Pool heat can be produced by different sources, i.e. solar, gas/LP heater, and heat pump. What kind of heater you need or want really depends on what your expectations are. I have no experience with solar, and in my region it's really not an option for me.
As far as the size of heater you need. That depends on how fast you want to raise the water temperature, and what is the delta of the temp change (i.e. 70 degrees to 80 degrees for a 10 degree difference, or 65-85 for a 20 degree difference).
It takes 1 BTU to raise 1 pound of water 1 degree. So, 1 BTU will raise 8.33 gallons of water 1 degree. Say you want to increase the temperature 20 degrees. That's 8.33 x 20 =166.6 BTU per gallon x 13,500 gallons = 2,249,100 BTUs. With a 100,000 BTU heater, it would take about 22 hours to get that 20 degree rise.
Depending on the weather, you can lose several hundered thousand BTU per day. A 100,000 BTU heater will burn ~1 therm of gas per hour. Running a natural gas heater 8 or so hours a day plus the initial start up could run 300 hours or 300 therms of gas. Average is $1 per therm so $300 a month depending on the weather?
A heat pump can maintain that same 80 degree water much for about $175 a month assuming 12 cents per kWhr national avg.
A few questions about how you use your pool. Do you use want to use it mostly on weekends or will it be used during theweek as well? Do you want to maintain to pool temp at say 80 degrees, or if only part time use, would you want to lower the temp to closer to ambient, and only raise the temp when you want to use it? Also, how long do you want to extend your swim season? Early May to late september could be possible with a heat pump without much issue.
As posted above, a 250,000 BTU heater would be better.
