An analytical solution has previously been published for flat capillary barriers in uniform layered rock with a quasi-linear relative permeability function. This solution can be extended to curved surfaces with a radius of curvature that is much greater than the characteristic length of the relative permeability curve α−1 because water flows around the cavity in a relatively thin boundary layer. A simple formula is obtained for the onset of seepage into cylindrical cavities. Published solutions to the quasi-linear seepage problem, which take the form of infinite sums of Bessel functions, agree with this formula in appropriate limits. Studies of seepage into a proposed radioactive waste repository have used a finite difference model to solve the equations of unsaturated flow explicitly. Using the boundary layer solution, the discretization error in such numerical models is calculated. The numerical estimate of the diversion capacity of a capillary barrier is shifted away from the exact solution by a factor of αd cos ϕ/sinh(αd cos ϕ), where 2d is the grid spacing and ϕ is the slope of the cavity wall.
 At Yucca Mountain, Nevada, USA, construction of a nuclear waste repository in the unsaturated zone has been proposed. Waste would be placed in drifts excavated in fractured tuff bedrock. Water seepage into the drifts at Yucca Mountain will be limited by capillary forces that impede the flow of water out of narrow fractures into the large openings. The plans for Yucca Mountain have stimulated research on the seepage of water into cavities excavated in unsaturated rock.
 In the unsaturated zone, percolating water that encounters a cavity but does not enter it must flow laterally around the opening. It is difficult to calculate seepage in this environment because the rock through which the diverted water flows is spatially heterogeneous and the equations of unsaturated flow are highly nonlinear.
 The seepage problem at Yucca Mountain has been analyzed with a stochastic finite difference model [Bechtel SAIC LLC, 2004a, 2004b, 2004c] that explicitly simulates the nonlinearities and heterogeneities of the flow system in three dimensions. This approach has several important advantages: (1) the model can be calibrated against seepage experiments that necessarily have different geometry from the postemplacement percolation, (2) spatial variability of permeability (which both increases the mean seepage rate and contributes to its spatial variability [Ho and Webb, 1998]) is taken into account, and (3) alternative drift geometries can be analyzed. However, it also has some disadvantages: (1) factors controlling the results can be hard to isolate; (2) discretization errors can affect the results; and (3) large computational loads can limit the ability to assess alternatives.
 An alternative approach is to use analytical solutions. These require simplifying assumptions, such as uniform properties, simple two-dimensional geometry, and a simple form for the relative permeability curve. However, they offer greater physical insight into the meaning of results, especially the dependence of seepage rates on system parameters. While the limitations of analytical solutions (the requirements of simple geometry and uniform properties in particular) make it impossible for them to replace detailed simulations, use of both approaches in combination yields greater understanding than can be obtained from either alone.
2. Cylindrical Cavity
 In previous work [Ross, 1990a], an analytical solution was developed for flow along a planar capillary barrier, a sloping plane separating an upper fine-grained layer and a lower coarse-grained layer. The solution yields a simple formula for the diversion capacity, the maximum amount of water that can be diverted before percolating water breaks through the barrier. The analysis uses the quasi-linear relationship
between the relative permeability kr and the capillary pressure ψ. In this solution, there is a wet boundary layer above the interface in which the potential ψ and the effective saturation Se differ from the values they take in the remainder of the overlying fine-grained porous medium. The thickness of the barrier depends on the parameter α, which is a measure of the strength of capillary forces.
 Intuitively, this solution should also apply to the capillary barrier created by a cylindrical opening in a porous medium if the radius of the opening is large compared to the thickness of the boundary layer. (The open cavity can be thought of as one big pore, the limiting case of a coarse-grained porous medium.) This intuition can be quantified by applying it to flow around an open cylinder and comparing the result to the exact solution derived by Philip et al.  for unsaturated flow around a cylindrical cavity. The Philip solution has previously been applied to the problem of drift seepage at Yucca Mountain [Ross, 1990b; Hughson and Dodge, 2000; Or et al., 2005].
 Consider a point on the surface of the opening whose coordinates are (X, Y) in a coordinate system with origin at the center of the opening. (See Figure 1.) The slope of the wall at this point is ϕ. A uniform water flux q percolates downward from an upper boundary at infinity. If no water enters the drift between the center line and (X, Y), the amount of water (per unit length of drift) that moves horizontally across the vertical line x = X is Q = qX.
 For an arbitrary relative permeability curve kr(ψ), an upper bound on the amount of water that can move horizontally as a result of the capillary barrier is given by equation (13) of Ross [1990a] or equation (6–151) of Warrick :
where Ks is the saturated hydraulic conductivity of the medium above the capillary barrier. Inserting tan ϕ = X/Y and Q = qX gives
This condition is most stringent when Y takes its maximum value ℓ, the radius of the tunnel. It becomes
 Inserting the quasi-linear relative permeability (1) and integrating gives
Noting that kr is always less than unity and Kskr(−∞) = q, we have
which can be rearranged as
Philip et al.  obtain a general solution for unsaturated flow around circular openings. The solution is given in terms of a variable ϑ = Kskr/q, whose maximum value on the cavity surface is ϑmax. The solution for ϑ is expressed as a function of a dimensionless variable s = αℓ/2. For large s, Philip et al.'s equation (84) gives
The condition for the absence of seepage is that the porous medium above the cavity does not fully saturate. This is equivalent to kr < 1, which gives
in which the first two terms are identical to the formula (7) derived from Ross [1990a].
