Much of our understanding of groundwater flow is based on experiments in the nineteenth century conducted by a French engineer named Henry Darcy. Darcy studied the movement of water through a porous medium. It was not until the mid20th century that researchers began to develop different approaches that could be used to estimate groundwater inflow to tunnels in rock [4,13].
Numerous researchers have proposed various analytical, empirical, semiempirical, and numerical methods/procedures for estimating steadystate groundwater inflows into rock tunnels [1, 3, 6, 7, 10, 1113]. These methods can produce varying results. The accuracy of all these methods is highly dependent on the quality of field data that characterize the geological and hydrogeological systems above, below and through which the tunnel is being constructed [13].
Goodman et al. developed mathematical approaches using computer and physical models to describe groundwater inflow into rock tunnel for a variety of situations. In most of these cases, they characterized fractured rock systems as a hydraulically equivalent and homogenous medium by utilizing data from water pressure tests from drill holes. From these drill hole tests, they approximated the permeability of the rock formation [5, 13].
Heuer (1995) developed a semiempirical procedure/method for estimating steadystate, groundwater inflows in rock tunnels. He used data collected from water pressure tests or packer tests to generate histograms. Testing was conducted with inflatable straddle packers at 10 to 20 foot (3 to 6 meter) spacing throughout the borehole depth and below the water table. The primary geologic factors that affect groundwater inflows were the presence of imperfections or defects in the rock mass through which water may flow. Heuer divides these factors into two categories: point sources of local, large inflow such as lava tubes or major solution features, and distributed features that are common throughout the rock mass such as interconnected joints and fractures, bedding planes, and fault systems (i.e., sometimes referred to as discontinuity features). Heuer focuses exclusively on these distributed features [5, 7,13].
Heuer (1995) did not distinguish between hydraulic conductivity (i.e., K) and permeability (i.e., ki) in his work. This assumption is reasonable since the medium of study is groundwater (i.e., as opposed to petroleum hydrocarbons) and temperatures in the subsurface do not vary greatly in the continental United States. It should be recognized that the hydraulic conductivity and permeability differ by the following relationship:
where K is hydraulic conductivity, k or ki is intrinsic permeability, ρ is the density of water, g is gravity, and μ is the dynamic viscosity. Heuer (1995) referred to “equivalent permeability” or ke, as the characteristic of the rock mass derived from standard analysis of packer test data. Equivalent permeability describes the rate of water flow through a fractured rock mass, as opposed to, the rate of water flow through a relatively homogeneous, isotropic, porous medium such as sand where water flows through interconnected voids between the sand particles (i.e., intrinsic permeability). Unless otherwise denoted, this paper and the analyses included herein address issues related to equivalent permeability [7, 8, 13].
In this paper, only the longterm, steadystate case will be considered. For practical analysis purposes, the steady state case can be reduced to two conceptual models that contribute groundwater inflow to rock tunnels. These two models include the vertical recharge with a nearby water source at constant head (e.g., a tunnel overlain by a lake or reservoir) and radial flow with a recharge source far away. More specifically, for the vertical recharge flow condition to be established, the tunnel must be near or within two diameters of the top of bedrock and a water body must be present and whose piezometric surface cannot be depressed or drawn down. Therefore, the vertical recharge conditions can most closely be associated with shallow rock tunnels. For the radial flow condition, the tunnel is typically several tunnel diameters below the top of rock with limited hydraulic communication with overlying surface water. Therefore, the radial flow condition can most closely be associated with deep rock tunnels. These two models are conceptually shown in Figure 1 and can be applied using the following equations for the vertical recharge and radial flow cases [5, 7, 1315].
Vertical Recharge (Case A)
where z = thickness of rock cover between tunnel and water source, z < 20r (i.e., 10 tunnel diameters).
Radial Flow (Case B)
where z > 20r (i.e., 10 tunnel diameters), H = water head from piezometric surface, K = mass permeability or hydraulic conductivity, qs = volumetric inflow per unit length of tunnel, Ro = radius of influence, distance to which piezometric head is influenced by the tunnel. Heuer’s semiempirical procedure is summarized at the end of Table 2 and utilizes the chart shown in Figure 2 [7, 14,15].
2.1 Proposed Modifications
In 2005, Heuer provided additional case study analyses and commented on his previous works. In this paper, he discussed and analyzed the Elizabethtown Tunnel Project in New Jersey. The tunnel crosses under the Delaware and Raritan Canal and the Raritan River which would indicate that it should be analyzed for the vertical recharge condition, but Heuer decided to analyze some parts for the vertical recharge conditions while other parts for the radial flow condition. Heuer estimated that water inflow into the tunnel would be 66 gpm (gallons per minute) or 246 L/min (liters per minute) as summarized in Table 1. The observed inflow rate was 100 gpm (378 L/min). The observed inflow was 1.5 times higher than the estimated value derived from Heuer’s 2005 analysis [7, 13].
Table 1: Summary of SteadyState(SS), Groundwater Inflows [13]
Project Name


Elizabethtown

Toledo


Geological Formation:

Red Shale

Dolomite

Variables

Parameter



L

Tunnel Length (ft)

1,225

1,500

ro

Tunnel Radius (ft)

4.25

4.5

z

Distance/Top of rock (ft)

15

23

Soil Conditioning

Grouted

No

No

Full Length Completed


Yes

Yes

Flow Condition


Vertical

Vertical

Heuer's Estimate based on 2005 Paper

SS Inflow Rate (gpm)

66

50

Steady State Groundwater Inflow Estimate

SS Inflow Rate (gpm)

99

30

utilizing Semiempirical Procedure (1995)




Observed Inflows (Qs)

SS Inflow Rate (gpm)

100

30

In Heuer’s 2005 paper, he compared this case study to other case studies where the radial flow condition exists and noted that the procedure also underestimated the groundwater inflows. He also noted that the observed inflows for all case studies cited were approximately 1.5 times larger than estimated values on average, and suggested, in order to improve the accuracy of estimates, using the upper boundary of each equivalent permeability bin when estimating the normalized steady state inflow intensity in Figure 2. This is equivalent to adjusting the 1995 analysis procedure upwards by a factor of 1.5. Heuer decided to apply this “correction factor” to both flow conditions [8, 13].
Heuer’s semiempirical method appeared to underestimate groundwater inflows for both flow conditions and with the same relative magnitude. However, case study analyses in the following sections will show that this is not correction [8, 13].