coupled 3d finite element modeling of surface processes...

International Symposium on Strong Vrancea Earthquakes and Risk Mitigation Oct. 4-6, 2007, Bucharest, Romania

COUPLED 3D FINITE ELEMENT MODELING OF SURFACE PROCESSES AND CRUSTAL DEFORMATION: A NEW TECHNICAL APPROACH AND PARAMETER STUDIES

D. Kurfeß 1, O. Heidbach 1

ABSTRACT We want to present a new technique to model the complexity of interactions between fluvial transport and geodynamic processes. The capabilities of numerical models of the Earth have immensely grown over the past years. On the one hand, several computer models simulat-ing surface processes (erosion and sedimentation) on geological scales have been devel-oped. On the other hand, simulation programs using the finite element method can give us a better understanding of complex mechanical processes within the earth. However, only few software models address the coupling between long-range mass redistribution on the Earth's surface and the complexity of crustal deformation. We are trying to fill this gap by a new ap-proach: we have currently developed a new 3D simulation tool combining landscape evolu-tion modeling and mechanical finite element modeling of the lithosphere. The technical con-cept behind this tool will be illustrated, and 3D model examples showing the active coupling between endogenous and exogenous processes will be presented. The influence of varying parameters, concerning both the geomechanical behavior and surface transport processes, on landform and tectonics will be demonstrated. Embedded in the work of the CRC 461-project A6, our research will give a better understanding of the origin of surface uplift and subsidence patterns in Vrancea.

INTRODUCTION The evolution of landscapes in nature is mainly controlled by surface processes (erosion and sedimentation) and tectonic processes as well as the interaction between these two (e.g., Burbank & Pinter, 1999, Bishop, 2007, and references therein). The transportation of eroded material by rivers over distances up to several thousand kilometers and the deposition of these sediments lead to long-range mass redistribution on Earth's surface. This changes the loads acting on the lithosphere, thus influencing tectonic processes. Vice versa, vertical movement of rock by tectonic processes changes Earth's topography that controls the pat-tern of channeled water flow, thus affecting river erosion as the most pervasive of erosive forces (Molnar, 2003). A better understanding of this interplay and feedback between ex-ogenous and endogenous processes is one of the key issues in geosciences (Molnar & Eng-land, 1990). Existing numerical landscape evolution models use different approaches to simulate the tec-tonic behavior of the subsurface (Coulthard, 2001). Some use a homogenous elastic plate to mimic the lithosphere (Braun & Sambridge, 1997, Garcia-Castellanos, 2002). Others include different rheologies that are either uniform for the complete subsurface or vary with depth (e.g., Garcia-Castellanos et al., 1997). A third group of models compute the deformations within the Earth by a 2D plane-strain approach on a vertical cross-section (e.g., Beaumont et al., 1992, Pysklywec, 2006, Burov & Toussaint, 2007).

1 Geophysical Institute, University of Karlsruhe, Hertzstrasse 16, 76187 Karlsruhe, Germany

D. Kurfeß & O. Heidbach 22

In our work the simulation of surface processes is combined with a truely three-dimensional model of the subsurface. This approach allows for spatially distributed material inhomogenei-ties, different rheologies, anisotropy, structures with complex three-dimensional geometry and faults with Coulomb friction. For this purpose we have developed a new simulation tool: CASQUS – a geoscientific software extension to the commercial finite element software

package ABAQUS. It integrates the erosion and sedimentation transport routines of the surface processes model CASCADE written by Braun & Sambridge (1997) into the powerful

ABAQUS/Standard finite element solver (ABAQUS, Inc., 2004). CASQUS allows for ge-omechanical analyses of complex models of the subsurface and additionally takes the feed-back effects of surface processes into account. We have started a process study to analyze the effect of different processes and model fea-tures on the sensitive coupling between erosion/sedimentation and tectonics. With CASQUS we quantify the influence of river network evolution at an escarpment on the mechanical re-sponse of an elastic lithosphere. We also study the influence of isostasy and the occurrence of faults on erosion rates. A better understanding of these basic interactions is important for the future application of CASQUS to the Vrancea region in Romania. Underneath the south-eastern Carpathians seismic tomography revealed a high-velocity body that can be identified as the slab of a for-mer subduction zone. In project A6 of CRC 461 we are working on a fully coupled 4D finite element model of this region including the Earth's mantle, the crust and surface processes (see Heidbach et al., 2007, in this volume). It will provide insights into the geodynamical state of the subduction. One of the problems is: Is the subduction slab already detached, or is it still coupled with the crust? CASQUS will give an answer to the question how much of the present-day uplift and subsidence rates can be explained by surface processes and the feedback to the subsurface. A comparison of GPS measurements and ero-sion/sedimentation rates computed by CASQUS will give an estimate of how much of the present day uplift/subsidence signal can be explained by surface processes only. In return we learn how much of this signal is originated from processes in the mantle and the coupling between slab and crust.

