I do not know what argillaceous rocks you are dealing with. Is it in a lacustrine facies sequence or a marine one? Is it an ephemeral or perennial lake? Many questions that have to be answered prior to give you a. tailormade answer. So, my answer at this level can only be rather general. Even if you might not be experienced in the study of microfossils you should have a keen eye on them and try and get in touch with a micro-paleontologist.
As far as the inorganic material is concerned there are two options left. You can study the clay minerals and find out whether there is a sequence like that arranged in order of distance from the shoreline: kaolinite group - illite/mica-chlorite-smectite-hormites/ palygorskite (only in strong evaporitic environments).
The second string to the bow is the heavy mineral assemblage which tends to show up with strongly stable HM (zircon, tourmaline, rutile) far off the coast whereas more labile ones, e.g., amphibole, pyroxene, or olivine that are restricted to the marginal facies of the terrestrial basins. In both pathways there some temporal and regional side effects which must not be ignores.
This is a rather general statement of mine and I would like to express a stark warning, too; in case of the usage of inorganic parameters a meticulous approach based upon knowledge and experience has to be taken.
If the succession you are dealing with is a carbonate, you can recognize the muddy texture based on the size of the mud crystals which is coarser in proximal areas because the proximal deposits are more susceptible to diagenetic alteration. Furthermore, you can compare the mudstone faces in proximal and distal areas according to the amount and presence of iron oxide and organic matter; based on the depth of water in the distal area and the latitude we can have very small euhedral dolomite rhombs. In addition, some fabrics like lamination due to the microbial activities and fenestral or bird-eyes fabric can be expected in the proximal mudstone.
I will suggest that it is impossible to determine this purely from thin section. Some idea of the stratigraphic context is needed to make an interpretation with reasonable certainty. Interbedded rocks and associated surfaces will provide better paleoenvironmental evidence.
Thank you Harald G. Dill and Umid Kakemem for your suggestions. These samples are in a Permian carbonate succession with various regressive and transgressive events. I grouped the mudstone samples into two groups based on presence of bioturbation, fossils, OM, and whether or not they look recrystallized (interestingly enough my "most" likely shallower facies are all recrystallized).
I refresh my suggestions with a slight modification. The marginal aspect with an increased detrital input will still exist in this sediment. In this case I would place also emphasis on the geochemistry of trace elements like zirconium (Zr) which is a good indicator of an increase in the detrital input. I used it very often successfully. On the opposite end and typical of a basinward facies is boron (B). While B is not part of routine XRF programs, Zr does so.
A good marker is the V/Cr ratio which increases with an increase of the reducing regime. Less sensitive but also helpful is the Ni/Co ratio. If you draw chemologs you will certainly get a chance to correlate them based on these elements and you can directly take a decision which way the detrital input will increase and find an antithetic correlation with the redox regime based on the V/Cr ratio that response to the bituminous organic matter.