Lithologic Variations in Slope 

Development Theory

GEOLOGICAL SURVEY 

CIRCULAR 485
Lithologic Variations in Slope 

Development Theory

By Adrian E. Scheidegger

GEOLOGICAL SURVEY CIRCULAR 485

Washington 7964
United States Department of the Interior

STEW ART L. UDALL, SECRETARY

Geological Survey

THOMAS B. NOLAN, DIRECTOR

Free on application to fhe U.S. Geological Survey, Washington, D. C. 20242
CONTENTS

Page Page

Abstract ____________________________ 1 Slope with 

Introduction. ________________________ 1 Soft layer _______________ 5

The basic slope program _____________ 2 Soft bottom _____________ 5

Slope with Pore water pressure.---- 7

Caprock _________________ 2 References _____-____---_----_---_- 8

Resistant layer ___________ 5

ILLUSTRATIONS

Page 

Figure 1. Graph showing basic slope development _____________________________________ 2

2 6. Graph showing slope with 

2. Caprock________________________________________ 4

3. Resistant layer__________________________________ 6

4. Soft layer ______ ________________________________ 6

5. Soft bottom _______________.____-_-_-____--__--__ 7

6. Pore water pressure.____________________________ 7

TABLES

Page 

Table 1. Basic slope program _______________________________________________________ 3

2-6. Slope with 

2. Caprock.______________________________________________________ 4

3. Resistant layer_________________________________________________ 5

4. Soft layer ______________________________________________________ 6

5. Soft bottom ____________________________________________________ 7

6. Pore water pressure.___________________________________________ 8

III
Lithologic Variations in Slope Development Theory

By Adrian E. Scheidegger

ABSTRACT

This paper presents a comprehensive review and amplification of the writer's earlier slope development theory. In 

particular, the influence of lithology on evolving slope profiles is investigated and calculations are made for various 

conditions, such as presence of caprock, soft bottom, and 

hard and soft intermediate layers.

INTRODUCTION

The understanding of the basic principles 

involved in the development of slopes is crucial for the understanding of the evolution of 

a landscape.

There are basically two methods by which 

the development of a slope can be studied. 

The first is the inductive method, in which 

measurements in the field are made in the 

hope to discover regularities which can later 

be formulated as "laws." In very complex 

phenomena, this approach may be hopelessly 

confusing. The second method, the deductive 

one, considers only one variable at a time, 

and makes a reasonable assumption regarding the physical principles involved in its 

change. The consequences of the assumptions are then deduced and, finally, the deductions are tested by determining whether the 

predicted features are found anywhere in 

nature. As Wolman (1963) stated: "The emphasis on principles...directs attention to interrelationships and hopefully lessens the 

tendency to observe, measure, and record 

everything simply because it is there."

With a view to providing a theoretical basis 

for work in the field, the writer (Scheidegger, 

1961) proposed a simple mathematical model 

of slope erosion. The aim of the present 

paper is to elaborate on the earlier procedure by investigating the effect of lithologic 

variations along the slope profile upon the 

appearance of the latter, as calculated from 

the writer's theory.

The basic mathematical slope development 

model described earlier (Scheidegger, 1961) 

does not attempt to describe in detail the 

"microscopic" aspects of erosion; rather, it 

attempts to describe the overall effect of 

many contributing causes. It is assumed (a) 

that the speed, with which denudation proceeds 

is proportional to the declivity of the slope 

and (b) that the action of the denudation is 

normal to the tangent of the slope profile. 

These are very simple as sumptions, but it is 

believed that they may describe some actual 

phenomena correctly; they lead to a nonlinear partial differential equation.

In the present context, the term "slope" 

should be regarded in a very general sense. 

It may refer to a mountainside, to the ungraded part of a headwater stream, or to the 

profile of a promontory. Under conditions 

where the aforementioned assumptions are 

satisfied, the slopes will develop as calculated here. Hopefully, such conditions will be 

found in nature.

Before proceeding with the investigation of 

the influence of lithology in slope development theory, the deduction of the original 

partial differential equation will be recapitulated. In fact, the assumptions denoted by 

(a) and (b) above immediately lead to

(1)

where y is the vertical height above some 

base line, T is time, and x is the horizontal 

abscissa of the slope point under consideration (arbitrary units of x andy ). If a lithological resistance factor a is introduced, the 

differential equation becomes
LITHOLOGIC VARIATIONS IN SLOPE DEVELOPMENT THEORY

This equation is solved easily on an electronic computer by the same method as that 

applied to the basic differential equation.

