I am located in Molalla Oregon and this weed has grown like crazy in my outdoor riding arena. The ground is hard and dry and mostly sandy-gravel. I do not want it to spread into my horse pasture or paddocks. Help
This is purslane -- Portulaca oleracea -- in its classic late-season (dry, hot weather) look.
There is good, general information on this previous identification answer about purslane (as previously linked by @GardenerJ in comment). Some basic information: it is edible by humans (and, to me, very tasty), it is very easy to pull/weed-out (especially in loose/rocky/sandy soil), it is ubiquitous and difficult to completely eradicate, but it is an annual.
Your question has the added context of horses. Since you have horses, caution is warranted. Though it does not seem to be on Wikipeida's list of plants poiosonous to equines, it is on the ASPCA list as toxic to horses. The cited reason however (calcium oxalate) is dubious to me; this is also in common yellow wood-sorrel and similar common weeds.
Standard-style disclaimer: I'm not an equine expert. You should consider doing more research to determine toxicity. Perhaps consider Pets.SE?