Abstract
Lateral metrescale periodic variations in porosity and composition are found in many dolomite strata. Such variations may embed important information about dolomite formation and transformation. Here we show that these variations could be fossilized chemical waves emerging from stressmediated mineralwater interaction during sediment burial diagenesis. Under the overlying loading, crystals in higher porosity domains are subjected to a higher effective stress, causing pressure solution. The dissolved species diffuse to and precipitate in neighbouring lower porosity domains, further reducing the porosity. This positive feedback leads to lateral porosity and compositional patterning in dolomite. The pattern geometry depends on fluid flow regimes. In a diffusiondominated case, the low and highporosity domains alternate spatially with no directional preference, while, in the presence of an advective flow, this alternation occurs only along the flow direction, propagating like a chemical wave. Our work provides a new perspective for interpreting diagenetic signatures in sedimentary rocks.
Introduction
Owing to their geological importance, dolomite formations have been intensively characterized, both physically and chemically. Recent highresolution sampling (with a sampling interval of 0.3 m) has revealed complex nested patterns of lateral porosity and permeability variations in dolomite strata^{1} (Fig. 1). Three scales of lateral variability are documented, including a nearrandom component at <0.3 m, a shortrange correlation structure of 4.0–6.7 m and longer periodic structures with wavelengths up to 12.2 m (ref. 2). The porosity usually oscillates in a range between 10 and 25%, while the permeability could vary by a factor of ~100 (Supplementary Fig. S1). Trace element contents and isotopic compositions vary in a similar manner^{1} (Supplementary Fig. S1). These oscillations appear to be common in many dolomite strata^{3,4,5,6,7}.
A mechanistic understanding of the origin of such oscillations is of manifold significance. First of all, these oscillations may embed important information about dolomite formation and transformation, and a mechanistic understanding is required to decode the embedded information. It has been suggested that such oscillations might have formed during or after dolomitization^{1,8}. However, the actual mechanism remains unknown. Furthermore, the existence of these variations raises a serious question about the validity of the existing field sampling schemes that tend to use sparse spot samples as proxies for interpreting ancient geologic environments^{1}. A reliable interpretation must account for the lateral heterogeneity in an individual dolomite stratum, based on a clear understanding of the underlying physical and chemical processes. In addition, a mechanistic understanding is also crucial for developing predictive tools for better quantification of spatial heterogeneity in a subsurface reservoir. Such tools are highly desirable for many reservoir engineering applications, as such heterogeneity is known to have a direct impact on pore connectivity, fluid flow patterns and therefore the overall performance of a subsurface reservoir^{4,6}.
In this article, we propose a mechanism that can account for all the main features of lateral physical and chemical variations in dolomite. The proposed mechanism is based on the concept of geochemical selforganization and chemical waves^{9,10}. Using a linear stability analysis, we show that the observed lateral variations could autonomously emerge from stressmediated mineral dissolution and precipitation during sediment burial diagenesis. These variations provide rich information about ancient geologic environments for dolomite formation and transformation. The proposed mechanism is expected to be universal for all sediments. Our work provides a new perspective for interpreting petrophysical and chemical signatures in sedimentary rocks.
Results
Effective stress–porosity feedback
Selforganization requires a positive feedback among physical and chemical processes involved in a system^{11}. To illustrate the concept, let us consider a postdolomitization scenario, in which a layer of dolomite with uniform depositional properties (that is, a uniform facies) and a thickness H undergoes burial diagenesis under a vertical loading σ_{0} (Fig. 2). The change in porosity (φ) can be described by:
where t is time; v_{m} is the molar volume of dolomite [CaMg(CO_{3})_{2}] (cm^{3} mol^{−1}); and R_{d} is the mineral dissolution rate (mol cm^{−3} bulk rock s^{−1}). The first term on the right side represents the porosity reduction due to mechanical compaction. The second term denotes the porosity increase due to mineral dissolution. The dissolution rate R_{d} is assumed to follow the firstorder kinetics:
where k is the reaction rate constant (cm^{3} solution cm^{−2} s^{−1}); S is the specific reactive surface area (cm^{2} cm^{−3} solid); C is the concentration of dissolved dolomite species (mol cm^{−3} solution) (for simplicity, they are here represented by a single equivalent species); and C_{s} is the solubility of dolomite, assumed to be a function of the effective stress σ (bar) to which mineral grains are subjected^{12}; Δv is the molar volume change of dolomite dissolution reaction (cm^{3} mol^{−1}); R is the gas constant (cm^{3} bar mol^{−1} K^{−1}); T is the temperature (K); and C_{e}^{0} is the solubility of dolomite under no stress. The term (1−φ) is the volumetric fraction of solid in the bulk rock (cm^{3} solid cm^{−3} bulk rock).
