. 1999; Goode and Senior 2000) in which estimated values ranged from 138:1 to over 10^{5}:1.

The accuracy of simulations that use an orthogonal FD grid is limited by the assumption that the principal directions of the conductivity tensor are parallel to the coordinate axes used to define the grid. Significant errors in flow can result if the orientation of the conductivity tensor does not coincide with the model coordinate axis (Hoaglund and Pollard 2003). The resulting errors in flow are a function of the dip, the direction of the hydraulic gradient, and the anisotropy (ratio of bedding-parallel to cross-bedding hydraulic conductivity). The errors are small (< 20%) for dip angles less than 10°, but can reach 100% for dip angles greater than 50° (Hoaglund and Pollard 2003). Weiss (1985) had previously developed a FD numerical method to address this issue by using a non-orthogonal, curvilinear coordinate system, but the method has not been incorporated in widely used FD models such as MODFLOW. The LVDA package in MODFLOW that represents variable-direction anisotropy could, in principal, be extended to three dimensions, but this modification had not been included in MODFLOW by 2008.

In this study, the FE model SUTRA was used to simulate groundwater flow in the folded rocks within the Shenandoah Valley where anisotropic hydraulic behavior has previously been observed in tracer and aquifer tests. Strike and dip directions were computed from bedding surfaces that were interpolated from generalized structural cross-sections. SUTRA was also used to mimic the application of FD models using the uniform-strike and variable-strike approaches defined above (Fig. 2b and c) by defining the orientation of the conductivity tensor to be horizontal and parallel to the strike direction of the bedding. The application of the inclined-grid and stair-step-grid approaches to this hydrogeologic setting would be very difficult and was not attempted. Simulation results obtained from the different representations of the pattern of bedding were compared to evaluate differences in predicted groundwater ages and in the delineations of the capture zone of a municipal water supply.

## Hydrogeologic setting

^{2}Shenandoah Valley, which is part of the Great Valley section of the Appalachian Valley and Ridge province that extends from southern New York to central Alabama (Fig. 1). The valley is underlain by Cambrian to Devonian carbonate and clastic rocks that were folded and faulted during the Alleghanian orogeny (about 300 million years ago) to form the Massanutten syncline (Fig. 4). The syncline underlies the central part of the valley and contains a section of strata as much as 5-km thick, with the core comprised of clastic rocks that underlie Massanutten Mountain. The valley is bounded to the east by Mesoproterozoic to Early Cambrian metamorphic rocks of the Blue Ridge anticlinorium, and to the west by the North Mountain thrust fault. The regional Staunton-Pulaski thrust fault divides the valley into two large structural blocks, each characterized by a distinct pattern of bedding.

The valley is drained northward by the Shenandoah River and Opequon Creek which are tributaries to the Potomac River (Fig. 4). The aquifer system appears to be near steady state (no change in storage) because long-term well hydrographs indicate little trend in water levels (other than seasonal fluctuations), groundwater withdrawals account for less than 3% of the total recharge to the valley and there is little anthropogenic influence from dams or reservoirs. Mean annual precipitation in the valley ranges from 91.5 to 110 cm (Nelms et al. 1997). Recharge in 20 drainage basins in the valley was estimated through a linear regression that related base flow to the proportions of four rock classes underlying each basin (Yager et al. 2008). Recharge rates are lowest in clastic rocks (13.9 cm/year) and highest in western-toe carbonates (39.5 cm/year; Yager et al. 2008). The western toe carbonates are overlain by colluvial gravel as much as 150 m thick which enhances recharge and dissolution of the carbonate rock. Median transmissivity values estimated from specific capacity data indicate that the western-toe carbonate is the most transmissive rock unit (110 m^{2}/day), followed by carbonate rocks (25 m^{2}/day), clastic rocks (10 m^{2}/day), and metamorphic rocks (5 m^{2}/day; Yager et al. 2008).

