| Cover | Sep. 11, 2023 |
| Cover [Abstract] | |
| Document Type | 8-K |
| Amendment Flag | false |
| Document Period End Date | Sep. 11, 2023 |
| Entity File Number | 811-22528 |
| Entity Registrant Name | First Trust Energy Infrastructure Fund |
| Entity Central Index Key | 0001513789 |
| Entity Tax Identification Number | 45-2751056 |
| Entity Incorporation, State or Country Code | MA |
| Entity Address, Address Line One | 120 East Liberty Drive |
| Entity Address, Address Line Two | Suite 400 |
| Entity Address, City or Town | Wheaton |
| Entity Address, State or Province | IL |
| Entity Address, Postal Zip Code | 60187 |
| City Area Code | 630 |
| Local Phone Number | 765-8000 |
| Written Communications | false |
| Soliciting Material | false |
| Pre-commencement Tender Offer | false |
| Pre-commencement Issuer Tender Offer | false |
| Entity Emerging Growth Company | false |