Lateral porosity differentiation occurs when the chemical reaction term, v_{m}R_{d}, overtakes the compaction rate, ((1−φ)/H)dH/dt, in equation (1) (Fig. 3). In this case, a positive feedback between mineral dissolution and stress becomes possible: crystal grains in a higher porosity domain within the layer are subjected to a higher effective stress than those in a lower porosity domain, thus resulting in pressure solution. The dissolved species diffuse to and precipitate in neighbouring lower porosity domains, further reducing the porosity in these domains. As shown below, this positive feedback leads to a lateral porosity differentiation within the dolomite layer. The differentiation would tend to occur in a consolidated sediment stratum comprising rigid mineral grains, in which mechanical compaction would be relatively small. For a rigid grain, such as a dolomite crystal, the influence of the exerting stress would extend to the whole grain, and dissolution thus occurs across the whole grain surface, especially on the pore surface, where the dissolved species can be easily transported away.
Stressmediated reactive transport
For simplicity, we assume that the mechanical compaction term is negligible as compared with the chemical reaction term. The porosity evolution of the dolomite layer is depending on three unknowns (C, P and φ ) and governed by
where P is the hydraulic head (cm). The effective diffusion coefficient D(φ) (cm^{2} s^{−1}) and the hydraulic conductivity K(φ) (cm s^{−1}) are described by respectively^{13,14,15}, where D_{0} is the diffusion coefficient of dissolved species in solution, and both m (≈2) (ref. 14) and K_{0} are constants.
The effective stress to which mineral grains are subjected at a spatial point (X, Y) is expected to depend on the overlying loading σ_{0}, the local porosity and the porosity distribution in the neighbourhood (that is, the rock texture in the neighbourhood). We define a texture descriptor ψ(X,Y,t) as a weighted average of porosity in the neighbourhood:
where K_{r} is a kernel function with The kernel function is isotropic and independent of location, and vanishes as X−X′→∞ or Y−Y′→∞. Based on this symmetry argument, by expanding the term φ in equation (6) into a Taylor's series and collecting the linear and secondorder terms, we obtain (see Methods):
where B is a constant. We then assume that the effective stress σ is proportional to the overlying loading σ_{0} and some function of ψ, f(ψ).
where α is a proportionality constant. To ensure that mineral grains in a higher porosity domain is subjected to a higher effective stress, it is required that One reasonable choice for function could be: f(ψ)=1/1−ψ. The effective stress would then be inversely proportional to the ratio of crosssectional area of solid to bulk rock. Given a typical porosity value in dolomite (~15%) (Fig. 1), the function chosen can be further simplified: f(ψ)=1/(1−ψ)≈1+ψ.
