Using the Total Apparent Spacing Approach to Optimize Foundation Drain Orientations

Uplift pressure is critical to both the sliding and overturning stability of concrete dams founded directly on rock. The uplift pressure beneath such a dam is a function of the hydraulic gradient, which is assumed to vary linearly from headwater at the heel of the dam to zero or tailwater at the toe of the dam (Figure 1a), assuming a homogeneous and isotropic condition where hydraulic conductivity does not change significantly over the dam site, which is not often the case in the real world. In the case where hydraulic conductivity does change, the relationship between uplift pressure and headwater and tailwater pressure is non-linear. Studies done by Grenoble et al. (1995) showed that rising reservoir levels can differentially deform discontinuities in the foundation and cause the hydraulic conductivity to increase at the heel of the dam and decrease at the toe (tapered joint). A non-linear response may produce an uplift pressure distribution that is greater than the conventional linear assumption for uplift (Ebeling and Pace, 1996a). Non-linear responses of uplift pressures to reservoir level have been analytically modeled (Ebeling and Pace, 1996b) and physically observed in several dams (Stone and Webster Engineering Corporation, 1992). The expansion and contraction of downstream faces due to seasonal ambient temperature changes may cause greater changes in uplift pressure than changes in headwater levels (Stone and Webster Engineering Corporation, 1992).

Installation of drains beneath the dam can alter the hydraulic gradient and help to reduce the uplift pressure (Figure 1b). Drains are considered to be the most reliable and cost-effective way of reducing foundation uplift pressures, especially for structures founded on rock (USACE, 2005). The effectiveness of the drainage system depends on the depth, size, and spacing of the drains; the character of the foundation; and the facility with which the drains can be maintained.

The hydraulic conductivity of a jointed rock mass is commonly controlled by its secondary permeability, so in many civil engineering applications, bedrock discontinuities dominate the drain design procedures. Assuming other discontinuity characteristics such as aperture, length, and filling are relatively uniform, foundation drain holes drilled at the optimum orientation and inclination with respect to the orientation and spacing of the predominant discontinuities should relieve uplift pressure most efficiently. However, it should be noted that the severity of uplift pressure may be independent of discontinuity orientation (Grenoble and Amadei, 1990) as discontinuity aperture and filling may control uplift pressures more than discontinuity orientation (Murphy et al., 2002).

When other discontinuity parameters are not relatively uniform and result in an anisotropic distribution of foundation hydrostatic pressures, optimization of drain hole orientations based solely on the orientation and spacing of the predominant discontinuities is not appropriate. Anisotropic hydrostatic pressures may occur in foundations where multiple episodes of structural deformation due to plate convergence or rifting have produced discontinuity sets having significantly varying discontinuity parameters. Variations in stress regimes may affect discontinuity aperture and length, and preferential flow of fluids during structural deformation may result in precipitation of minerals that completely fill certain discontinuity sets. Drilling foundation drain holes at the optimum orientation and inclination with respect to the orientation and spacing of the predominant discontinuities is appropriate only when discontinuity parameters are relatively uniform, a case more likely to exist within bedrock foundations that have experienced only minor structural deformation. Site reconnaissance and subsurface investigation should seek to identify and characterize those discontinuities that affect the hydraulic pressures in the foundation, which could include less prevalent discontinuities and, in some cases, even one single discontinuity.

Several authors have addressed aspects of drain hole design. Cook et al. (2007) discussed the state of the art and presented suggestions for improvement in the design of landslide horizontal drains. Two approaches for optimizing borehole orientation have been presented in Environmental & Engineering Geoscience. Zhou and Maerz (2002) developed an approach to find the optimum drilling direction for maximizing a borehole’s interception of discontinuity sets. The basic concept involves finding a drilling direction that minimizes the linear sampling bias index (LSBI), which is a function of the relative angle between the orientation of the borehole and the mean orientation of the poles of each of the discontinuity sets. The optimum drilling direction is the direction along which the LSBI is minimized. Haneberg (2009) observed that while the LSBI minimization approach produces correct results, it suffers from multiple shortcomings that inhibit its utility. Haneberg (2009) presented an approach that builds upon the approach taken by Zhou and Maerz (2002) by incorporating changes to make it potentially more useful and geometrically meaningful. Haneberg (2009) described this approach to optimize a borehole’s orientation by minimizing the linear sampling angular deviation and provided examples produced using a series of functions written for the computer program Mathematica (Haneberg, 2004).

