Prograding and Retrograding Hypo- and Hyper-Pycnal Deltaic Formations into Quiescent Ambients

Abstract

Sediment transport from mountainous rivers into a quiescent ambient with simultaneous formation of deltas is reviewed. The focus is restricted to flow in vertical cross-sections with no changes perpendicular to the plane of the flow. The bed load transport in the river is derived for quasi-steady situations using sediment mass balance and the Mohr-Terzaghi shear stress-pressure relation with the angle of internal friction, $\varphi$, as the essential frictional parameter. The emerging model is a diffusion equation for the upper surface of the moving sediment layer and corresponding boundary conditions. Its diffusivity is expressible in terms of the hydraulic discharge, the densities of the sediment and the turbid water, the angle of internal friction and a parameter characterizing the bed-parallel sediment velocity in terms of the average velocity in the turbid layer. When this river flow enters quiescent water, two different classes of deltas can be formed. When the entering water is either neutrally buoyant or lighter than the ambient water, the sudden reduction in tractive force along the bed generates a conspicuous avalanching flow to depth. This leads to steep-sloped foreset deposits with delta fronts inclined by the angle of internal friction. Such so-called Gilbert-type deltas are governed by a jump requirement of the sediment flux across the shore line and the geometry of the receiving basin. When the inflowing discharge is denser than the receiving ambient water, it will dive down as a turbulent under-current. The basal sediment transport in this subaqueous density current is analogous to the subaerial case and again described by a diffusion equation with similarly determined diffusivity. The combined dual subaerial-subaqueous sedimenting process is mathematically very similar to a (generalized) Stefan problem, e.g. the freezing of an ice cover on a lake. We present (mostly analytical) solutions for (1) bedrock-alluvial transitions, (2) overtopping failure of a dam, (3) topset-foreset diffusion processes for hypo- and hyper-pycnal deltas. Laboratory experiments demonstrate the adequacy of the models.

This chapter is a near-reprint of ‘A tutorial on prograding and retrograding hypo- and hyper-pycnal deltaic formations into quiescent ambients’ [25]. Hutter thanks Prof. P. Rutschmann for granting permission of reproduction.

Appendices

Appendix A: Derivation of the Sediment Flux Boundary Condition at the Plunge Point of a Gilbert-Type Delta

In this appendix an explicit derivation of the flux boundary condition (31.42) at the plunge point of a hypopycnal delta will be given. The derivation follows Kostic and Parker [30] but in the notation of this chapter.

Consider Fig. 31.36. With reference to this figure the front surface of the foreset delta can be described as

$$\begin{aligned} \hat{\zeta }(X, t) = \hat{\zeta }_{s} - \tan \phi \left( X - s(t)\right) ,\quad \text {where}\quad \hat{\zeta }_{s} = \zeta (s(t), t). \end{aligned}$$

(31.167)

Fig. 31.36

Definition sketch for a Gilbert-type deltaic deposition on a non-erodible basement of slope angle$\alpha _{1}$. The origin of the $(X, Z)$-coordinates is at the intersection of the basement, $Z = b(X)$ and the lake surface at time $t = 0, Z = Z_{\ell }(0)$, the plunge point is at $[X = s(t), Z = \hat{\zeta }_{s}(t)]$ and the front of the wedge is at $[X = u(t), Z = \hat{\zeta }_{toe}(t)]$. The sediment flow through the plunge point from the topset is $q_{s}(s(t), t)$ and the conservation of mass of the sediment, expressed in formula (31.169) is explained in the inset. The river water depth at the plunge point is $h_{s}(t)$ and the level of the lake may vary with time, $Z = Z_{\ell }(t)$

If this is evaluated at the toe of the alluvial deposit,

$$\begin{aligned} \hat{\zeta }_{b} = \hat{\zeta }_{s} - \tan \phi \left( u(t) - s(t)\right) \end{aligned}$$

(31.168)

is obtained.

If conservation of mass is formulated for a sediment element as shown in the inset of Fig. 31.36, then one may deduce

$$\begin{aligned} \begin{array}{ll} n_{s}\displaystyle {\frac{\partial \, \hat{\zeta }}{\partial t}}\text {d}X &{}= \bar{q}_{s}(X) - \bar{q}_{s}(X + \text {d}X)\\ &{}\simeq \bar{q}_{s}(X) - \bar{q}_{s}(X) - \displaystyle {\frac{\partial \bar{q}_{s}}{\partial X}}\text {d}X = - \frac{\partial \bar{q}_{s}}{\partial X}\text {d}X , \end{array} \end{aligned}$$

or, since the solid volume fraction is assumed to be constant,

$$\begin{aligned} \frac{\partial \, \hat{\zeta }}{\partial t} = - \frac{\partial (\bar{q}_{s}/n_{s})}{\partial X} = - \frac{\partial q_{s}}{\partial X}. \end{aligned}$$

(31.169)

In the above, $\bar{q}_{s}$ is the sediment flux at a certain volume fraction, whereas $q_{s}$ is the corresponding effective flux.