Scaling and asymptotic analysis
Equations (2–8) form a closed system that describes lateral porosity evolution in a single dolomite layer during burial diagenesis. To delineate the relative importance of each term in the equations, we scale the equations by choosing appropriate time and length scales, a typical dissolved species concentration and a typical hydraulic head so that each variable in the equations has a magnitude of one. From equations (2), (7) and (8), we choose the typical dissolved species concentration ([Cmacr ]) to be the solubility of dolomite under a uniformly distributed initial porosity (φ_{o}) and a given overlying loading (σ_{o}): where . As porosity evolution due to mineral dissolution and precipitation is of our main interest, the relevant time scale would be the one over which a significant change in porosity could occur. The time scale (T) is thus chosen to be: obtained by scaling equations (2) and (5). Similarly, the length scale (L) and the typical hydraulic head ([Pmacr]) and to make both the diffusion term and the advective term in equation (3) comparable with the chemical reaction term so that they can interplay with each other. Accordingly, the other variables and the related mathematical operators can be scaled as follows: y=Y/L, Equations (2–8) can then be cast into a set of dimensionless scaled equations:
where These equations show that the system under consideration has dual time scales: T (the long time scale) and ɛT (the short time scale), which is a common feature of many rock–water interaction systems. To study the porosity evolution, we have to use the long time scale T. Given the typical value of v_{m} (=64 cm^{3} mol^{−1}, ref. 16) and [Cmacr ] (≈10^{−6} mol cm^{−3}, ref. 17), the parameter ɛ is typically very small (<10^{−4}). Therefore, the lefthand side terms in equations (10) and (11) are negligible. These three equations thus can provide solutions to porosity evolution on long time scales, given the three unknowns (C, P and φ) of the general case.
Linear stability analysis
We now perform a linear stability analysis of equations (9–11) to determine the geometry and wavelength of lateral porosity oscillations (details in Methods). We first solve the equations for their steadystate solution: φ_{s}=φ_{0}, c_{s}=1 and where v is a scaled fluid flow velocity. We then linearize the equations and introduce a small perturbation around the steadystate solution. Finally, we obtain the following dispersion equations describing the growth rate (ζ) of a perturbation of a wave number vector (ω_{1}, ω_{2}):
where η=f′(φ_{s}). The real part of ζ, Re(ζ), represents the actual growth rate of a perturbation, while the imaginary part Im(ζ) indicates a temporal oscillation of a spatial pattern. A positive Re(ζ) value means that the perturbation with the corresponding wave number tends to grow with time, leading to the emergence of a spatially repetitive pattern^{18}. The wave number vector (ω_{max,1} ω_{max,2}) corresponding to the maximum Re(ζ) dictates the periodicity (or the spacing) of a pattern actually formed. As indicated in equation (12), the preferred wave number vector (ω_{max,1} ω_{max,2}) is independent of both θ and η. Thus, the choice of the actual form of function f(ψ) in equation (8) has no effect on the periodicity of the pattern formed as long as that is, as long as the positive feedback between the effective stress and the porosity is maintained.
The linear stability analysis demonstrates that periodic lateral oscillations in porosity can autonomously emerge from the stressinduced instability of sediment burial diagenesis, as indicated by the existence of positive Re(ζ) values for certain waves numbers (Fig. 4). In a noflow case, Re(ζ) is a function only of with no directional preference, implying the formation of a spatially isotropic pattern. As Im(ζ)=0 with no flow, the chemical wave formed would be stationary. In contrast, in a flow case, ω_{max,2}=0, that is, an infinite wavelength in Ydirection, indicating that any periodic oscillations in the Ydirection are suppressed, causing a symmetry breakdown. The repetitive patterns can thus form only along the flow direction. With flow, Im(ζ) can be nonzero, implying that the porosity at each spatial point would oscillate with time. In that case, the stripes of low and highporosity domains would slowly move downstream along the flow path, forming a propagating chemical wave. A careful evaluation of equation (12) indicates that the transition from an isotropic pattern to a stripe pattern occurs when the scaled flow velocity (v) is greater than 0.1, corresponding to an actual velocity of ~2 cm per year with the following estimated parameter values: D_{0}=10^{−4} cm^{2} s^{−1}, k=10^{−10} cm^{3} solution cm^{−2} s^{−1}, S=10 cm^{2} cm^{−3} solid and φ_{0}=0.15 (see below for a detailed discussion on the range of parameter values). Therefore, careful characterization of lateral heterogeneity in a dolomite layer may provide rich information about paleohydrologic environments for dolomite formation and transformation.