Groundwater flows in the Shenandoah Valley through a set of interconnected flow systems. The foliations described above form a three-dimensional network of fractures that is approximately orthogonal and aligned with the bedding of the rock in areas of carbonate and clastic rocks, while in metamorphic rocks, the network is more irregular. This pervasive fracture network forms a well-connected regional flow system throughout the valley. The direction of maximum hydraulic conductivity in sedimentary rocks is assumed to be oriented within bedding planes, with the direction of minimum hydraulic conductivity perpendicular to the bedding. This arrangement causes the anisotropic hydraulic behavior apparent in preferential flow parallel to the strike of bedding, as observed in tracer and aquifer tests (Jones 1991; Burbey 2003). The fracture openings in the carbonate rocks may also have been widened by dissolution, causing preferential flow within bedding planes. The direction of maximum hydraulic conductivity within bedding planes could, therefore, be either parallel or perpendicular to strike, with the medium hydraulic conductivity having a lower value.

The regional flow system is intersected locally by faults that offset the bedding and act as barriers to flow. Intense fracturing and cleavage associated with some faults also provide conduits for vertical flow, however, and have been correlated with spring locations in carbonate rocks of the valley (Harlow et al. 2005). In addition, dissolution along fractures in some carbonate rocks has created narrow karst channels that funnel groundwater flow. Sinkholes provide discrete points of recharge, and springs serve as discrete points of discharge. These discrete features (faults and karst) create local flow systems that are hydraulically connected to the regional flow system. Geochemical evidence indicates that groundwater discharge to springs in the valley is generally near saturation with respect to calcite (L.N. Plummer, US Geological Survey, personal communication, 2007). In contrast, groundwater in areas dominated by discrete fracture flow is typically undersaturated with respect to calcite, because travel times are too short to allow dissolution reactions to reach saturation (White 1988). Thus, discharge to springs in carbonate rock in the Shenandoah Valley appears to be dominated by the regional flow system, rather than by discrete fractures.

## Bedrock structure

Form lines drawn parallel to bedding illustrate the folded strata and are generalized due to the scale and spacing of the cross sections. Form lines on each section were connected through interpolation parallel to the strike direction to create “form surfaces”, which provide spatial continuity throughout the valley. The interpolation was performed using procedures (Richard Winston, US Geological Survey, personal communication, 2006) written for SutraGUI (Winston and Voss 2004), a graphical interface developed for SUTRA that is based on the Argus ONE program (Argus Interware 1997). The resulting stack of form surfaces provides a three-dimensional representation of bedding. Due to complexity, the form surfaces are generalized for clastic rocks that underlie Massanutten Mountain and for carbonate rocks to the east. Structure within the metamorphic rocks is complex and not explicitly represented.

## Groundwater flow model

Groundwater flow in the Shenandoah Valley was simulated with three-dimensional models constructed using SUTRA with an irregularly connected mesh of hexahedral elements. Steady-state simulations were used to compute the hydraulic-head distribution and to estimate the rate of groundwater flow because the aquifer system is assumed to be near steady-state, as indicated by long-term well hydrographs and the relatively small rate of groundwater withdrawals. The models represent the aquifer system in fractured bedrock with a constant fluid density and a constant saturated thickness. Downward flow from the saturated overburden is represented by recharge to the underlying fractured bedrock.

Parameter values in alternative models of groundwater flow in the Shenandoah Valley

Model | ||||
---|---|---|---|---|

A | B | C | D | |

Parameter | Variable strike and dip | Uniform strike | Variable strike | Isotropic |