An explicit expression for $q_{s}$ is obtained, if Eq. (31.169) is integrated from $X = s^{-}$ to $X = u(t)$. This integration is composed of an ‘integration’ from $X = s^{-}(t)$ to $X = s^{+}(t)$ plus the integration from $X = s^{+}(t)$ to $X = u(t)$. Thus,

$$\begin{aligned} q_{s} |^{s^{+}(t)}_{s^{-}(t)} = q_{s}(s^{+}(t)) - q_{s}(s^{-}(t)) = - \int _{s^{-}(t)}^{s^{+}(t)} \frac{\partial \hat{\zeta }}{\partial t}\,\text {d}\,X = 0. \end{aligned}$$

Here, the integral on the far right vanishes because of continuity requirements for $\hat{\zeta }(\cdot )$. It follows that the flux $q_{s}$ is continuous across the plunge point. Therefore, we may write

$$\begin{aligned} \begin{array}{l} \displaystyle {\int _{s^{-}(t)}^{u(t)} \frac{\partial q_{s}}{\partial X}}\text {d}X = \underbrace{q_{{s}_{\mid u(t)}}}_{=0} - q_{s^{-}\mid s(t)} = \int _{s^{-}(t)}^{u(t)}\,{\frac{\partial \, \hat{\zeta }}{\partial \,t}}\text {d}\,X \\ \qquad \qquad \qquad = -q_{s\mid s^{-}(t)} = -\displaystyle {\int _{s^{-}(t)}^{u(t)}} {\frac{\partial \hat{\zeta }}{\partial t}}\text {d}X, \end{array} \end{aligned}$$

(31.170)

in which integration can now be restricted to $X > s(t)$. Moreover, it was assumed that the sediment flux at the toe of the frontal surface of the delta vanishes, which is realistic. With $\hat{\zeta }$ as given in (31.167) one may write

$$\begin{aligned} \begin{array}{ll} \displaystyle \frac{\partial \,\hat{\zeta }}{\partial \,t} &{}= {\dfrac{\text {d}}{\text {d}\,t}}\hat{\zeta }\left( s^{-}(t), t\right) - {\dfrac{\partial }{\partial \,t}}\left( \tan \phi (X - s(t))\right) \\ &{}= \displaystyle {\frac{\text {d}}{\text {d}\,t}} \hat{\zeta }\left( s^{-}(t), t\right) + \tan \phi {\frac{\text {d}s}{\text {d}\,t}} - \frac{\text {d}(\tan \phi )}{\text {d}\,t}(X - s(t)) \end{array} \end{aligned}$$

(31.171)

in which

$$\begin{aligned} \begin{array}{ll} \displaystyle {\frac{\text {d}}{\text {d}\,t}}\left( \hat{\zeta }(s^{-}(t), t)\right) &{}= {\dfrac{\partial \,\hat{\zeta }}{\partial t}}_{{{\mid _{s(t)}}}} + {\dfrac{\partial \,\hat{\zeta }}{\partial X}}\dot{s}(t) \\ &{}= \displaystyle {\frac{\partial \,\hat{\zeta }}{\partial t}}_{{{\mid _{s(t)}}}} - \tan {\alpha _{1}}_{{{\mid _{s^{-}(t)}}}}\dot{s}(t). \end{array} \end{aligned}$$

(31.172)

Here, $\tan \alpha _{1}$ is the slope of the sediment bed in the topset of the plunge point. Substituting (31.172) into (31.171) and the resulting expression for $\partial \,\hat{\zeta }/\partial \,t$ into (31.170) yields

$$\begin{aligned} \begin{array}{l} {q_{s}}_{{{\mid _{s(t)}}}} = \displaystyle {\int _{s(t)}^{u(t)}}\left\{ {\dfrac{\partial \,\hat{\zeta }}{\partial \,t}}_{{ {\mid _{s(t)}}}} - (\tan \alpha _{1} - \tan \phi )\dot{s}(t) -(\tan \phi )^{\cdot }(X - s(t))\right\} \text {d}X \\ = \left\{ {{\dfrac{\partial \,\hat{\zeta }}{\partial \,t}}}_{{{\mid _{s(t)}}}} + (\tan \phi - \tan \alpha _{1}){\dfrac{\text {d} s}{\text {d}\,t}}\right\} \left( u(t) - s(t)\right) - \dfrac{1}{2}(\tan \phi )^{\cdot }(u(t) - s(t))^{2}. \end{array} \end{aligned}$$