The wavelength of porosity oscillations (L_{p}=2πL/ω_{max}) can be estimated from (see Methods):
for a noflow case
for a flow case with v>0.1
Parameter β characterizes the degree of sediment consolidation (that is, the mechanical strength of bulk sediment). The wavelength increases with sediment mechanical strength, diffusion coefficient and flow velocity, and decreases with mineral dissolution rate. For a noflow case, taking the typical parameter values for dolomite: β=~1.0, D_{0}=~10^{−4}−10^{−3} cm^{2} s^{−1} (accounting for a possible effect of elevated temperature), k=10^{−10}−10^{−8} cm^{3} solution cm^{−2} s^{−1} (using a dolomite dissolution rate of 10^{−14}–10^{−16} mol cm^{−2} s^{−1} (ref. 17), and assuming a typical concentration of dissolved species of 10^{−6} mol cm^{−3} (ref. 17)), S=10−100 cm^{2} cm^{−3} solid and φ_{0}=0.15, we estimate the wavelength of the pattern to be 0.3–30 m, consistent with field observations^{1,2}. Accordingly, the time scale for the predicted porosity differentiation is estimated to be 5×10^{2}–5×10^{5} years. For a flow case with a typical flow rate of 0.1–20 cm per year^{19}, the scaled flow rate (v) is estimated to be 0.0001–1. However, as discussed above, the effect of flow on pattern formation becomes significant only when v>0.1. Given that in equation (15), the scale of a pattern formed in a flow case is expected to be on the same order of magnitude as that in a noflow case, but the geometry of the pattern will be very different. With or without flow, as the sediment undergoes progressive burial diagenesis, newly formed chemical waves may superimpose on the relics of early ones, resulting in complex nested lateral porosity variations.
Discussion
Petrographic observations support our model predictions. Cathodoluminescence images of Mississippian dolomites that exhibit lateral patterns suggest that cement overgrowths on dolomite crystals are common in lowporosity samples while highporosity samples from the same bed show little dolomite cementation or even dissolution (Supplementary Fig. S2). These features are typically attributed to rock–water interaction processes, but why the abundance of cements would vary laterally on the metre scale is not explained by that traditional interpretation. Our model provides a logical explanation for these lateral variations. Differential mineral dissolution and cementation in different domains are a natural outcome of the proposed selforganization phenomenon. Furthermore, as mentioned above, a stressinduced chemical wave may evolve with time and propagate along a flow path. We would then expect that multiple stages of cementation or dissolution may occur at the same locality in a dolomite layer even under a singleflow regime with a constant fluid composition. This seems consistent with the widespread occurrence of luminescent zonation of dolomite cement. Such zonation has long been interpreted to record varying compositional and/or Eh changes of advecting fluids^{20,21}.
Our model also provides a reasonable explanation for the associated variations in chemical and isotopic signatures^{1} (Supplementary Fig. S1). In higher porosity domains, dolomite dissolves, releasing the contained trace elements into pore water. The released trace elements are reincorporated into newly precipitated dolomite cements in lower porosity domains. Consequently, the elements with partitioning coefficients >1 tend to be enriched in these cements, while those with partitioning coefficients <1 are depleted^{22}. Bulkrock geochemical analyses should thus show higher Fe and Mn, and lower Sr, in lower porosity domains. These predictions seem relatively well confirmed by field observations given the dynamic nature of the system that our model reveals. As shown in Supplementary Figure S1, the contents of trace elements in dolomite oscillate with the similar periods as porosity, with Fe and Mn varying roughly antipathetically and Sr sympathetically with porosity^{1} (Supplementary Fig. S1). Similarly, given the temperature dependence of oxygen isotopic fractionation^{23} and the temperature increase in burial diagenesis, lower porosity domains are predicted to be depleted in ^{18}O relative to higher porosity domains, roughly as observed (Supplementary Fig. S1).
The emergence of lateral chemical waves seems inevitable as long as a sediment stratum is subjected to a vertical loading. This explains why lateral periodic porosity oscillations are common in dolomite. The proposed mechanism is expected to operate in all stages of sediment burial diagenesis, and its effect on general rock–water interactions still needs to be clarified. For example, during initial dolomitization of a bed, the stressinduced porosity differentiation should occur both in the limestone domain ahead of a reaction front and in the dolomite domain behind the front as dolomitization proceeds. Field and petrographic observations show that stress or crystallization force has an important role in dolomitization^{8,24}. How the proposed mechanism affects the overall dolomitization process and modifies the original dolomitization signatures is certainly a topic that warrants further research.
