http://www.treehugger.com/clean-water/6-grasses-for-low-maintenance-drought-resistant-lawns.html

Zoysia or St. Augustine

Growing grass under an oak tree is no problem in San Antonio, but there is no seed that will do the job. You must be new in town, so welcome to the south.

St Augustine is the only grass that will do well under your shade. If you look, you already have it. It's the golden tan stuff in the deepest shade. If you start watering NOW, it might come back for you. Once it does it will take over completely. St Aug stretches out about 15 feet per year in the spring and the fall when the temps are in the 70s. If you are patient it will cost you nothing to have a full St Augustine lawn. If you are impatient, you can put St Augustine sod down now and have an instant lawn. Milburgers Nursery inside Loop 1604 on Bulverde Road is the place to get sod. In the full sun the best variety is Floratam. In the shade, you have to go by their availability and recommendation.

I would begin the watering routine with a soaker hose. Wrap the soaker around the tree so you get into the center of the canopy and grass area. Connect a regular hose to the soaker and but not to the faucet yet. Turn the faucet on to a trickle of 1 cup per minute. Seriously. Put a cup under the faucet and adjust the drop until it's about 1 cup per minute. Then connect the soaker hose hose. Leave that running day and night for as long as you want, but at least 5 days. The top of the soaker hose will not get wet, but the soil will. Within a few days you should see some green coming back from the dormant St Augustine. If you don't then you're going to need some sod. I had some golden tan St Aug under an oak tree like that last summer, and it's all green now, so I'm hoping yours isn't all the way gone.

Zoysia is a choice, but the only one that does well in the shade is not available any more. Zoysia for the sunny part is also a choice, but it comes with a lot of flaws that seem to drive people back to St Augustine.

If you insist on seed, bermuda is the only seeded grass that will survive in San Antonio. Unfortunately you have too much shade under the tree for that, but it will thrive out in the sunny areas. The best bermuda is the hybrid TIF 419. It only comes as sod. The rest of the seeded varieties are all common bermuda and do not have the same behavior as TIF 419. Milbergers will have TIF, too.