NUMERICAL MODELING TECHNIQUE

CASQUS combines the finite element software ABAQUS used for geomechanical analy-ses with the surface processes model CASCADE. These two parts of the CASQUS program

communicate with each other via the topography of the Earth's surface. ABAQUS recog-nizes changes in topography due to surface processes as changes of gravitational loads. It responds to these changes by deformations within the geomechanical model, leading to up-lift or subsidence of rocks, until the equilibrium of forces is reached. In return, CASCADE identifies changes in topography due to vertical movement of rocks and adapts the computa-tion of surface processes to the new conditions. So the interaction in both directions is di-rectly implemented in CASQUS, which makes a fully coupled, automatic simulation possible.

CASCADE's surface processes routines are integrated in ABAQUS via a user subroutine

that is compiled at the start of an ABAQUS analysis. No user input is needed during the analysis; no interruptions occur after the model has been defined once and the simulation has been started (Fig.1).

International Symposium on Strong Vrancea Earthquakes and Risk Mitigation

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Figure 1. Flow of information in CASQUS and program flow. CASQUS integrates the sur-

face processes model CASCADE into the finite element program ABAQUS in order to provide a simulation tool that is able to model the interaction between erosion/sedimentation and tectonic processes in an integrated approach.

Finite Element Modeling with ABAQUS

ABAQUS is a finite element analysis software product for the numerical solution of various types of mechanical and thermo-mechanical problems (ABAQUS, Inc., 2004). With the finite element method structures of airplanes are analyzed, for example, as well as the deforma-tion behavior of human bones or the stability of boreholes. The solution to these static me-chanical problems is always based on the solution of the equation describing the equilibrium of forces in a continuous medium:

∂σij/∂xj + fi = 0 , i,j=1,2,3 . (1)

Here σσσσ is the stress tensor, f is the body force per unit volume and x1, x2 and x3 are the co-ordinates in a Cartesian reference frame (Davis & Selvadurai, 1996). The finite element method solves this equation numerically by partitioning the model volume into smaller subvolumes, the so-called finite elements (Zienkiewicz, 1977). This discretiza-tion can be arbitrary and irregular, i.e., it is not bound to a rectangular or equidistant grid. So three-dimensional models including structures of geometrically complex shape can be ana-

lyzed. In commercial finite element software packages such as ABAQUS a wide range of rheologies is implemented, as well as inhomogeneous and anisotropic material properties and contact definitions. A contact definition is used where subvolumes interact on contact with each other; for example, to model Coulomb friction for a tectonic fault. A growing num-ber of geoscientists uses finite element software for geomechanical modeling on tectonic scales (Beaumont et al., 1992, Chéry et al., 2001, Khazaradze et al., 2002, Heidbach & Drewes, 2003, Chéry et al., 2004, Wu, 2004, Dyksterhuis et al., 2005, Hetzel & Hampel, 2005, Hampel & Hetzel, 2006, Hergert & Heidbach, 2006, Pysklywec, 2006, Buchmann & Connolly, 2007). CASCADE's Geomorphic Model For the mathematical formulation of surface processes in CASCADE Braun & Sambridge (1997) follow an approach that Kooi & Beaumont (1994) used before: They assume that on tectonic time scales and large spatial scales landscape evolution results from the interplay between two processes, long-range sediment transport by rivers and short-range gravita-tional mass movement (hillslope processes).


