We observed the L1506 filament, which is located in the Taurus molecular complex, with the Herschel PACS and SPIRE instruments. Our aim is to prove the variation in grain properties along the entire length of the filament. In particular, we want to determine above which gas density this variation arises and what changes in the grain optical properties/size distribution are required. We use the 3D radiative transfer code CRT, coupled to the dust emission and extinction code DustEM, to model the emission and extinction of the dense filament. We test a range of optical properties and size distributions for the grains: dust of the diffuse interstellar medium (interstellar PAHs and amorphous carbons and silicates) and both compact and fluffy aggregates. We find that the grain opacity has to increase across the filament to fit simultaneously the near-IR extinction and Herschel emission profiles of L1506. We interpret this change to be a consequence of the coagulation of dust grains to form fluffy aggregates. Grains similar to those in the diffuse medium have to be present in the outer layers of the cloud, whereas aggregates must prevail above gas densities of a few 1000 H/cm3. This corresponds to line-of-sights with visual extinction in the V band of the order of 2 to 3. The dust opacity at 250 microns is raised by a factor of 1.8 to 2.2, while the grain average size is increased by a factor of 5. These exact numbers depend naturally on the dust model chosen to fit the data. Our findings agree with the constraints given by the study of the gas molecular lines. Using a simple approach, we show that the aggregates may have time to form inside the filament within the cloud lifetime. Our model also characterises the density structure of the filament, showing that the filament width is not constant along L1506 but instead varies by a factor of the order of 4.
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