Front door leak

Have a leakage problem with my front door. Probably pretty conventional, but I've put a sketch of the bottom of the door at . Inside on the left, outside on the right. When it rains I find water collects between the sill and the bottom of the door at X. There is very little gap between door and sill so it may get there from some sort of capilliary effect from spashes up under the shield near the bottom of the door.
Occasionally the water will leak into the house but usually the water just stays trapped in the gap. Trouble is this has caused the bottom of the door at T to swell up this winter - doubt if there's much in the way of waterproofing under there. I've had to take a millimetre or two off the sill at X to overcome the sticking, but what options do I have to prevent it happening again?
The shield is a little high and could probably be moved down by about 3mm without catching the sill, if that would help, although I'm not sure how solidly it is fixed.
Malcolm (Fantrace).
Be happy.
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