(31.173)

The surface point $\hat{\zeta }_{\mid s(t)}$ is given by the level of the lake surface and the water depth above the sediment as follows: $\hat{\zeta }_{\mid s(t)} = Z_{\ell }(t) - h_{\mid s(t)}$. Consequently,

$$\begin{aligned} \frac{\partial \, \hat{\zeta }}{\partial t}_{{{\mid _{s(t)}}}} = \frac{\partial }{\partial t}\left( Z_{\ell } - h\right) _{{{\mid _{s(t)}}}} = \dot{Z}_{\ell }(t) - \frac{\partial h}{\partial t}_{{{\mid _{s(t)}}}}. \end{aligned}$$

(31.174)

Substituting this into (31.173) yields a first variant of the final formulae for $q_{s}$:

$$\begin{aligned} \begin{array}{ll} {q_{s}}_{{{\mid _{s(t)}}}} = \left\{ \left[ \left( \dot{Z}_{\ell } - {\dfrac{\partial \,h}{\partial \,t}}_{{{\mid _{s(t)}}}}\right) \right. \right. &{}\!\!\!\Biggl . + \left( \tan \phi - \tan \alpha _{1}\right) {\dfrac{\text {d} s}{\text {d}\,t}}\Biggr ]\left( u(t) - s(t)\right) \\ &{}\Biggl .-\dfrac{1}{2}\big (\tan \phi \big )^{\cdot }\big (u(t) - s(t)\big )^{2}\Biggr \}. \end{array} \end{aligned}$$

(31.175)

Sometimes it is more convenient to additionally use the trigonometric relation

$$\begin{aligned} \left( u(t) - s(t)\right) = \dfrac{\hat{\zeta } - \hat{\zeta }_{toe}}{\tan \phi }. \end{aligned}$$

(31.176)

We then obtain

$$\begin{aligned} \begin{array}{ll} {q_{s}}_{\mid \,s(t)} = \left\{ \left[ \left( \displaystyle {\frac{\dot{Z}_{\ell }(t) - {\dfrac{\partial \,h}{\partial \,t}}_{{{\mid _{s(t)}}}}}{\tan \phi }}\right) \right. \right. &{} + \Biggl .\left( 1 - {\dfrac{\tan \alpha _{1}}{\tan \phi }}\right) {\dfrac{\text {d} s}{\text {d}\,t}}\Biggr ]\left( \hat{\zeta }_{s} - \hat{\zeta }_{toe}\right) \\ &{} \left. -\displaystyle \frac{1}{2}(\tan \alpha _{1})^{\cdot }{\frac{\left( \hat{\zeta }_{s} -\hat{\zeta }_{toe}\right) ^{2}}{(\tan \phi )^{2}}}\right\} . \end{array} \end{aligned}$$

(31.177)

Even though $\tan \alpha _1 <(\ll ) \tan \phi$, it is not justified in general, to ignore the expression $\tan \alpha _{1}/\tan \phi$ in the above formulae. However, it is justified to ignore the term associated with $(\tan \phi )^{\cdot }$. Ignoring also $(\partial \,h/\partial \,t)_{\mid s(t)}$ and $\dot{Z}_{\ell }(t)$ yields

$$\begin{aligned} q_{s\mid s(t)} = \left( \hat{\zeta }_{s} - \hat{\zeta }_{toe}\right) \frac{\text {d}\,s(t)}{\text {d}\,t}. \end{aligned}$$

(31.178)

Formula (31.42) with $\sigma = 0$ is obtained from (31.175), if the river water depth is ignored, $(\partial \,h/\partial \,t)_{\mid s(t)}\simeq 0$ and $\tan \alpha _{1}$ is ignored in comparison to $\tan \phi$.

Appendix B: Characteristics of Error Functions

In this Appendix we collect a number of properties of mathematical expressions which are connected to the error function. These have been collected and/or derived in Carslaw and Jaeger (1959) [8]

• Definition of the error function and complementary error function

  $$\begin{aligned} \mathrm{{erf}}(x)&= {\frac{2}{\sqrt{\pi }}}\int _{0}^{x} \exp (-\xi ^2)\mathrm{{d}}\xi , \end{aligned}$$

  (31.179)
  $$\begin{aligned} \mathrm{{erfc}}(x)&= 1 - \mathrm{{erf}}(x) = {\frac{2}{\sqrt{\pi }}}\int _{x}^{\infty } \exp (-\xi ^2)\mathrm{{d}}\xi , \end{aligned}$$

  (31.180)

