The mathematics of early diagenesis: From worms to waves

The changes that sediments undergo after deposition are collectively known as diagenesis. Diagenesis is not widely recognized as a source for mathematical ideas; however, the myriad processes responsible for these changes lead to a wide variety of mathematical models. In fact, most of the classical models and methods of applied mathematics emerge naturally from quantification of diagenesis. For example, small-scale sediment mixing by bottom-dwelling animals can be described by the diffusion equation; the dissolution of biogenic opal in sediments leads to sets of coupled, nonlinear, ordinary differential equations; and modeling organisms that eat at depth in the sediment and defecate at the surface suggests the one-dimensional wave equation, while the effect of waves on pore waters is governed by the two- or three-dimensional wave equation. Diagenetic modeling, however, is not restricted to classical methods. Diagenetic problems of concern to modern mathematics exist in abundance; these include free-boundary problems that predict the depth of biological mixing or the penetration of O-2 into sediments, algebraic-differential equations that result from the fast-reversible reactions that regulate pH in pore waters, inverse calculations of input functions (histories), and the determination of the optimum choice in a hierarchy of possible diagenetic models. This review highlights and explores these topics with the hope of encouraging further modeling and analysis of diagenetic phenomena.