#### 2.1. Streamflow Routing

[8] Most modeling applications use shallow water equations to simulate free-surface flows in natural channels in one or two dimensions (1-D or 2-D) [*Tchamen and Kahawita*, 1998; *Heniche et al.*, 2000]. However, flow initiation is difficult to model because the shallow water equations are discontinuous through zero depth and, in their standard forms, are difficult to use for problems involving zero depths. Several studies have addressed the problem of wetting and drying in stream channels and have developed techniques for simulating dry stream conditions with the shallow water equations [*Tchamen and Kahawita*, 1998; *Heniche et al.*, 2000; *Valiani et al.*, 2002; *Niswonger et al.*, 2005].

[9] Methods used to simulate zero depths with the Saint-Venant equations are difficult to apply when considering high seepage loss rates at the streamflow front during flow initiation in a dry channel. This is partly due to the relatively high seepage loss rates at the streamflow front and the added nonlinearity caused by the interdependencies between seepage losses, stream wetted perimeter, and stream depth. Consequently, this study uses a kinematic wave approximation of the 1-D form of the shallow water (Saint-Venant) equations to simplify the simulation of flow in initially dry streams. The 1-D kinematic wave equation can simulate wetting and drying of stream channels during high streambed seepage loss without any special numerical treatment. However, the kinematic wave equation neglects the effects of dynamic waves, only is applicable to subcritical flow conditions, and does not consider backwater effects [*Woolhiser*, 1974]. The applicability of the kinematic wave equation to the streams studied here was evaluated with the kinematic wave number of the observed flow as demonstrated by *Woolhiser* [1974]:

where

*k*_{w}is the kinematic wave number,

*S*_{0}is the channel slope,

*L*_{0}is the channel reach length (L),

*h*_{0}is the normal flow depth (L),

*Fr*is the Froude number.

Kinematic wave numbers greater than 10 indicate that the kinematic wave model is applicable.

[10] The kinematic wave approximation to the Saint-Venant equations assumes that waves possess only one velocity at each point, such that the continuity equation governs wave movement [*Lighthill and Whitham*, 1955]:

where

*Q*is the streamflow (L^{3}T^{−1}),

*t*is time (T),

*x*is the distance along the longitudinal axis of the stream (L),

*A*is the cross-sectional area (L^{2}),

*A*_{d}is the cross-sectional area of channel depressions with dead storage (L^{2}),

*q*is the seepage loss per unit length along the longitudinal axis (L^{2}T^{−1}),

*q*_{d}is the rate at which dead storage is filled. *q*_{d} is equal to , which is nonzero during flow initiation and zero after channel depressions are filled (L^{2}T^{−1}).

[11] If streamflow is adequately represented by a single-valued function of the channel cross-sectional area, *A*, and location, *x*, then the wave celerity, *c*, is equal to the slope of the flow concentration curve [*Lighthill and Whitham*, 1955]:

[12] The typical representation of momentum in the kinematic wave equation assumes that gravitational forces are balanced by frictional forces:

where *S*_{f} is the friction slope and *S*_{0} is the channel slope.

[13] Thus, momentum can be represented by Manning's equation in combination with equation (4):

where

*C*_{m}is a constant, which is 1.0 for cubic meters per second and 1.486 for cubic feet per second (units depend on the units used in the model),

*n*is Manning's roughness parameter, and

*R*is the hydraulic radius of the stream, which is equal to the stream cross-sectional area divided by the wetted perimeter, (L).

[14] The wave celerity (equation (3)) can be calculated by taking the derivative of equation (5) with respect to the channel area, *A*. Because the hydraulic radius, *R*, is a function of *A*, a relationship must be defined to relate *A* and *R*; however, there is no universal function for this relationship that applies to all channels. Consequently, the derivative of equation (5) is calculated numerically, and the relationship between *A* and *R* is calculated by integrating their values over specified cross sections perpendicular to the stream axis.

[15] Equation (2) was solved numerically by using an implicit four-point, finite difference solution technique [*Fread*, 1993]. The spatial derivative in equation (2), ∂*Q*/∂*x*, is approximated by a forward difference quotient that includes time weighting of the quotient at the previous time step. The time derivative in equation (2), ∂*A*/∂*t*, was approximated by a forward difference quotient centered between the end of the previous and current time steps. The finite difference approximations are solved iteratively using Newton's method [*Burden and Faires*, 1997].

