sw[h_] := Piecewise[{{1, h >= 0}, {E^(0.01*h), h < 0}}]

krw[h_] := Piecewise[{{1, h >= 0}, {E^(0.01*h), h < 0}}]

sol = NDSolveValue[{sw'[h[t, y]]*D[h[t, y], {t, 1}] - D[krw[h[t, y]]*D[h[t, y], {y, 1}], {y, 1}] == NeumannValue[1, y == 0] + NeumannValue[0, y == 1], h[0, y] == 0}, h, {y, 0, 1}, {t, 0, 10}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement"}}];