SEARCH

SEARCH BY CITATION

Keywords:

  • fluvial sediments;
  • Stochastic modeling;
  • macrodispersion;
  • scaling;
  • bed forms

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] This study explores the link between spatial variability within fluvial sedimentary strata and river channel geometry. This link is then used to determine the macrodispersion coefficients for solute transport in groundwater flow in river deposits. In doing that we combine concepts from sedimentology, geostatistics, and the stochastic-Lagrangian theory of subsurface transport. It is proposed to analyze the spatial variability of river sediments in terms of transition probabilities. The transition probabilities can be determined from the averages and variances of the lengths of stratasets and their volumetric fractions using concepts developed by Carle and Fogg (1996) and Ritzi (2000). Strataset length scales are shown to correlate well with the geometry of the river channel and its bed forms and can be determined indirectly from aerial photogrammetry, and, more directly, through surface ground penetrating radar surveys. Stratal dimensions can also be related, using Lagrangian transport analysis, to macrodispersivity of subsurface transport. This allows relationships between the geometry of river channels and subsurface transport parameters of associated fluvial deposits. For demonstration the longitudinal macrodispersivity in a compound bar deposit in the gravelly, braided Sagavanirktok River was investigated.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] This paper explores the link between the heterogeneity of subsurface river deposits and river geometry and dynamics. The primary motivations for this study are to (1) better understand these links, (2) provide the tools needed for systematic investigations and intersite comparisons, and (3) alleviate the difficulties associated with hydrogeological characterization (i.e., mapping the spatial distribution of sediment types (facies) and their corresponding hydraulic properties such as permeability and porosity). The possibility of estimating subsurface flow and transport parameters from geometric features of rivers is appealing because it allows deriving prior (in a statistical sense) estimates of such parameters in ungauged basins.

[3] The geometry, water flow and sediment transport of rivers affect the character of deposits in channel belts, which in turn determines the spatial variability of hydrogeologic properties (e.g., permeability and porosity) of fluvial deposits. The hydrogeologic properties of fluvial sediments are determined by grain size variations within individual strata, and by the geometry of sets of strata of different scales [Jordan and Pryor, 1992; Davis et al., 1993; Webb and Anderson, 1996; Davis et al., 1997; Ritzi et al., 2000; Willis and White, 2000; Bridge, 2003; Lunt et al., 2004a, 2004b; Bridge and Lunt, 2006]. Knowledge of these different scales of strataset has been derived from investigation of modern rivers for many years [Cant and Walker, 1978; Allen, 1982; Bristow, 1993; Willis, 1993; Collinson, 1996; Miall, 1996; Best et al., 2003] (summarized by Bridge [2003]). Recent advances in our ability to monitor river bed morphology using photogrammetric techniques [i.e., Lane et al., 2001; Westaway et al., 2000] and to observe the resulting deposits using ground-penetrating radar combined with trenches or cores [Bridge et al., 1998; Regli et al., 2002; Jol and Bristow, 2003; Best et al., 2003; Skelly et al., 2003; Lunt et al., 2004a, 2004b; Wooldridge and Hickin, 2005; Sambrook Smith et al., 2005; Mumpy et al., 2006] have allowed quantitative, comprehensive assessment of the formation of fluvial deposits.

[4] The geometry of the deposits of braided rivers and its relation to subsurface solute transport properties are the focus of this paper. Research effort in this direction has been contemplated by several researchers. The National Center for Earth Dynamics (NCED), for example, has identified the understanding of subsurface architecture as one of its major research thrusts. In particular, NCED assigns high priority to understanding how surface channel properties, spatial patterns, and temporal evolution interact with net deposition to create the architecture of sedimentary deposits, and how these deposits are modified by subsequent erosion and biochemical alteration. The ability to address these issues depends on (1) development of databases suitable for empirical studies and hypothesis testing; and (2) the availability of tools for coherent analysis of these databases. Work in support of the first element has been going on for decades, as mentioned above. Scaled laboratory experiments [i.e., Schumm et al., 1987; Ashmore, 1982, 1991; Ashworth et al., 1999; Paola, 2000; Moreton et al., 2002; Sheets et al., 2002; Doeschl-Wilson and Ashmore, 2005] have also proved useful, but experimental studies cannot be expected to recreate the full range of bed forms seen in rivers, and consequently the smaller scales of heterogeneity in fluvial deposits. Both field and experimental data should be applied to constrain numerical simulations of fluvial processes and deposits [i.e., Paola et al., 1992; Mackey and Bridge, 1995; Sun et al., 1996; Karssenberg et al., 2001; Hickson et al., 2005]. Work in support of the second element is at an early stage. It is proposed here that data analysis can be undertaken using geostatistical tools such as presented here and in earlier studies, including Rubin [2003], Ritzi et al. [2004], and Dai et al. [2005].

[5] The paper is organized as follows. Section 2 provides background information on the geometry of river deposits. In section 3, the suitability of transition probability concepts for analyzing the spatial variability of river deposits is evaluated. Section 4 demonstrates an application of these statistical tools for estimating subsurface solute transport properties. In section 5, discussion and conclusions are presented.

2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[6] This section examines the geometry of bed forms and cross sets in rivers. It also evaluates similarity and persistence of geometrical characteristics across scales and across rivers. If such similarity and persistence can be identified, then this would suggest the possibility for generalizing the findings from individual field studies. We start with a few general observations, and proceed to evaluate them against observations.

[7] It is well known that there is a hierarchy of different bed form sizes in rivers [Vanoni, 1974; Allen, 1982, Jordan and Pryor, 1992; Bridge, 2003; Ritzi et al., 2004]. Small-scale bed forms such as sand ripples, dunes, and bed load sheets are superimposed on larger-scale features such as unit bars. Unit bars are commonly amalgamated to form compound bars such as point bars and braid bars (Figure 1). It is also well established that the dimensions of dunes and bars in rivers are related to river channel geometry [Simons et al., 1965; Allen, 1976; Parker, 1976; Nelson, 1990; Yalin, 1992; Julien and Klassen, 1995; Karim, 1999; Bridge, 2003; Wu and Yeh, 2005]. Dune length and height are primarily proportional to flow depth, with variations in height and length due to the effects of unsteady flows [Allen, 1976; Nasner, 1978; Nelson and Smith, 1989; Mohrig and Smith, 1996; Kleinhans, 2002; Wilbers and ten Brinke, 2003], and the length of unit bars and compound bars are proportional to channel width [Parker, 1976; Fredsoe, 1978; Engelund and Fredsoe, 1982, Fujita and Muramoto, 1985; Tubino et al., 1999; Wu and Yeh, 2005].

image

Figure 1. Aerial photograph of the Sagavanirktok River, Alaska, showing a compound bar cut by a cross-bar channel, lobate unit bars, and superimposed dunes.