Both processes are simulated based on numerical solutions of differential equations. These equations are numerically implemented on irregular grids, which means that the same dis-cretization can be used that defines the Earth's surface in a tectonic finite element model. Another advantage of CASCADE compared to other landscape evolution models is the fact that it concentrates on first-order surface processes describing mass redistribution on spatial scales of the order of (101..103) km and time scales of (105..106) years. This coincides with

the scales that are used for tectonic ABAQUS models. The governing differential equa-tions behind the two types of surface processes implemented in CASCADE will shortly be in-troduced in the following; more details are given by Beaumont et al. (1992), Kooi & Beau-mont (1994), Braun & Sambridge (1997) and Beek & Braun (1998). Fluvial Transport Rivers incise the landscape and transport sediments from higher to lower ground where they are deposited. One can define a local equilibrium sediment carrying capacity qf

eqb as a measure of how much sediment a river is able to transport maximally given a point on the landscape. In CASCADE its value depends proportionally on the river discharge per unit width qr and on the slope in the direction of river drainage dh/dl, qf

eqb = - Kf qr dh/dl . (2) Here Kf is a nondimensional transport coefficient, giving consistent dimensions (volume per unit width per time) for the fluxes q. The fluvial discharge qr results from conservation of wa-

ter over the upstream catchment area AC, qr=∫AC νR dA, where νR is the mean precipitation

rate. The approach of Kooi & Beaumont (1994) includes a formulation for fluvial mass re-moval that is not a priori transport-limited like in some previous surface processes models (Willgoose et al., 1991a, Willgoose et al., 1991b), which means rivers are not forced to carry at capacity. Instead, the disequilibrium between the actual sediment flux qf of a river and its sediment transport capacity qf

eqb controls the rate of erosion (where qf<qfeqb) or sedimenta-

tion (where qf>qfeqb). Mathematically the temporal height change of a point of the landscape

is given by

∂h/∂t = - 1/lf (qfeqb-qf) , (3)

where lf is the erosion or deposition length scale. It is a measure of how easily the river sub-strate can be eroded or how fast sediments are deposited, thus affecting how fast the river locally tends towards equilibrium. In the numerical implementation of CASCADE the length scale for deposition is not a parameter, implying that it is much smaller than any channel segment length of the surface discretization. So in CASCADE streams never carry more load than their carrying capacity, i.e., the model of Braun & Sambridge (1997) is transport-limited in the sense that qf

eqb is the upper bound for qf. Hillslope Processes Observable fluvial transport only occurs where a significant amount of water drains off. Addi-tionally, hillslope processes that are mainly driven only by gravity can feed the rivers from their flanking slopes. In the CASCADE model the cumulative effect of the different types of these processes (such as weathering, slope wash, rain splash, mass wasting and soil creep) is implemented as a linear diffusion equation:

∂h/∂t = Ks ∇2h , (4)

where Ks as the diffusion constant is a function of both climate and lithology, controlling the rate of the short-range processes.


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PROCESS STUDY Model Setup In the following we will present 3D finite element models that serve to analyze the effects of isostasy and faults on the coupling between surface and tectonic processes. We will analyze three models: (1) Model A considers only surface processes and no mechanical response of the subsurface. (2) Model B also includes the mechanical response of an elastic lithosphere in combination with isostasy. (3) Model C is the same as model B with the only difference that it additionally considers the effect of a tectonic fault.

Figure 2. Model setup for the process study, shown as a 2x vertically exaggerated side view

of the 3D finite element model. Young's modulus E and density ρ for the elastic crust and the elastic lithospheric mantle are shown in the figure; Poisson's ratio is 0.25 for the complete lithosphere. Isostasy is modeled by elastic foundations that mimic the buoyancy of the asthenosphere, symbolized by springs. The vertical side boundary faces are fixed in normal direction, symbolized by rollers. The model as shown here is extruded by 100 km perpendicular to the viewing plane (in Y-direction). Slip on the fault is controlled by Coulomb friction with a friction coeffi-cient of 0.4. The fault changes its dip from ~60° at the front side (Y=0) shown here