The mathematical solution representing 

the denudation of an originally straight slope 

bank for a variety of conditions will be given 

here. A point of reference also will be provided by reproducing the basic solution (a= 1) 

of the slope problem.

THE BASIC SLOPE PROGRAM

In order to provide a point of reference 

for the calculations reported here, the wellknown case for the evolution of a straight 

slope bank consisting of homogeneous material will be presented first. For conformity 

with the results to be reported later, the 

solution of the differential equation has been 

recalculated for a slope bank with the following initial configuration:

y = 4* for 0<x<0.25 

y = 1 for 0.25^x^1.0 (3)

The steps in x were in units of 0.01. The 

time steps were originally chosen as 0.01, 

but these were modified by the computer 

program to satisfy the criterion of the characteristics. The present computer program 

has been described in detail previously by 

the writer (Scheidegger, 1962).

The results of the calculations, which will 

represent a reference solution, presenting 

the slope profile for the same time values as 

will be used later for the further calculations, are shown in table 1 and figure 1. The 

units for x (horizontal coordinate), y (vertical coordinate), and T (total time elapsed) 

are arbitrary, and may be so chosen as to 

suit a particular problem.

SLOPE WITH CAPROCK

The first modification of the basic slope 

development problem to be presented is that 

of a horizontal layer of caprock. This can 

easily be accomplished by setting in equation 2

a = 1.0 

a= 0.1

for 

for

Thus, the caprock is 10 times more resistant 

to being eroded than the body of the slope.

It is again assumed that initially, the slope 

represents the appearance of a straight slope 

bank as given by equation 3. As time proceeds, a characteristic profile develops, as 

is shown numerically in table 2 and graphically in figure 2.

The calculated profile is a very good representation of slope profiles that are found in

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 1. Graph showing basic slope development.
SLOPE WITH CAPROCK 

Table 1. Basic slope program

X

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

.66

.68

.70

.72

.74

.76

.78

.80

.82

.84

T = 0.0 T=0.015

0 0

.08 .010

.16 .035

.24 .071

.32 .12

.40 .18

.48 .24

.56 .32

.64 .39

.72 .47

.80 .55

.88 .63

.96 .71

1.00 .79

.87

.95

1.00

1.00

T= 0.030

0

.003

.014

.032

.058

.090

.13

.17

.22

.28

.34

.40

.48

.55

.63

.71

.79

.87

.94

1.00

1.00

1.00

T= 0.045 :

0

0

.0058

.017

.033

.054

.080

.11

.15

.18

.23

.27

.33

.38

.44

.50

.57

.64

.71

.78

.86

.94

1.00

1.00

1.00

r = 0.60

0

0

.0023

.0085

.019

.034

.053

.076

.10

.13

.16

.20

.24

.28

.33

.38

.43

.48

.54

.60

.66

.73

.80

.87

.94

1.00

1.00

1.00

T=0.075

0

0

0

.0041

.011

.022

.036

.053

.074

.097

.12

.15

.18

.21

.25

.29

.34

.38

.43

.48

.53

.58

.64

.70

.76

.82

.89

.96

1.00

1.00

1.00

T = 0.90

0

0

0

.0018

.0060

.013

.024

.038

.054

.073

.094

.12

.14

.17

.20

.24

.27

.31

.35

.39

.43

.48

.53

.58

.63

.68

.74

.80

.86

.92

.98

1.00

1.00

1.00

T=0.120

0

0

0

0

.0014

.0044

.0098

.018

.028

.041

.056

.073

.092

.12

.14

.16

.19

.21

.24

.27

.31

.34

.38

.42

.46

.50

.54

.58

.63

.68

.73

.78

.83

.88

.94

.99

1.00

1.00

1.00

T = 0.150

0

0

0

0

0

.0011

.0033

.0074

.014

.022

.032

.045

.059

.074

.092

.11

.13

.15

.18

.20

.23

.25

.28

.31

.35

.38

.41

.45

.49

.52

.56

.60

.65

.69

.73

.78

.83

.88

.93

.98

1.00

1.00

1.00

nature. If the profile were assumed to be 

symmetrical with regard to some vertical

axis, the form that 

that of a mesa.

would evolve would be
LITHOLOGIC VARIATIONS IN SLOPE DEVELOPMENT THEORY

O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

X

Figure 2. Graph showing slope with caprock. 