Download figure to PowerPoint

[8] Each of the different scales of bed form in rivers is associated with a different scale of strataset (Figure 2). For example, small-scale cross strata are associated with ripples, medium-scale cross strata are associated with dunes, and large-scale inclined strata are associated with unit and compound bars [Rubin and Hunter, 1982; Jordan and Pryor, 1992; Bridge, 2003; Ritzi et al., 2004]. Furthermore, it is being recognized that the mean height and length of bed forms are proportional, respectively, to the mean thickness and length of associated stratasets [Leclair and Bridge, 2001; Lunt et al., 2004a, 2004b]. This means that the geometry of stratasets formed by dunes and bars is related to the geometry and dynamics of the river.

image

Figure 2. Illustration of different scales of strataset. An across-stream section through the channel belt shows that it is made up of compound bar deposits. This, in turn, contains unit bar deposits and cross-bar channel fills. The internal structure of a unit bar deposits is shown in the lower diagram. The cross section through the bed form and cross set shown in the inset correspond to (a) a vertical profile along the white flow direction arrow and (b) the cross set inside the shaded rectangle. Modified from Lunt et al. [2004a] with permission from Blackwell Publishing.

Download figure to PowerPoint

[9] The nature of the relationship between bed form height and strataset thickness is the subject of a theory first outlined by Paola and Borgman [1991], modified by Bridge and Best [1997], and experimentally verified by Leclair and Bridge [2001].

2.1. Length Scales and Ratios in the Sagavanirktok River

[10] The bed forms and deposits of the Sagavanirktok River, Alaska (Figures 1 and 2), are used as an example of how strataset geometry is related to bed form geometry and dynamics. The nature of channel evolution and deposits of the Sagavanirktok river were studied using aerial photographs taken since 1949, trenches, more than 100 km of ground-penetrating radar (GPR) data, and 17 cores [Lunt et al., 2004a, 2004b; Lunt and Bridge, 2004]. The Sagavanirktok River is a sandy-gravelly braided river with a channel belt width of 2.4 km, maximum bankfull flow depth of 3.8 m, valley slope of 0.0013 m/m, bankfull discharge of around 600 m3 s−1, and mean grain size of 4 mm in the reach studied.

[11] The dimensions of bed forms of different scale (e.g., dunes and bars) were measured using aerial photographs, and depth mapping of the river bed based on GPR profiles taken through the active channels (Figure 1). The dimensions of the various deposits were measured over rectilinear grids, superimposed on GPR lines and trenches. Medium-scale cross sets (formed by dunes) could only be measured in trenches, however, there is significant other information that was obtained from GPR images, such as shown in Figure 3. The majority of deposits comprise simple sets of large-scale strata, formed by unit bar deposition, which increase in thickness toward their downdip margins and contain cross strata that increase in dip in the same direction (Figure 2). Cross-bar channel fills have concave-upward erosive bases, length-to-thickness ratios of approximately 2:1, and may contain small unit bar deposits. Compound sets of large-scale strata (formed by compound bars) can be recognized primarily by their erosive boundaries that truncate numerous simple large-scale sets, and commonly contain concave upward cross-bar channel fills on their upper surfaces.

image

Figure 3. The 110 MHz GPR profile showing a compound bar deposit with an erosional base (thickest white line; arrowed) made up of a number of unit bar deposits. The top of the compound bar deposit has been truncated. Modified from Lunt et al. [2004a] with permission from Blackwell Publishing.

Download figure to PowerPoint

[12] The dimensions of bed forms reported by Lunt et al. [2004a, 2004b] are maximum length and width, because they were taken crest to crest or trough to trough in the horizontal direction, and trough to crest, vertically. Dunes were found to have a mean maximum length of 15 m and unit bars have a mean maximum length of 345 m (or 3.5 channel widths) and mean width of 66.2 m (or 0.7 channel widths). The length and thickness of the stratasets were used to determine the scaling relationships between bed forms and their deposits (Figures 4 and 5) .

image

Figure 4. Histograms of the length:thickness ratio of different scales of strataset. Modified from Lunt et al. [2004a] with permission from Blackwell Publishing.

Download figure to PowerPoint

image

Figure 5. The wavelength:height ratios of bed forms are related to the length:thickness ratios of their associated deposits. Data are taken from published studies mentioned in the text. Modified from Bridge and Lunt [2006] with permission from Blackwell Publishing.

Download figure to PowerPoint

[13] The ratios of maximum length to thickness of the deposits were found to increase from medium-scale to compound large-scale deposits (Figure 4), and suggest that simple relationships between the dimensions of morphological features in active rivers can be used to determine the dimensions of deposits formed by that river.

2.2. Data From Other Rivers

[14] Data from other published studies of modern rivers show that the relationships between the geometry of bed forms and deposits observed in the Sagavanirktok River are not unique. These data sets are from modern braided fluvial deposits, and include the Jamuna River [Best et al., 2003], the Niobrara River [Skelly et al., 2003], the Fraser River [Wooldridge and Hickin, 2005], and the South Saskatchewan River [Sambrook Smith et al., 2005]. These rivers range in channel belt width from around 600 m to over 15km, and the mean grain size varies from 0.04 mm to over 15 mm. A comparison of the length to thickness ratios of bed forms and deposits from all these rivers (Figure 5) shows that the dimensions of bed forms (and hence deposits) vary in different sized rivers, as would be expected. As bed forms such as bars and dunes scale with the channel dimensions, larger rivers give rise to deposits with larger lengths and heights. However, for rivers of all sizes and bed material types, the wavelength-to-height ratio of bed forms is very similar to the length-to-thickness ratio of deposits. This holds true over the entire range of bed forms for which data are available. Importantly, this suggests that the dimensions of subsurface deposits can be predicted from a limited amount of information about the formative river. Also, it may be possible to predict the relative dimensions of different scales of strataset. This will be expanded in further detail in section 4.

3. Transition Probabilities for Characterization of River Deposits

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[15] There is strong evidence indicating that the dimensions of fluvial deposits are determined primarily by the dimensions of their formative bed forms. It also appears, from our previous discussion, that these dimensional relationships are transferable between rivers. Additional work is needed in support of these ideas, and this is turn requires quantitative tools for analyzing the structures observed in river sediments. Such tools must (1) be able to represent complex, hierarchical structures, with different patterns of spatial variability at each hierarchy, (2) be formulated in statistical terms, to be able to correctly represent the uncertainty associated with spatial variability, on one hand, and be easily applied in data-sparse situations, on the other, (3) address asymmetries in the spatial correlation patterns, such as in the case of upward fining or coarsening grain size trends in fluvial deposits [Carle and Fogg, 1996], and (4) be able to represent site-specific features yet allowing comparison between and synthesis of multiple data sets.