The proposed mechanism is expected to operate also in other sedimentary rocks, in particular in limestones and sandstones dominated by compaction and pressure solution diagenesis. Lateral porosity and geochemical variations may characterize such rocks, but to our knowledge no previous workers have ever examined sandstones or limestones dominated by burial diagenesis for such features. The proposed mechanism may also apply to other observations in sandstones. For example, it was found that carbonate concretions in the Great Estuarine Group (Middle Jurassic) sandstones postdate the pressure solution of shell against quartz and feldspar but predate further compaction^{25}. The concretions may thus be spatially distributed in accordance with our model. Careful mapping of calcite cementation in concretions of the Frontier Formation sandstone in central Wyoming does show lateral periodic variations^{26}. The length of the concretions is ~4.2 m, and the width is ~5.3 m. Concretion centres are approximately Poisson distributed within the sandstone. Isotopic data indicate that calcite cement is precipitated near a burial depth of 1.5 km (ref. 26). The fraction of calcite cement varies horizontally on scale of a few metres to 10 m. The scale of the pattern is consistent with our model prediction.
Our model can be extended to sediments comprising multiple mineral phases with different mechanical competency. We believe that the proposed mechanism may be responsible at least partly for concretion formation in shale^{25}. Qualitatively, it works as follows: Carbonate minerals, being more mechanically competent, are expected to be the main framework to support the overlying loading. Carbonate crystals in a clayrich domain would be subjected to a higher effective stress, causing pressure dissolution. The dissolved species would then diffuse to and precipitate in neighbouring carbonate mineralrich domains. Similar to the dolomite case, this positive feedback would lead to the formation and the lateral separation of concretions in a shale unit. To fully model such processes, a constitutive model able to account for detailed stress partitioning among different mineral phases will be required^{27,28}. But, nevertheless, the model presented here allows for a rough estimation of the spacing of the concretions. As indicated in equation (14), the typical scale of the pattern depends on D_{0}, k, S, β and φ_{s}. The first three parameters are expected to have similar values to those for the dolomite case. However, the last two parameters could be quite different. A shale formation should have a smaller β value (that is, lower mechanical strength) and a smaller porosity value than a dolomite stratum. Let us assume that the parameters β and φ_{s} are 10 and 3 times smaller than those for dolomite, respectively. The spacing of carbonate concretions in shale is then estimated to range from half a decimetre to a few metres, which is roughly consistent with field observations^{25}.
The foregoing discussion is focused on the emergence of lateral heterogeneity in a single layer of dolomite. Our model would apply, however, to every layer in a vertical sequence and predicts lateral differentiation of porosity within every layer due to stressmediated mineralwater interactions during burial diagenesis. The magnitude of each layer's response will depend on the attributes of the layer, including bed thickness, initial porosity, mineralogical composition, hydraulic conductivity and rheology, yet any layer that is subjected to stress should respond and differentiate porosity. Field data^{1,2,4,5} show that the vertical variation in average properties of each layer in a dolomite sequence (and all sedimentary rocks for that matter) is related to vertical facies changes, but that the range of variations within different facies is often similar in magnitude. The mechanism proposed herein would be one means for creating those similar intrafacies ranges, and would further suggest that lateral intrafacies ranges that were heretofore considered random variations might, in part, relate to structured and predictable selforganizing chemical waves. A question for future study is whether the porosity differentiations in a set of vertically stacked sediment layers would proceed cooperatively through stress coupling and vertical mass transfer during burial diagenesis.
