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Subsidence Modeling Techniques An Overview of Graphical,
Analytical,
and Numerical Methods
Paul E. Nelson
RESPEC Rapid City, South Dakota, USA
Fall 2005 Meeting 25 October 2005
Nancy, France
Technical Paper
SOLUTION MINING RESEARCH INSTITUTE
105 Apple Valley Circle Clarks Summit, Pennsylvania 18411
USA Country Code: 1 Voice: 570.585.8092 Fax: 570.585.8091
E-mail: [email protected] www.solutionmining.org
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ABSTRACT
Ground subsidence is a consequential result of underground
mining. Associated with ground subsidence is surface tilt and
extensional and compressional strains which may cause damage to
pipelines, roads, railways, buildings, and other structures.
Therefore, it is essential to understand and predict the effects
that an underground opening may have at the ground surface. The
earliest subsidence modeling methods used empirical relations
relating subsidence measurements to mine geometry and graphical
techniques to apply the empirical relation toward making subsidence
predictions. Other subsidence modeling methods, and most currently
used methods, are based on theoretical models of material response
(e.g., elasticity).
INTRODUCTION
One result of creating an underground excavation is vertical
subsidence at the ground surface. Associated with subsidence is
surface tilt and extensional and compressional strains which may
result in damage to pipelines, roads, railways, buildings, and
other structures. Therefore, it is essential to understand and
predict the effects that an underground opening may have at the
ground surface.
The earliest methods used to predict subsidence were developed
by the European coal mining
industry. What may be the earliest formula to predict ground
subsidence is [Kratzsch, 1983]:
coszv M= (1)
where:
vertical displacement at the ground surface
mine height
dip angle of the mining seam
zv
M
=
=
=
which states that the vertical subsidence is equal to the height
of a vertical section through the mining seam.
Subsidence prediction techniques have evolved significantly
since Equation 1 was derived.
The modern discipline of subsidence engineering is based on the
theories of solid mechanics and typically incorporate numerical
techniques. The increased sophistication has allowed modeling of
increasingly complex geometries and stratigraphies. Current
numerical techniques also allow the prediction of time-dependent
subsidence above excavations created in creeping materials.
The underlying assumptions used in the prediction of ground
subsidence can be based on
empirical relations or material behavior models such as elastic,
plastic, or stochastic theories,
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while the technique used to make subsidence predictions may be
graphical, analytical, numerical, or use influence functions. For
example, ground displacement predictions may be made by creating a
computer model of the underground opening and surrounding
stratigraphy. The computer model is then used to make subsidence
predictions based on elastic rock behavior. This paper is organized
to first give a basic understanding of the assumptions upon which
most subsidence modeling techniques are based and then discusses
the modeling techniques themselves.
Ground subsidence, as discussed in this paper, should be
differentiated from catastrophic
subsidence events such as sinkholes (catastrophic is used here
in the sense of the magnitude of surface displacements). Modern
subsidence modeling techniques assume that the rock mass behaves as
a continuum; that is, the rock beds deform uniformly. Sinkholes
result in discontinuous displacements that are much more difficult
to model.
EMPIRICAL METHODS
Empirical methods were developed in the coal mining regions of
Europe and are some of the earliest techniques used to predict
ground subsidence. The empirical method uses data gained from
observations (i.e., subsidence measurements) to develop an equation
or set of equations to make predictions about the effect of future
mining. In effect, an equation (the empirical relation) is fit to
past subsidence measurements. An example (one method of many) is
the Polish profile-curve method which uses the following
exponential equation to match subsidence data [Kratzsch, 1983]:
2
nr
z z Maxv v e
= (2)
In Equation 2, z Maxv is the maximum measured subsidence, r is
the distance from the mining panel center to the point where
subsidence is to be calculated, and n is expressed as:
2z MaxvnR c
= (3)
where:
radius of the critical area
average roof displacement.
R
c
=
=
The example given above fits an exponential equation to measured
subsidence data. However, other empirical relations may incorporate
logarithmic, quadratic, or trigonometric functions.
Empirical methods are typically developed for a particular
mining district with its own
unique mine geometry, stratigraphy, and rock properties and are
continually modified as new
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data are collected. Although an empirical relation may
accurately predict subsidence for a particular mining district, the
empirical relations are developed to suit the ground response for a
particular set of data (and stratigraphy). Therefore, the empirical
relations developed for one mining district may not accurately
predict subsidence at another region. Also, the empirical relations
cannot be developed until measurable subsidence has already
occurred. Thus the potential effects of subsidence as a result of
mining at a new site cannot be predicted accurately a priori.