to 74° at the back side (Y=100 km). The geometric setup of the 3D finite element models is illustrated as a side view in Fig.2. The initial topography of the models is dominated by a mountain range with a plateau limited by an escarpment. Little noise in elevation is added to the smooth topography to allow rivers to develop. The crust and the lithospheric mantle are implemented as elastic layers; the rheological parameters are given in Fig.2. Elastic foundations at the bottom of the litho-spheric mantle mimic the buoyancy of the asthenosphere. Once the models are loaded by gravity, compaction of the lithospheric material occurs. Addi-tionally, the plate bends to compensate the gravitational load of the mountain range (Fig.3). Bending stresses and the increased buoyancy below the mountain range now keep the model in mechanical equilibrium. Here we use the concept of a symmetrical model setup, where the Y-Z-plane defining the left side boundary of the model is the plane of symmetry. In order to save computation time we do not model the part located left of this plane of symme-try. This can be imagined as the left symmetric counterpart of the 200 km long plate shown in Fig.2 and 3. Like all side faces, this model boundary is fixed in normal direction. The reac-tion forces that act at this side mimic the mechanical effect of the missing second half of the plate. After this first step of the finite element analysis when gravity is compensated by isostasy and bending stresses, topography is nearly the same for all three finite element models (A, B, and C).


Figure 3. Finite element model in isostatic equilibrium (2x vertically exaggerated). Gravita-

tional loads are compensated by buoyancy and bending stresses. Erosion and sedimentation processes have generated a river network at the surface.

In the second step, CASQUS starts the computation of surface processes. Water is precipi-tated onto the model surface, eroding material in the mountain range and transporting sedi-ment load to lower ground, thereby establishing river networks (Fig.3). When water and sediment load reach the right edge of the model that marks the line of lowest surface eleva-tion, they are transported out of the model (Fig.3). I.e., conservation of mass does not hold here for the complete model. The mean precipitation rate is uniformly distributed. The sur-face transport parameters according to Eqs.2, 3, and 4 are the same for model A, B, and C:

stream erosion times net precipitation rate Kf⋅νR=0.03 m/a, diffusion constant Ks=0.3 m2/a, al-

luvial erosion length scale lf,a=10 km and for bedrock lf,b=100 km (Braun & Sambridge, 1997). The numerical surface processes computation is iterated by time steps of 80 a. Results & Discussion Some differences evolve for the three different finite element models A – C in the second analysis step. Model A simulates only surface processes during this step, mechanical cou-pling of the subsurface to the surface is suppressed. I.e., there is no tectonic uplift or subsi-dence. For model B and C the time-dependent computation of the feedback between surface and tectonic processes is active. During a period of 1.5 Ma CASQUS alternately computes the mass redistribution due to surface processes, the mechanical response of the litho-sphere, again surface processes with a topography changed by tectonic uplift or subsidence and so on. The only difference between model B and C is that the fault shown in Fig.2 is only active in model C and does not exist in model B. The landforms and river patterns that have evolved after 1.5 Ma model time differ only slightly from each other, though some characteristic differences can be observed (Fig.4). The cumulative outflow of sediment material out of the model shows a clearer trend (Fig.5). For model A we observe less sediment outflow than for models B and C. This is what we ex-pect as for the uncoupled model A there is no tectonic uplift increasing erosion in the moun-tain range. Model B and C also differ significantly from each other. Slight changes in topog-raphy at the fault zone can change the evolution of river profiles drastically, leading to differ-ent water and sediment discharge.


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The long-range mass redistribution by rivers strongly affects the behavior of the subsurface. The elastic deflection of the lithospheric plate is controlled by the gravitational loads, and these are changed by surface mass redistribution. Thus a spring-back of the lithosphere be-low the mountain range where erosion takes place and material is removed is as expected (Fig.6). The rock uplift here is in the range of a few 100 meters after 1.5 Ma of erosion; the maximum value at X=0 is approximately 300 m. In regions of lower terrain sediment is de-posited, and the plate subsides. The difference between model B and C is not significant af-ter 1.5 Ma. Apparently, model A cannot provide results to this problem as it does not include any mechanical representation of the subsurface. The net surface uplift and subsidence for the model time period of 1.5 Ma is shown in Fig.7. The coarse pattern does not significantly vary among the three models A – C. From the combination of these results with the results for rock uplift, however, we infer that exhuma-tion of rocks at the mountain range for model B and C exceeds the exhumation values for model A by approximately 300 m (England & Molnar, 1990). This gives a difference in ex-

humation rates in the order of 300 m/1.5 Ma=0.2 mm/a, which is a remarkable value com-

pared to the maximum river incision rates in model A that do not exceed 0.5 mm/a. Though the feedback process between erosion/sedimentation and tectonics may not influ-ence the shape of the Earth's surface significantly, the subsurface may be highly affected by this coupling. The process study has shown that CASQUS is able to simulate this feedback appropriately and that phenomena found in nature are mapped in the model. Future models with more complex topographies and subsurface structures may give more complex surface uplift/subsidence patterns, where the effect of rock uplift due to feedback processes may play a more important role at the surface.