Vertical diagenetic differentiation within or between sediment layers is evident in many sedimentary rocks. For example, stylolites are commonly observed in carbonate rocks. These evenly spaced dissolution seams exhibit complex columntosocket interdigitations, with the lateral extension up to tens of metres and the interseam spacing up to ~1 m (ref. 29). They are usually parallel to bedding, if formed owing to a burial stress, or, in some occasions, oblique to bedding, if caused by a tectonic stress. The formation of stylolites has been attributed to the vertical structural differentiation and mass transfer induced by pressure solution^{29}. In principle, vertical mass transfer could occur between two neighbouring sediment layers, further differentiating mineralogical composition between the layers and therefore enhancing preexisting beddings. This process may partly be responsible for the formation of chert–shale couplets in bedded chert. Major and rareearth element data from Franciscan assemblage and Claremont Formation bedded chert sequences, along with physical observations such as the presence of rare and highly corroded radiolarians in shale interbeds, suggest a dominantly diagenetic origin of chert–shale couplets^{30}, contradicting the traditional depositional theories. By incorporating appropriate sediment rheology and vertical mass transfer, our model will be able to simulate any threedimensional complexity formed in sedimentary rocks during stressinduced burial diagenesis.
Finally, an accurate prediction of the scale and geometry of lateral porosity and permeability variations is of great interest for reservoir engineering. Numerical simulations show that fingering, sweep efficiency and breakthrough time in enhanced oil recovery are sensitive to such variations^{2,31,32}. Such effects may also be important for subsurface carbon dioxide sequestration and storage^{33}. A linear stability analysis indicates that horizontal permeability variations may enhance supercritical CO_{2} dissolution in deep brine aquifers^{34} and dramatically change the qualitative behaviour of the buoyancy flow of CO_{2} in such environments^{35}. Furthermore, lateral porosity variations may create localized stagnant domains that may affect the transport of chemical species in subsurface environments. Smallscale porosity variations have been found to have a significant impact on radionuclide transport in dolomite formations^{36}. Our work indicates that a large portion of such variations may originate from selforganized sediment diagenesis and therefore are probably predictable.
Methods
Curie principle
By expanding the term φ in equation (6) into a Taylor's series, we obtain:
In an isotropic medium subject to a vertical uniaxial loading, according to the Curie principle^{37}, the influence of a surrounding point (X′, Y′) on the point (X, Y) should be also isotropic. That is, replacing X−X′ with X′−X or Y−Y′ with Y′−Y will not change the scalar texture descriptor and therefore the function value of σ, implying that the second, third and sixth terms on the righthand side of equation (16) must vanish. Let A and B denote the coefficients of the remaining terms:
We then obtain equation (7).
Derivation of dispersion equation
Let ɛ→0 in equation (9–11)):
Linearizing the above equations around the steadystate solution with respect to perturbations δφ, δc and δp, we obtain:
Assuming that the perturbations have the form:
we obtain the following homogeneous algebraic equations:
The condition for a nontrivial solution of is:
Solving the equation for ζ, we obtain the dispersion equations (12) and (13).
Estimation of pattern scale
For a noflow case (v=0), from equation (12), we have
The preferred wave number ω_{max} can be found by taking :
We then have:
As φ_{s}^{m} ≪(1−φ_{s}) ≈ 1 and β is on the order of unity, we obtain:
or
The term is added to convert the unit of cm^{3} solid to the unit of cm^{3} solution.
For a flow case with a large v, as ω_{max,2}=0 (Fig. 4), from equation (12), we have
Similarly, the preferred wave number ω_{max,1} can be found by taking :
Solving for we obtain:
We then have:
for a large v.
Adding the term to convert the unit of cm^{3} solid to the unit of cm^{3} solution, we obtain:
Additional information
How to cite this article: Wang, Y. & Budd, D.A. Stressinduced chemical waves in sediment burial diagenesis. Nat. Commun. 3:685 doi: 10.1038/ncomms1684 (2012).
References
Budd, D. A., Pranter, M. J. & Reza, Z. A. Lateral periodic variations in the petrophysical and geochemical properties of dolomite. Geology 34, 373–376 (2006).
Pranter, M. J., Zulfiquar, R. & Budd, D. A. Reservoirscale characterization and multiphase fluidflow modeling of lateral petrophysical heterogeneity within dolomite facies of the Madison Formation, Sheep Canyon and Lysite Mountain Wyoming USA. Petroleum Geosci. 12, 29–40 (2006).
