The first blast occurred at 17:35 local time (13:35 GMT), the second following seven minutes later.
"No casualties have been reported so far," Guram Gubaz said.
An investigation has been launched. Abkhazian police suspect the incident could have been a terrorist attack aimed at Russian railroad troops.
Moscow announced the deployment of around 300 unarmed railroad troops in the self-proclaimed republic on May 31 to repair railroad tracks. The deployment met a furious reaction from Georgia, which accused Moscow of preparing for military intervention.
The Russian Defense Ministry said last week an attempt to carry out a terrorist act on June 13 along a section of the Tamysh-Ochamchira track was aimed against railroad troops in Abkhazia.
Abkhazia broke away from Georgia in the early 1990s following the collapse of the Soviet Union. Between 10,000 and 30,000 people were killed in the subsequent hostilities. The two sides signed a ceasefire in 1994 in Moscow.
Peace talks between Abkhazia and Georgia broke off in July 2006 when Tbilisi sent troops into Abkhazia's Kodori Gorge and established an alternative Abkhaz administration there.