where K is the saturated hydraulic conductivity (mm h-1), 0s is the saturated-soil water content (m3 m-3), 0t is the soil initial water content (m3 m-3), h' is the matric potential at the wetting front (mm), and Z is the cumulative depth of infiltration (mm).
However, the equation that more precisely describes the flow in porous media is the Richards equation. For border or basin irrigation, the one-dimensional form is appropriate, whereas, for furrow irrigation, the two-dimensional form would be required:
where C(h) is 30/dh, 9 is soil water content, h(9) is the pressure head, K(h) is the hydraulic conductivity, t is time, x is the horizontal distance, and zis the vertical distance from soil surface (positive downward).
The use of Eq. (5.133) in the continuity equation (5.126) not only increases the complexity of the solution of the flow equations but also requires much more detailed and accurate information on the hydraulic soil properties. However, the information provided by the corresponding model would be more detailed and, hopefully, better represent the dynamics of the irrigation process [21-23].
A great deal of effort has been put forth to develop numerical solutions for both the continuity and momentum equations (5.126) and (5.127) and the Richards and Green-Ampt infiltration equations (5.132) and (5.133). The current approaches to solutions of Eqs. (5.126) and (5.127) are the method of characteristics, converting these equations into ordinary differential ones; the Eulerian integration, based on the concept of a deforming control volume made of individual deforming cells; (c) the zero-inertia approach, assuming that the inertial and acceleration terms in the momentum equation (5.127) are negligible in most cases of surface irrigation; and the kinematic-wave approach, which assumes that a unique relation exists to describe the Q = f (y) relationship. A consolidated review and description of those solutions, using the Kostiakov equations (5.128) or (5.129) for the infiltration process, are given by Walker and Skogerboe .
These solutions are incorporated in the computer programs SIRMOD  and SRFR [25, 26]. SRFR solves the nonlinear algebraic equations adopting time-space cells with variable time and space steps and also includes a full hydrodynamic model adopting the Kostiakov infiltration equation (5.129). The model adapts particularly well to describe level furrows and basins as well as the impacts of geometry of furrows on irrigation performance. With the same origin , a menu-driven program, BASIN, for design of level basins has been developed . These user-friendly programs correspond to the present trends in software development, which make complete design tools available to users.
These computer models can be used for most cases in irrigation practice, for both design and evaluation. Nevertheless, there are many other developments in modeling recently reported in literature, mainly relative to improvements in zero-inertia and kinematic wave models (comments in ).
For many problems in the irrigation practice, the simple volume balance equation  can be appropriate for sloping furrows and borders:
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