This paper describes how the total apparent spacing (TAS) approach optimizes drain hole orientation using a minimization process based on discontinuity spacing and orientation. The TAS approach seeks to minimize the total apparent spacing of discontinuities along the drain hole and to maximize the number of discontinuities encountered by the drain hole. Beyond the foundation drain hole orientation example provided herein, use of the TAS approach to strategically orient a drill hole so as to intersect the maximum number of discontinuities along the drill hole for a given drilling length may be useful for other purposes within the realms of geotechnical exploration, rock engineering, and foundation treatment.

Fractures occur in response to stresses that exceed the strength of the rock to resist the resulting force. Fracture-producing stresses may be tensile, compressive, shear, or a combination of stress types. Stresses may vary in strength and orientation over time and space, so fractures may occur in multiple sets having varying orientations and characteristics within a region. Once formed, fractures can be modified as a result of tectonic and thermal stress and strain, changes in fluid pressure, and physical, chemical, and biological processes such as erosion and valley stress relief, dissolution or precipitation of minerals, and degradation by roots (NASEM, 2020).

Planar bedrock discontinuities, which may not necessarily be flat, often occur in sets of discontinuities having semi-parallel orientations or falling within a range of similar orientations. Horizontal to gently dipping sedimentary rocks with limited tectonic stresses often feature two nearly orthogonal joint sets that are both roughly perpendicular to a set of bedding joints developed along bedding planes. One joint set is composed of “strike joints,” which generally strike in about the same direction as bedding strike, while the other set is composed of “dip joints,” which generally strike roughly parallel to the bedding dip direction. Additional discontinuity sets are often present in sedimentary rocks subjected to higher tectonic or other external stresses. A similar blocky jointing pattern may occur in igneous rocks having flow banding. “Flat-lying joints” are approximately horizontal and strike parallel to the flow lines indicated by the preferred orientation of the long axes of the rock crystals. A second joint set is composed of “Q joints” or “cross joints” that strike nearly perpendicular to the flow lines. A third set is composed of “S joints” or “longitudinal joints” that dip steeply and strike parallel to the flow lines (Krynine and Judd, 1957). In igneous rock masses lacking flow banding, local and regional stress and strain distributions tend to define the fracture pattern. “Tensile joints” form generally parallel to the maximum stress trajectory, and “shear joints” form at an angle of roughly 20 to 35 degrees to the maximum stress trajectory (NASEM, 2020).

The pattern of discontinuities within a rock mass, specifically the typical orientation and spacing of discontinuities, must be determined in order to use the TAS approach. The orientation of relatively planar discontinuities can be expressed by an azimuth (either strike or dip direction) and a dip angle. Discontinuity spacing and orientation data can be collected using one or more of several available survey methods. Properly conducted surveys can furnish data with a high probability of accurately representing the typical orientation and spacing of the discontinuities within a rock mass.

When suitable representative outcrops are available, either naturally or by means of excavation, discontinuity spacing and orientation can be measured directly by traditional methods (e.g., linear scan line or window sampling) using one or more tools such as a measuring tape and a compass, pocket transit, smart phone, or tablet. Corrections for observation bias have been developed for scan-line sampling (Terzaghi, 1965) and for borehole sampling, which may be considered as a subsurface scan line (Mauldon and Mauldon, 1997). Observational bias may be removed by orienting scan lines perpendicular to one another and by recording the trace length, orientation, and point of intersection of every discontinuity along the scan lines (La Pointe and Hudson, 1985). For window sampling, the window should be as large as possible to minimize sampling bias. Ideally, at least two windows of similar size should be established on adjacent rock exposures having different (preferably orthogonal) orientations (Priest, 1993). Window sampling reduces the observation bias due to the discontinuity orientation and length associated with scan-line sampling; however, bias may remain as a result of curtailment of the discontinuities due to outcrop cover.

