I found a pair of waterproof pants abandoned on the trail, they were fine except for several large rips that the previous owner had tried to fix with ductape.
How can I patch these rips?
Duct tape works as a temporary repair.
A: remove the duct tape. B: trying various solvents, remove any residual glue. You can try solvents first on the duct tape itself. Solvents to try include alcohols, mineral spirits/varsol, naphtha (camping fuel), gasoline, ether, butane, acetone and various propritary gunk removers. Acetone in particular may disolve the waterproof coating. Experiment on a lower cuff.
Note that most of these are flamable, some explosively so. (Ether, acetone) Do this outside, away from spark and flame. You have been warned.
C: Wash the garment to remove remaining solvent. An extra cycle without soap will reduce soap residues.
D: Rinse the area with isopropyl alchohol 99% and blot dry with paper towels. This will reduce residual grease.
A: Most camping stores carry repair tape. This is rip-stop nylon with an adhesive back. Put a magazine, or slick finish ad in the garment so that any exposed stick bits can't glue the leg together. Apply the tape to the tears.
I have found that ironing it down helps the adhesive stick better.
B: On the internet you will find various recipes for DIY seam sealer made from a solvent and silicon seal. Make up a batch of this, and using a foam paint brush run a tiny bead along the edge of your patch. This reduces the chance of the patch lifting. Turn the garment inside out, and do the same to the inside edges.
C: If the garment is single layer construction (Your pic makes it seem lined...) you may want to apply the patch from the inside. The tear will be more visible from the outside, but the colour of the patch will show less. If the garment is lined. You may also want to patch the lining the same way.
Turn the garment inside out. Sew up the gash. This will shorten the fabric there by about 1/2, so it will have a pucker. Use two passes: A straight stitch about 1/4" from the edge to fasten the tear, then a zig-zag stitch with one side off the edge of the fabric will bind the loose edge.
Apply liquid seam sealer to the inside edge.
If you have access to someone who is good with a sewing machine:
A: Scrounge some fabric similar in weight and colour to the garment. B: Cut patches from the fabric. C: Sew patches in place. D: Seal edges and stitching.
This will give the best looking repair, but will at minimum cost you a nice bottle of wine for the seamstress/tailor
Many dry cleaners/clothing cleaners will do repairs, often for very modest fees. Get them to sew up the tears, and then apply seam sealer as above.
Sew it back together and tape the seams.
Working with Goretex or any other type of waterproof material is the exact same as any other type of fabric. The only difference is that you tape the seams after you're done.
With those rips there you could simply stitch them closed with a broad and tight zigzag stitch, and then seal it with one strip of seam tape on the back. If the rip is pretty shredded you may need to buy a Goretex patch, or cut a patch of some other waterproof material and seal everywhere you sew.
For what it's worth, you could forgo the taping and call it good just stitching them back together, I've done this a number of time with my Goretex gaiters. It won't be perfectly waterproof, but it'll still breathe the same and repel water.