For decades, the giant Groningen gas field beneath the flat, green farmland in the north of this country counted among the greatest prizes for Exxon Mobil Corp. and Royal Dutch Shell PLC. Then the earthquakes started. The exploitation of Groningen—the biggest gas field in Europe—has been causing tremors for over two decades, rattling a bucolic province with no previous history of quakes and exposing two of the world’s biggest energy companies to a criminal probe and rising reconstruction bills. Amid a public outcry, the Dutch government has imposed increasingly strict limits that have more than halved Groningen’s gas production since 2013. Now, authorities are proposing another 10% cut in hopes of further reducing earthquakes. And a Dutch public prosecutor is preparing to open a criminal investigation into responsibility for the earthquakes.