Maximum hydraulic conductivity, m/day | ||||

Carbonate rock | 3.4 | 3.7 | 4.0 | 1.0 |

Clastic rock | 0.26 | 0.26 | 0.23 | 0.04 |

Metamorphic rock | 0.13 | 0.17 | 0.14 | 0.08 |

Western-toe carbonate rock | 1.1 | 1.3 | 1.2 | 0.77 |

Anisotropy ratios | ||||

Maximum:medium | ||||

Carbonate rock | 1 | 1 | 1 | 1 |

Clastic rock | 1 | 1 | 1 | 1 |

Metamorphic rock | 1 | 1 | 1 | 1 |

Western-toe carbonate rock | 1 | 1 | 1 | 1 |

Maximum:minimum | ||||

Carbonate rock | 17 | 7.8 | 8.0 | 1 |

Clastic rock | 17 | 7.8 | 8.0 | 1 |

Metamorphic rock | 1 | 1 | 1 | 1 |

Western-toe carbonate rock | 1 | 1 | 1 | 1 |

Power function | ||||

Decay factor: m | 1.1E-03 | 1.3E-03 | 1.2E-03 | 6.0E-04 |

Depth threshold, m | ||||

Carbonate rock | 150 | 150 | 150 | 150 |

Clastic rock | 50 | 50 | 50 | 50 |

Metamorphic rock | 70 | 70 | 70 | 70 |

Western-toe carbonate rock | 150 | 150 | 150 | 150 |

Stream leakance, day | ||||

Potomac River | 4.E-03 | 4.E-03 | 4.E-03 | 4.E-03 |

Other stream channels | 40 | 40 | 40 | 40 |

### Model design and discretization

^{2}and includes the watersheds of the Shenandoah River and Opequon Creek, with the exception of areas west of the North Mountain Fault. The model domain is discretized using an irregularly connected, FE mesh in the areal plane that is extended vertically to the bottom of the model domain. Areally, the mesh appears as a set of quadrilateral elements with maximum lateral lengths of 1 km, with smaller elements (500 m) used to better represent the bedrock geology in the narrow valley east of Massanutten Mountain. A generalized stream network was delineated and used to constrain the mesh generation (Fig. 7).

The approach chosen here to simulate groundwater flow was based on the resolution of the simulation model and the capability of available modeling programs. The 1-km lateral-mesh spacing was chosen to provide a model that could represent the entire valley with a computation time that was sufficiently short to allow model calibration. The lateral-mesh resolution is fine enough to represent regional trends in topography and the principal drainage channels, but is too coarse to represent discrete features (karst and minor faults) that may be locally important. The choice of lateral-mesh spacing was, therefore, a compromise that considered both the regional nature of the model and the available field data. The current groundwater model, therefore, was designed to represent fracture flow through the regional flow system, and not local flow systems controlled by discrete features.

#### Model boundary conditions

All of the lateral and bottom boundaries of the model domain were assumed to be impervious to groundwater flow. Perennial stream channels were represented by head-dependent flow boundaries that utilized modifications of the SUTRA code (A.M. Provost, US Geological Survey, personal communication, 2006) permitting a separate conductance term to be specified for each specified-head node and allowing only outflow from boundary nodes when the calculated head is higher than the elevation of the stream channel (similar to a drain boundary in MODFLOW). The top model boundary represented recharge using the rates presented earlier that were estimated through linear regression relating base flow to the proportions of the four rock classes.

#### Hydraulic conductivity

Hydraulic-conductivity values were estimated through model calibration for each of the four rock units and a power function was used to decrease hydraulic conductivity with increasing depth below a specified threshold. Conductivity tensors for metamorphic and western-toe carbonate rocks were assumed to be isotropic, while conductivity tensors for carbonate and clastic rocks were assumed to be anisotropic. In the upper model unit (Fig. 8), the elements belonging to each rock unit are defined from the bedrock geology at land surface (Fig. 4). Elements associated with the lower three units correspond to clastic rocks in unit 2, either carbonate or western-toe carbonate rocks in unit 3, and metamorphic rocks in unit 4. A minimum unit thickness of 1 m was specified in parts of the model domain where a rock unit is absent to maintain hydraulic continuity of each unit throughout the domain; this 1-m unit is assigned the value of hydraulic conductivity of the unit below it.

*K*with increasing depth below land surface and used the following equation:

*K*

_{depth}is the hydraulic conductivity [L/T] at depth

*d*[L] below a threshold depth

*D*and λ is a decay factor [L

^{−1}]. Threshold depths

*D*were assigned for each rock unit based on the mean depth of wells completed in the unit. The depth-decay factor λ was estimated through nonlinear regression and the minimum value of

*K*

_{depth}was limited to 10

^{−12}m/s.