Newer methods use computer software to measure virtual discontinuity surfaces within a digital terrain model of the rock outcrop developed by means of digital photogrammetry or light detection and ranging (LiDAR) surveys (Haneberg et al., 2006; Tonon and Kottenstette, 2007; Haneberg, 2008; Kemeny and Turner, 2008; Kottenstette and Shaffner, 2008; Shaffner et al., 2009; Kemeny et al., 2011; and Watts, 2011). Using both digital photogrammetry and LiDAR imaging may be useful in some circumstances (Otoo et al., 2011). Although useful for measuring locations and orientations of discontinuities, photogrammetry may not provide critical geological information that can be gleaned only by human examination of the discontinuities (Gates and Haneberg, 2012; Battulwar et al., 2021). Traditional field characterization and input from experienced professionals may enhance the reliability of laser scanning (Farny, 2017).

When suitable outcrops are not available, discontinuity spacing and orientation can be measured using oriented core drilling techniques or downhole imaging tools (e.g., optical and acoustic sondes) that provide digital images of the borehole wall. Provided borehole instability does not preclude the use and retrieval of the imaging sonde, borehole imaging serves as a powerful tool for measuring fracture orientations and spacings, better characterizing zones of broken core, and unraveling the mysteries of core losses.

The survey methods used should provide an understanding of the variability in orientations and dip angles of the discontinuities. This understanding is very important in ultimately selecting the typical discontinuity orientation and dip angle. Larger variability in the data will make the selection of the typical discontinuity parameters even more important. Moreover, discontinuity data should be collected as close to the site as possible. Discontinuity data from an off-site location should be compared to available information from the site to determine its applicability.

The TAS approach for identifying the optimum foundation drain hole orientation is based on the following general assumptions:

1. Jointing is the primary pathway of water flow and pressure through the rock mass forming the dam foundation.

2. Surveys are properly conducted and accurately represent the typical discontinuity spacing and orientation of the predominant discontinuity sets within the dam foundation.

3. The spacing between individual discontinuities within a set is generally uniform throughout the dam foundation; consequently, the average discontinuity spacing of each set can be used.

4. The orientations of discontinuities within a set are generally uniform throughout the dam foundation; consequently, the average discontinuity orientation of each set can be used.

5. The discontinuity sets are persistent throughout the dam foundation.

6. A drain hole direction trending parallel to the plane representing the average orientation of a discontinuity set has an infinite apparent spacing and is excluded as an optimum foundation drain hole direction.

7. The discontinuity sets are relatively uniform in aperture, length, and filling. The TAS approach may be modified by adding weighting factors when this is not the case.

If discontinuity spacing, orientation, and persistence are significantly non-uniform, then consideration should be given to subdividing the foundation into regions or areas of similar discontinuity properties and performing separate evaluations.

For a drain hole to drain water most effectively from a jointed rock mass, the drain hole must be oriented so that it intercepts the discontinuity network most frequently; i.e., the highest number of discontinuities is intercepted per unit length of drill hole. Since the discontinuity network consists of a certain number of predominant discontinuity sets, a drain hole will intercept each predominant discontinuity set of the discontinuity network at a certain average distance along the length of the drain hole. Optimizing the drain hole orientation thus requires a minimization of the average distances between joint sets intercepted by the drain hole. The dot product formula provides a means to accomplish this minimization.

The designated predominant discontinuity sets should be those that significantly contribute to pressure in the foundation of the structure. When the characterization of foundation conditions determines that multiple discontinuity sets contribute to pressure in the foundation, but that their contributions vary significantly, consideration should be given to assigning weighting factors to the predominant discontinuity sets or to subdividing the foundation into regions or areas of similar discontinuity properties and performing separate evaluations.

Although discussed in terms of “joints” and “joint sets,” the formulas below are applicable to the broader category of structural breaks encompassed by the terms “discontinuity” and “discontinuity sets.”

The dot product (or scalar product) of two three-dimensional Cartesian vectors

Let be the normal vector (pole) of a plane P that represents the average joint orientation in a joint set. Let the magnitude of (denoted ⁠) represent the orthogonal (true) average spacing of the joints within the set. Let be an arbitrary vector representing the proposed drain hole orientation starting at the origin and connecting the origin to an arbitrary point on plane P. The magnitude of (denoted ⁠) represents the apparent average spacing of the joints within the set along the drain hole (Figure 2).