[16] These requirements are generally met by various geostatistical tools, and as a result, the description of sedimentary deposits and their flow properties using geostatistical methods has become common in both the hydrogeological and reservoir characterization literature [Johnson and Dreiss, 1989; Johnson, 1995; Gelhar, 1993; Jussel et al., 1994; North, 1996; Deutsch and Journel, 1998; Weissmann and Fogg, 1999; Ritzi et al., 2000; Strebelle, 2002; Rubin, 2003].

[17] Indicator variograms and transition probabilities were found to be particularly useful. Johnson and Dreiss [1989], Rubin and Journel [1991], Johnson [1995], Rubin [2003], Ritzi et al. [2004], and Dai et al. [2004] have all employed indicator variograms to characterize such complex, hierarchical deposits. The variogram describes the degree of correlation between two points over a range of separation distances. The form of the resulting variogram can be interpreted to reveal information about the correlation length scales. Transition probabilities [e.g., Carle and Fogg, 1996] describe the spatial correlation structure of a region in a similar way. In this case, the probability of ‘transitioning’ from one class to another (e.g., from a unit bar deposit to a channel fill) is calculated over a range of separation distances. The advantage of transition probabilities is that they allow the direction of transition to be incorporated. For instance, the probability of passing from a unit bar deposit to a channel fill may be different from the probability of the reverse transition. This is particularly useful in characterizing fluvial sediments, in which the flow direction has a strong impact on the permeability distribution in the deposits.

[18] Geostatistical tools are efficient in reducing complex spatial data into a set of summary statistics. However, it can be shown that they are also related to the underlying depositional processes. Ritzi [2000] related the ranges and types of autotransition probability and variogram models used for stratigraphic characterization to the variability in facies lengths. He showed that as this variability increases, the indicator correlation range increases whereas the indicator variogram model evolves from a periodic linear structure to aperiodic spherical structure to an aperiodic exponential structure. These concepts are useful because they allow, as we will show below, translation of the length distributions, such as reviewed in section 2, into spatial characterization tools, and eventually into flow and transport parameters.

[19] Consider Ik(x) to be an indicator space random function (SRF) in a two-level hierarchical river deposit with N mutually exclusive sedimentary facies, with k varying from 1 to N. A two-level hierarchy implies an assemblage of different facies, with the permeability in each of the N types of facies being also spatially variable within the facies. This implies variability at two levels: at the lower level, there is variability within the facies, and at the higher level, there is variability in the organization of the various facies in space. Ik assumes the values of either 1 or 0 with probabilities Pk and (1−Pk), respectively: Ik(x) is equal to 1 if x is within a unit of type k, and 0 otherwise. Pk(x) = Pr[Ik(x) = 1], where Pr denotes probability. Pk is equal, in a stationary domain, to the volume fraction of geological unit k, and must satisfy the constraint:

  • equation image

The indicator is used here to describe the spatial arrangement of the geological units. Intrafacies variability for variables such as permeability can then be characterized using continuous space random functions (SRFs), with different types of SRFs attached to different facies [Rubin, 1995]. Higher-order hierarchies were investigated by Dai et al. [2004, 2005].

[20] The transition probability tensor tjk(h), with j, k = 1, N, and h the lag vector between x and x′, is defined by the conditional probability [cf. Ritzi, 2000, equation (2)]:

  • equation image

where Pr[Ik (x + h) = 1, Ij(x) = 1] is the joint (bivariate) probability for the facies at x+h and x to be of types k and j, respectively. The cross-transition probability is tjk(h) with jk. Of particular interest is the autotransition probability tkk(h):

  • equation image

because it can be related, following Carle and Fogg [1996] and Ritzi [2000], to the statistical distribution of the lengths of geological unit of type k.

[21] The autotransition probability can be related [see Ritzi, 2000, equation (3)] to the indicator variovariogram, γkk(h) = 〈[Ik(x) − Ik(x + h)]2〉/2, through the identity:

  • equation image

[22] The indicator variogram model is defined through spatial variability models based on the variance of Ik and a vector of correlation ranges, ak,i, i = 1,..., 3, where k = 1,..., N, defines the facies type, and the second subscript, i, denotes the Cartesian direction in space. The correlation range is defined as the lag distance hi where the correlation between the indicators, in the ith direction, vanishes.

[23] Carle and Fogg [1996] defined another length scale, equation imagek,i, the mean length of unit k in the ith direction (where i denotes a Cartesian coordinate, with i = 1,2 denoting the horizontal axes. Ritzi [2000] showed that if the unit types occur in equal numbers, then equation imagek,i can be related to the correlation range ak,i through the identity:

  • equation image

where ϕ = 1, 1.5, or 3 for linear, spherical, or exponential models, respectively. If the unit types do not occur in equal numbers, (5) needs to be modified by an embedding coefficient [Ritzi, 2000]. The choice of model depends on the coefficient of variation (Cv) of ℓk,i, as shown by Ritzi [2000]. A linear model is used if Cv < 0.2, a spherical model is used for Cv ∼ 0.7, and an exponential model used for Cv ∼ 1.

[24] Ritzi [2000] showed that for N = 2, the variograms of the two facies are equal, i.e., γ11 = γ22. As a result, the ranges of the indicators I1 and I2 are identical, which indicates, following (5), that the ratio of average lengths is equal to the ratio of their volumetric fractions:

  • equation image

with P1 = 1 − P2.

[25] To examine this relationship, let us consider the spatial distribution of compound sets of large-scale strata. Lunt et al. [2004a, p. 391] in their study of the Sagavanirktok River refer to compound sets of large-scale strata truncated laterally by major channel fills. Compound sets and channel fills were found to have average maximum lengths of 627.9 and 37.9 m, respectively. The average maximum lengths were found to be larger than the average lengths by a factor of 1.26, with coefficient of correlation of 0.82, leading to equation image1,1 and equation image1,2 values of equal approximately to 498.3 m and 30.0 meters, respectively. The coefficient of variation of the maximum lengths were found to be equal to 0.37 and 0.48, for compound stratasets and channel fills, respectively, suggesting that the coefficients of variation of the average lengths are much larger, possibly of the order of 1. This in turn indicates, according to Ritzi [2000], that γ11 and γ22 are exponential, and that ϕ of equation (5) is equal to 3. Combining (1) with (6), we obtain that the volumetric fractions of the compound sets and the channel fills are P1 = 0.943 and P2 = 0.057, respectively, which is in excellent agreement with GPR- and trench-based based estimates. We obtain from (5) that a1,1 = a2,1 = 85 meters.