Conductivity tensors for clastic and carbonate rocks were aligned with the strike and dip of the bedding in model A (variable strike and dip), with maximum and medium conductivity (*K*_{max} and* K*_{med}) parallel to bedding and minimum conductivity (*K*_{min}) perpendicular to bedding. Angles required to define the conductivity tensor in model A were interpolated for each mesh element from the stack of form surfaces that provided a three-dimensional representation of bedding within the folded rocks underlying the valley. The interpolation was performed using three SutraGUI functions (Richard Winston, US Geological Survey, personal communication, 2006) that transformed the angles into the coordinate system utilized by SUTRA.

The conductivity tensors for clastic and carbonate rocks were oriented horizontally (with the* K*_{med}-direction oriented vertically) in the three alternative models (B, C and D) that used SUTRA to mimic the application of FD grids to the Shenandoah Valley (Fig. 2). In model B (uniform strike), all* K*_{max}-directions were oriented parallel to the N30°E axis of the valley throughout the model domain, while in model C (variable strike) the *K*_{max}-directions were equivalent to those used in model A (variable strike and dip). In model D (isotropic), the *K*_{max}, *K*_{med} and *K*_{min} values were equal and no preferred flow direction was specified. The conductivity tensors for western-toe carbonate and metamorphic rocks were isotropic in all four models.

*K*

_{max}and the directions of strike and dip in model A are portrayed in perspective three-dimensional views of the model domain cropped along section D–D’ (Fig. 9). The predominant strike direction of the bedding is N30°E, but the strike directions in the model range from N30°W to S30°W (Fig. 9b). The directions N30°E (yellow) and S30°W (blue) are parallel and distinguish bedding that is inclined southeastward from bedding that is inclined northwestward, respectively. These areas cover most of the model domain and are associated with clastic and carbonate rocks. Narrow bands of equal-strike direction correspond to undulations in the bedding. The area with a strike direction of 90°E (green) is defined in SUTRA for areas that are isotropic and corresponds to western-toe carbonate and metamorphic rocks. Dip angles range from 0 to 60°E and are generally steepest in carbonate rocks along the flanks of the basin (Fig. 9c). Dip angles of 0° (red) are assigned to western-toe carbonate and metamorphic rocks, and areas where the bedding is horizontal.

### Model calibration

The groundwater flow models were calibrated through nonlinear regression with UCODE (Poeter et al. 2005) to estimate hydraulic-conductivity values using measured groundwater levels in 354 wells and discharges at 23 gaging stations in the Shenandoah Valley. The estimated parameters included* K*_{max} of each of the four rock classes, the* K*_{max}:*K*_{min} ratio for carbonate and clastic rocks, and the decay factor in the power function used to decrease hydraulic conductivity with depth (Table 2). Note that in variable-strike-and-dip model A, *K*_{max}:*K*_{med} for carbonate and clastic rocks is the ratio of strike-parallel to dip-parallel conductivity and *K*_{max}:*K*_{min} is the ratio of strike-parallel to cross-bedding conductivity.

Water-level measurements were obtained from the USGSs National Water Information System (NWIS) database for well data. The measurement interval was defined by the top and bottom of water-bearing zones intercepted by the well, or the open interval in the well, if the water-bearing zones were not noted. The measurement depth in the model was assumed to be midway between the top and bottom of the measurement interval, unless the water level was below the top of the interval, in which case the measurement depth was set midway between the water level and the bottom. The average measurement-interval length was 4 m with an average depth of 55 m for the 249 wells in Virginia, and 68 m at 31-m depth for the 105 wells in West Virginia, where detailed well completion data were not available. About 40% of the Virginia wells were measured in 1984 or 1985 when water levels were near the long-term average, while the measurement date was unknown for the remainder. Nearly all the measurements in West Virginia were made in 2003 or 2004 when water levels were lower than normal.

Base-flow records exceeded 10 years for 19 streamflow-gaging stations and were less than 2 years at the other 4 stations. The smaller watersheds are nested within the larger watersheds, and drainage-area sizes range from 36 to 7,876 km^{2}. Weights assigned to the head and flow observations in the regression were chosen to reflect measurement error, and to insure that the weighted head residuals and weighted flow residuals influenced the regression equally. Weights assigned to head observations were equivalent to a 3-m standard deviation in measurement error. Weights assigned to flow observations were proportional to the estimated base flow and were equivalent to coefficients of variation ranging from 1 to 4%. These values underestimated the actual rating-curve error, but served to increase the influence of flow measurements, so that the both head and flow data had approximately equal weight in the regression.