THEORETICAL MODELS
Theoretical models involve the application of solid mechanics
(in the case of elasticity or plasticity) or statistics (in the
case of a stochastic model) to derive a set of equations describing
the displacement of a surface over an underground opening. The
theoretical models assume that the rock mass behaves as a
continuum. This assumption implies that the rock behaves as if it
is composed of infinitesimally small particles, all with the same
properties. Thus deformations are assumed continuous (i.e., no
discontinuities) throughout the material. However, inhomogeneous
materials and anisotropic material properties can be simulated with
some numerical techniques; the finite element method is an
example.
Elastic Theory
An elastic material will deform under a given load and return to
its original shape upon removal of the load. Although this often
does not exactly describe the behavior of a typically porous and
fractured material such as rock, an elastic model can often
reasonably simulate the ground surface response to the excavation
of an underground opening in many situations. In addition to
assuming that the rock behaves as a continuum, it is assumed that
volume is conserved. That is, the volume of the subsidence bowl
equals the closed volume of the mine or cavern.
Several elastic models exist dependent upon how the underground
excavation is treated. For example, Maruyama [1964] derives the
following equation from the closing of a volume in a semi-infinite
half-space:
( ) ( )1
2 2 2 2( , ) tan
2ub UVD UVD UV
Z x yDU D V D
=
+ + (4)
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where:
( , ) ultimate subsidence at any point ( , ) on the ground
surface
vertical closure of the opening
depth to the top of the opening
uZ x y x y
b
D
=
=
=
and:
2 2 2U V D = + + (5)
The notation is used to represent an equation of the form:
( ) ( ) ( ) ( ) ( )2 2 2 1 1 2 1 1, , , , ,f U V f U V f U V f U
V f U V= + (6) where:
1
2
1
2
2
2
2
2
c
c
c
c
WU X x
WU X x
LV Y y
LV Y y
=
= +
=
= +
(7)
where (Xc, Yc) are the coordinates of the center of the
underground opening represented by a parallelepiped where L and W
are its length and width, respectively. Equations 4 through 7 are
the basis of the subsidence calculations made by the Solution
Mining Research Institute (SMRI) sponsored subsidence-modeling
computer program SALT_SUBSID [Nieland, 1991].
The linear behavior of an elastic material allows the
superposition of subsidence predictions made by the closing of
multiple excavations. It is important to note that elastic theory
does not follow the rule of the angle of draw. That is, elastic
models predict finite displacements outside the area defined by the
angle of draw.
Plastic Theory
Because of discontinuities in the rock mass (e.g., fractures and
void spaces), rock typically does not behave as a perfectly elastic
material. This is because the discontinuities present in the rock
absorb some of the movement (in addition to the elastic response)
upon application of a load. This movement is then not recovered
upon removal of the load.
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An example of an equation derived for the prediction of
subsidence above a plastic rock body is given by Kratzsch [1983].
This equation is:
0.5 0.5
tanh tanh2z
aM x l x lv
H H
+ =
(8)
where:
ratio of vertical subsidence to mine closure
room height
depth to mine horizon
distance from the center of subsidence
opening width
a
M
H
x
l
=
=
=
=
=
and:
4
z mV =
(9)
where:
rate of subsidence
kinematic stiffness of the rock body.
z mV =
= (10)
Stochastic Theory
The stochastic theory is a statistical approach that treats the
rock body as a cohesionless medium. A rock mass that contains
numerous and pervasive fractures may behave as a material with zero
cohesion, much like a sand.
The stochastic theory states that when a volume of rock is
removed (an extraction element), it is replaced by an equal volume
of rock lying above. The resultant particle movement at the ground
surface is predicted by a bell-shaped probability curve
representing the subsidence profile. This is demonstrated by Figure
1. A probabilistic approach to predicting surface subsidence is not
a commonly used but may be the basis of certain particle flow
computer codes.
SUBSIDENCE MODELING TECHNIQUES
Subsidence modeling is based on either an empirical relation or
a theoretical model. However, there are several techniques for
utilizing the model or empirical relation to perform predictive
subsidence calculations. These include (1) graphicalthe use of
graphs, nomograms,
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Figure 1. Representation of Probabilistic Particle Movement in a
Cohesionless Material (After Kratzsch [1983]).
or gridded overlays to estimate subsidence; (2) influence
functions; (3) analyticalthe direct calculation of subsidence at a
point; and (4) numericalthe use of numerical models (e.g., finite
element).
Graphical Techniques
Graphical methods and empirical relations are some of the
earliest techniques developed to predict surface subsidence.
