Appendix C Brine Inflow to Excavations in the Salado
Permeability Of Wipp Salt Anhydrite And Interbeds
Observation of brine seeps into excavations (both tunnels and boreholes) at the WIPP has led to the assumption that the Salado salt has a finite, but extremely low (˜ 10^{21}  10^{23} m^{2}) permeability in the classical sense of Darcy flow. There are several inconsistencies in this model. An alternative model in which undisturbed salt is assumed to contain brine only in unconnected porosity appears in better agreement with underground observations at WIPP. In this model, "damage" to the salt (i.e., development of the disturbed rock zone, DRZ) resulting from introduction of the excavation, leads to the development of a local zone of interconnected porosity characterized by a finite permeability. It is the release of the brine in the previously unconnected pores by the stressinduced damage that produces the brine seeps—at least in the initial stages (Borns, 1995).
McTigue (1995a,b) has examined this alternative model in detail he notes:
Data for flow to boreholes in WIPP Room D are represented well qualitatively by the classical model. However, the capacitance indicated by fitting model calculations to the field data is orders of magnitude larger than that expected on the basis of the compressibilities of salt and brine. That is, the decay of the brine flux into the boreholes takes place over a time scale that is much too long to be explained in terms of the processes assumed in the development of the classical model. The classical model invokes an unbounded domain of interconnected porosity. This concept is contradicted by both mechanical arguments and geochemical observations. (McTigue 1995b).
The mechanical arguments from the same reference point out that salt creeps at vanishingly small deviatoric stress (i.e., stress difference). Thus, over long time frames, interconnected porosity can not be sustained. Brine will be confined to local domains of isolated porosity containing fluid at a pressure equal to (minus) the mean [i.e., lithostatic] stress in rock.
Geochemical observations, however, indicate that
…brine chemistry is highly variable, even among samples separated by distances of the order of tens of centimeters. If these brines were derived from an interconnected pore network, one would expect that molecular diffusion would have eliminated any significant contrasts in brine composition over the very long existence of the formation. … Over a period of 230 Ma the diffusion length, L_{a}, is…of the order of hundreds of meters. The observation that compositional differences persist in the brines over short length scales and very long time, then, suggests strongly that the brine in undisturbed salt is local, isolated domains. (McTigue, 1995b)
McTigue (1991) later analyzes brine seepage rates into Room Q and again concludes that this experiment confirms the view that undisturbed salt is impermeable.
Borns (1995) notes that measured changes in the electrical resistivity in the DRZ around Room Q indicate an initial desaturation (as the DRZ develops)
followed by a resaturation over a period of approximately three years. This could suggest that Darcy flow may occur in the salt, but it could also be the result of the progressive development of the DRZ, which eventually connects the anhydrite layers hydrologically to the room excavation.
The McTigue and Borns observations can be reconciled by a model in which classical Darcy flow occurs in the anhydrite marker beds (e.g., MB 139), but the salt between these layers is impermeable.
The following calculations indicate the order of magnitude of brine inflow predicted on the basis of assuming
 an impermeable salt containing anhydrite interbeds that exhibit Darcy flow;
 salt with a small, but finite, permeability, allowing classical Darcy flow radially into a circular excavation. (In this case, the permeable interbeds are assumed to be an integral part of the homogeneously permeable salt.)
OneDimensional Flow In Anhydrite Interbeds In Impermeable Salt
The instantaneous rate of fluid inflow q, at some time t, per meter of tunnel length* from a horizontal permeable layer (Figure C1), thickness d and of infinite extent (onedimensional flow) can be calculated (Churchill, 1972) from the formula shown in (C1):
where k = kpg/µ is the brine conductivity, d is the thickness of the layer, Ø_{0} is the initial head, and c = k/S_{s} is the diffusivity. (Other symbols are defined in Tables C1 and C2.) In the case of a constant head at a finite distance L from the tunnel along the layer, the instantaneous rate of inflow can be written (Churchill and Brown, 1987) in dimensionless form:
where ¯q = Lq/dkØ_{0} and τ = tc/L^{2}. The dimensionless rates of inflow, from an infinite layer and from a layer with a fixed head at the finite distance L, as a function of dimensionless time, are shown in Figure C2.
The total inflow Q per meter of the tunnel—one side of the tunnel only—from an infinite layer can be obtained by integrating (C1):
The curves shown in Figure C3 are calculated by using the parameters for an anhydrite layer in salt, as shown in Table C1. Total inflows (C1) from an infinite layer for the base case mentioned above and two cases used by McTigue (1991) (parameters shown in Table C2) are compared in Figure C4.
