It depends on what type of

It depends on what type of landscape you have.  If it's grassy where the seams are you can just fill in the foam with ground foam foliage.  Or, you can spackle the seams with light weight spackle.  I've used both methods with success.

and if the gaps are large

one can get  foam in a spray can.

Controlling spray can Foam

This can be tricky and some what difficult.  Recommend you do a test run before an attempt for real.  I have enjoyed seeing the results of filling a house car with spray foam, not good...

Non-.expandable foam..

Gorilla Glue

Gorilla glue can also be used to fill gaps and attach foam.   Is is a bit "stiffer" than the extruded foam, but can be sanded, carved, and even cut with a hot knife.  It is what I used when my foam mounted track developed a hump in it. I used a grit saw to carve underneath the roadbed and allow the track to sag.  Then I squirted a line of Gorilla glue into the gap on each side and placed a straightedge on top of the track with weights of top.   The glue foamed out the sides and made a very solid (and flat) repair that could then be shaped to blend with the surroundings.

Toni

I'd cut...

I would cut it so there is no seam.  The 12" section you will use to add elevation somewhere.  You stated the extrusion is $9.00 a sheet.  You will spend more than that in time and materials trying to get rid of the seam.  If the sheet has any warp to it,you will have to be super careful to glue it down to the supporting material you have underneath.  It just seems like a pretty big waste of time to try to save that small piece which you will probably use later to form some type of land feature anyway.  My two cents...

-bill

Thanks

Ok, I decided to cut it into 4'x18" sections to make it easier with no seams. Anyway to prevent the croners from being chipped off or edges being scraped up?

Blue Foam

What are you putting the 1/2" blue foam over?   2" thick blue or pink foam is sufficiently rigid that you can put it right on top of an open grid as your sub-roadbed, and avoid using MDF, plywood, or a hollow core door altogether, and you can make a grade about 4' long with just a couple of pieces of scrap foam pushed underneath for intermediate supports.  Its only drawback is that you can't easily attach stuff (switch machines, etc.) on the underside of your layout. 1" foam is fairly rigid, but needs more support, and 1/2" foam is pretty flexible, so that it needs something else underneath it continuously.    I've used all three thicknesses separately and together for sub-roadbed and for building up rough forms to cover with plaster cloth and/or sculpta-mold for hills and valleys. 

Im placing the foam over 3/8"

The idea is to get the two

The idea is to get the two mating surfaces as close to the same plane as possible.  However, you will have some variance, and their edges may not be square, so there will be an obvious gap between them.  What I do is to make some attempt to plug the gap so that whatever you use over the gap doesn't get wasted down where the sun don't shine....filling the gap.  For example, using ground foam for that can get rather pricey.

I would place a bead of DAP acrylic latex caulk, or anything similar that you have handy and open already, and draw a wet finger across the top of the bead.  Let it cure.  After that, you can make it look really good, if flat and somewhat unrealistic, by sprinkling sifted dirt or a suitable terrain material, even zip texturing, over the entire area.  The gap will disappear and you can then cover the sprinkled material with ground foam.  But I did say it will look unrealistically flat.

So, you can place shallow blobs of something, more sifted dirt, here and there, or build a bit of a berm, say, and cover that with ground foam and brushes.

Crandell

Tape

I used the two-inch thick blue styrofoam sheets on a previous layout. To hide the seams between foam sheets or layout sections, I used two-inch wide masking tape to cover the joint. Make sure the foam is pretty clean of dust and dirt or the tape won't stick well. The tape can be painted the same as the foam.

I used a traditional grid for benchwork and knew the foam would not stay put using just caulk. I cut the foam to fit against the backdrop, which was easy in most places where it was straight. Any overhanging foam was cut even with the front of the benchwork. The fascia was then installed to keep the foam in place with the top of the fascia even with the top of the foam. The second image down on this webpage may give you an idea of that technique.

Eric

Feathered Gorilla or duct tape

I finally started the benchwork for my shelf switcher and have some seams between foam sheets / sections.  I'm going to try Gorilla or duct tape and then cover that with ground foam, etc.

To break up the straight edges, I'm going to randomly scallop or feather the tape with scissors or a box cutter.  Might have to put the tape on a sheet of acrylic to work with it.

If that doesn't work, it's back to acrylic caulk - should be flexible enough not to crack if the sections flex.

Thanks. Lots of ideas to

Thanks. Lots of ideas to work.

If you have a seam use Some

Durham's Water Putty from your local big box store.  It mixes up easily and then use your trusty putty knife (I have one inch, three inch and six inch wide blade)  

I also use Durham's Water Putty in my ground goop, along with appropriate colored earth from the model site; gray latex paint (yes the area earth is a sun bleached gray hue) and enough water to get the consistency needed for the ground cover of the site.  I have most of my track laid down the dirt roads of 1895 Colorado Springs.

Use a piece of adhesive

Use a piece of adhesive drywall. tape over seam, then cover with any of the materials above. I use fusionfiber, seeing how we make it. ) Its flexible enough so if there is movement it will not chip.

Have done a number of portable modules for clubs with a scenery flap that hides the seam between portable modules.

.