[26] Once the lengths equation imagek,i and the volumetric fractions, Pk, k = 1, ..,N, are defined, the indicator variogram and the autotransition probabilities become readily available. The lengths equation imagek,i can be related to the correlation ranges. Pk can be related to the indicator variance, which is in fact equal to Pk(1 − Pk), and the variance of the lengths can be related to the shape of the variogram, based on the work of Ritzi et al. [2000], as discussed earlier. Since the equation imagek,i generally have different values in different directions, a statistically anisotropic variogram for each k should be expected. With the variograms defined, the autotransition probabilities can also be derived, for example, using (4).

[27] In order to provide the complete geostatistical characterization that would be needed for simulating a sedimentary deposit with many facies, it is necessary to go beyond autotransition probabilities, and augment the previous identities and definitions with cross-transition probabilities. The cross-transition probabilities can be obtained from observations. However, it is useful to investigate the relationship between the cross-transition probabilities and the autotransition probabilities, in order to obtain accurate definitions of the cross-transition probabilities, particularly in cases where the quality and/or number of field data are insufficient for determination of correlation models. In this case, two relationships can be used as starting points. The first is the identity [see Carle and Fogg, 1996, equation (17b); Dai et al., 2005, equation (B2)]

  • equation image

and the second one is the identity:

  • equation image

which again holds for any i. In these equations and subsequently, i, j, k = 1, N.

[28] These identities must be translated into relationships between the autotransition and cross-transition probabilities. Unique relationships between the autotransition and cross-transition probabilities exist for N = 2 (as will be shown below). In the more general case of N > 2, equations (7) and (8) can be used as constraints.

[29] Equation (8) can be simplified using several auxiliary relationships. Using equation (2), each of the summed up terms on the left-hand side of (8) can be expressed using cross-transition probabilities, as follows:

  • equation image

with no summation over repeated indices here or subsequently, unless where explicitly mentioned. The right hand-side of (8) can be expressed using autotransition probabilities, as follows (see Rubin [1995, equation [8]] for a similar derivation):

  • equation image

Substituting equations (9) and (10) into (8) yields the following identity:

  • equation image

[30] To demonstrate the usefulness of these relationships, consider the case of N = 2. In this case, equation (6) yields

  • equation image

whereas (11) yields, for i = 1:

  • equation image

An identical equation is obtained for i = 2. Introducing (4) into (13) confirms that γ11 = γ22, as suggested earlier. This result can also be obtained by combining (7) with the definition of conditional probability [see also Carle and Fogg, 1997, equation (10)]:

  • equation image

[31] For the case of N = 2, (7) and (11) provide a complete bivariate statistical characterization once any of the autotransition probabilities, either t11 or t22, is defined. For N > 2, (7) and (11) translate into a set of transition probability equations that can be transformed into variogram equations or covariance equations, using (4). These equations can be used as constraints when calculating transition probabilities (or variograms) from data.

[32] These results are significant because they provide a simple way to obtain a complete (for N = 2), or constrained (for N > 2), bivariate characterization, using the statistics of the facies' lengths as a starting point. As was shown in section 2 and originally by Lunt et al. [2004a], these statistics can be obtained in some cases using GPR [Neal, 2004; Rubin and Hubbard, 2005]. In other cases, as we will show later, they can be obtained from analysis of the channel and bar dimensions.

[33] Let us assume that an exponential indicator variogram was identified, given by γ = Pk (1 − Pk)[1 − exp (−∣h∣)/λI], for both k = 1, 2. λI is the integral scale, which in the exponential case is equal to 1/3 of the correlation range. Other variogram models can also be employed, as appropriate (see Rubin [2003, chapter 2] for a list of models and relationships between integral scales and correlation ranges). The autotransition probability tkk(h), k = 1,2, can be obtained from equation (4) and the cross-transitional ones, t12(h) and t21(h), can be obtained from (12). With these expressions defined, the complete bivariate statistical distribution of the indicators has been obtained, employing only a small number of parameters.

4. Scaling and Solute Macrodispersion in Braided River Deposits

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[34] Models relating indicator statistics to subsurface solute transport parameters are well documented [cf., Rubin, 1995, 2003; Ritzi et al., 2004; Dai et al., 2004]. Hence, on the basis of the relationship between equation imagek,i and the river geometry on one hand, and models relating the spatial statistics of the indicators to transport parameters on the other, various aspects of the river morphology and dynamics can be related to subsurface transport properties.

4.1. River Geometry and Facies Lengths

[35] Lengths of unit bars were found to vary between 2 and 7 times the formative channel width, regardless of channel pattern [Bridge, 2003]. Measurements collected from the Sagavanirktok River (Figure 6) show that

  • equation image

where ℓ is the unit bar length and CW denotes channel width, and β1 = 3.5. The estimation error is given by ɛ1, and the estimation error variance is σ12 = 4247 m2. In general, unit bar lengths can be measured directly from aerial photographs, or in the case of opaque water in sediment-laden flows, can be related through (15) to measurements of channel width. The height of unit bars is generally equivalent to the bankfull depth of channels [Tubino et al., 1999; Bridge, 2003] but may be measured to within 0.1 m, from LIDAR imagery or stereo photogrammetry [Lane et al., 2001, 2003]. The relationship between the dimensions of river channels and unit bars is combined with a relationship between the dimensions of unit bars and their deposits. This will allow the dimensions of deposits to be inferred from river channel geometry.

image

Figure 6. Channel width versus unit bar length from the Sagavanirktok River.

Download figure to PowerPoint

[36] The dimensions of these deposits in the Sagavanirktok River are primarily based on the interpretation of GPR profiles, and compare well with limited field data from the studies used to develop Figure 5 as discussed in section 2. The mean ratio between unit bar deposit length (equation image1,1) and unit bar length is given by

  • equation image

with β2 = 0.25 for the Sagavanirktok River. Thus we obtain that equation image1,1 = β1β2CWa + ɛ2, with CWa the average channel width. Because β1β2 ≈ 1, we can approximate (16) with equation image1,1CWa, subject to estimation error.

4.2. Solute Macrodispersion in Braided River Deposits

[37] There is a significant body of work relating macrodispersion of solutes in the subsurface to correlation models of spatial variability such as discussed in section 3 (see summary by Rubin [2003, chapter 10]). One of the cases discussed by Rubin [2003], following earlier work by Rubin [1995], is that of macrodispersion in geological formations with bimodal heterogeneity, such as would be described by (1) with N = 2. This implies that the flow domain is composed of two types of facies, each characterized by a different model of spatial variability of hydraulic conductivity. The log conductivity in this domain is modeled using the hybrid SRF:

  • equation image

where Y denotes the log conductivity SRF, Y1 and Y2 denotes the log conductivity SRFs in each of the two facies, respectively. The means, variances, and integral scales of the log conductivity facies are denoted by mi, σi2 and λi, and the volumetric fractions are given by P1 and P2 = 1 − P1, for i = 1, 2, respectively. P = P1 will be used in our subsequent discussion.