Let represent the average pole to Joint Set 1 and represent the proposed drain hole orientation. In spherical coordinates, the unit vectors are and ⁠. Converting to Cartesian coordinates produces and ⁠.

Hence, d₁ is the apparent joint spacing for Joint Set 1 when drilling with azimuth θ and plunge φ.

where k = number of selected joint sets,

n_i = true joint spacing of the ith joint set,

φ_i = joint dip angle of the ith joint set,

θ_i = joint dip direction of the ith joint set,

φ = drain hole inclination, and

θ = drain hole azimuth.

It should be noted that the angular measurements in Eq. 12 reference spherical coordinates as used in mathematics, so θ and θ_i refer to the angular measurement from the x axis in a counterclockwise direction within the X-Y plane (with east = 0 degrees), and φ and φ_i refer to the angular measurement from the z axis (with zero degrees representing vertically upward and 180 degrees representing vertically downward). If joint dip direction and drain hole azimuth are measured in compass bearings (0 to 360 degrees with north = 0 degrees), then the compass bearing should be subtracted from 90 degrees to convert to θ and θ_i. If joint dip angle and drain hole inclination are measured from the horizontal plane (with 90 degrees representing vertically upward and −90 degrees representing vertically downward), then the angle should be subtracted from 90 degrees to convert to φ and φ_i.

Equation 12 provides a formula to minimize the sum of the apparent spacings of all selected joint sets along a theoretical drain hole, which is the purpose of the TAS approach. Application of the TAS approach will typically involve the following activities:

1. Perform the research and site investigations needed to characterize the discontinuities within the rock mass forming the dam foundation and to group the discontinuities into sets.

2. Identify the discontinuities or sets of discontinuities that significantly influence uplift.

3. Assess the TAS approach assumptions to determine whether the TAS approach is applicable.

4. If the variation in the parameters of the selected discontinuity sets indicates the TAS approach is not applicable, then consider subdividing the foundation into zones for separate evaluations and/or assigning weighting factors to the selected discontinuity sets.

5. Determine the best-fit orientations of the selected discontinuity sets.

6. Determine the range of favorable drain hole orientations.

7. For each possible favorable drain hole orientation, use Eq. 11 to calculate the apparent spacing of each joint set and sum the apparent joint spacings along the drain hole per Eq. 12. A computer spreadsheet will facilitate these calculations.

8. Use the calculated results to identify the optimum drain hole orientation.

The level of effort devoted to applying the TAS approach should be commensurate with the relative importance of the drain design in the overall dam design or stability analysis. For example, if reduction in uplift pressure by foundation drains is not being factored into the stability analysis (a conservative approach), then a less rigorous application of the TAS approach may be warranted. Rehabilitation of Gilboa Dam provides an example of the application of the TAS approach.

During the rehabilitation of Gilboa Dam in New York State, the TAS approach was used to optimize drain hole orientation.

Gilboa Dam is located in the Catskill Mountains in southern Schoharie County, NY (Figure 3). The dam is situated approximately 125 mi (200 km) north-northwest of Manhattan and 37 mi (60 km) southwest of Albany, NY. Gilboa Dam impounds Schoharie Reservoir, which is operated by the New York City Department of Environmental Protection, and which is a key component of the city’s water supply system. Schoharie Reservoir has a normal storage volume of approximately 20.8 billion gallons (7.88 × 10⁷ m³) and covers 1,159 acres (469 hectares) at normal pool. The reservoir has a watershed area of approximately 314 mi² (813 km²).

The eastern portion of Gilboa Dam consists of a 160-ft-high (48.8 m), 1,324-ft-long (404 m) concrete spillway founded on rock and having a crest elevation of 1,130 ft (344.4 m). Due to the presence of a deep, pre-glacial channel in the bedrock, the western portion of Gilboa Dam consists of a 180-ft-high (54.9 m), 700-ft-long (213 m) earth embankment with a crest elevation of 1,150 ft (351.5 m). A retaining wall extends downstream from the junction of the concrete spillway and earth embankment (Figure 4).