*K*

_{max}values for the four rock units differ by about an order of magnitude, with the largest values estimated for carbonate rocks and the smallest value estimated for metamorphic rock (Table 2). Although the estimated

*K*

_{max}value for western-toe carbonates rocks (1.1 m/day) is less than those estimated for other carbonate rocks (3.4 m/day), the

*K*

_{min}value is larger (1.1 m/day and 0.2 m/day, respectively) because the western-toe carbonate is assumed to be isotropic in model simulations. The estimated decay factor λ results in about an order of magnitude decline in hydraulic conductivity with 1 km of land surface for each rock unit (Fig. 11). Hydraulic conductivity values decrease more slowly with depth beneath the carbonate rock areas where

*K*

_{max}is larger and the threshold depth is deeper (Fig. 9a). Simulated groundwater velocities are higher (greater than 5 m/day) in lowlands within carbonate areas and lower (less than 1 m/day) in uplands in clastic and metamorphic areas (Fig. 12). Simulated velocities generally decline to less than 0.5 m/day within 1 km of land surface, except beneath carbonate areas where velocities of 0.5 m/day persist to 2 km. Velocities were computed using an effective porosity estimated with groundwater age simulations, as discussed in the following section.

## Comparison of alternative models

*K*

_{max}:

*K*

_{med}, 7.8:1), which is less than one-half the corresponding ratio (

*K*

_{max}:

*K*

_{min}, 17:1) estimated for model A (Table 2). There is little difference in model error between models A, B and C (Table 3). Hydraulic conductivity values estimated for the isotropic model D are smaller than those estimated for the other models and the model error is slightly larger, mainly due to increased error in areas of clastic rock.

Standard error (m) in head observations for groundwater flow models of the Shenandoah Valley

Rock unit | Number of observations | Model | |||
---|---|---|---|---|---|

A | B | C | D | ||

Variable strike and dip | Uniform strike | Variable strike | Isotropic | ||

Carbonate | 226 | 21.1 | 20.6 | 20.7 | 20.5 |

Clastic | 63 | 18.2 | 19.7 | 18.8 | 27.5 |

Metamorphic | 16 | 22.0 | 23.0 | 22.5 | 20.5 |

Western-toe | 49 | 18.2 | 18.1 | 18.4 | 18.1 |

All | 354 | 20.2 | 20.2 | 20.1 | 21.6 |

### Groundwater age distributions

Groundwater age was computed through steady-state simulations in which the solute concentration was interpreted as age. The initial concentration (age) and concentration of recharge were specified as zero, and a zero-order (linear increase in time) solute production of unit strength (1 concentration unit/year) was used to represent aging. Longitudinal and transverse dispersivities were specified as 800 and 30 m, respectively, to facilitate convergence of the transport solution and reduce numerical oscillations. Results of transport simulations presented herein are largely insensitive to smaller specified values of longitudinal dispersivity.

The simulated ages are inversely proportional to the value of effective porosity specified in the models, which has not been estimated for the modeled area. A basin-wide effective porosity (1.3 × 10^{−4}) was estimated by adjusting the flow-weighted average age of groundwater discharge in model A to the 20-year residence time previously computed for the Potomac watershed by Michel (1992), an estimate based on long-term records of tritium content in surface water. The simulated age distributions are only examined qualitatively to distinguish younger waters from older waters, however, because of uncertainty regarding the variability of effective porosity within rock units and with depth below land surface.

Spatial distributions of young groundwater (less than 50 years) are similar for all models (Fig. 13), except along the Passage Creek valley, where simulated groundwater ages are up to 40 years older in models B (uniform strike) and C (variable strike). Massanutten Mountain coincides with a groundwater divide in the flow distribution simulated by models B and C, partitioning westward flow to the North Fork from eastward flow to the South Fork. Flow is isotropic in the vertical plane in models B and C and the degree of vertical hydraulic connection is uniform throughout the model domain, so flow is largely controlled by topography, rather than by the dip of bedding, as in model A. Groundwater flow simulated with model D (isotropic) is also controlled by topography (Fig. 13d). Young water penetrates deeper beneath uplands along North Mountain and Massanutten Mountain than in the other models because the hydraulic conductivity decreases more slowly with depth, as a result of the smaller estimated decay factor (Table 2).