Table 2. Slope with caprock

X

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

T = 0 

(0 steps)

0

.080

.16

.24

.32

.40

.48

.56

.64

.72

.80

.88

.96

1.00

1.00

T= 0.015 

(96 steps)

0

.011

.035

.072

.12

.18

.24

.32

.39

.47

.55

.63

.71

.93

1.00

1.00

1.00

T= 0.030 

(192 steps)

0

.0031

.014

.033

.060

.092

.13

.18

.23

.28

.34

.41

.48

.55

.63

.72

.99

1.00

1.00

1.00

T = 0.045 

(288 steps)

0

0

.0062

.017

.034

.056

.082

.11

.15

.19

.23

.28

.33

.39

.44

.51

.57

.64

.73

1.00

1.00

1.00

T = 0.060 

(384 steps)

0

0

.0025

.0089

.020

.035

.055

.077

.10

.13

.17

.20

.24

.29

.33

.38

.43

.49

.55

.61

.67

.80

1.00

1.00

1.00

T= 0.075 

(480 steps)

0

0

0

.0043

.011

.022

.037

.055

.075

.099

.13

.15

.19

.22

.26

.30

.34

.38

.43

.48

.53

.59

.65

.71

.96

1.00
SLOPE WITH SOFT BOTTOM

SLOPE WITH RESISTANT LAYER

The influence of a horizontal resistant 

layer on the shape of a developing slope profile will now be calculated.

The following values for a in equation 2 

will be assumed:

a= 0.1 for 0.4<y <0.5

a = l .0 for all other values of y

Naturally, the operation begins with a straight 

slope bank as given in equation 3. The results of the calculation are shown numerically in table 3 and graphically in figure 3.

SLOPE WITH SOFT LAYER

The reverse condition to that considered 

in the previous paragraph is that of a slope

with a horizontal soft layer. It is assumed 

that

a = 1.0 for 0.4 ^y <0.5 

a « 0.1 otherwise

and equation 2 is recalculated for these 

values of a. The results are shown numerically in table 4 and graphically in figure 4.

SLOPE WITH SOFT BOTTOM

The condition of a slope with a soft bottom 

is now considered, and it is assumed that

a= 0.1 for 

a= 1.0 for

This condition is, in a way, the reverse of 

that representing a slope with caprock. The

Table 3. Slope with resistant layer

X

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

.52

.54

.56

T = 0 

(0 steps)

0

.080

.16

.24

.32

.40

.48

.56

.64

.72

.80

.88

.96

1.00

1.00

1.00

T = 0.015 

(96 steps)

0

.011

.035

.073

.12

.18

.25

.48

.52

.56

.60

.66

.73

.80

.87

.95

1.00

1.00

1.00

T= 0.030 

(192 steps)

0

.0031

.014

.033

.060

.092

.13

.18

.23

.45

.52

.57

.61

.66

.71

.76

.82

.89

.96

1.00

1.00

1.00

T= 0.045 

(288 steps)

0

0

.0062

.017

.034

.056

.082

.11

.15

.19

.23

.43

.52

.57

.62

.67

.71

.76

.81

.87

.93

.98

1.00

1.00

1.00

T = 0.060 

(384 steps)

0

0

.0025

.0089

.020

.035

.055

.077

.10

.13

.17

.20

.25

.45

.52

.57

.62

.67

.72

.76

.81

.86

.92

.97

1.00

1.00

1.00

T = 0.075 

(480 steps)

0

0

0

.0044

.012

.022

.037

.055

.075

.099

.13

.15

.19

.22

.27

.49

.53

.57

.62

.67

.72

.77

.81

.86

.91

.96

1.00

1.00

1.00
LITHOLOGIC VARIATIONS IN SLOPE DEVELOPMENT THEORY

Resistant layer

'XXXXXXXXXXXX " ~

O.I 0.2 0.3 0.4 0.5 O.I 0.2 0.3 0.4

Figure 3. Graph showing slope with resistant layer. Figure 4. Graph showing slope with soft layer.

