MADRID, Spain -- Rafael Nadal began his bid for a sixth Madrid Open title in simple fashion on Wednesday, easing past youngster Carlos Alcaraz 6-1, 6-2 to reach the last 16.

Nadal, who returned to number two in the world last week after his thrilling win over Stefanos Tsitsipas in the Barcelona final, strolled past his 18-year-old opponent in the Spanish capital.

Ranked 120 in the world, Alcaraz walked out to fans singing 'happy birthday' -- and later received a cake -- as he celebrated his birthday playing against his childhood idol.

Alcaraz battled bravely and even broke his compatriot Nadal's serve in the second set but was predictably overwhelmed by one of the game's greats.

"The victory in Barcelona helped me a lot. Things have got better since," said Nadal, reserving special praise for his vanquished opponent.

"He has a lot of potential. He's young and a good guy. He already has a great level of tennis today, but I really believe that he's going to be a fantastic player in the near future."

"I wish him all the very best," added Nadal. "(As a) Spanish player and Spanish (tennis) fan, I really believe that we need somebody like him, and it's great to have him here."

Nadal will take on Alexei Popyrin in the next round after the Australian knocked out rising star Jannik Sinner 7-6 (7/5), 6-2.

Tsitsipas, the 2019 runner-up, demolished Benoit Paire 6-1, 6-2 in less than an hour, with the Frenchman hampered by an abdominal injury.

The Greek fourth seed picked up his joint ATP-leading 27th win of the season and will meet Norway's Casper Ruud for a quarter-final berth.

Earlier second seed Daniil Medvedev, returning to the circuit after testing positive for Covid-19 in mid-April, picked up his first win on clay for two years when he came back from a set down to beat Alejandro Davidovich Fokina 4-6, 6-4, 6-2.

The Russian has always struggled on clay.

In four attempts, he has failed to get past the first round of the French Open. Nor has he won a match in Rome while this was his first win in Madrid in his third appearance.

"I love clay," he said ironically into the camera at the end of the match.

In total, the 2021 Australian Open finalist has only 11 wins on the surface although he will have the chance to make that 12 when he takes on Cristian Garin in the last 16.

Fifth seed Alexander Zverev swept past Kei Nishikori in straight sets 6-3, 6-2.

World number nine Diego Schwartzman was dumped out by Aslan Karatsev 2-6, 6-4, 6-1. In-form Russian Karatsev will take on Alexander Bublik in the next round.

- Barty faces surprise package Badosa -

In the women's draw Ashleigh Barty will take on Paula Badosa in the semi-finals after beating Petra Kvitova 6-1, 3-6, 6-3 on Wednesday.

World number one Barty is chasing her fourth singles title of the year after her triumph in Stuttgart 10 days ago.

In the last four she faces the woman who knocked her out in the quarter-finals in Charleston in early April.

On Wednesday, the Australian came out on top against two-time Wimbledon champion Kvitova, the winner in Madrid tournament in 2011, 2015 and 2018.

The match appeared settled when Barty surged 3-0 ahead in the deciding set.

Kvitova broke back in game five to close to 3-2. But, after saving two break points, she lost her serve for a second time in the set in game six.

Barty then saved three break points on her way to going 5-2 up and held again two games later to see out the match.

Wildcard Badosa became the first Spanish woman in the tournament's 12-year history to make the semis with her 6-4, 7-5 win over eighth seed Belinda Bencic.

The 23-year-old's win over the world number 11 was her third against a player in the top 20 following her victories over Bencic and Barty in her run to the semi-finals in Charleston.

Badosa kissed the clay court in celebration of a win that will shoot the world number 62 up the WTA rankings.

Anastasia Pavlyuchenkova defeated Karolina Muchova 7-6 (7/4), 7-6 (7/2) and will play Aryna Sabalenka for a spot in the final.

Fifth seed Sabalenka advanced when her doubles partner Elise Mertens retired from their quarter-final with a thigh injury while trailing 6-1, 4-0.

© Agence France-Presse