Okavango Fan, Botswana. Image from NASA; https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

1. CHAPTER 33: MODELING OF FLUVIAL FANS AND BAJADAS IN SUBSIDING BASINS The morphodynamics of fluvial fans and bajadas in subsiding basins are considered in this chapter. The fans studied here do not end in standing water. It is assumed that all the sediment fed into the basin is consumed in filling the hole created by subsidence. Okavango Fan, Botswana. Image from NASA; https://zulu.ssc.nasa.gov/mrsid/mrsid.pl Bajada in Glendale, USA From the Web.

2. SOME DEFINITIONS x = down-channel coordinate  = channel sinuosity xb = down-basin coordinate = x/ Bb = basin width (may change with xb) Lb = basin length Bbf = bankfull channel width  = subsidence rate The analysis presented here is based on the original work of Paola et al. (1992) and Heller and Paola (1992). Additional material may be found in Parker et al. (1998a, 1998b).

3. SUBSIDENCE RATES Subsidence can be driven by tectonic movement, compaction of sediment under its own weight or some combination of the two. Here tectonic subsidence is considered for simplicity. Long-term tectonic subsidence rates of 10 mm/year or more have been recorded (Leeder, 1999). A particularly well-documented case is Ridge Basin, a pull-apart basin along the San Andreas Fault Zone, California, USA (Allen and Allen, 1990). During the relevant period of active subsidence some 13.5 km of sediment was deposited at an estimated rate of 3 mm/year. The subsidence rate must have been at least near this value. Subsidence rates can be much smaller than this and still have profound effects on river profiles. The effect of a given subsidence rate on river profile in a basin increases with basin area. This is illustrated in more detail in the next slide.

4. ACCOMODATION SPACE, UNDERFILLING AND OVERFILLING OF BASINS As in the analyses of Chapters 24 and 25, it is assumed that the mean annual load of bed material sediment is transported by a bankfull flow continuing for fraction If of any year. The bed material load at the flood (bankfull) flow is Qtbf, and the mean annual bed material load is IfQtbf. In analogy to Chapter 25, it is assumed that sediment is carried in the channel(s) traversing the basin, but deposited uniformly across the basin width as a result of channel migration, avulsion and overbank deposition. Also in analogy to Chapter 25, it is assumed that for every 1 unit of bed material load deposited in the basin  units of wash load are deposited. Now let this sediment flow into a basin of area Ab subsiding at (constant, uniform) rate . The rate at which “accomodation space” (volume available to store sediment) is created by subsidence is given as Ab. Any deposit that formed would have pores, so the actual rate of creation of storage space for sediment is (1-p)Ab. Note that even a small subsidence rate  can create considerable accomodation space if basin area Ab is sufficiently large.

5. ACCOMODATION SPACE, UNDERFILLING AND OVERFILLING OF BASINS contd. When wash load is included, the annual supply of sediment available for deposition is given as If(1+)Qtbf. If If(1+)Qtbf = (1-p)Ab the basin is perfectly filled with sediment, resulting in no net vertical movement of the sediment surface, even though the basement continues to subside. If If(1+)Qtbf < (1-p)Ab then the basin underfills with sediment, and the sediment surface continues to move down, though at a rate that is less than basement subsidence. If on the other hand If(1+)Qtbf > (1-p)Ab then the basin overfills with sediment, and the sediment surface will move upward even though the basement continues to subside.

6. BASIN FILLING Perfect filling with no change in elevation of deposit surface Underfilling; deposit surface elevation moves downward in time Overfilling; deposit surface elevation moves upward in time.

7. EXNER EQUATION OF SEDIMENT CONSERVATION The formulation is the same as that of Chapter 25, except that allowance is made for subsidence. As in Chapter 4, let base denote some basement elevation, and let the subsidence rate  be given as Sediment is transported in the channel within bankfull width Bbf, but is allowed to deposit over the basin width Bb. The formulation for sediment conservation becomes Reducing with the relations it is found that or

8. EXNER EQUATION OF SEDIMENT CONSERVATION contd. As in Chapter 25, for every 1 unit of bed material load deposited it is assumed that  units of wash load is deposited in the basin, so that As a result the final form of Exner becomes

9. BOUNDARY CONDITIONS The boundary conditions on the Exner equation are as follows: Specified sediment feed rate at the upstream end of the basin: No escape of sediment from the basin: The above boundary condition implies a completely closed basin in terms of sediment. It can easily be modified to allow for some drainage of sediment through a river at the downstream end of the basin.