Table C1. Base case values of parameters used to calculate brine inflow into a tunnel from a horizontal anhydrite layer.
permeability 
κ 
10^{18} 
(m^{2}) 
brine viscosity 
μ 
1.8 · 10^{3} 
(Pa·s) 
specific weight 
ρ 
1230 
(kg/m^{3}) 
initial head 
Ø_{0} 
650 
(m) 
fixed head distance 
L 
1000 
(m) 
layer thickness 
d 
1 
(m) 
specific storage 
S_{s} 
1.5 · 10^{7} 
(1/m) 
conductivity 
κ 
6.69 · 10^{12} 
(m/s) 
diffusivity 
c 
4.45 · 10^{5} 
(m^{2}/s) 
Table C2. Values of parameters used in "base case" and by McTigue to calculate brine inflow from an anhydrite layer into a tunnel.

Base case 
McTigue case 1 
McTigue case 2 

κ 
10^{18} 
10^{21} 
10^{22} 
(m^{2}) 
μ 
1.8 · 10^{3} 
1.60 · 10^{3} 
1.60 · 10^{3} 
(Pa s) 
ρ 
1230 
1230 
1230 
(kg/m^{3}) 
Ø^{0} 
650 
815 
815 
(m) 
d 
1 
1 
1 
(m) 
κ 
6.69 · 10^{12} 
7.67 · 10^{15} 
7.67 · 10^{16} 
(m/s) 
c 
4.45 · 10^{5} 
6.20 · 10^{8} 
10^{10} 
(m^{2}/s) 
Radial Flow Into Excavations In Permeable Salt
The Laplace transform of the instantaneous rate of inflow ˜q into a circular opening in an infinite, homogeneous, porous medium (Figure C5) is as follows (Detournay and Cheng, 1988):
where K_{0} and K_{1} are the modified Bessel functions of the second king, of order zero and one respectively.
Inversion of (C4) is performed numerically. The total inflow Q (per meter along the tunnel) for k = 6.25 · 10^{14} m/s (corresponding to К = 10^{20} m^{2}), radius of tunnel a = 1.524 m, and an initial hydrostatic pressure p_{0} = 6 MPa, is shown in Figure C7.
Bredehoeft Calculation (Bredehoeft, 1988)
Around 1987, observations of brine inflow into underground excavations at the WIPP led Dr. J.D. Bredehoeft, then a member of the NRC WIPP committee, to suggest that the Salado formation at WIPP could have finite, albeit low, permeability in the classical Darcy sense and that the Salado was possibly saturated with brine. In defining the Salado, he (Bredehoeft, 1988) did not distinguish between the salt and anhydrite interbeds. He calculated, numerically, the daily brine inflow into rectangular excavations for several assumed values of hydraulic conductivity k. [For WIPP brine the intrinsic permeability κ (m^{2}) = 0.16 · 10^{6}k (m/s). Thus k = 6.25 10^{14} m/s corresponds to k = 10^{20} m^{2}, etc.].
Figure C6 compares Bredehoeft's results (curves 14) with those obtained analytically by using (C4). It is seen that the two sets of results are consistent. The principal difference is due to the higher range of brine conductivities assumed by Bredehoeft. The values used in the current calculations are based on more recent values used by DOE.
Using a Darcy flow model to interpret recent insitu measurements, Beauheim et al., (1993a) conclude that "the vertically averaged hydraulic conductivities of the tested intervals (primarily anhydrite interbeds) range from about 1·10^{14} to 2·10^{12} m/s (permeabilities of 2·10^{21} to 3·10^{19} m^{2}). Storativities of the tested intervals range from about 1·10^{8} to 2·10^{6} and values of specific storage range from 9·10^{8} to 1·10^{5} l/m. …" Anhydrite interbeds appear to be one or more orders of magnitude more permeable than the surrounding halite, primarily because of subhorizontal beddingplane fractures present in the anhydrites.
Freeze and ChristianFrear (in preparation) attempting to interpret the brine inflow results for Room Q, indicate that a value of "farfield permeability of 5·10^{22} m^{2} with a bulk rock compressibility of 5.4·10 ^{12} l/Pa" fits the apparent brine inflow rates from two to five years. Interpretation was complicated by the absence of very early time (06 months) brine inflow data—due to malfunctioning of the room seals. Thus, it seems that the hydraulic conductivity of (i) Salado halite (interpreted in terms of Darcy flow model) is of the order of 10^{13} to 10^{16} m/s (permeability 10^{20} to 10^{23} m^{2}) and (ii) anhydrite interbeds is 10^{12} to 10^{14} m/s (permeability 10^{19} to 10^{21} m^{2}). Since the anhydrite interbeds are relatively thin layers within the halite, an overall value for conductivity (permeability) of the Salado would tend to be at the lower end of the combined anhydrite—halite
range (i.e., say 10^{14} m/s), resulting in the inflows in the range of curves 4, 5, and 6 in Figure C6.