[38] To develop a macrodispersion model, we assume that the groundwater flow is two-dimensional in the horizontal plane and that the it is at steady state. The flow is assumed to be uniform in the mean, and the average fluid velocity is taken equal to U. The Cartesian coordinate system is aligned such that the x1 axis and the mean velocity are in the same direction. The model also assumes that the conductivity is not correlated across sedimentary facies, and that the spatial correlation is anisotropic in the horizontal plane. This is a simplistic model, but general and sufficiently meaningful for demonstrating the significance of the previous derivations. Other models (e.g., models allowing for three-dimensional heterogeneity) can be also employed [see, e.g., Rubin, 2003; Dai et al., 2004].

[39] Under these assumptions and definitions, the longitudinal macrodispersion coefficient is given by [Rubin, 1995]:

  • equation image

where t is elapsed traveltime, τj = tUj, and the rest of the coefficients are given in Table 1. The parameters in Table 1 are expressed in terms of length scales that can be identified through an analysis of field data. An analysis of the spatial variability of the log conductivity within each of the facies will yield the parameters m1, m2, σ12 and σ22, and the integral scales, λ1 and λ2. Dai et al. [2005] presented, in their equation (23), a generalized form of (18), applicable for higher-order hierarchies.

Table 1. Coefficients for Equation (18)
jαjηjσj2
1λ1P2σ12
2equation image1P(1 − P12
3λ2(1 − P)2σ22
4equation image1P(1 − P22
5λI(m1m2)2P(1 − P)

[40] Each of the terms in (18) identifies a different mechanism contributing to solute macrodispersion in saturated groundwater flow. The j = 1,3 terms represent the effects of spatial variability of the permeability within each of the facies, whereas the j = 2,4 terms represent the effects of the irregular distributions of the facies in space. The j = 2,4 terms contribute to macrodispersion even in the absence of differences between the mean log conductivities of the two facies. The effects arising from the difference in log conductivities of the two facies are captured by the j = 5 term. An extended discussion of this topic is provided by Rubin [1995] and Dai et al. [2004].

[41] The main challenge in application of (18) for subsurface transport modeling is to estimate its parameters, and in particular, the indicator statistics, from field data. Several methods for analyzing the spatial variability of indicators were reported. Hubbard and Rubin [2000], Hubbard et al. [2001], and Chen and Rubin [2003] have used GPR technology to determine the intrafacies variability. Lunt et al. [2004a, 2004b] demonstrated that GPR can be used for large-scale facies delineation and hence for identification of the spatial variability of the indicators.

[42] The possibility of estimating λI from the river morphology is explored here, looking at an assemblage of unit bar deposits and cross-bar channel fills as an example. We adopted an exponential spatial covariance for I, based on our discussion of the variability of unit lengths provided in section 2. We denoted the average length (in the horizontal plane) and volume fraction of the facies k by equation imagek,1 and Pk. Under these conditions, the integral scale λI in the horizontal direction, being equal to 1/3 of the range, is given by:

  • equation image

for any k. In deriving (19), it is recalled that in the exponential case, ϕ of equation (5) is equal to 3 and the integral scale is equal to 1/3 of the correlation range. Introducing (15) and (16) into (19), we obtain

  • equation image

Using equation (18) and (20) and the coefficients given in Table 1, it can be seen how the river planform geometry can be related to the longitudinal macrodispersion in a compound bar deposit, composed of unit bar and channel fill deposits. Different macrodispersion coefficients can be assigned to units with different proportions.

[43] For demonstration, we consider a case built in part on parameters from the Sagavanirktok River, including β1 = 4, β2 = 0.24, m1m2 = ln(1.2), σ12 = σ22 = 1.0, and U = 0.03 m/d. λ1 = 0.5 m and λ2 = 2.0 m are assumed values. Figures 7a and 7b show DL for P = 0.95 (as in reality), and 0.6, respectively, with P representing the volumetric fraction of the unit bar deposits.

image

Figure 7. Variation of DL with time according to equation (17). The proportion of unit bar deposits, P, simulated in Figure 7a is 0.95 and in Figure 7b is 0.6. CWa denotes average channel width.

Download figure to PowerPoint

[44] To better understand the results plotted in Figure 7, consider the large time, asymptotic limit of DL [Rubin, 1995, equation (19)]:

  • equation image

Equation (21) shows that DL depends linearly on the various length scales through αj (see Table 1). λI increases in value as CWa increases, as shown in equation (20), and this leads to a larger DL. Because CWa determines the length of unit bar deposits, an increase in CWa will correspond to an increase in equation imagek,1. Following (19), this will result in a larger integral scale and a larger macrodispersivity. DL increases as P decreases from 0.9 to 0.6, due to a more frequent occurrence of contrasts in hydraulic conductivity, as well as the increase in volumetric fraction of the facies with a larger integral scale. As Table 1 indeed shows, P = 0.6 leads to larger impact of the terms j = 2, 4 and 5.

[45] The macrodispersion coefficient depends on time, and its asymptotic, large time limit is attained only after 200 and 2000 days in our examples. For a given P, the early time behavior does not depend on the magnitude of CW. From Figure 7, the effect of differences in CWa starts to become noticeable only after 1000 days. If the mean velocity is 0.03 m/d, then after 1000 days the center of the plume would not have traveled more than 30 m, and would be in the same unit bar deposit for all CWa considered.

[46] Following equation (6), we have that equation image2,1 = (1 − P)equation image1,1/P. With P = 0.6, the cross-bar channels are longer, compared to the case of P = 0.95, which is why the macrodispersion coefficient would keep growing.

5. Discussion and Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information

[47] Depositional processes in river channels create complex heterogeneity, which is characterized by multiple length scales. One way to model such heterogeneity is through a hybrid geostatistical model that combines discrete space random functions, in the form of indicators, and continuous variables. The indicators in this hybrid model describe the juxtaposition of the various facies, whereas the continuous random variables describe the distribution, within the facies, of variables such as permeability. Models built on this concept were presented by Rubin [1995, 2003], Rubin and Journel [1991], Dai et al. [2004, 2005], and Ritzi et al. [2004].

[48] In this study, we employ the hybrid geostatistical models to characterize the spatial variability of the river deposits, on one hand, and to develop macrodispersion coefficients for solute transport, on the other. Thus we end up with a link between the river channel geometry and subsurface solute transport.