[16] Streamflow must be defined at the upstream boundary for the duration of the simulation period. Finite difference nodes that represent the stream are numbered as *k* = 1, *N*, where *N* is the total number of finite difference nodes, with *k* = 1 at *x* = 0. *x* denotes distance along the stream axis from the upstream end of the model. *k* = *N* at *x* = *L*, where *L* is the location at the downstream end of the model. Streamflow is routed for times *j* = 1, *M*, where *M* is the total number of time steps in the simulation. The boundary condition at *x* = 0 is specified on the basis of the measured flow at *k* = 1 (*Q*_{1}) for all *M* time steps. The boundary condition at *k* = *N* is defined as *Q*_{N} = *constant* and initial conditions are defined as *Q*^{j=1,M} = *constant* for all stream nodes in the model, where *constant* can be zero or nonzero.

#### 2.2. Seepage Model

[17] A relatively simple approach was adopted for simulating transient seepage loss coupled to the kinematic wave equation for channel flow. Transient infiltration into a dry streambed begins at high rates driven by large capillary potential gradients. Following several hours of streamflow inundation, the capillary potential gradients subside and infiltration decreases to a steady rate. These two periods of infiltration were simulated using the equation developed by *Philip* [1957]:

where

*I*is the cumulative infiltration (L),

*t*_{w}is the time since channel inundation by streamflow (T),

*S*(θ_{0}, θ_{i})is sorptivity (LT^{−1/2}), which is a function of physical sediment characteristics and the initial water content (θ_{i}) and wetting front water content (θ_{0}), and *A*_{g} is the coefficient for the gravity component of infiltration in Philip's model.

[18] It is prudent to mention that Philip's equation has limitations in its applicability because of truncation of terms in the analytical solution to the partial differential equation [*Parlange*, 1973]. Because the problem addressed herein only concerns instantaneous ponding, the main limitation of Philip's equation is its accuracy during gravity dominated infiltration. However, because early time infiltration has the largest effect on the SFV, and because the parameters in the infiltration equation are fitted, the Philip equation provides a simple equation for testing the new approach. Future studies on this topic might benefit from adopting more accurate infiltration equations.

[19] Streams exert pressure on the channel bottom surface that can enhance seepage loss. The effects of stream depth on infiltration have been incorporated into analytical infiltration equations [*Philip*, 1958a, 1958b; *Parlange*, 1973; *Freyberg et al.*, 1980]. The approach of *Philip* [1958a and 1958b] was adopted whereby stream depth has the effect of enhancing sorptivity, *S*. In order to incorporate the effects of stream depth into the seepage model, the following representation of sorptivity was adopted [*Smith et al.*, 1993]:

where

- θ
_{s} is the saturated water content of the sediment,

- θ
_{i} is the initial water content of the sediment,

*ψ*is the capillary pressure potential (L), and

*ψ*_{i}is the capillary pressure potential corresponding to the initial sediment water content (L).

[20] Incorporating the effects of stream depth in a similar manner to *Philip* [1958b], the sorptivity is

where

*h*_{c}is the stream depth for an unsaturated flow column that is calculated on the basis of the kinematic wave equation (L), and

*K*_{r}(*ψ*)is the relative hydraulic conductivity as a function of capillary pressure potential.

[21] The *Brooks and Corey* [1964] model was chosen to define the relative hydraulic conductivity as

where

*ψ*_{a}is the air entry potential (L), and

*η*is the Brooks-Corey exponent.

[22] The *Brooks and Corey* [1964] model also was used to relate the sediment water content to capillary pressure:

where

- θ
is the water content corresponding to the capillary pressure *ψ*, and

- θ
_{r} is the residual water content.

[23] Infiltration is calculated separately for each stream section that spans the distance between adjacent finite difference nodes and for each unsaturated flow column within the stream cross section (Figure 1). The wetting front depth, *z*, for a particular unsaturated flow column is calculated assuming *I* = *P**_{c}*z*, where *P*_{c} is the wetted perimeter of the unsaturated flow column (Figure 2). The vertical saturated hydraulic conductivity minus the unsaturated hydraulic conductivity at antecedent capillary pressure, *K*_{s} − *K*(*ψ*_{0}), is used to represent the gravity term coefficient, *A*_{g}, from equation 8. Representing *A*_{g} with *K*_{s} − *K*(*ψ*_{0}) has been shown to be a reasonable approximation [*Hsu et al.*, 2002]; however, it also has been shown that *A*_{g} will be slightly less than *K*_{s} − *K*(*ψ*_{0}) because of the effects of entrapped air [*Bouwer*, 1966]. The seepage loss is calculated for a discrete time period between times *t*_{w,j−1} and *t*_{w,j}, Δ*t*_{w}, according to

where

[24] is the volume of infiltrated water per unit area of the unsaturated flow column during the time period Δ*t*_{w}.

