(Solved): A farmer has a row irrigation system as shown in cross-section below. The soil is bounded on both ...

A farmer has a row irrigation system as shown in cross-section below. The soil is bounded on both sides by drainage ditches which allow water to be removed from the soil through lateral groundwater flow. The water in the drainage ditches is maintained at a depth $$H_0$$. The irrigation rate $$I$$ is applied at the top of the soil profile. The water which is not evaporated via transpiration by the crops recharges ( at a rate $$R$$ ) the unconfined aquifer below (which is bounded by a horizontal impermeable layer at depth $$D$$ ). The crops being grown by the farmer have a rooting depth of $$d$$ and have an average evapotranspiration rate $$E$$. The farmer must decide the irrigation rate such that the rootzone remains unsaturated (i.e. the water table remains below the rooting depth throughout the entire length $$L$$ ). The soil has a saturated hydraulic conductivity $$K$$. Assuming steady-state conditions with no change in storage in the unsaturated zone: a) Write an expression relating the irrigation, evapotranspiration, and recharge rates. b) Determine the expression for the water table profile beneath the crops as a function of $$x$$ (and in terms of $$R,K,H_0$$, and $$L$$ ). c) Based on your answers to parts a) and b), write the expression for the maximum irrigation rate the farmer can apply to maintain unsaturated conditions in the rootzone. d) Assuming $$H_0=1.0\mathrm{m},L=6\mathrm{m},D=4\mathrm{m},\mathrm{d}=1\mathrm{m},K=3.0{\mathrm{cm}}^{?1}{\mathrm{y}}^{?1}$$, and $$E=8\mathrm{mm}\mathrm{day}$$, what is this irrigation rate in $${\mathrm{mm}}^{?1}{\mathrm{day}}^{?1}$$ ?