[49] On the basis of data collected at the Sagavanirktok river in Alaska [Lunt et al., 2004a, 2004b], it is shown that a geostatistical model for the river deposits can be identified based on GPR surface surveys, augmented by measurements taken at trenches. The link between the hybrid model and the GPR survey is based on identifying the distribution of facies lengths using GPR, on one hand, and an analysis of the relationship between the statistical properties of these lengths and geostatistical models, on the other [Carle and Fogg, 1996, 1997; Ritzi, 2000].

[50] An analysis of the length to thickness ratios of bed forms and deposits from several rivers (Figure 5) shows that the dimensions of bed forms (and hence deposits) vary in different sized rivers, as would be expected. The data suggests that for rivers of all sizes and bed material types, the wavelength-to-height ratio of bed forms is very similar to the length-to-thickness ratio of deposits. This suggests that the dimensions of subsurface deposits and the relative dimensions of different scales of strataset can be predicted from a limited amount of information about the formative river. This has potentially wide-reaching implications, because the different scales of strataset in the subsurface determine important hydrogeologic parameters such as macrodispersion coefficients for subsurface solute transport. This idea is demonstrated using data from the Sagavanirktok river study, for the case of transport in a compound bar deposit composed of unit bar and channel fill deposits.

[51] The field examples used here are primarily from braided rivers and their deposits. However, the relationships between channel geometry and deposit dimensions presented here will hold for a wide variety of fluvial settings. Subsurface studies commonly take great pains to distinguish the deposits arising from different fluvial planforms, which must be interpreted from sparse data. Generally, it is difficult to distinguish meandering from braided river deposits from subsurface data [Bridge and Tye, 2000; Bridge, 2003; Lunt and Bridge, 2004a, 2004b] and the only true diagnostic is the recognition of a complete braid bar, which is impossible to do using core data alone. However, measurements of unit bar length and channel width from field, experimental and theoretical studies [e.g., Lewin, 1976; Tubino et al., 1999; Lanzoni, 2000] indicate that the ratio between these lengths in meandering rivers is similar to those of braided rivers. Therefore, although our discussion is based on braided river deposits, we believe it applies equally to meandering river deposits.

[52] Our approach relies upon the assumption that the river channel geometry has not varied significantly over the recent past, so that the formative river channel geometry would have been similar to that observed today. Application of our methods to subsurface deposits that were formed by rivers that are significantly different from their present-day counterparts requires information about the width and depth of formative channels, which must be determined from geophysical or borehole data in representative locations. Bridge and Tye [2000] presented a method for deriving formative channel dimensions from subsurface data.

