The burial rate is the thickness of sediments given in meter and being laid down on a rock sequence of whatever type or any geological feature (e.g. fault zone). It is the consequence of a sedimentation and subsidence process. The burial rate can be measured at outcrop, in drill holes or by any geophysical method providing depth-related information, e.g. seismics.
If you want to get a gradient or a rate of burial sensu stricto you have to measure the thickness as a function of time in the course of subsidence. There may also be "negative" values in such a depth-time graph when uplift and erosion take place.
The best examples for this issue you may find in publications released by the petroleum industry.
if you want to know the maximum burial depth you can measure the overconsolidation ratio (OCR) with an odometer test. Perhaps it is useful to measure the OCR along a profile to get information about the burial history. There are several publications dealing with mudrock on this topic e.g.:
Mohajerani, M.; Delage, P.; Monfared, M.; Tang, A. M.; Sulem, J.; Gatmiri, B.: Oedometric compression and swelling behaviour of the Callovo-Oxfordian argillite
Dao, L. Q.; Cui, Y. J.; Tang, A. M.; Delage, P.; Li, X. L.; Sillen, X.: Anisotropy in Oedometer Test on Natural Boom Clay
There are several standards about this topic, too. But they deal with soil and not with rock e.g.:
DIN 18135:2012-04: Baugrund - Untersuchung von Bodenproben - Eindimensionaler Kompressionsversuch (Soil - Investigation and testing - Oedometer consolidation test) (in German)
There are also the possibilities to measure the conodont alteration index (CAI) and the quantification of the transition during diagenesis from smectite to illite. The CAI measures the maximum temperature which was experienced by the sediment. But you can recalculate the pressure with the knowledge of the geothermal gradient.
The sediment burial rate can be estimated for the sediment transported by any drainage basin by hydrographic survey method for example , Indus river at Tarbela reservoir it is 1 percent of total reservoir capacity.
by accident I found the following paper related to your question:
John C. Tipper (2016): Measured rates of sedimentation: What exactly are we estimating, and why?; Sedimentary Geology Volume 339, 15 June 2016, Pages 151–171