Minor:
Ok, it will be a dry well. Forget the words French Drain. Agree, use perforated pipe. However, that sock is not needed, costs mre and is likely to plug. Water runs out and doesn't run in. It is water flow that carries along dirt and the direction is out.
The main info you need to provide is what type of soil is this going into? If it is clay soil, it may not work well or at all. This design should be the same as one would have for disposing of sewage water from a septic tank. If you really want it to work and need to figure sizes of dry well, check with a local Soil Tester, Sanitarian or similar profession who size waste water drain fields.
A crude way to figure size is to run an old fashioned infiltration test and do some calculations. Dig a hole to the depth you want (and the shallow depth you show looks not enuff) and set a ring, say a 5 gallon pail with the bottom cut out. Fill water inside it and outside it to same level always. (outside so your inside infiltration is only down, not sideways). Fill maybe 8 or 10 inches high. Let it soak for a few hours and keep inside and outside filled. Then (assuming a steady state), measure the drop in inches per hour. That can be done with a pointed measuring stick, against a board across the top. Then, that is a rough estimate of how well the dry well will work on a cubic foot per square foot of area for a given period of time, say one day. You might assume a rainfall of 3 or 4 inches per day for the extreme. An hourly rainfall of one inch could be used to figure volume of trench needed as a minimum. Don't forget the stone will take up maybe 65 percent or more of the total volume. To cut down on the length of trench, just make it wider. Play it safe and daylight the end of the pipe, even it it is up. Put a screen there to keep animals out.
However, keep in mind about freeing of ground. If the surface is not snow covered, frost depth can go well below 4 feet down in sand and gravel and a few feet in fat clay. If water table is close to surface, then this dry well method won't work very well.