Figure C7 shows the total inflow per meter length of excavation for the lower permeability results (curves 5 and 6) shown in Figure C6. It is seen that after 100 years, a total of about 200 liters (or 0.3 m^{3}) of brine has flowed into the excavation. If a 10m excavation width (or ''width" of waste material) is assumed, this amount of brine would occupy a height of 2 cm over the 10m width. If the compacted waste is assumed to occupy 70% of the original space (i.e., leaving 30% voids), then the height of brine would increase to (2/0.3) cm, or approximately 6 cm of the height of compacted waste. However, since the DRZ would exist for at least a substantial part of the 100 years, then much of this brine would gravitate into the fractured rock in the DRZ underlying the waste. Brine from MB 139, below the excavation, would also flow into the same fractured rock below the floor. It seems probable, therefore, that much of the brine will not come into contact with the waste but instead will flow downgradient in the DRZ below the waste. This suggests that with appropriate repository design, rooms could be maintained in a "brine humid" condition rather than a "brineflooded" condition. According to Brush, gas generation caused by corrosion of the steel waste drums in brine humid conditions is negligibly small.
Should there be a more significant rate of gas generation, pressure in the rooms would tend to rise, thereby reducing brine inflow—and the associated gas generation rate. Although it is possible, under some conditions, for gas to flow out of the room [e.g., along the (finite permeability) marker beds] at the same time as brine flows in, a recent DOE study by Webb and Larson (1996) indicates that such "countercurrent" flow is unlikely for the 1° dip of the marker beds at the WIPP until the gas pressure in the excavation is close to lithostatic pressure. Any gas at lower pressure will directly reduce the hydraulic pressure differential and, hence, reduce the brine inflow into the excavation as well. This, in turn, will limit corrosion and the associated gas generation. If the gas pressure reaches (and "attempts" to exceed) lithostatic, the permeability of the anhydrite interbeds would increase substantially since fracturing of the interbeds would occur and countercurrent flow (gas out, brine in) in the 1° updip section of the anhydrite interbed would result in increased brine inflow into the excavation. From curves presented by Webb and Larson (1996) in inflow appears to be or the order of twice the inflow rate observed for the unpressurized (i.e., open) excavation. With the low brine inflows calculated earlier and the associated low gas generation rates, it seems unlikely that gas pressure could approach lithostatic before the wastefilled excavations had closed to their residual (compacted) volumes.
Shaft Seals  Approximate Check On Pa Flow Calculations
If Darcy flow is assumed,
where q is the volumetric flow rate (m^{3}/s), k is the hydraulic conductivity of the (porous) seal (m/s), dl is the length (m) across which dØ (m) is measured, and A is the crosssectional area (m^{2}) normal to the flow direction.
The hydraulic conductivity may be determined from the properties of the porous (seal) material and the saturating fluid. Thus,
where κ is the intrinsic permeability of the porous (seal) material (m^{2}); ρ is the fluid density (kg/m^{3}); g is the gravitational acceleration (m/s^{2}); and μ is the fluid viscosity[Pa·s].^{†}
The term "hydraulic conductance" C (m^{2}/s) is sometimes used, where
and L is the length of the porous material (i.e., the shaft seal in this case).
where Ø is the fluid pressure (head) differential (m) over the length L, and
where υ_{z} is the Darcy velocity^{‡} or specific discharge (m/s) of the fluid in the porous medium (seal) with a head differential of Ø (m).
Using PA assumed values (i.e., a shaft seal length L = 100 m; total area of the four shaft seals A = 100 m^{2}; seal permeability κ of 1·10 ^{16} m^{2}; brine viscosity of 0.0018 Pa·s; fluid density of 1230 kg/m ^{3}; and gravitational acceleration of 9.792 m/s^{2}), yields (U.S. Department of Energy, 1995d, p. 5):
At the time the seal is first installed, the repository will be essentially dry and fluid flow through the seal initially will be downward—primarily from the Rustler formation overlying the Salado. Flow upward through the seal will begin only when the fluid pressure exerted at the bottom of the shaft seal exceeds the hydrostatic pressure due to the height of water in the shafts.
