[22] 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* [1997] 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

[23] 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 2*d* 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 *k*_{r} < 1 [see *Bechtel SAIC LLC*, 2004a, pp. 6–35]. Lateral flow through the other layers is calculated at positions −*z* = 3*d*, 5*d*, …. If *k*_{r}(*z*) is strongly nonlinear on a scale of 2*d*, there will be a significant discretization error. This is the case when *α*^{−1} is comparable to or smaller than *d*.

[24] 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*, 3*d*, 5*d*, …. Comparison of these two solutions provides an estimate of the magnitude of the discretization error.

[25] 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.

[26] This makes it convenient to use a coordinate system in which *z* = 0 at the drift wall. In these coordinates, *ψ*(0) = 0 and *k*_{r}(0) = 1. Figure A1 shows the forces on an element of water. Above a capillary barrier with *q* ≪ *K*_{s}, 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:

[27] 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 *v*_{x} which is *v*_{h} = *v*_{x} cosϕ. Substituting this relationship and (A1) into equation (A3) results in

Inserting equation (A2) into the quasi-linear relationship (1) gives

Combining these equations and integrating over the boundary layer gives the total diverted flow in the continuum model:

[28] In a discretized model, the integral in equation (A6) is replaced by a sum over grid block centers:

[29] 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 sinh*x* = *x* + *x*^{3}/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.

[30] 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

[31] In numerical models, the relationship between relative permeability and suction derived by *van Genuchten* [1980] 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 *α*_{vG}) 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, *α*_{vG} must be replaced with an equivalent value of *α*.

[32] For this purpose, the quasi-linear relationship *k*_{r} = *e*^{αψ} must be fitted to the van Genuchten permeability-suction curve, using *α* as a fitting parameter. In the van Genuchten model, *k*_{r} and *α* are both expressed in terms of the effective saturation *S*_{e} (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*.

[33] Making the substitution and solving equation (A10) for *S**_{e} yields

Combining these last two equations with (A9) yields