When it rains (and we get lots of it) we have up to 4 inches of standing water for several days in the back third of our back garden--it seems to be lower lying than the rest of our garden. None of our surrounding neighbours have standing water so the water is probably running from their yards into our lower lying yard. We've had some very expensive suggestions involving drainage pipes and sump pumps. Do you have any less expensive solutions? Also, we apparently have about 3 yards of clay below the surface in this area.
How Do I Get Rid of Water in a Low Lying Garden?
Certified GKH Gardening Expert
It sounds like you may have a clay layer of soil under the topsoil. Heavy clay can act almost like a layer of plastic below the topsoil, not allowing any water to trickle down through the layers.
Unfortunately, there are no really easy ways to correct this. One common practice is to build up the yard, filling in the spots some with soil and reseeding. French drains are another common solution, but for the amount of water your yard holds, I would talk to a contractor about this as he/she would be better qualified to determine that.
Building a berm is another alternative to help redirect drainage in the area away from your backyard. These articles will have more information on this: https://www.gardeningknowhow.com/special/spaces/creating-and-using-berms-in-the-landscape.htm
Your only other option is to learn to garden with it. There is a type of gardening style, called rain gardens, where people actually work to create conditions like what you have in your yard. It uses plants and trees that not only do well in those conditions but also help to suck up excess water. You may be able to turn your yard into a rain garden. Here is more information: http://www.uri.edu/ce/healthylandscapes/raingarden.htm