[53] To test our proposed approach further, a comprehensive database is needed to allow analysis of the relationship between river channel geometry and deposit geometry in a wide range of environments, using common tools and measures. With such a database, it could be possible to borrow correlation models from well-documented river valleys for use in ungauged, or minimally sampled basins, and to derive statistical priors for parameters of fluvial deposits such as macrodispersion. The development of such a database requires a unified approach for characterizing river basins and fluvial deposits, such as proposed here.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information
  • Allen, J. R. L. (1976), Computational models of dune time-lag: General ideas, difficulties and early results, Sediment. Geol., 15, 153.
  • Allen, J. R. L. (1982), Sedimentary Structures: Their Character and Physical Basis, vol. 1, 593 pp., Elsevier, New York.
  • Ashmore, P. E. (1982), Laboratory modeling of gravel braided stream morphology, Earth Surf. Processes, 7, 22012225.
  • Ashmore, P. E. (1991), How do gravel-bed rivers braid? Can. J. Earth Sci., 28, 326341.
  • Ashworth, P. J., J. L. Best, J. Peakall, and J. A. Lorsong (1999), The influence of aggradation rate on braided alluvial architecture: Field study and physical scale modelling of the Ashburton River gravels, Canterbury Plains, New Zealand, in Fluvial Sedimentology VI, edited by N. D. Smith, and J. Rogers, Spec. Publ. Int. Assoc. Sedimentol., 28, 333346.
  • Best, J. L., P. Ashworth, C. Bristow, and J. Roden (2003), Three-dimensional sedimentary architecture of a large, mid-channel sand braid bar, Jamuna River, Bangladesh, J. Sediment. Res., 73, 516530.
  • Bridge, J. S. (2003), Rivers and Floodplains, 491 pp., Blackwell, Malden, Mass.
  • Bridge, J. S., and J. L. Best (1997), Preservation of low-amplitude planar laminae due to migration of low-relief bed waves over aggrading upper-stage plane beds: Comparison of experimental data with theory, Sedimentology, 44, 253262.
  • Bridge, J. S., and I. A. Lunt (2006), Depositional models of braided rivers, in Braided Rivers, edited by G. S. Sambrook Smith et al., Spec. Publ. Int. Assoc. Sedimentol., 36, 1150.
  • Bridge, J. S., and R. S. Tye (2000), Interpreting the dimensions of ancient fluvial channel bars, channels, and channel belts from wireline-logs and cores, AAPG Bull., 84, 12051228.
  • Bridge, J. S., R. E. L. Collier, and J. Alexander (1998), Large-scale structure of Calamus river deposits revealed using ground-penetrating radar, Sedimentology, 45, 977985.
  • Bristow, C. S. (1993), Sedimentary structures exposed in bar tops in the Brahmaputra River, Bangladesh, in Braided Rivers, edited by J. L. Best, and C. S. Bristow, Geol. Soc. Spec. Publ., 75, 277289.
  • Cant, D. J., and R. G. Walker (1978), Fluvial processes and facies sequences in the sandy braided south Saskatchewan River, Canada, Sedimentology, 25, 625648.
  • Carle, S. F., and G. E. Fogg (1996), Transition probability-based indicator statistics, Math. Geol., 28, 453476.
  • Carle, S. F., and G. E. Fogg (1997), Modeling spatial variability with one- and multidimensional continuous Markov chains, Math. Geol., 29(7), 891918.
  • Chen, J., and Y. Rubin (2003), An effective Bayesian model for lithofacies estimation using geophysical data, Water Resour. Res., 39(5), 1118, doi:10.1029/2002WR001666.
  • Collinson, J. D. (1996), Alluvial environments, in Sedimentary Environments and Facies, 3rd ed., edited by H. G. Reading, pp. 3782, Elsevier, New York.
  • Dai, Z., R. W. Ritzi Jr., C. Huang, Y. Rubin, and D. F. Dominic (2004), Transport in heterogeneous sediments with multimodal conductivity and hierarchical organization across scales, J. Hydrol., 294(1–3), 6886.
  • Dai, Z., R. W. Ritzi Jr., and D. F. Dominic (2005), Improving permeability variograms with transition probability models of hierarchical sedimentary architecture derived from outcrop analog studies, Water Resour. Res., 41, W07032, doi:10.1029/2004WR003515.
  • Davis, J. M., R. C. Lohmann, F. M. Phillips, and J. L. Wilson (1993), Architecture of the Sierra Ladrones Formation, central New Mexico: Depositional controls on the permeability correlation structure, Geol. Soc. Am. Bull., 105, 9981007.
  • Davis, J. M., J. L. Wilson, F. M. Phillips, and M. B. Gotkowitz (1997), Relationship between fluvial bounding surfaces and the permeability correlation structure, Water Resour. Res., 33, 18431854.
  • Deutsch, C., and A. Journel (1998), GSLIB:Geostatistical Software Library and User's Guide, 2nd ed., Oxford Univ. Press, New York.
  • Doeschl-Wilson, A. B., and P. E. Ashmore (2005), Assessing a numerical cellular braided-stream model with a physical model, Earth Surf. Processes Landforms, 30, 519540.
  • Engelund, F., and J. Fredsoe (1982), Hydraulic theory of alluvial rivers, Adv. Hydrosci., 13, 187215.
  • Fredsoe, J. (1978), Meandering and braiding of rivers, J. Fluid Mech., 84, 609624.
  • Fujita, Y., and Y. Muramoto (1985), Studies on the process of development of alternate bars, Bull. Disaster Prev. Res. Inst., 35, 5586.
  • Gelhar, L. W. (1993), Stochastic Subsurface Hydrology, 390 pp., Prentice-Hall, Upper Saddle River, N. J.
  • Hickson, T. A., B. A. Sheets, C. Paola, and M. Kelberer (2005), Experimental test of tectonic controls on three-dimensional alluvial facies architecture, J. Sediment. Res., 75, 710722.
  • Hubbard, S., and Y. Rubin (2000), Integrated hydrogeological-geophysical site characterization techniques, J. Contam. Hydrol., 45, 334.
  • Hubbard, S., J. Chen, J. Peterson, and Y. Rubin (2001), Hydrogeological characterization of the South Oyster Bacterial transport site using geophysical data, Water Resour. Res., 37(10), 24312456.
  • Johnson, N. M. (1995), Characterization of alluvial hydrostratigraphy with indicator variograms, Water Resour. Res., 31, 32173227.
  • Johnson, N. M., and S. J. Dreiss (1989), Hydrostratigraphic interpretation using indicator geostatistics, Water Resour. Res., 25, 25012510.
  • Jol, H. M., and C. S. Bristow (2003), Ground Penetrating Radar in Sediments, Geol. Soc. Spec. Publ. 211, 326 pp.
  • Jordan, D. W., and W. A. Pryor (1992), Hierarchical levels of heterogeneity in a Mississippi River meander belt and application to reservoir systems, AAPG Bull., 76, 16011624.
  • Julien, P. Y., and G. J. Klassen (1995), Sand-dune geometry of large-rivers during floods, J. Hydraul. Eng., 121, 657663.
  • Jussel, P., F. Stauffer, and T. Dracos (1994), Transport modeling in heterogeneous aquifers: 1. Statistical description and numerical generation of gravel deposits, Water Resour. Res., 30, 18031817.
  • Karim, F. (1999), Bed-form geometry in sand-bed flows, J. Hydraul. Eng., 125, 12531261.
  • Karssenberg, D., T. E. Törnqvist, and J. S. Bridge (2001), Conditioning a process-based model of sedimentary architecture to well data, J. Sediment. Res., 71, 868879.
  • Kleinhans, M. G. (2002), Sorting Out Sand and Gravel: Sediment Transport and Deposition in Sand-Gravel Bed Rivers, Neth. Geogr. Stud., vol. 293, 317 pp., R. Dutch Geogr. Soc., Utrecht, Netherlands.
  • Lane, S. N., J. H. Chandler, and K. Porfiri (2001), Monitoring river channel and flume surfaces with digital photogrammetry, J. Hydraul. Eng., 127, 871877.
  • Lane, S. N., R. M. Westaway, and D. M. Hicks (2003), Estimation of erosion and deposition volumes in a large gravel-bed, braided river using synoptic remote sensing, Earth Surf. Processes Landforms, 28, 249271.
  • Lanzoni, S. (2000), Experiments on bar formation in a straight flume: 1. Uniform sediment, Water Resour. Res., 36, 33373349.
  • Leclair, S. F., and J. S. Bridge (2001), Quantitative interpretation of sedimentary structures formed by river dunes, J. Sediment. Res., 71, 713716.
