Ostapenko and Kichenoka, who are seeded fifth in the tournament, lost 7-6 (7-0), 2-6, 7:10 to the first-seeded Suwei Xie from Taiwan and Elise Mertens from Belgium.
Ostapenko earned 215 WTA doubles ranking points for reaching the quarterfinals.
In the first round match, Ostapenko and Kichenok beat Estonian Ingrid Neela and Czech Linda Noskova 6-1, 7-5, and in the second they beat Czech Marketa Vondrousova and British tennis player Katie Boulter 6-3, 7-6 (8:6).
In the singles competition, Ostapenko lost in the round of 16.
The tournament in Madrid takes place on clay courts.
