All the examples in the NHibernate documentation for Oracle have the following setting for the driver_class:

<property name="connection.driver_class">
  NHibernate.Driver.OracleClientDriver
</property>

This will use the Microsoft Oracle client. The documentation states that you can also use the client from Oracle (ODP.NET), but it does not state how you can achieve this.

Apparently you have to use the NHibernate.Driver.OracleDataClientDriver as driver_class in your NHibernate config file:

<property name="connection.driver_class">
  NHibernate.Driver.OracleDataClientDriver
</property>