Kerans, C., Lucia, F. J. & Senger, R. K. Integrated characterization of carbonate ramp reservoirs using Permian San Andres Formation outcrop analogs. Am. Assoc. Petroleum Geologists Bull. 78, 181–216 (1994).
Eisenburg, R. A., Harris, P. M., Grant, C. W., Goggin, D. J. & Conner, F. J. Modeling reservoir heterogeneity within outer ramp carbonate facies using an outcrop analog, San Andres Formation of the Permian Basin. Am. Assoc. Petroleum Geologists Bull. 78, 1337–1359 (1994).
Grant, C. W., Goggin, D. J. & Harris, P. M. Outcrop analog for cyclicshelf reservoirs, San Andres Formation of the Permian Basin: stratigraphic framework, permeability distribution, geostatistics, and fluidflow modeling. Am. Assoc. Petroleum Geologists Bull. 78, 23–54 (1994).
Jennings, J. W. Jr. Spatial Statistics of Permeability Data from Carbonate Outcrops of West Texas and New Mexico: Implications for Improved Reservoir Modeling, Report of Investigation 258 (Bureau of Economic Geology, University of Texas, 2000).
Bribiesca, E. A Lateral Variations in Petrophysical, Geochemical and Petrographic Properties of an Avon Park (Middle Eocene) Dolograinstone, Gulf Hammock Quarry, Florida (M.S. thesis University of Colorado: Boulder, 2010).
Merino, E. & Canals, A. Selfaccelerating dolomiteforcalcite replacement: selforganized dynamics of burial dolomitization and associated mineralization. Am. J. Sci. 311, 573–607 (2011).
Merino, E. & Wang, Y. Geochemical selforganization in rocks: occurrences, observations, modeling, testing – with emphasis on agate genesis In NonEquilibrium Processes and Dissipative Structures in Geoscience, SelfOrganization Yearbook Vol. 11: (eds Krug, H. J. & Kruhl, J. H.) 13–45 (Duncker & Humblot, 2001).
Wang, Y., Xu, H., Merino, E. & Konishi, H. Generation of banded iron formation by internal dynamics and leaching of oceanic crust. Nat. Geosci. 2, 781–784 (2009).
Nicolis, G. & Prigogine, I. SelfOrganization in NonEquilibrium Systems (Wiley, 1977).
Maliva, R. G. & Siever, R. Diagenetic replacement controlled by force of crystallization. Geology 16, 688–691 (1988).
Boudreau, B. P. Diagenetic Models and Their Implementation: Modeling Transport and Reactions in Aquatic Sediments (Springer, 1997).
Oelkers, E. H. Physical and chemical properties of rocks and fluids for chemical mass transport calculations. Rev. Mineral. 34, 131–191 (1996).
Dullien, F. A. L. Porous Media: Fluid Transport and Pore Structure (Academic Press, 1992).
Robie, R. A., Hemingway, B. S. & Fisher, J. R. Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascals) Pressure and at Higher Temperatures (US Geological Survey, 1978).
Pokrovsky, O. S. & Schoot, J. Kinetics and mechanism of dolomite dissolution in neutral to alkaline solutions revisited. Am. J. Sci. 301, 597–626 (2001).
Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview, 2001).
Wangen, M. Physical Principle of Sedimentary Basin Analysis (Cambridge, 2010).
Barnaby, R. J. & Rimstidt, J. D. Redox conditions of calcite cementation intrepred from Mn and Fe contents of authigenic calcites. Geologic. Soc. Am. Bull. 101, 795–804 (1989).
Fraser, D. G., Feltham, D. & Whiteman, M. Highresolution scanning proton microprobe studies of micronscale trace element zoning in a secondary dolomite: implications for studies of redox behavior in dolomites. Sediment. Geol. 65, 223–232 (1989).
Wang, Y. & Xu, H. Prediction of trace metal partitioning between minerals and aqueous solutions: a linear free energy correlation approach. Geochim. Cosmochim. Acta 65, 1529–1543 (2001).