10. MORPHODYNAMIC FORMULATION: SAND-BED RIVERS The formulation in Chapter 24 based on the Engelund-Hansen (1967) sediment transport relation is also used here.

11. MORPHODYNAMIC FORMULATION: GRAVEL-BED RIVERS The formulation in Chapter 24 based on the Parker (1979) sediment transport relation is also used here; where p = 11.2, form* = 0.0487 and

12. CASE OF A STEADY-STATE BALANCE BETWEEN SEDIMENT DEPOSITION AND SUBSIDENCE Now let  be decomposed into a downstream value d(t) plus a deviation dev(x,t) from this downstream value; so that dev satisfies the boundary condition The Exner equation then becomes

13. CASE OF A STEADY-STATE BALANCE BETWEEN SEDIMENT DEPOSITION AND SUBSIDENCE contd. A steady state is realized for the conditions The Exner equation of sediment continuity then becomes The sediment bed material load declines downstream as it is consumed in filling the hole created by subsidence. The case corresponds to an underfilling basin, and the case corresponds to an overfilling basin.

14. STEADY-STATE BALANCE BETWEEN SEDIMENT DEPOSITION AND SUBSIDENCE contd. The effect of shift/avulsion across the width of the basin and flow intermittency is to greatly magnify the effect of subsidence as opposed to a hypothetical case where all the sediment deposits in the channel, and the river is always in flood. To see this, it is useful for the moment to set  and dd/dt = 0 and assume constant river width Bbf. (assumptions that are not generally used in the subsequent analysis). If If < 1 and deposition across the entire basin is allowed, Exner takes the form If, on the other hand the river is always in flood and sediment only deposits in the channel, Intermittency and basinwide deposition thus amplify  by the factor (Bb/Bbf)/If.

15. STEADY-STATE BALANCE BETWEEN SEDIMENT DEPOSITION AND SUBSIDENCE contd. Apply the upstream boundary condition to get Apply the downstream boundary condition to get

16. STEADY-STATE BALANCE BETWEEN SEDIMENT DEPOSITION AND SUBSIDENCE contd. The value of is computed as follows; or thus If the fan is of precisely the right length, then, drops to zero and the hole created by subsidence is perfectly filled by sediment. If Lb is not sufficiently long (e.g. if the basin is restricted at its downstream end), any excess sediment beyond that required to fill the hole causes the the basin to aggrade uniformly in space and at a constant rate in time (overfilled basin). If the basin is too long, or subsides sufficiently rapidly relative to sediment supply, can become negative (underfilled basin). Volume rate input of sediment into the hole Volume rate of creation of space to store sediment in the hole

17. A BAJADA UNDERGOING PISTON-STYLE SUBSIDENCE A bajada may have many sediment sources over a given width Bb. Piston-style subsidence is such that  is spatially constant. Bajada in western China. Image from NASA; https://zulu.ssc.nasa.gov/mrsid/mrsid.pl Bb

18. A BAJADA UNDERGOING PISTON-STYLE SUBSIDENCE contd. A bajada can be approximated as a rectangular basin, so that Bb = constant in xb. In such a case Qtbf refers to the sum of the load of all the rivers flowing into the bajada over width Bb. The subsidence rate  is assumed to be constant (piston-style subsidence). For this case the solution reduces to: That is, the bed material load declines linearly to zero at the end of the basin. Note that if subsidence perfectly balances sediment input then

19. A BAJADA FORMED BY SAND-BED RIVERS In a bajada formed by sand-bed rivers the wash load is likely to be silt. The relations for bankfull characteristics of sand-bed rivers were found to be or thus Between the relation for slope and the equation below it is found that

20. A BAJADA FORMED BY SAND-BED RIVERS contd. Recalling that S denotes down-channel bed slope, Now defining the relation can be cast in the form

21. A BAJADA FORMED BY SAND-BED RIVERS contd. Integrating yields the parabolic profile Since by definition dev(Lb,t) = 0, it follows that in which case the final solution takes the form

22. A BAJADA FORMED BY SAND-BED RIVERS contd. The final forms for a bajada formed by sand-bed rivers are as follows.

23. A BAJADA FORMED BY GRAVEL-BED RIVERS In a bajada formed by gravel-bed rivers the wash load is likely to be sand. The corresponding forms for a bajada formed by gravel-bed rivers are as follows.