 The planned drifts at Yucca Mountain will have a radius ℓ of 2.75 m [Bechtel SAIC LLC, 2004b], and in the surrounding fractured tuff α−1 is a few centimeters [Trautz and Wang, 2002]. Thus αℓ is on the order of 100. Not only does the formula obtained from the more physically interpretable model of Ross [1990a] coincide with the leading terms in the series expansion of the solution obtained by Philip et al. , but for the parameter values applicable to Yucca Mountain it differs from the more exact Philip solution by less than one part in 1000. Even the second term in equation (10) can be dropped without appreciable error.
 An additional benefit of the boundary layer analysis is that it provides a quantitative description of the distribution of lateral flow within the boundary layer. This information is useful for numerical modeling because it provides guidance about grid design. Appendix A presents an explicit calculation of the grid discretization error in a finite difference model of seepage into cavities.
3. Cavity With Asperity
 One of the idealizations in the above solutions is the perfectly smooth cylindrical cavity walls. Seepage into cavities of other shapes has been analyzed by Selker  and Philip , but these studies also address cavities with smooth walls. An extension of the quasi-linear analysis of Philip et al.  to irregularly shaped cavities is provided, however, by Hughson and Dodge .
 The approach used by Hughson and Dodge  is analogous to an approach often used to study noisy physical systems: A single representative perturbation with parameterized dimensions is added to the system. In this case, the perturbation is a triangular protrusion of rock into the circular drift. This perturbation, referred to as an asperity, is located at the crown of the drift, where the effect on seepage will be greatest. Hughson and Dodge conclude that “As the vertical dimension of the [asperity] becomes a significant fraction of the α−1 characteristic length, the effect of the boundary perturbation comes to dominate threshold percolation flux.”
 There are several reasons to apply boundary layer analysis to the same problem. First, solutions are calculated by Hughson and Dodge  only for 4 ≤ s ≤ 16, while values of s on the order of 50 are of greatest interest for Yucca Mountain. Boundary layer analysis, which represents a limit of large s, is relevant here. Second, the simple analytic relations that come out of boundary layer analysis are often easier to use.
Figure 2 shows the asperity geometry analyzed by Hughson and Dodge . The seepage threshold is reached when ψ = 0 at the tip of the asperity. At steady state, water cannot accumulate in the asperity; thus in the absence of seepage the upward suction force must be equal to or greater than the downward gravitational force at every point along the wall surface. For this to be true, the potential difference between the tip of the asperity and the point where it intersects the cylindrical surface of the drift wall must be greater than the height of the asperity; that is, ψ < −h. Equation (14) of Ross [1990a] provides a bound on the diversion capacity at this point:
 The amount of water that must be diverted out of each side of the asperity is qw, where 2w is the width of the asperity. A necessary (but not sufficient) condition for the absence of seepage is then
Inserting sin ϕ = w/ℓ and rearranging gives
where ℓ′ = ℓ cos ϕ is the distance from the center of the cavity to the (flat) base of the asperity. This result indicates that the presence of an asperity at the top of the cavity shifts the seepage threshold by an amount that depends on the ratio of the asperity height to α−1 and is, for asperities that are narrow compared to the tunnel radius, nearly independent of the asperity width.
 The ratio of equations (14) and (7) is a lower bound on the shift (expressed as a ratio) in the seepage threshold. This ratio is
Hughson and Dodge  calculated equivalent ratios for selected values of s and w. These ratios are displayed in their Figures 7 and 8. Table 1 compares the ratios calculated by Hughson and Dodge with values of (15). (Hughson and Dodge present a range of capillary strengths; the weakest capillarity case they address, s ≡ αℓ/2 = 16, is presented in the table because boundary layer analysis is most accurate in the limit of weak capillarity.) For the smallest values of w and the largest values of h, there is good quantitative agreement, as would be expected because these values approach the limit in which the boundary layer approximation is an exact solution. For larger w, the lower bound on the shift predicted by (15) is indeed smaller than the more exact ratio calculated by Hughson and Dodge.
 The agreement between the two approaches provides more confidence in the results. It also provides additional support for the conclusion of Hughson and Dodge quoted above about the conditions under which asperities control seepage behavior.
 Quasi-linear boundary layer analysis predicts seepage in a way that is simple and whose physical meaning is easily understandable. It agrees with more complicated analytical seepage solutions in the appropriate limits. Equation (7), the criterion for the onset of seepage that is derived from the analytical boundary layer solution, is a useful complement to numerical simulations and facilitates their physical interpretation. Boundary layer analysis can also be used to quantify the discretization error in numerical models, which remain necessary to deal with the complex geometries and heterogeneous properties that are found in practical problems.