  • Lewin, J. (1976), Initiation of bed forms and meanders in coarse-grained sediment, Geol. Soc. Am. Bull., 87, 281285.
  • Lunt, I. A., and J. S. Bridge (2004), Evolution and deposits of a gravelly braid bar and a channel fill, Sagavanirktok river, Alaska, Sedimentology, 51, 415432.
  • Lunt, I. A., J. S. Bridge, and R. S. Tye (2004a), A quantitative, three-dimensional depositional model of gravelly braided rivers, Sedimentology, 51, 377414.
  • Lunt, I. A., J. S. Bridge, and R. S. Tye (2004b), Development of a 3-D depositional model of braided river gravels and sands to improve aquifer characterization, in Aquifer Characterization, edited by J. S. Bridge, and D. Hyndman, Spec. Publ. SEPM Soc. Sediment. Geol., 80, 139169.
  • Mackey, S. D., and J. S. Bridge (1995), Three dimensional model of alluvial stratigraphy: Theory and application, J. Sediment. Res., Sect. B, 65, 731.
  • Miall, A. D. (1996), The Geology of Fluvial Deposits, Springer, New York.
  • Mohrig, D., and J. D. Smith (1996), Predicting the migration rates of subaqueous dunes, Water Resour. Res., 32, 32073217.
  • Moreton, D. J., P. J. Ashworth, and J. L. Best (2002), The physical scale modelling of braided alluvial architecture and estimation of subsurface permeability, Basin Res., 14, 265285.
  • Mumpy, A. J., H. M. Jol, W. F. Kean, and J. L. Isbell (2006), Architecture and sedimentology of an active braid bar in the Wisconsin River based on 3D ground penetrating radar, in Stratigraphic Analysis Using Ground Penetrating Radar, edited by G. S. Baker, and H. M. Jol, Spec. Pap. Geol. Soc. Am., in press.
  • Nasner, H. (1978), Time-lag of dunes for unsteady flow conditions, paper presented at 16th International Conference on Coastal Engineering, Hafenbautech. Ges., Hamburg, Germany.
  • Neal, A. (2004), Ground-penetrating radar and its use in sedimentology: Principles, problems and progress, Earth Sci. Rev., 66, 261330.
  • Nelson, J. M. (1990), The initial instability and finite-amplitude stability of alternate bars in straight channels, Earth Sci. Rev., 29, 97115.
  • Nelson, J. M., and J. D. Smith (1989), Mechanics of flow over ripples and dunes, J. Geophys. Res., 94, 81468162.
  • North, C. P. (1996), The prediction and modeling of subsurface fluvial stratigraphy, in Advances in Fluvial Dynamics and Stratigraphy, edited by P. A. Carling, and M. R. Dawson, pp. 395508, John Wiley, Hoboken, N. J.
  • Paola, C. (2000), Quantitative models of sedimentary basin filling, Sedimentology, 47, suppl. 1, 121178.
  • Paola, C., and L. Borgman (1991), Reconstructing random topography from preserved stratification, Sedimentology, 38, 553565.
  • Paola, C., P. L. Heller, and C. L. Angevine (1992), The large-scale dynamics of grain-size variation in alluvial basins, 1: Theory, Basin Res., 4(2), 7390.
  • Parker, G. (1976), On the cause and the characteristic scales of meandering and braiding in rivers, J. Fluid Mech., 76, 457480.
  • Regli, C., P. Huggenberger, and M. Rauber (2002), Interpretation of drill core and georadar data of coarse gravel deposits, J. Hydrol., 255, 234252.
  • Ritzi, R. W. (2000), Behavior of indicator variograms and transition probabilities in relation to the variance in lengths of hydrofacies, Water Resour. Res., 36, 33753381.
  • Ritzi, R. W., D. F. Dominic, A. J. Slesers, C. B. Greer, E. C. Reboulet, J. A. Telford, R. W. Masters, C. A. Klohe, J. L. Bogle, and B. P. Means (2000), Comparing statistical models of physical heterogeneity in buried-valley aquifers, Water Resour. Res., 36, 31793192.
  • Ritzi, R. W., Z. Dai, and D. F. Dominic (2004), Spatial correlation of permeability in cross-stratified sediment with hierarchical architecture, Water Resour. Res., 40, W03513, doi:10.1029/2003WR002420.
  • Rubin, D. M., and R. E. Hunter (1982), Bed form climbing in theory and nature, Sedimentology, 29, 121138.
  • Rubin, Y. (1995), Flow and transport in bimodal heterogeneous formations, Water Resour. Res., 31, 24612468.
  • Rubin, Y. (2003), Applied Stochastic Hydrogeology, 416 pp., Oxford Univ. Press, New York.
  • Rubin, Y., and S. S. Hubbard (2005), Hydrogeophysics, 520 pp., Springer, New York.
  • Rubin, Y., and A. G. Journel (1991), Simulation of non-Gaussian space random functions for modeling transport in groundwater, Water Resour. Res., 27, 17111721.
  • Sambrook Smith, G. S., P. J. Ashworth, J. L. Best, J. Woodward, and C. J. Simpson (2005), Morphology and facies of sandy braided rivers: Some considerations of scale invariance, in Fluvial Sedimentology VII, edited by M. D. Blum, S. B. Marriott, and S. F. Leclair, Spec. Publ. Int. Assoc. Sedimentol., 35, 145158.
  • Schumm, S. A., M. P. Mosley, and W. E. Weaver (1987), Experimental Fluvial Geomorphology, John Wiley, Hoboken, N. J.
  • Sheets, B. A., T. A. Hickson, and C. Paola (2002), Assembling the stratigraphic record: Depositional patterns and time-scales in an experimental alluvial basin, Basin Res., 14, 287301.
  • Simons, D. B., E. V. Richardson, and C. F. Nordin (1965), Sedimentary structures generated by flow in alluvial channels, Spec. Publ. Soc. Econ. Paleontol. Mineral., 12, 3452.
  • Skelly, R. L., C. S. Bristow, and F. G. Ethridge (2003), Architecture of channel-belt deposits in an aggrading shallow sandbed braided river: The lower Niobrara River, northeast Nebraska, Sediment. Geol., 158, 249270.
  • Strebelle, S. (2002), Conditional simulation of complex geological structures using multiple-point statistics, Math. Geol., 34(1), 121.
  • Sun, T., P. Meakin, and T. Jossang (1996), A simulation model for meandering rivers, Water Resour. Res., 32, 29372954.
  • Tubino, M., R. Repetto, and G. Zolezzi (1999), Free bars in rivers, J. Hydraul. Res., 37, 759775.
  • Vanoni, V. A. (1974), Factors determining bed forms of alluvial streams, J. Hydraul. Div. Am. Soc. Civ. Eng., 100, 363377.
  • Webb, E. K., and M. P. Anderson (1996), Simulation of preferential flow in three-dimensional, heterogeneous conductivity fields with realistic internal architecture, Water Resour. Res., 32, 533545.
  • Weissmann, G. S., and G. E. Fogg (1999), Multi-scale alluvial fan heterogeneity modeled with transition probability geostatistics in a sequence stratigraphic framework, J. Hydrol., 226, 4865.
  • Westaway, R. M., S. N. Lane, and D. M. Hicks (2000), The development of an automated correction procedure for digital photogrammetry for the study of wide, shallow, gravel-bed rivers, Earth Surf. Processes Landforms, 25, 209226.
  • Wilbers, A. W. E., and W. B. M. ten Brinke (2003), The response of subaqueous dunes to floods in sand and gravel bed reaches of the Dutch Rhine, Sedimentology, 50, 10131034.
  • Willis, B. J. (1993), Ancient river systems in the Himalayan foredeep, Chinji village area, northern Pakistan, Sediment. Geol., 88, 176.
  • Willis, B. J., and C. D. White (2000), Quantitative outcrop data for flow simulation, J. Sediment. Res., 70, 788802.
  • Wooldridge, C. L., and E. J. Hickin (2005), Architecture and evolution of channel bars in wandering gravel-bed rivers: Fraser and Squamish rivers, British Columbia, Canada, J. Sediment. Res., 75, 844860, doi:10.2110/jsr.2005.066.
  • Wu, F.-C., and T.-H. Yeh (2005), Forced bars induced by variations of channel width: Implications for incipient bifurcation, J. Geophys. Res., 110, F02009, doi:10.1029/2004JF000160.
  • Yalin, M. S. (1992), River Mechanics, 219 pp., Elsevier, New York.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Geometry of River Deposits and Its Relation to River Channel Geometry and Dynamics
  5. 3. Transition Probabilities for Characterization of River Deposits
  6. 4. Scaling and Solute Macrodispersion in Braided River Deposits
  7. 5. Discussion and Conclusion
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
wrcr10756-sup-0001-t01.txtplain text document0KTab-delimited Table 1.

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.