Mathematical modeling on the Karkheh reservoir stresses and its application to the Dalpari fault

In this research, a new mathematical modeling on strength changes due to reservoir elastic stresses across the preexisting fault plane is introduced. The method has been applied to the Dalpari fault, which is one of the potential seismic sources in the vicinity of the Karkheh reservoir. In this method the distribution of total stress across the fault cannot be determined because the initial stress is unknown; the pore pressure due to the reservoir is also not considered. The mathematical modeling method has been explained briefly in the following.
The lake first is divided into small rectangles of sides a and b by two sets of orthogonal straight lines, one set conveniently east-west and the other north-south. The mean water depth h in each rectangle with area S is estimated, and the water pressure on the floor of the rectangle is replaced by a vertical force F =? gS h at the center of rectangle. It is clear that rather smaller rectangles lead to more precise modeling, hence, each rectangle with increasing h is divided into some parts. The water pressure of the lake is simulated by a set of point forces F which applied in the -X3 direction and acting on the rectangles. We define now a mathematical model of the single force F in the elastostatic fields using the delta function conception: The point force F is defined as:


The component of displacement at point due to in the direction, , is given by:


where j=3, for the water pressure of the lake, and is the distance from origin to point P. The general three dimensional relationships between nine Cartesian strain component and three Cartesian displacements are given by:

These nine terms constitute the infinitesimal strain tensor, a symmetric tensor with six independent quantities. The stress tensor is given by stress-strain relationships based on constitutive law called Hooke`s law is given by:


Using the conception of the stress tensor and well-known relationships in elastostatic theory the various stress parameters such as shear and normal stresses due to reservoir can be determined at the point P in a plane with normal n. In this way, we would be able to achieve the strength values due to the reservoir across the specified preexisting fault plane. The shear stress ( ) and strength ( ) due to the reservoir across the preexisting fault plane are, respectively, as follows.




is angle measured in the preexisting fault plane between resolved shear stresses due to reservoir and ambient causes, and it is measured from the direction of the latter coefficient of friction along the fault plane, is coefficient of friction across the preexisting fault plane. The earthquakes near new reservoirs is classified into the following three cases on the basis of positive, negative and zero values of ; a reservoir based on this classification may stabilize some parts and destabilize other parts of the same nearby fault surface.
Case I: induced or reservoir assisted natural earthquakes; > 0. This situation will arise when 0 ? ? < 90 and the reservoir stresses have a net destabilizing influence on some parts of fault plane in which has a suitably large component in the same direction as initial tectonic shear stress, therefore the earthquake occurs earlier than its natural time.
Case II: natural tectonic earthquakes despite the inhibiting influence of the new reservoir; < 0. This situation will arise when 90 Case III: natural tectonic earthquakes with no influence of the reservoir; = 0. The reservoir exerts neither a stabilizing nor a destabilizing influence on the fault plane. The earthquake occurs neither hindered nor assisted by the reservoir; its occurrence time is the same as the natural time. Hence there is again a natural earthquake near the reservoir.
We have applied these concepts to the Karkheh reservoir and discussed fault stability analysis on the Dalpari fault plane. For this, we identify the strength equation as the main operational relation and list here the inputs required for its evaluation. Firstly, the location and orientation of the segment of the Dalpari fault in the vicinity of the reservoir should be specified. An estimate of the coefficient of friction, Poisson's ratio, shear and Young's modulus in the crustal rocks beneath and around the reservoir should be available also. Secondly, we should specify the direction of resolved shear stress on the fault due to ambient crustal stresses. If the fault plane solution for the earthquake is available, or can be modeled, then the direction of slip on the chosen nodal plane may be taken as the estimate of the desired shear stress direction. Thirdly, we should have estimates based on computations of the elastic stresses due to the Karkheh reservoir on the Dalpari fault plane. It is observed that in the Zagros high angle reverse faults (dips>30) appeared to be more common than low angle trust (dips<30); there is a peak in the distribution in the range 30-60 and very few nodal plane dips corresponding to low angle thrusts which would plot in the ranges 0-30 and 60-90. It is observed that in this region the seismogenic depths vary from 4 to 20 km, with typical uncertainties being ± 4 km. As a result, due to unavailable valuable information needed about the Dalpari fault plane parameters, we have considered different dips between 15° to 60° and the maximum depth 15 km for analysis.
The Karkheh reservoir is situated on the hillwall part of the Dalpari fault. Based on the analysis it is observed that the reservoir may exert a stabilizing influence and delay the time of earthquake occurrence associated with this segment. These observations are in agreement with the theoretical stress analysis due to the reservoir in this segment. This results in a more crustal stability at all dips of the surface segment of the Dalpari fault. The possibility of induced earthquake may occur at the 15° dip of the hidden segment of this fault and at depths shallower than 2.5 kilometer, which continues until nearby down the dam site. In this segment the maximum reservoir induced strength in the direction of the slip vector fault is estimated about 0.12 bar at a depth of ~1 kilometer.