APPLICATION TO THE VRANCEA REGION & CONCLUSION The process study is important for a thorough understanding of the coupling between sur-face processes and tectonic processes as simulated by CASQUS. By using CASQUS sur-face processes are included in the fully coupled 4D finite element (FE) model of the Vrancea region. The integration of all three model parts (mantle, crust and surface processes) into the fully coupled model is still work in progress. The technical integration of surface processes into the crustal FE model, however, is finished and working. A test run of this integrated surface – crustal Vrancea model yielded preliminary results. In the test run the coupling between surface processes and the mechanical response was computed for 60,000 years. The crustal FE Vrancea model contains the crust and the upper

mantle down to a depth of 80 km. Its dimensions are 550 km from NNE to SSW times

380 km from WNW to ESE (Fig.8). Both the crust and the mantle are purely elastic, with dif-ferent tectonic blocks having different elastic material properties and different densities. These density variations and the surface topography lead to stress variations and deforma-tions in the crust due to gravity. The vertical side boundaries are fixed in normal direction. A horizontal plane in 80 km depth defines the fixed lower boundary of the model. No isostasy or viscous behaviour is included yet. The surface transport parameters are adjusted very similarly to the ones used in the process study (see previous section). Surface uplift/subsidence is shown in Fig.9. As expected, the elastic response to surficial mass redistribution is negligibly small without isostasy. Only very local patterns of uplift and subsidence can be observed that evolve solely due erosion and the accumulation of sedi-ments. No coherent regional surface uplift or subsidence due tectonic uplift or subsidence is generated by surficial mass redistribution (Fig.9). The simulated values of the mean uplift or subsidence rate in 3 km below sea level range from +0.005 mm/year to -0.006 mm/year (see Heidbach et al., 2007, in this volume).


As indicated by the process study, however, the fully coupled model that contains deeper parts of the mantle and also addresses viscous behaviour and isostasy may give a more pronounced vertical surface uplift/subsidence signal. The results of the complete, fully cou-pled 4D model will then be compared with the vertical signal components of GPS measure-ments (Heidbach et al., 2007, in this volume). This is in order to address the question to what extent the tectonic response to mass redistribution due to erosion and sedimentation processes contributes to the GPS signal. Other processes that may contribute to this signal are processes within the mantle or the elastic rebound of the lithosphere due to a recent de-tachment of the subduction slab. The part of the vertical GPS signal that cannot be ex-plained by surface processes and the corresponding mechanical response of the subsurface has to be attributed to these processes originated in the mantle. This will give a clearer pic-ture of the geomechanical state of the subduction.

ACKNOWLEDGEMENTS This research is funded by the Deutsche Forschungsgemeinschaft (DFG) within the frame-work of the Collaborative Research Center (CRC) 461: "Strong Earthquakes – A Challenge for Geosciences and Civil Engineering".

Figure 4. Surface topography for the three conceptional models A, B, and C after 1.5 Ma of

coupled surface processes / tectonic simulation. Slight differences in the drainage pattern evolve in the area of the main river at the escarpment. The fluvial fan in

front of the main drainage is more pronounced for model A, while more sediment material has been transported out of this region in models B and C.


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Figure 5. Cumulative sediment volume outflow representing the net flux of material out of

the model. The outflow for model B and C exceed the one for model A.

Figure 6. Deflection of the elastic lithosphere as measured at its bottom. Erosion and re-

moval of sediment material in the mountain range lead to a rebound of the plate associated with rock uplift. In the lower terrain sediment is accumulated, leading to subsidence.


Figure 7. Surface uplift/subsidence within the 1.5 Ma model time. The coarse pattern is the

same for all three models. One river branch that developed in model B does not evolve when the fault is active in model C.


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Figure 8. Model setup of the crustal Vrancea model (3x vertically exaggerated). The differ-

ent tectonic blocks with varying material parameters are painted in different colors. The main faults are also included.

Figure 9. Surface uplift/subsidence after 60 ka for the crustal Vrancea model (Fig.8), com-

puted by the combination of surface processes modeling and geomechanical finite element modeling. Significant vertical movements evolve only locally.


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