Construction of Gilboa Dam began in 1919 and was completed in 1927. A major rehabilitation of Gilboa Dam began in 2006 and was completed in 2014. The rehabilitation included installation of 80 post-tensioned rock anchors in the spillway, placement of 57,000 yd³ (43,580 m³) of concrete buttress on the downstream side of the concrete dam, a new reinforced concrete lining in the spillway channel and plunge pool, installation of a new 9-ft-diameter (2.7 m) low-level outlet, and addition of a gallery with foundation drains. In 2015, the project was named the Association of State Dam Safety Officials National Dam Rehabilitation Project of the Year.

Bedrock in the vicinity of Gilboa Dam is part of the Catskill clastic wedge and consists generally of thick sandstone beds and thinner beds of siltstone, shale, and mudstone, which are locally mapped as the Moscow Formation of the Hamilton Group (Fisher et al., 1961). The sediments forming these Middle Devonian rocks appear to have been deposited within a migrating, deltaic shore zone between marine sediments to the west and non-marine (fluvial) sediments to the east (Bridge and Willis, 1994; Bridge and Jarvis, 1998; and Bridge, 2000). Gilboa is the site of one of the world’s oldest fossil forests, and during the original construction of Gilboa Dam, tree stump fossils were sent to several museums (Bartholomew and Brett, 2003; Hernick, 2003). Excavation of a former quarry downstream of the dam during the dam rehabilitation provided an opportunity for researchers to observe and map a portion of the fossil forest (Stein et al., 2012). The area was glaciated, and Glacial Lake Grand Gorge and Glacial Lake Schoharie occupied the Schoharie Valley during the Wisconsinan stage (Rich, 1935; Titus, 2003).

Core borings at Gilboa Dam penetrated a sandstone-dominated portion of the Moscow Formation. Bridge and Willis (1994, pp. 1441–1443) described the Gilboa Dam section as two 35-m-thick sandstone-dominated sequences separated by a 5-mr-thick mudstone-rich interval with sediments of the two sandstone bodies showing mainly wave and tidal current influence lower down and offshore-directed current influence higher up, which reflects deposition in water becoming increasingly shallow and closer to land. In addition to sandstone, dam rehabilitation core borings encountered minor beds of calcareous siltstone, shale, and mudstone, some of which are visible in outcrops below the dam (Figures 5, 6, and 7). Analyses of several finer-grained marker beds using the three-point method indicated the overall bedrock strike is N80°W, and the overall dip is 1.5 degrees to the south-southwest.

Rock outcrops along Schoharie Creek below Gilboa Dam were examined as part of the reconnaissance effort for the dam rehabilitation. During a detailed line survey consisting of five traverses, the orientations of 45 discontinuities were measured, and notes were taken on their properties (e.g., length, continuity, roughness, infilling, thickness, waviness, and moisture). Discontinuity orientation data are summarized in Tables 1, 2, and 3. Little variation was observed in discontinuity properties, as might be expected in relatively flat-lying beds having only minor deformation due to plate convergence during the Alleghenian orogeny, so discontinuity sets were not weighted, and the dam site was not subdivided into zones during the foundation drain design process.

Most measured discontinuity surfaces were vertical, and all but one had a dip of 75 degrees or more. The great circles and poles corresponding to the 44 steeply dipping measurements are plotted in a stereonet in Figure 8. Rocscience Dips software was used to determine the global best-fit orientations of the three identified joint sets, which expressed in terms of strike and dip are N43°E/89°NW, N75°W/88°NE, and N5°W/89°SW. The three joint sets were observed on both the east and west sides of the Schoharie Creek gorge below the dam (Figures 9 and 10). The three identified joint sets appear to be of regional significance, as they bear similarities to joint systems depicted in the Gilboa and Schoharie 15-minute quadrangles (Isachsen and McKendree, 1977a) and to brittle structures delineated in the Gilboa 15-minute quadrangle (Isachsen and McKendree, 1977b). The northerly strike of the third joint set is parallel to the overall trend of Schoharie Creek through the gorge containing the Schoharie Reservoir and Gilboa Blenheim Reservoir, a power supply reservoir located downstream of Gilboa Dam.

d_n = the apparent joint spacing of joint set n,

φ_n = the plunge of the pole to joint set n,

θ_n = the trend of the pole to joint set n,

φ = the plunge of the traverse, and

θ = the trend of the traverse.

