{ FIXED [Well ID] : MIN( if [production amount] > 0 then [Date] END) }

That will give you the min date where there is actual production.

You'll have to translate that to your own actual field names, of course.

And you might want to consider what to do with a negative number for output.  Even though the business rules say it can't happen, you can't count on data following the rules.  You might want to do <> (not equals) zero rather than greater than zero to catch that.

If you can't use LOD, then you'll use a table calc.

WINDOW_MIN( if SUM([production]) > 0 then ATTR([DATE]) END )

But that will put the MIN value on all dates for a given Well.  And you'll have to have all dates on the sheet for the table calc to evaluate across them.

Given the layout of your first screen shot, table(across) for the table calc setting would to the trick, but you might need to specify dimensions and a restart dimension to evaluate it from well to well.

Do you have a sheet layout in your example workbook where you want this displayed, and where would you like it displayed?  (I haven't opened the workbook yet.)

See attached.

Addendum.

If you want to display each well without displaying the individual dates, then see the attached.

In copy 2 I dragged the date field to details.  Now you see that you get x-many marks per row, even though you're not displaying any value.  That's because the date pill forces marks.  And you need to have the date on the sheet like this for the table calc to evaluate each day's production.  So right click on the [First Production Date] pill on ROWS and select Edit Table Calc.  Here you can see that I selected individual dimensions, and told it to evaluate for each well along the date dimension (in that order) and to restart for each well.  Without this setting tableau was till doing table across, and each "across" is just each individual date for each well.

So I still have 10 dots.  Go to Sheet 3.

Here I created a calc called INDEX and put that on filters, and set the table calc setting the same as the date calc.  Once it was set that way, I selected for index=1 and now we just see the first mark for each well.  (Since the table calc value is the same for all marks within a well, we only need to see one, and the first one is as good as any.)  If you look at the tooltips for each mark, you'll see that the actual date being displayed for each mark is Nov 10.

See attached.