Appendix A:: Grid Discretization
 When a numerical model is used to solve differential equations such as those of unsaturated flow, the main approximation is the discretization of continua. Webb  compared the Ross [1990a] analytical solution with numerical analyses and found that computational limitations prevent use of arbitrarily fine grid spacing. In such circumstances, the error introduced by discretization can be difficult to evaluate. Boundary layer analysis provides a means of overcoming this difficulty because it can be used to calculate the discretization error analytically.
A1. Magnitude of Error
 The published numerical solution for seepage into heterogeneous tuff at Yucca Mountain [Bechtel SAIC LLC, 2004b] uses a finite difference model in which flow between grid blocks is calculated from the material properties along a line between the block centers. The boundary at the drift wall, where ψ is set to zero, coincides with the interface between blocks. However, the relative permeability for lateral flow through the layer of grid blocks closest to the drift wall is calculated from properties “measured” inside the rock at −z = d, where 2d is the grid spacing. (The z axis points downward, for consistency with the notation of Ross [1990a].) At this location, the fractures are partially drained and kr < 1 [see Bechtel SAIC LLC, 2004a, pp. 6–35]. Lateral flow through the other layers is calculated at positions −z = 3d, 5d, …. If kr(z) is strongly nonlinear on a scale of 2d, there will be a significant discretization error. This is the case when α−1 is comparable to or smaller than d.
 The quasi-linear equations are solvable for both the lateral flow through a continuum and the lateral flow through a stack of layers with relative permeabilities corresponding to locations −z = d, 3d, 5d, …. Comparison of these two solutions provides an estimate of the magnitude of the discretization error.
 The first step in the calculation is to determine how the capillary pressure ψ varies with position in the boundary layer. In a continuum model of drift seepage, the fracture network is fully saturated at the point on the drift wall where seepage occurs. As one moves away from the wall, moisture content, suction potential, and relative permeability vary continuously.
 This makes it convenient to use a coordinate system in which z = 0 at the drift wall. In these coordinates, ψ(0) = 0 and kr(0) = 1. Figure A1 shows the forces on an element of water. Above a capillary barrier with q ≪ Ks, the flow parallel to the boundary varies slowly with position within the boundary layer and the variation of potential in the x direction is negligible compared with its variation in the z direction. In the absence of seepage, the component of flux normal to the boundary is negligible, and the z component of gravity must balance the pressure gradient:
 The next step is to calculate the diversion capacity, the maximum flux of water that can move in the x direction, parallel to the drift wall. From equation (12) of Ross [1990a], the diversion capacity is
Because the flow normal to the wall is zero, the horizontal flux is simply the horizontal component of vx which is vh = vx cosϕ. Substituting this relationship and (A1) into equation (A3) results in
Combining these equations and integrating over the boundary layer gives the total diverted flow in the continuum model:
 In a discretized model, the integral in equation (A6) is replaced by a sum over grid block centers:
 Comparing equations (A6) and (A7) shows that the ratio between the diversion capacity in the discretized and exact models is
Because the Taylor series expansion of the hyperbolic sine is sinhx = x + x3/3! + …, the fractional discretization error is (αd cos ϕ)2/6 plus higher-order terms. When αd is on the order of unity or larger, the discretization error is large because most of the flow passes through a single layer of grid blocks adjacent to the interface.
 Seepage into cylindrical cavities is controlled by the diversion capacity at the top of the cavity where ϕ = 0, so discretization error shifts the seepage threshold by a ratio of dα/sinh (dα).
A2. Van Genuchten Relative Permeability
 In numerical models, the relationship between relative permeability and suction derived by van Genuchten  is frequently used. Like the quasi-linear model, this model contains a parameter called α with units of inverse length that measures the strength of capillary forces. However, because it uses a different permeability-suction curve, the van Genuchten α (referred to here as αvG) does not correspond exactly to the quasi-linear α [Rucker et al., 2005]. When the formulas derived in the previous section are used to estimate discretization error, αvG must be replaced with an equivalent value of α.
 For this purpose, the quasi-linear relationship kr = eαψ must be fitted to the van Genuchten permeability-suction curve, using α as a fitting parameter. In the van Genuchten model, kr and α are both expressed in terms of the effective saturation Se (a parameter that does not enter the calculations directly because it drops out of the steady state flow equation) and an exponent m. The relationship takes the form
As Rucker et al. discuss in detail, the method of comparing the two curves is problem-specific; they should be fitted to match in the relevant region of potential. As noted above, discretization error is a concern when d is comparable to or larger than α−1 and most of the water that is diverted around a cavity flows through the layer of grid blocks immediately adjacent to the cavity wall. In a finite difference model, most of the flow will then be calculated using the value of relative permeability determined along a surface located at a distance d from the drift wall. The suction potential on this surface is ψ = −d cos ϕ. At the top of a cylindrical cavity, where seepage occurs first, the potential simplifies to ψ = −d. The equivalent quasi-linear α should be determined from the van Genuchten relative permeability (1) at that point:
where S*e is the value that, when substituted in equation (A10), yields ψ = −d.
 Making the substitution and solving equation (A10) for S*e yields
Combining these last two equations with (A9) yields
 This work was supported by the Electric Power Research Institute under subcontract to Monitor Scientific LLC. I thank Dafina Edwards for help with calculations.