### Capture zones of a municipal water supply

Capture zones of the Martinsburg water supply in Berkeley County, West Virginia (WV) were delineated through reverse-flow simulations using all four models. The water supply, which yields 17,000 m^{3}/day (200 L/s) derived from groundwater, was represented in these simulations by pumping from a single well. Reverse-flow simulations are achieved by multiplying all specified heads and all sources by negative one (−1). The water-production well is represented as an injection well in the reverse-flow simulation and recharge rates are specified as negative values. Further, the drain boundaries representing discharge to streams are replaced by specified head boundaries with negative head values. These changes have the effect of routing flow in a reverse direction from inflow points at production wells and along streams to discharge points at land surface. Inflow from the well was labeled with a conservative tracer of unit strength and tracked backwards through steady-state transport simulations to delineate an approximate capture zone. Longitudinal and transverse dispersivities were specified as 800 and 30 m, respectively, as in the simulations of groundwater age already presented.

This procedure is similar to backward tracking using particles (Zheng and Bennett 2002), but the simulation of dispersion results in larger delineated areas. The dispersion coefficient represents heterogeneity in aquifer hydraulic conductivity that creates uncertainty in the position of the capture zone. A normalized concentration distribution can be strictly interpreted as a location-probability distribution (Neupauer and Wilson 2004) for a similar simulation with somewhat different boundary conditions. Qualitatively, the simulated concentration distributions presented herein may be interpreted as the relative likelihood that water recharging parts of the capture zone will reach the well. Water recharging at locations of higher concentrations are more likely to reach the well than water recharging at locations of lower concentrations. The steady-state simulations did not represent release of groundwater from storage due to declining water levels, which would have resulted in smaller areas before steady conditions are reached.

Plan views of capture zones delineated with all models are similar (Fig. 15), but the perspective views indicate that the capture zone delineated with model A is shallower (Fig. 16). The model-A capture zone extends downdip along the bedding of carbonate rock and its vertical extent is limited by a low *K*_{min} value (0.2 m/day) that limits cross-bedding flow (Fig. 16a). In contrast, capture zones delineated with the other models (in which flow is vertically isotropic) extend to the bottom of the flow system and do not reflect the dip of the bedding (Fig. 16b–d). Although most of the simulated flow to the well is through the upper 150 m of the model domain, it is unreasonable to assume that water from depths of 5 km will discharge to the well, as predicted by models B, C and D.

## Discussion

Variable-strike-and-dip model A provides a distinctive characterization of the flow system in the Shenandoah Valley that reflects the underlying bedrock structure. The varying direction of the conductivity tensor is an important control on the simulated pattern of flow in folded fractured rock. It limits the depth of simulated flow where bedding is near horizontal and allows flow to penetrate more deeply where the bedding is inclined. This produces a flow system that is consistent with conceptual models of flow through folded fractured rock. In contrast, flow fields in models that do not explicitly represent the dip of bedding are more strongly controlled by topography and the location of model boundaries. These models overestimate the magnitude of vertical flow in areas where bedding is horizontal (e.g. beneath Massanutten Mountain) or inclined (e.g. beneath the Martinsburg water supply).

*K*

_{max}:

*K*

_{min}) estimated for sedimentary rock in the Shenandoah Valley with this regional model are relatively small (17:1 in model A and 8:1 in models B and C), so differences in the simulated flow systems are subtle. Little information is available concerning the degree of anisotropy, however, and the uncertainty in estimated value is relatively large, so it is possible that the true anisotropy is larger. Models A (variable strike and dip) and B (uniform strike) were recalibrated using a specified anisotropy of 100:1 to magnify the differences in the simulated flow systems. Simulated groundwater age distributions are shown in Figs. 17 and 18, and the delineated capture zones are shown in Figs. 19 and 20.