#### 2.3. Coupled Streamflow Routing and Seepage Loss

[25] Equation (11) is coupled to equation (2) by relating the cumulative infiltration, *I,* to the seepage term, *q*, from equation (2):

where *q*_{c} is the seepage loss for a single inundated unsaturated flow column. The total seepage loss, *q*, for a stream section is equal to the sum of the individual seepage losses, *q*_{c}, for all inundated unsaturated flow columns.

#### 2.4. Channel Drying During Streamflow Cessation

[26] After cession of streamflow, water drains from the streambed by a process often referred to as internal drainage. Internal drainage determines the sorptivity during reinitiation of flow (equation 8). Internal drainage was simulated using a simplified approximation of the process that assumes similarity of the infiltration front (i.e., the infiltration front remains nearly rectangular). Similarity is expressed by the solution of the absorption equation [*Smith et al.*, 1993]. However, greater than a few meters below the streambed surface, gravity results in internal drainage that does not produce similarity in shape of the water content profile. Because the model is used to represent the shallow streambed sediment only, absorption is dominant in the model domain where streambed redistribution is calculated.

[27] The model used to simulate internal drainage is based upon mass conservation. It assumes a nearly rectangular infiltration front that decays over the profile equally as a function of time following the cessation of flow. This approach has been used by several researchers in the past. *Gardner et al.* [1970], *Dagan and Bresler* [1983], *Milly* [1986], and *Smith et al.* [1993] provide a more complete description of the approach. *Smith et al.* [1993] compared the simplified redistribution model to results from a numerical solution of Richards' equation and found the results to be within 10% for a variety of cumulative infiltration values and soil types; however, this error increases with the number of redistribution cycles.

[28] During a streamflow hiatus following a period of flow, the sediment water content in the streambed decays according to the following equation [*Smith et al.*, 1993]:

where

- θ
_{0} is the streambed water content during the streamflow hiatus,

*t*_{d}is the time since the previous flow event ceased (T),

- θ
_{i} is the initial water content prior to the streamflow event,

*K*_{i}is the unsaturated hydraulic conductivity for water content θ_{i} (L/T),

*K*_{0}is the unsaturated hydraulic conductivity for water content θ_{0} (L/T),

*I*_{z}is the cumulative infiltrated water during the hiatus, (L),

*β*is a scaling constant to account for deviations in the shape of the infiltration front—a value of 1 represents a rectangular front; a value of 0.9 is typical and was used in this study,

*p*is the fraction of the infiltration front depth that is elongating, (2 was used in this study to represent half of the front elongating), and

*G*(θ_{i}, θ_{0})is the capillary drive at the front of the infiltration front, and is calculated directly from sorptivity (L).

Equation (13) was solved using an adaptive time step size Runge-Kutta-Fehlberg method [*Patel*, 1994].

[29] Seepage loss in the stream section is calculated for separate compartments in the direction perpendicular to the channel to consider variable hydraulic properties and sediment saturations in the thalweg, stream banks, and floodplain (Figure 2). The benefit of dividing the channel into separate columns is that during continuous low-flow conditions seepage losses can occur because of gravity potential gradients, and if flooding occurs, capillary pressure gradients in the stream bank and floodplain can be considered. Internal drainage also can occur within the riverbank while infiltration occurs beneath the thalweg (Figure 2).

[30] The modeling procedure for estimating seepage losses and streambed hydraulic conductivity consists of assigning reasonable estimates or measured values for all model parameters, and then adjusting *K*_{s} until the simulated SFV matches the measured values for a particular reach of the stream (a stream reach can include multiple stream sections). The process is then repeated for each downstream reach. The equations presented herein were added to the Streamflow Routing (SFR2) package [*Niswonger and Prudic*, 2005] of MODFLOW-2005 [*Harbaugh*, 2005] as an additional option for simulating routing down channels and infiltration through initially dry channels. The program can be attained through the U.S. Geological Survey's public Web page http://water.usgs.gov/software/ground_water.html.