However, graphical techniques tend to be quite labor intensive. One
graphical approach employs a set of graphs from which key points on
the subsidence surface are located (Figure 2). Another graphical
technique uses a circular, sectored overlay to calculate the amount
of subsidence at a point by measuring the percentage of the mine
area that falls within each sector (Figure 3).
Influence Functions
Influence functions are based on several assumptionsone of which
is superposition which states that the subsidence at a point can be
calculated by summing the elementary subsidence troughs for each
and every extraction volume located within that points cone of
influence (defined by the angle of draw). The previous statement
follows another assumption which states that a subsidence trough is
delimited by outward extending lines which make some angle with
vertical (the angle of draw). Influence functions are also assumed
to possess rotational
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Figure 2. Graph (a) Developed for the Angle-of-Intersection
Method Used to Locate Key Points of the Subsidence Trough in the
Dip Direction (b) and in the Strike Direction (c) After Kratzsch
[1983].
symmetry and equivalence (all extraction elements of the same
size and at the same depth have an equal amount of influence at the
ground surface). Lastly, influence functions assume that volume is
conserved, which states that a specified volume of closure at depth
is expressed at the surface with an equal subsidence volume.
Influence functions derive their name from the concept that the
closure of an extraction
element results in a finite amount of subsidence (influence) at
a given point on the ground surface. The magnitude of the influence
(i.e., subsidence), kz, may be given as a function of the solid
angle ( ) , the radial distance from the point of measurement to
the center of the subsidence trough (d), or as the distance from
the extraction element to the point of measurement (f). These
functions are represented in Figure 4 where the angle noted by
represents the angle of draw. Influence functions are derived from
either empirical relations or a theoretical material model (e.g.,
elasticity).
(b) (c)
(a)
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Figure 3. Sectured Overlay Used to Predict Subsidence at Point P
(After Kratzsch [1983]).
Figure 4. Key Parameters Commonly Used to Describe the Influence
(Subsidence) at Point P as a Result of Excavating Volume dA.
d
f
P
dA
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An example of an influence function (which is a function of the
distance from the cavern axis, d) is provided by Reitze [2000] for
the prediction of subsidence ( ), ,zv x y z , at any point above a
solution-mined cavern.
( )2
2tan
, ,o u
Max
d
Z Zz zv x y z v e
= (11)
and
2tanMax
E
z o u
kVv a
Z Z= (12)
where:
maximum subsidence
distance from the cavern axis
depth of the cavern roof
depth of the cavern floor
limit angle
transmission factor (0 1)
convergence factor (0 1)
final volu
Maxz
o
u
E
v
d
Z
Z
a a
k k
V
=
=
=
=
=
=
=
= me.
The advantage of using an influence function is that the
extraction volume is discretized into small extraction elements.
Therefore, irregular mine or cavern geometries and multiple mining
levels are easily managed. A disadvantage of influence functions is
that they ignore the interaction between extraction elements unless
accounted for in the function itself.
Analytical Techniques
Analytical techniques simply solve for subsidence at a point by
substituting the appropriate values into an equation derived from a
material behavior model. Analytical techniques are similar to
influence functions, but violate one of the assumptions made by
influence functions; specifically, the assumption regarding the
angle of draw. Analytical techniques based on an elastic material
model predict subsidence outside the area delineated by the angle
of draw. Similar to influence functions, when superimposing
subsidence any interaction between excavations is ignored.
Analytical techniques are best used in conjunction with a
computer when calculating
multiple subsidence valuesat several hundred grid points while
developing a contour plot, for example. SMRIs subsidence modeling
program SALT_SUBSID uses this approach.
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Numerical Techniques
Numerical techniques include finite element, finite difference,
and boundary element methods. The numerical techniques noted above
involve subdividing (discretizing) the area of interest (the model
region) into discrete areas or volumes (elements or zones) to
create a mesh or grid. Suitable material properties are then
assigned to each of the elements (or zones) along with appropriate
boundary and initial conditions. Mines or caverns are represented
as void(s) in the model.
A typical numerical model involves the following tasks:
1. Discretization of the model region
2. Assign constitutive models and material properties to the
elements
3. Apply appropriate boundary conditions (e.g., fixed or
rollered)
4. Apply appropriate initial conditions (i.e., in situ stress
and temperature fields)
5. Step to a solution
6. Extract the results.
Calculating the subsidence is a simple task that involves
extracting the displacements of the points lying on the plane of
the model that represents the ground surface. Horizontal strains
can also be directly output from the computer model.
The advantage of numerical techniques is that complex geometries
and stratigraphies can be
accurately represented. Most numerical programs include elastic,
plastic, and viscoelastic (creep) constitutive models that can be
applied to individual elements. Thus material inhomogeneities can
be modeled. Directional anisotropy can also be modeled as well as
discontinuities such as faults (given appropriate properties).
