I think this is Worthy of Magic, by Sage Ra, published on FanFiction.net.
Harry discovers he has magic at a young age. He snaps and kills the Dursleys due to his ill-treatment.
In Part 1, "The Void", Harry murders the Dursleys after being punished for beating Dudley in a maths test:
"Hi," he said. "My Aunt and Uncle are dead, if you came looking for them. It felt re~ally good killing them." It was the longest sentence Harry had ever spoken.
He is then sent to a correctional facility, where Dumbledore finds him
and enrols him into Hogwarts.
In Part 2, "Convicted", Harry is at the Janus Home for Criminal Children, and meets Dumbledore for the first time.
Harry had fallen into a comfortable routine at The Janus Home for Criminal Children. [...] Today was a strange day for Harry. His routine had been broken.
Instead of meeting Beverly and giving her the silent treatment, he was
to meet a Professor from a school for the gifted. [...] "You have a
place in my school Harry. Your parents, who were a witch and wizard by
the way, enrolled your name when your were three months old. It will
be your home for seven years, until you're ready to join the magical
world as an adult."
Harry has no friends, except for Hermione who couldn't leave him alone.
In Part 4, "Hogwarts", Harry avoids making friends, but Hermione is insistent.
They tried to befriend him. They tried to talk to him. [...] Learning
about magic was all he cared about. [...] "What do you want?" Harry
asked with disinterest. [...] "To be friends," she said firmly. Harry
blinked. She had been persistent because she wanted to be friends?
"Friends?" Harry echoed strangely. It was an entirely foreign concept
Harry is fascinated by runes. He searches for a rune that can turn
anything to sand. He is drawn towards the desert rune.
Runes seem to be mentioned quite a lot throughout, as well as deserts - for example, in Part 24, "Two Problems", there's specific mention of the desert rune.
This is an unfinished fan fic, maybe even on hiatus.
I glanced through the last one but without reading the whole thing, I couldn't tell you if it counted as an ending or not, so I'll leave that one for you!
It's pretty lengthy, with at least 80k words, without romance/pairings.
There are 57 parts on FanFiction.net, it's now at 147K words. I can't tell if there's any romance, but there are none mentioned in the summary or tags.
Fits into insane Harry/horror/supernatural categories.
Category on FanFiction.net is Horror/Adventure.