First, an average apparent joint spacing was calculated for each joint set for each traverse having at least two measurements for the joint set (Table 4). The traverse trend was determined from field measurement with a Brunton compass. In cases where the traverse trend was not constant, a weighted average trend was calculated based on the lengths of each traverse segment. Next, the average true joint spacing was calculated, and then a weighted average was calculated for each joint set based on the lengths of the traverses. The weighted calculated average true joint spacing for Joint Sets 1, 2, and 3 was 3.7 ft (1.1 m), 9.4 ft (2.9 m), and 6.8 ft (2.1 m), respectively.

Based on the principal component and average joint spacing of each joint set, the derived equations were used to calculate the apparent spacing between joints of each set for drain holes trending in an upstream direction and plunging from 45 to 70 degrees from horizontal. An Excel spreadsheet was used to perform the calculations in 1-degree increments of azimuth and plunge, so 4,706 possible drain hole orientations were evaluated. A partial plot of the results indicates that the shallowest plunge (45 degrees) is the most favorable of the hole inclinations considered (Figure 11).

The most favorable azimuths are clustered around 142, 249, 71, and 203 degrees, in that order. Holes having these azimuths and plunging 45 degrees are ranked 1st, 15th, 77th, and 203rd when the results are sorted by total apparent spacing, i.e., the combined apparent spacing of each joint set within the proposed drain hole. The calculations indicate a drain hole trending 142 degrees has the optimum total apparent joint spacing and will intercept joints belonging to Joint Sets 1, 2, and 3 every 5.2 ft (1.6 m), 20.9 ft (6.4 m), and 17.1 ft (5.2 m), respectively, on average (Figure 12).

As part of a peer review of the proposed foundation drain design, including the favorable drain orientations identified using the TAS approach, an alternate analysis was performed using the LSBI approach of Zhou and Maerz (2002). Using the LSBI approach, the most favorable drain hole azimuths were found to be clustered around 140, 74, 251, and 199 degrees, in that order (Figure 13). Table 5 provides a comparison of the results for favorable drain hole azimuths using the two approaches.

To optimize borehole inclination, the LSBI approach calculates the LSBI based on the angle between the borehole inclination and the average dips of the joint sets projected onto a vertical east-west plane (0 to 180 degrees). Drain holes inclined 45 to 90 degrees fall in the range of inclinations from 45 to 135 degrees. Similar to the TAS approach, the LSBI approach indicates that shallower inclinations are more favorable (Figure 14), which intuitively makes sense given the near-vertical average orientations of the three joint sets. Both the TAS and LSBI approaches are thought to be computationally similar in the application of the respective approach and produce comparable results. The TAS approach uses a formula that factors in both borehole azimuth and inclination to calculate the total apparent spacing along the borehole, while the LSBI approach calculates a separate LSBI for the borehole azimuth and for the borehole inclination.

For a given drain hole orientation, drain hole length influences the reduction of the uplift pressures the most, followed by drain hole spacing and drain hole diameter, in that order (da Silva, 2005). Drain hole length has a large influence on the values of the uplift pressures up to approximately half the value of the upstream reservoir depth. From this point onwards, the influence of the drain hole length decreases up to a value equal to the reservoir depth. After that, its influence is very small (da Silva, 2005). Current design standards suggest that foundation drains should be installed to a vertical depth of two-thirds to three-quarters of the depth of the existing grout curtain (USACE, 1995). The depth of the foundation drains should never exceed the depth of the grout curtain (ASCE and USCOLD, 1967). Geologic setting, engineering parameters, and operational and performance objectives also need to be considered in evaluating the design depth of the drain holes.

At Gilboa Dam, the final drain hole design consisted of two rows (A and B) of drain holes. Row A drain holes were drilled S38°E at an angle of 45 degrees from vertical and 27 ft (8.2 m) into rock. Row B drain holes were drilled vertically 10 ft (3 m) into rock. Given the calculated apparent joint spacings, each angled Row A drain hole would be expected to intercept at least five, and possibly six, joints belonging to Joint Set 1 and at least one, and possibly two, joints belonging to Joint Sets 2 and 3 (Figure 12). The vertical Row B drain holes would be expected to intercept mainly bedding joints dipping 1.5 degrees to the south-southwest.

