When creating and ADO.NET Entity Connection String you get something like
You don't need to create instance of ObjectContext. Because in that case you must reference assemblies where metadata are stored and for performance is not also very good.
There is simpler way to get provider connection string in StoreConnection property.
using (var ec = new EntityConnection(connstr.ConnectionString))
{
var sqlConn = ec.StoreConnection as SqlConnection;
sqlConn.Open();
}