Table 4. Slope with soft layer

T -0.03 T = 0.06 T= 0.09 T = 0.12 T = 0.15

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

0

.080

.16

.24

.32

.40

.48

.56

.64

.72

.80

.88

.96

1.00

1.00

0

.044

.11

.19

.27

.35

.40

.41

.44

.65

.75

.83

.91

.98

1.00

1.00

0

.028

.080

.15

.22

.30

.37

.40

.40

.42

.52

.72

.85

.94

1.00

1.00

0

.020

.060

.11

.18

.55

.33

.39

.40

.40

.41

.43

.61

.81

.95

1.00

1.00

0

.015

.046

.092

.15

.21

.29

.36

.40

.40

.40

.41

.42

.50

.70

.89

1.00

1.00

0

.011

.037

.075

.12

.18

.25

.32

.38

.40

.40

.40

.41

.41

.43

.59

.79

.96

1.00

1.00
LITHOLOGIC VARIATIONS IN SLOPE DEVELOPMENT THEORY

results of the calculations are shown numerically in table 5 and graphically in figure 5.

It is observed that the development of the 

slope is completely characterized by the decay in the bottom layer, which results in 

virtually parallel slope recession, the speed 

of which is determined by the speed of ero- 

.sion in the bottom layer.

SLOPE WITH PORE WATER PRESSURE

A final modification was made in the calculation procedure by setting

a= 1 - 0.9y

This represents a slope in which the resistance to erosion varies linearly from 

bottom to top; at the bottom it is lowest, at

0 O.I 0.2 0.3 0 

X

Figure 5. Graph showing slope with soft bottom. Figure 6. Graph showing slope with pore water pressure.

Table 5. Slope u'ith soft bottom

X

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

T =0

0

.080

.16

.24

.32

.40

.48

.56

.64

.72

.80

.88

.96

1.00

1.00

T =0.03

0

.0030

.015

.10

.25

.35

.43

.51

.59

.67

.75

.83

.91

.98

1.00

1.00

T = 0.06

0

0

.0025

.0092

.024

.17

.34

.46

.54

.62

.70

.78

.86

.94

1.00

1.00

T =0.09

0

0

0

.0019

.0065

.016

.47

.28

.44

.56

.65

.73

.81

.89

.97

1.00

1.00

T =0.12

0

0

0

0

.0015

.0048

.011

.025

.15

.35

.53

.67

.76

.84

.92

.99

1.00

1.00

T = 0.15

0

0

0

0

0

.0012

.0037

.0086

.017

.045

.25

.45

.63

.77

.87

.95

1.00

1.00
LITHOLOGIC VARIATIONS IN SLOPE DEVELOPMENT THEORY

Table 6. Slope with pore water pressure

X

0.00

.02

.04

.06

.08

.10

.12

.14

.16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

.52

.54

.56

.58

T =0

0

.080

.16

.24

.32

.40

.48

.56

.64

.72

.80

.88

.96

1.00

1.00

T=0.03

0

.0030

.014

.034

.062

.098

.14

.20

.26

.34

.44

.57

.73

.92

1.00

1.00

1.00

1.00

T=0.06

0

0

.0025

.0090

.020

.036

.057

.082

.11

.15

.19

.24

.29

.36

.43

.53

.67

.94

1.00

1.00

1.00

1.00

T=0.09

0

0

0

.0019

.0063

.014

.026

.040

.058

.080

.10

.13

.17

.20

.24

.29

.34

.41

.48

.57

.71

.95

1.00

1.00

1.00

1.00

T =0.12

0

0

0

0

.0015

.0047

.011

.019

.031

.045

.061

.081

.10

.13

.16

.19

.22

.26

.30

.35

.41

.48

.55

.66

.83

1.00

1.00

1.00

1.00

1.00

T =0.05

0

0

0

0

0

.0012

.0036

.0080

.015

.024

.035

.049

.065

.083

.10

.13

.15

.18

.21

.25

.28

.33

.37

.43

.49

.56

.65

.80

.99

1.00

the top highest. This condition could be correlated with the possibility of the slope being 

water soaked. Then, one might argue that 

the erodability of the material varies just in 

the manner indicated.

The results of the calculations are shown 

numerically in table 6 and graphically in 

figure 6. It is seen that the slope development is intermediate between that of a slope 

with caprock and that of a slope with soft 

bottom, just as would be expected.

REFERENCES

Scheidegger, A.E., 1961: Mathematical models of slope development: Geol.Soc. America Bull., v. 72, p. 37-50.

____1962, Some modifications of the slope 

development problem: Royal Astron. Soc. 

Geophys. Jour., v. 7, p. 40 53.

Wolman, M.G., 1963, Review of "Theoretical 

Geomorphology": Geograph. Rev., v. 53, 

no. 2, p. 331.

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