The drain hole spacing will be controlled by the apparent spacing of joint sets along the drain hole, which has already been determined, and the apparent spacing of joint sets along the gallery where the drain holes are to be drilled. In order to avoid gaps between drain holes, the drain hole spacing should not exceed the apparent joint spacing along the gallery of each joint set multiplied by the number of joints belonging to the joint set theoretically penetrated by the drain hole.

φ_n = joint dip angle of joint set n,

θ_n = joint dip direction of joint set n,

φ = gallery inclination, and

θ = gallery azimuth.

d_n = apparent spacing of joint set n along the drain hole.

At Gilboa, the gallery trends approximately 253 degrees and plunges about 3 degrees from horizontal. Table 6 provides a summary of the maximum drain hole spacing calculations assuming a drain hole length of 27 ft. For Joint Sets 1, 2, and 3, the maximum drain hole spacings are 38.5, 22.9, and 11.0 ft, respectively. Joint Set 3 controls the maximum drain hole spacing since the strike of Joint Set 3 has the smallest angular deviation from the drain hole orientation.

Drain spacing must account for deterioration of foundation drains over time due to clogging, incrustation, mineralization, etc., as well as consideration of non-linear uplift distribution. Major hydraulic structures must be designed with both resiliency and redundancy, and a reliable and robust foundation drainage system is one of the most important aspects of maintaining a stable dam. A drain spacing of 10 ft (3 m) was selected for the Row A drain holes. The above calculation does not apply to Row B drain holes since they were drilled vertically. A spacing of 10 ft (3.0 m) was also selected for the Row B drain holes.

Nominal 3-in.-diameter (7.6 cm) drain holes were typically drilled with percussive rotary methods using water flush at a spacing of 10 ft (3 m) (Figure 15). Where exploratory holes were required, NQ wireline double-tube rock coring was specified. At each drilling location, one hole was drilled S38°E at an angle of 45 degrees from vertical and 27 ft (8.2 m) into rock, and a second hole was drilled vertically 10 ft (3 m) into rock. In total, 248 drain holes were drilled at 124 drilling locations. The drain holes were drilled prior to the creation of the dam gallery during construction of the downstream concrete buttress (Figures 16 and 17).

Uplift was considered in the design of the site structures at Gilboa Dam. The spillway, spillway channel, plunge pool concrete panels, and retaining/training walls were designed assuming full uplift. Uplift of the concrete slabs in the spillway channel and plunge pool was addressed using passive anchor bars grouted into bedrock. Although the gallery foundation drains in the spillway and the underdrain system within the spillway channel and plunge pool were included in the design, the drains were not accounted for as an uplift reduction method since the gallery and spillway channel were designed to be below tailwater elevation during flood conditions.

Fractures develop in a rock mass in response to stresses that exceed the strength of the rock to resist the resulting forces. Discontinuities typically exhibit preferred orientations and often occur as sets, with each set consisting of regular joints sub-parallel to each other. In each set, discontinuities usually have the same general orientation and exhibit relatively common physical characteristics. Rock masses may contain multiple discontinuity sets, and three to four sets are typical of many rock masses.

Uplift is one of the primary loads that affect the stability of a concrete dam founded on rock. The hydraulic conductivity of a jointed rock mass is commonly controlled by its secondary permeability, and in most civil engineering applications involving rock foundations, the secondary permeability controls the drain design procedures. Drains are used to reduce dangerous uplift pressure beneath hydraulic structures and must be correctly placed in three-dimensional space to intersect hydraulically significant discontinuities that are capable of fracture flow.

Numerous methods, such as scan-line outcrop surveys and core hole imaging, can be used to systematically collect the data required to characterize the jointing pattern of a rock mass. Properly conducted surveys can provide data on the orientation, spacing, and physical condition of the discontinuity sets within a rock foundation or rock slope. These data can provide the a priori knowledge about the discontinuity structure of the rock foundation that is needed to effectively utilize the analytical approaches described herein. The TAS approach can be used to define an optimum drilling direction, that is, the correct orientation and inclination along which a borehole can be drilled to intersect as many discontinuities as possible for a given drilling length. This analytical tool has a direct application in the design of systematic drains installed in a jointed rock foundation.

Primary and secondary permeability may overlap in some rock foundations. In these circumstances, rock foundations may exhibit primary permeability that is generally similar to secondary permeability. The approaches presented herein do not explicitly consider the effect of primary permeability on seepage and, hence, cannot be used to properly design a foundation drainage system that is significantly affected by primary permeability. Flow nets, or other analytical or numerical methods, are required to design a foundation drainage system largely influenced by primary permeability.

