Introduction
The main mechanism of magma transport in the Earth's crust is the formation of cracks (dykes) along which magma rises to the surface\cite{Rubin_1995} . Basaltic magmas typically rise from depths of several tens of kilometers {REF}, for kimberlite magmas - up to 150-200 km \citep{LENSKY_2006}. Dyke widths can vary from centimeters to tens of meters, horizontal extend - from meters to kilometers. Magmaa ascent in dykes is controlled by the buoyancy forces and the tectonic stress field. Most dikes do not reach the surface, but are blocked at the level of neutral buoyancy [4], or with structural barriers in the form of stronger rock layers. As a result of repeated introduction of dikes into the near-surface (first kilometers) region of the earth's crust, it melts with the formation of magma foci, which can reach thousands of cubic kilometers, although usually the volume of foci is much smaller (kilometers-tens of kilometers). Foci of magmatic melt are recorded by seismic tomography using shear wave attenuation. They may have an irregular shape, but most often appear to be flattened bodies with vertical or horizontal strike. Under active volcanoes there can be several foci located at different depths [5].
The formation of magma chambers is simulated both on global geodynamic models [6] and in more detailed local models, where penetration and heat transfer between individual dykes and host rocks are considered [7, 8]. Models of the first type consider regions with a characteristic size of tens of kilometers and a grid spacing of several hundred meters. They cannot resolve the subtle heat exchange processes that occur during the real transport of individual portions of magma, but they allow one to estimate the size and position of magma chambers based on the global distribution of temperatures, rheological properties of rocks and stresses, as well as the consumption of magma between individual chambers.
In the models of the second type, the region into which magma is introduced, as well as the consumption of the latter, is set in advance based on the geological structure of the rocks and estimates of the time of formation of magmatic bodies. An example of the reconstruction of a real magmatic system is [9]. The model assumes horizontal introduction of dikes with lowering the underlying rock layer to the width of the dike. The heat equation is solved taking into account the heat of fusion of the rocks and the real temperature dependence of the concentration of crystals. An explicit scheme for solving the heat equation is used, which imposes a significant limitation on the time step.
In the model [8], the introduction of dikes can occur in an arbitrary direction. To determine the field of displacements, the rocks are considered a viscous fluid and the Navier-Stokes equation is solved. This approach is not justified for low temperatures, at which the behavior of the rocks is described by the relations of the theory of elasticity. In [10], the introduction of dikes is considered vertical. Rock movement is determined solely by kinematic relationships.