Also, the interaction between caverns can be simulated which is not
possible with influence functions or analytical techniques.
A disadvantage of numerical techniques is the effort required to
generate a model, with some
complex geometries requiring a significant amount of time to
generate. Also, estimates of the stratigraphy and rock properties
may be absent or expensive to obtain. Of course, simplifications to
the model and assumptions of material properties can be made at the
expense of accuracy. However, it is possible to calibrate a
numerical model to subsidence measurements by adjusting model
parameters until a reasonable match to the measured data is
obtained.
Advances in modeling software have made many numerical packages
very user friendly,
making it possible for most anyone to generate a model and
perform a subsidence simulation. However, a significant amount of
experience is required to judge whether the modeling results
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accurately represent reality. It is easy to get results from a
numerical model; the difficulty is determining whether the results
are correct or not.
TIME DEPENDENCY
Up until now, little mention has been made of time-dependent
surface subsidence with the exception of numerical models which can
simulate the ground response as a result of salt creep. All
equations noted in regard to influence functions and analytical
techniques predict final or ultimate subsidence.
Influence functions and numerical techniques can be used to
predict time-dependent subsidence with certain assumptions. A key
assumption associated with making time-dependent subsidence
predictions is that the spatial distribution of surface subsidence
rates is proportional to the spatial distribution of surface
subsidence. This states that the amount of subsidence at specific
points of the ground surface is, at any time, proportional to the
ultimate subsidence. This approach allows the surface subsidence to
be predicted at any time by multiplying the ultimate subsidence by
a time-dependent function. This approach will be explained more
fully in the description of SALT_SUBSID that follows.
SALT_SUBSID
Equation 4 is used in SALT_SUBSID to predict the ultimate
surface displacement at any point. However, an opening created in
salt closes as some function of time. Therefore, Equation 4 is
modified as follows to account for this time-dependency:
( ) ( ) ( ), , ,uZ x y t Z x y G t= (13) where:
( )
( )0
0
1 ; if 1 1
1 ; otherwise
N
N
E tss
E tss
y t y eGt
y t y e
+ >= +
where:
0, , , model parameters
extraction ratio ( 1 if a solution mine)
time since excavation.
ssy y N
E
t
== =
=
The model parameters are modified to match existing subsidence
measurements or estimated when no subsidence measurements are
available. One method of estimating the
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SALT_SUBSID parameters is with the use of a numerical model,
such as a finite element simulation. Numerous case studies have
been given testifying to the effectiveness of using SALT_SUBSID to
model surface subsidence rates (see Ratigan [2000]; Van Sambeek
[2000]; Cartwright and Ratigan [2005].
SUMMARY
Subsidence modeling has evolved from the use of empirical
relations and graphical techniques to predict surface subsidence
unique to a particular mining district to the use of sophisticated
numerical models allowing for the prediction of horizontal and
vertical surface displacements above complex geometries and
stratigraphies. However, the use of less sophisticated modeling
software, such as SALT_SUBSID, has also been shown to be widely
adaptable to a variety of conditions while providing reasonable
subsidence predictions.
REFERENCES
Cartwright, M. J. and J. L. Ratigan, 2005. Case HistorySolution
Mining a Cavern That Intersects a Plane of Preferred Dissolution,
Solution Mining Research Institute Fall Meeting, Nancy, France,
October 25. Kratzsch, H., 1983. Mining Subsidence Engineering,
Springer-Verlag Berlin Heidelberg, New York, NY. Maruyama, T.,
1964. Statistical Elastic Dislocations in an Infinite and
Semi-Infinite Medium, Bulletin of the Earthquake Research
Institute, Tokyo University, Vol. 42, pp. 289368. Nieland, J. D.,
1991. SALT_SUBSID: A PC-Based Subsidence Model, Users Manual,
RSI-0389, prepared by RE/SPEC Inc., Rapid City, SD, for the
Solution Mining Research Institute, Woodstock, IL. Ratigan, J. L.,
2000. Anomalous Subsidence at Mont Belvieu, Texas, Solution Mining
Research Institute Fall Meeting, San Antonio, TX, October 1518.
Reitze, A., 2000. Prediction of Ground Movement Above Salt Caverns
Using Influence Functions, Solution Mining Research Institute Fall
Meeting, San Antonio, TX, October 1518. Van Sambeek, L. L., 2000.
Subsidence Modeling and Use of Solution Mining Research Institute
SALT_SUBSID Software, Solution Mining Research Institute Fall
Meeting, San Antonio, TX, October 1518.