The formula used in the TAS approach will provide an infinite total apparent spacing in certain scenarios. For example, using the formula to optimize a scenario including a discontinuity set oriented exactly parallel to the proposed drain hole will have a solution of infinity, since the drain hole will theoretically never intercept a discontinuity set oriented exactly parallel to the drain hole. In this case, if the proposed drain hole orientation is deemed to be necessary for some reason, then a preferable solution might be to optimize the orientation of one set of drain holes based on the non-parallel discontinuity sets and add a second set of drain holes having an orientation optimized to intercept the parallel discontinuity set.

Likewise, when multiple sets of discontinuities exist and influence uplift, optimizing the drain hole orientation based on an analysis factoring in all the discontinuity sets may result in a compromise solution that may be less effective than a design incorporating a variety of drain hole orientations, each optimized to target a single set or group of discontinuity sets, e.g., one drain hole orientation optimized to intersect near-vertical discontinuity sets and one drain hole orientation optimized to intersect near-horizontal bedding joints.

Regardless of the resulting optimized drilling orientation developed by the TAS or alternate methods, the actual optimum orientation results may be limited by factors outside the numerical analysis. The physical characteristics of the site and the limitations of equipment and access must be taken into account prior to designating an orientation for boreholes in a drilling work plan, scope of work, or contract. The analysis may indicate that drain holes having a shallow plunge from horizontal are most favorable, but the mechanics of executing such an orientation may preclude the feasibility of drilling a drain hole having that orientation. The access and logistics of a site may also preclude using equipment that might otherwise be able to achieve the optimized orientation. Under these conditions, alternative scenarios must be considered to determine the best practical solution during the analysis and design phase, rather than after equipment and personnel have been mobilized. Drilling tolerance and the potential for deviation should also be considered when putting the results of these methods into a specific application. At greater inclinations from vertical, the borehole is more likely to deviate from the desired orientation, all other factors being equal.

Despite a rigorously executed site investigation program, considerable geologic uncertainties may still exist. Any site investigation program must be thoroughly evaluated for sampling bias and other deficiencies. Site investigation deficiencies and inadequacies must be thoroughly considered when finalizing any foundation drain design.

Conclusions based upon limited sampling data may introduce significant epistemic uncertainty into the geotechnical design process. This uncertainty should be fully recognized and considered, and sound engineering principles and judgement are still required when applying the analytical approaches described herein.

The TAS approach can be used to define an optimum drilling direction, that is, the direction along which a borehole can intersect as many discontinuities as possible for a given drilling length. Given a priori knowledge of the jointing pattern within a rock foundation, this approach provides a means to rationally design the optimal orientation, depth, and spacing of rock foundation drains. Drains properly installed at the optimal orientation, depth, and spacing will more effectively and efficiently diminish the hydraulic pressure beneath a concrete dam founded directly on jointed rock. When properly applied, the TAS approach can supersede the long-standing rules-of-thumb that have historically been used to design the spacing, depth, and orientation of foundation drains.

The TAS approach has numerous geotechnical exploration and rock engineering applications. Specific applications include the orientation of drill holes used for: geotechnical site investigations, permeation grouting, permeation grouting verification, pressure (packer) testing, installation of geotechnical instrumentation, groundwater sampling and production wells, rock slope investigation and design, installation of rock support anchors, and collection of rock core samples for engineering property testing, etc. The TAS approach may be applied strategically to intercept as many discontinuities as possible or to focus on intercepting those discontinuities of greatest significance, or those that pose the greatest risks due to structural or seepage concerns.

The authors thank Dr. Abdul Shakoor, former co-editor of Environmental and Engineering Geoscience, for his invitation to submit this manuscript to the journal after a presentation on the TAS approach at an Association of Environmental and Engineering Geologists Annual Meeting. The authors also thank the New York City Department of Environmental Protection for permission to use the Gilboa Dam rehabilitation project as a worked example; Jake Roman for his kind assistance on the mathematical aspects of the manuscript; and the peer reviewers for their many insightful comments and thoughtful suggestions for improving the manuscript.