The predicted change in pressure with time at the shaft bottom will depend on a complex interaction of several repository variables, such as the total amount of void space in the wastefilled rooms and all of the (backfilled or open) excavated space; rate of reduction in void space due to salt creep; rate of inflow of brine into the void space; rate of gas generation and room pressurization; loss in fluid pressure due to fluid flow from the wastefilled rooms to the bottom of the seal, etc. Depending on the particular combination of assumed values of these variables used in a calculation, the brine pressure at the shaft bottom may remain very small for a long time after sealing or may increase after a period of the order of several hundreds of years, when gas generated by brine corrosion of the steel waste containers develops an increasing pressure in the repository.
The relationship between seal permeability and flow through the seal over a given time [Eq. (C.8)] is linear, so the net cumulative flow, over a period of time through a seal of a given permeability can be determined simply from the average driving fluid pressure [i.e., the pressure Ø above hydrostatic at the bottom of the seal over the same time (This is 650 m for the waterfilled shaft.)]. If the pressure Ø is initially below hydrostatic (i.e., negative), then flow during this period will be negative, that is, downward through the seal.
It is seen from (C8) that the net cumulative flow over 10,000 years through a shaft seal of assumed permeability 10^{16} m^{2} will be 10,000 m^{3}, if the average (over 10,000 years) pressure head Ø at the shaft bottom is Ø = 10,000/211 = 40 m (above hydrostatic). For a seal of permeability of 10^{17} m^{2}, the average pressure head required to produce a cumulative upward flow of 10,000 m^{3} over 10,000 years will be Ø = 470 m above hydrostatic—(650 m) or a total average brine pressure head of 110 m. Similarly, for a seal permeability of 10^{18} m^{2}, a total average pressure head of 1120 m would produce a net cumulative brine outflow of 1,000 m^{3} in 10,000 years.
Figure 4.1 in Chapter 4 of this report shows the results of a number of calculations—each for a randomly chosen combination of repository parameter values (closure rate, brine inflow rate, etc.)—of the cumulative brine flow through the seals over 10,000 years, as a function of seal permeability. It is seen that, especially at the higher permeabilities, flow into the repository (i.e., the negative values) tends to be more likely. For a permeability of 10^{16} m^{2} or lower, the net flow is essentially zero.
Creep of the salt around the shaft will result in further consolidation of the 100mhigh seal of crushed, compacted salt, so that the permeability of the seal will tend eventually towards the permeability of intact salt. As noted above, intact pure salt likely to be impermeable. Where intact salt is assumed to have a finite permeability, this is usually taken as about 10^{21} ∼ 10^{23} m^{2}. Calculations indicate (Callahan et al., 1996) that a seal of crushed salt at a depth between 550 m ∼ 650 m, initially compacted to a permeability about 10 ^{16} m^{2} is likely to achieve a permeability of 10^{20} m^{2} or so within 100 years after placement.
Thus, it appears that seals consisting of 100mhigh layers of compacted crushed salt should provide an effective barrier to release of radionuclide contaminated brine to the accessible environment over 10,000 years or longer.
The height to which the brine would rise above the seals depends on the average porosity of the material in the shaft (see discussion of Darcy flow below). Thus, if the shaft above the seal was left completely open and the (total) crosssectional area (of the shafts) was assumed to remain constant at 100 m^{2}, a cumulative brine flow of 10,000 m^{3} would create a brine column 100 m high above the seal. If the average porosity of the material in the shaft was 1%, the brine would rise, hypothetically, 100 times higher. In the case of WIPP, the brine would rise to the top of the Salado and then would be able to flow laterally into the Culebra.) In both cases, the total volume of brine outflow would be essentially unchanged. For a given concentration of radionuclides in this same volume of brine, the radionuclide release will also be the same, independent of the porosity of the material through which it flows.
Darcy Velocity
The Darcy velocity (or specific discharge) is the volumetric rate of flow per unit area through which flow occurs. Hence, it has the same dimensions as velocity, but it is not the speed at which the fluid moves. To derive the average fluid velocity in the pore space from the Darcy velocity, it is necessary to divide the latter by the effective porosity of the rock through which flow occurs (i.e., if the effective porosity of the rock is 10% of the total volume of the rock, the average fluid velocity in the pores is ten times the Darcy velocity). The volume of fluid flowing across the crosssection over a given time, which is the relevant quantity for determination of radionuclide flow through the seals (i.e., this is strictly not a "release", since there are additional barriers, such as upper levels of the shaft and the Culebra, between the seals and the accessible environment)—given a known radionuclide concentration in the fluid—is the same for both calculations, that is, with or without porosity.
Conclusion
These approximate calculations confirm the general impression that effective sealing of the repository is achievable using the designs proposed by DOE.