Hi and welcome to GoY ... I'm adding this question to GoYpedia in case anything on the membranes page helps you.
... click on Weed Suppressing Membranes above...

I haven't seen Somerset's question, nor 'Phil's' advice, so I can't comment on that.

A membrane means a geotextile fabric of varying thickness and strength - heavy duty ones may be used under driveways before laying paving to suppress any weed growth, and lighter ones may be used on top of the soil, with bark or shingle spread over the top, again to suppress weed growth. I'm not sure what purpose the membrane suggested in the question you refer to was supposed to serve, not having seen either the question or the answer.

Regarding what you have said about your own problem, if the artificial turf lasted some years before becoming mouldy, its likely that a replacement of the same sort will also last some years. If you have a drainage problem in your back garden (waterlogged most of the year, for instance), then real turf won't do too well either. I can't comment on the best solution, because I don't have enough information about your own situation to make such a comment.

I will be honest with you, get rid of the artificial turf it al ways looks naff, deal with the underlying problem and sought the drainage out, that done you have to keep the dogs off the lawn otherwise it's going to be a waste of time and money, if you have to use the dogs for exercising on the lawn then a rethink of the garden needs to be taken, if your garden is big then you could divide the garden so the dogs have their space ie a shingled area which means you can have a lovely lawn.

Sorry, but I deair of the sector of the population that uses the words lovely and lawn in the same sentence - unless that sentence is "Wouldn't it be lovely not to have top faff around a with a lawn?"

Let's face it there really isn't anything that can be classed as a "lovely" lawn unless it's in the middle of "All England" and "Tennis Club" and the owners are spending 50 weeks of the year preparing the grass so that it looks good for a few days of the other two weeks.

So, basically, unless you have the money and the fleet of groundsmen/women and particularly if you have dogs, then don't waste time, energy and money on a lawn.
(ps before the grass brigade take umbrage, I have no problem with people who want a patch of grass that is allowed to grow and mix with daisies, buttercups, clover etc - it's just the stuff that scalped closer than the average skinhead's skull)

Rant over - I will take it out on the students in their exam tomorrow!