Kim, S. T. & O'Neil, J. R. Equilibrium and nonequilibrium oxygen isotope effects in synthetic carbonates. Geochim. Cosmochim. Acta 61, 3461–3475 (1997).
Maliva, R. G., Budd, D. A., Clayton, E. A., Missimer, T. M. & Dickson, J. A. D. Insights into the dolomitization process and porosity modification in sucrosic dolomites, Avon Park Formation (Middle Eocene), eastcentral Florida. J. Sediment. Res. 81, 218–232 (2011).
Raiswell, R. Evidence for surface reactioncontrolled growth of carbonate concretions in shale. Sedimentology 35, 571–575 (1988).
Dutton, S. P., White, C. D., Willis, B. J. & Novakovic, D. Calcite cement distribution and its effect on fluid flow in a deltaic sandstone, Frontier Formation, Wyoming. Am. Assoc. Petroleum Geologists Bull. 86, 2007–2021 (2002).
Dewers, T. & Ortoleva, P. J. Mechanochemical coupling in stressed rocks. Geochim. Cosmochim. Acta 53, 1243–1258 (2010).
Karato, S. I. Deformation of Earth Materials: An Introduction to the Rheology of Solid Earth (Cambridge, 2008).
Merino, E., Orteleva, P. & Strickholm, P. Generation of evenlyspaced pressuresolution seams during (late) diagenesis: a kinetic theory. Contrib. Mineral. Petrol. 82, 360–370 (1983).
Murray, R. W., Jones, D. L. & Buchholtz ten Brink, M. R. Diagenetic formation of bedded chert: evidence from chemistry of the chertshale couplet. Geology 20, 271–274 (1992).
Jennings, J. W., Ruppel, S. C. & Ward, W. B. Geostatistical analysis of permeability data and modeling of fluidflow effects in carbonate outcrops. SPE Reservoir Eval. Eng. 3, 292–303 (2010).
Pranter, M. J., Hirstius, C. B. & Budd, D. A Multiple scales of lateral petrophysical heterogeneity within dolomite lithofacies as determined from outcrop analogs: implications for 3D reservoir modeling. Am. Assoc. Petroleum Geologists Bull. 89, 645–662 (2005).
Doughty, C. & Pruess, K. Modeling supercritical carbon dioxide injection in heterogeneous porous media. Vadose Zone J. 3, 837–847 (2004).
Xu, X., Chen, S. & Zhang, D. Convective stability analysis of the longterm storage of carbon dioxide in deep saline aquifers. Adv. Water Resourc. 29, 397–407 (2006).
Saadatpoor, E., Bryant, S. L. & Sepehrnoori, K. 6th Annual Conf. Carbon Capture and Sequestration 7–10 May (2007).
Tidwell, V., Meigs, L. C., Christian Frear, T. & Boney, C. M. Effects of spatially heterogeneous porosity on matrix diffusion as investigated by Xray adsorption imaging. J. Contam. Hydrol. 42, 285–302 (2000).
De Groot, S. R. & Mazur, P. Nonequilibrium Thermodynamics (NorthHolland, 1962).
Acknowledgements
Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energy's National Nuclear Security Administration under contract DEAC0494AL85000. This work is partly supported by DOE Sandia LDRD Program. D.A. Budd's field and petrographic studies were supported by the industrial supporters of the AVID consortium. We would like to thank Enrique Merino of Indiana University for insightful discussions.
Author information
Authors and Affiliations
Contributions
Y.W. formulated the model and drafted the paper; D.A.B. provided the field data and measurements.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures S1S2. (PDF 444 kb)
Rights and permissions
About this article
Cite this article
Wang, Y., Budd, D. Stressinduced chemical waves in sediment burial diagenesis. Nat Commun 3, 685 (2012). https://doi.org/10.1038/ncomms1684
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms1684
This article is cited by

Zebra rocks: compaction waves create ore deposits
Scientific Reports (2017)

Selforganized ironoxide cementation geometry as an indicator of paleoflows
Scientific Reports (2015)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.