24. AN AXISYMMETRIC FAN UNDERGOING PISTON-STYLE SUBSIDENCE Axially symmetric fan: Bb = fxb where xb is a radially outward coordinate from the fan vertex and f is fan angle. xb Axisymmetric fan in western China. Image from Internet.

25. AN AXISYMMETRIC FAN UNDERGOING PISTON-STYLE SUBSIDENCE contd. Bed material load declines parabolically down the fan.

26. A FAN FORMED BY A SAND-BED RIVER The relations for bankfull characteristics of sand-bed rivers are again or thus Between the relation for slope and the equation below it is found that

27. A FAN FORMED BY A SAND-BED RIVER contd. Recalling that S denotes down-channel bed slope, Now defining the relation can be cast in the form

28. A FAN FORMED BY A SAND-BED RIVER contd. Integrating yields the cubic profile If subsidence perfectly balances deposition, the downstream bed elevation can be used as the datum for the elevation profile; In which case the final solution takes the form

29. A FAN FORMED BY A SAND-BED RIVER contd. The final forms for a sandy fan are as follows.

30. A FAN FORMED BY A GRAVEL-BED RIVER The corresponding forms for a fan formed by a gravel-bed river are as follows.

31. SLOPE PROFILES OF FANS AND BAJADAS

32. ELEVATION PROFILES OF FANS AND BAJADAS

33. EVOLUTION TOWARD STEADY STATE A basin may not yet have evolved toward its steady-state profile. The morphodynamic problem takes the form where in general Bb is a function of xb and e.g. for sand-bed river(s) The boundary conditions are or thus An appropriate initial condition completes the problem. The extension to a basin traversed by gravel-bed river(s) is straightforward.

34. NUMERICAL FORMULATION OF EVOLUTION TOWARD STEADY STATE: CASE OF FAN TRAVERSED BY SAND-BED RIVER(S) Discretize the domain as follows: Handle upstream b.c. with a ghost node at i = 1 where sediment is fed in at rate Qtbf,feed Handle downstream b.c. as

35. NUMERICAL FORMULATION contd. Sample initial condition: at t = 0, An implementation for both sandy and gravelly fans is given in RTe-bookSubsidingFan.xls.

36. CALCULATION FOR A SAND FAN with RTe-bookSubsidingFan.xls The subsidence rate is 5 mm per year. The mean aggradation rate that could be achieved in the absence of subsidence is 3.77 mm per year. So the basin will underfill at the ultimate steady state.

37. After 50 years

38. After 500 years

39. After 500 years: close to ultimate steady-state response Note that at the final steady state surface elevation drops everywhere at a constant rate due to underfilling.

40. REFERENCES FOR CHAPTER 33 Allen, P. A. and Allen, J. R., 1990, Basin Analysis Principles and Applications, Blackwell Science, Oxford, U.K., 451 p. Heller, P. L., and Paola, C., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 2: Application to syntectonic, Basin Research, 4, 91-102. Leeder, M. L., 1999, Sedimentology and Sedimentary Basins From Turbulence to Tectonics, Blackwell Science, Oxford, U.K., 592 p. Paola, C., Heller, P. L., and Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins, 1: Theory, Basin Research, 4, 73-90. Parker, G., Paola, C., Whipple, W. and Mohrig, D., 1998a, Alluvial fans formed by channelized fluvial and sheet flow: Theory, Journal of Hydraulic Engineering, 123(10), 985-995. Parker, G., Paola, C., Whipple, W., Mohrig, D., Toro-Escobar, C., Halverson, M., Skoglund, T., 1998b, Alluvial fans formed by channelized fluvial and sheet flow: Application, Journal of Hydraulic Engineering, 124(10), 996-1004.