• The above definitions imply

  $$\begin{aligned} \begin{array}{lll} \mathrm{{erf}}(0) = 0, \quad &{} \mathrm{{erf}}(\infty ) = 1, \quad &{} \mathrm{{erf}}(-x) = -\mathrm{{erf}}(x),\\ \mathrm{{erfc}}(0) = 1, \quad &{} \mathrm{{erfc}}(\infty ) = 0, \quad &{} \mathrm{{erfc}}(-x) = 2 - \mathrm{{erfc}}(x). \end{array} \end{aligned}$$

  (31.181)

• Both,

  $$\begin{aligned} \mathrm{{erf}}\left( \frac{x}{2\sqrt{Dt}}\right) \quad \mathrm{{and}}\quad \mathrm{{erfc}}\left( \frac{x}{2\sqrt{Dt}}\right) \end{aligned}$$

  satisfy the diffusion equation

  $$\begin{aligned} \frac{\partial \,f}{\partial \,t} - D \frac{\partial ^{2}f}{\partial \,x^{2}} = 0. \end{aligned}$$

• The $n$th integral complementary error functions are defined as

  $$\begin{aligned} \mathrm{i}^{n}\mathrm{{erfc}}(x)&\qquad = \qquad \int _{x}^{\infty }\mathrm{i}^{n-1}\mathrm{{erfc}}\,\xi \mathrm{{d}}\xi , \quad n = 2, 3, 4, \dots \nonumber \\ \mathrm{i}^{0}\mathrm{{erfc}}(x)&\qquad = \qquad \mathrm{{erfc}}(x), \end{aligned}$$

  (31.182)
  $$\begin{aligned} \mathrm{i}\mathrm{{erfc}}(x)&\qquad = \qquad \int _{x}^{\infty }\mathrm{{erf}}\,\xi \,\mathrm{{d}}\xi \nonumber \\&\mathop {=}\limits ^{\text {integr. by parts}} \xi \mathrm{{erfc}}(\xi )|_{x}^{\infty } - \frac{1}{\sqrt{\pi }}\int _{x}^{\infty }\underbrace{(-2\xi )\exp (-\xi ^2)}_{\frac{\mathrm{{d}}}{\mathrm{{d}}\xi } \left( \exp (-\xi ^2)\right) }\mathrm{{d}}\xi \end{aligned}$$

  (31.183)
  $$\begin{aligned}&\qquad = \qquad - x\,\mathrm{{erfc}}(x) + \frac{\exp \,(-x^2)}{\sqrt{\pi }} . \end{aligned}$$

  (31.184)

• The function $\mathrm{{erfc}}(x)$ exhibits the following properties:

  • (1)
    $$\begin{aligned}&\hat{\zeta }_{s}^{(2)}(x, t) = \frac{A}{D}\sqrt{Dt}\,\mathrm{{ierfc}}\left( \frac{x}{2\sqrt{Dt}}\right) , \quad x \geqslant 0,\nonumber \\ - \qquad&\lim _{x \rightarrow \infty }\hat{\zeta }_{s}^{(2)}(x, t) \rightarrow 0, \nonumber \\ - \qquad&\frac{\partial \,\hat{\zeta }_{s}^{(2)}}{\partial \,x} = \frac{A}{2D}\left( \mathrm{{erf}}\left( \frac{x}{2\sqrt{Dt}}\right) - 1\right) \\ - \qquad&\zeta _{s}^{2}(x, t) \quad {\text {satisfies the diffusion equation}} \nonumber \\&\frac{\partial \,\hat{\zeta }_{s}^{(2)}(x, t)}{\partial \,t} - D \frac{\partial ^{2}\,\hat{\zeta }_{s}^{(2)}(x, t)}{\partial \,x^{2}} = 0,\nonumber \end{aligned}$$

    (31.185)
  • (2)
    $$\begin{aligned}&\hat{\zeta }_{s}^{(3)}(x, t) = \frac{A}{D}\sqrt{Dt}\,\mathrm{{ierfc}}\left( \frac{|x|}{2\sqrt{Dt}}\right) , \quad x \leqslant 0,\nonumber \\ - \qquad&\lim _{x \rightarrow -\infty } \hat{\zeta }_{s}^{(3)}(x, t) \rightarrow 0 ,\nonumber \\ - \qquad&\frac{\partial \,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,x} = \frac{A}{2D}\left( -\mathrm{{erf}}\left( \frac{|x|}{2\sqrt{Dt}}\right) + 1\right) \\ - \qquad&\hat{\zeta }_{s}^{(3)}(x, t) \quad {\text {satisfies the diffusion equation}} \nonumber \\&\frac{\partial \,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,t} - D \frac{\partial ^{2}\,\hat{\zeta }_{s}^{(3)}(x, t)}{\partial \,x^{2}} = 0. \nonumber \end{aligned}$$

    (31.186)