The groundwater age distribution simulated with model A clearly depicts the influence of bedding in the western part of section D–D’ beneath the North Fork of the Shenandoah River (Fig. 17a). Model A simulates a wide area of older water (more than 20 years) that flows updip toward the North Fork from the east and west, and parallels the form lines shown in Fig. 6. In contrast, a much narrower zone of older water is simulated discharging to the North Fork with model B (Fig. 17b). Profiles of groundwater age with depth indicate that ages simulated with model B are much younger than those simulated with model A (Fig. 18). This is because anisotropy limits horizontal flow in model B and increases vertical flow. As a result, large differences are apparent in ages simulated by models A and B, especially beneath Massanutten Mountain near the center of section D–D’. Comparison of Figs. 14 and 18 indicates that ages simulated with model B are highly dependent on the anisotropy value, while ages simulated with model A, which are largely controlled by bedrock structure, are much less dependent on the anisotropy value.

Differences in the capture zones delineated with models A and B are exaggerated with the larger anisotropy value of 100:1 (Figs. 19 and 20)—the model-A capture zone is wider, shorter and shallower than the model-B capture zone. The direction of flow is uniform in model B, while flow directions in model A are variable and reflect the influence of stream-discharge boundaries (Fig. 19).

## Conclusions

Finite-element methods such as SUTRA, can be used to represent the generalized three-dimensional anisotropy in folded fractured-sedimentary rocks. In this study, hydraulic-conductivity tensors that conform to bedding planes were defined to duplicate the pattern of folded sedimentary units in a three-dimensional structural model of the Shenandoah Valley, USA. Varying directions of maximum and medium hydraulic conductivity conform to the strike and dip of bedding, respectively, while the minimum hydraulic-conductivity direction is perpendicular to bedding. Alternative models were also prepared in which the hydraulic-conductivity tensors are oriented horizontally and the dip of bedding is not represented. These models mimic the application of finite-difference models using uniform-strike or variable-strike approaches to represent horizontal anisotropy. The variable-strike and dip model produces a flow system that reflects the underlying bedrock structure, with a flow field that is significantly different from those produced by the alternative models.

Topography and the location of model boundaries are more dominant controls on flow systems simulated by alternative models that do not represent the dip of bedding. These models overestimate the magnitude of vertical flow where bedding is horizontal or inclined. It would be possible to vary the degree of vertical anisotropy specified in these models to represent changes in the dip of bedding, or the model grids could be deformed to reflect the orientation of bedding surfaces. It is more straightforward and accurate, however, to use finite-element models to represent the pattern of folding, once a three-dimensional model of bedrock structure has been developed.

For the Shenandoah Valley, estimates of the anisotropy ratio between strike-parallel and cross-bedding conductivity are about 10:1, irrespective of the representation of the conductivity tensor employed. A full description of parameter estimation is given in Yager et al. (2008). Both the magnitude and direction of the hydraulic-conductivity tensor are important controls on flow patterns simulated by groundwater models. Even though the magnitudes of principal values of conductivity were similar among the alternative representations of the conductivity tensor considered in this study, each utilized different tensor directions and these produced different flow patterns. Differences are most apparent in flow patterns and groundwater ages at depth, although the age of shallow groundwater in principal discharge areas is also different, especially for simulations in which the anisotropy ratio was increased to 100:1. The sensitivity of simulated groundwater ages in principal discharge areas to the degree of anisotropy suggests that including groundwater age data in model calibration could provide a better estimate of anisotropy than that obtained with the head and flow data available to this study.

## Acknowledgements

Information required to construct and calibrate the groundwater flow models described in this study was provided by G. Harlow, J. Eggleston, D. Hayes and J. Krstolic (USGS, Richmond, VA); and M. Kozar, K. McCoy and J. Atkins (USGS, Charleston, WV). R. Winston (USGS, Reston, VA) wrote procedures for SutraGUI that generated the strike direction and dip angles used to define conductivity tensors specified in the groundwater flow models. A. Provost, (USGS, Reston, VA) modified the implementation of specified heads in the SUTRA program.