Your question was clear at the get-go. The place to start is determining how much heat your mom needs. I don't know much about oil furnaces, but you need to quantify her heat needs, in btus or a modern equivalent. Preferably, total for a normal winter.
Then you get to determining how many btus you can realistically retrieve from your cisterns. One way to bump that up would be to increase your storage temperature. Several methods that would work. Make it very high, and you really need to insulate your cisterns. Heat storage in dirt around 70º works well, a lot higher does not.
Don't worry about a thin plastic tank reducing your heat transfer from the earth. The formula for conduction is q = A (k/t) ∆T, where A is area, k is thermal conductivity, t is thickness, and ∆T is the temperature differential. A very thin t means a low thermal conductivity isn't very important. The major factors are A and ∆T. Get them both large enough and you've got something interesting.
I design earth heat storage for Passive Annual Heat Storage (PAHS) where we need to determine how much heat is put into the storage, and how fast it can be retrieved. And no, it is not a system that readily retrofits an old house.
Your primary problem will be finding a ground source heat pump company able and willing to do the calculations you need.
The short answer: yes it can work, but the devil's in the details.
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Today's Featured Article - Old Time Threshing - by Anthony West. A lovely harvest evening late September 1947, I was a school boy, like all school boys I loved harvest time. The golden corn ripens well and early, the stoking, stacking,.... the drawing in with the tractors and trailers and a few buck rakes thrown in, and possibly a heavy horse. It would be a great day for the collies and the terrier dogs, rats and mice would be at the bottom of the stacks so the dogs, would have a busy time hunting and killing, all the corn was gathered and ricked in what we c
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