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The basic idea, which allows to simulate leaching in the asymmetrical model lies in dividing the cavern and the surrounding salt body into eight azimuthal sectors, 45°/22.5° each, with edges at the cavern axis. Each sector has its own cavern wall profile which evolves in time depending on leaching properties assigned to this sector, independently of cavern shape in the other sectors.
The cavern wall profile in every sector is sector-equivalent, i.e. at any depth the area of circle sector of 45°/22.5° with the sector-equivalent radius at this depth is the same as the area of horizontal cavern section at this depth within the limits of the relevant sector.
The conversion from a real, irregular cavern horizontal contour to 8/16 sector-equivalent radii, is simple and obvious. However, if the model working with sectors is created, a reverse procedure is necessary, converting 8/16 circle sectors to a continuous irregular contour.
This can be done by interpolation that satisfies two following conditions:
- sector-equivalent radii obtained from the interpolated contour must be the same as the starting equivalent radii
- at the boundary between two sectors, radius must have the average value of these two sectors.
One can say that division into 8/16 sectors is an approximation which loses some part of information, but the loss is definitely smaller than in the case of an axisymmetric model.
In the asymmetrical model, the source of difference between sectors is differentiation in leaching properties of rock salt. This differentiation cannot be determined by laboratory tests on core samples as the core represents borehole in the cavern axis, and azimuthal sectors are inaccessible for sampling. The differentiation can be established after echo sonar survey in the cavern only, using a special procedure named KorLog.
The KorLog-module is integral part of the package WinUbro. Its role is analogous to modify the leaching parameters of rock salt at successive depths, comparing the results of sonar logging with the WinUbro prediction at the same moment.
One must be aware of the fact that (excepting very special cases) WinUbro-code has no possibility of really asymmetrical simulation till the first sonar logging in the modeled cavern because the differentiation in leaching properties between the sectors is then unknown. So, the initial leaching stage from the borehole to the first sonar measurement is usually simulated by non-axial model with coefficients as in the axial model.
Time and depth approximations applied in the WinUbro are the following:
The time is approximated with an inhomogeneous step limited from above by a given maximum value.
The wall profile of each cavern sector is treated using the following rules of the UBRO model:
- cavern vertical contour (profile) can be described as radius being a function of depth - so one can assume that the depths of points at the profile are monotone, the possibility that a profile of non-monotone depths can occur is excluded, (no so-called "fingers" or "pockets" can be modeled).
- cavern profile is approximated by an open polygon (broken line), independently of depth approximation inside the cavern,
- inhomogeneous, time-dependent step is applied to profile approximation, where the vertices (approximation nodes) follow the cavern boundary and change their depths with time, and during simulation new vertices may be created as well as old ones may disappear,
- for cavern horizontal section at any depth, each sector of azimuthal approximation is represented by a single radius value that gives the area corresponding to area cavern section element within this sector and at this depth; this representing value is equal to the radius of polygon vertex if a vertex lies at given depth, otherwise it is interpolated between neighboring vertices.
During every time step the algorithm determining the cavern profile movement is called eight times, i.e. for each sector successively. To determine the displacement of individual sides of the cavern sector profiles, the algorithm takes leaching coefficients values corresponding to the sector and depth where the relevant profile side belongs.
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Other algorithm elements are following:
- brine concentration inside the cavern and flow velocity through the cavern are treated as functions of two variables: time and depth,
- depth inside the cavern, is approximated with a homogeneous, time-independent step, with fixed approximation nodes; brine concentration and flow are determined in calculation blocks (cavern slices) connected with these nodes,
- time and depth approximations are coupled together by means of so-called explicit Lelevier-type scheme; it implies certain limitation of the time step to avoid approximation instability in cavern of small diameter; at sufficiently large cavern dimensions the step becomes homogeneous and equal to its maximal value.
When new cavern profile (after time step interval) is determined in every of 8/16 cavern sectors, the volumes of successive cavern slices are calculated. Slice volume is a sum of 8/16 "pies", i.e. 45°/22.5°-sectors of truncated cones approximating the cavern in successive sectors.
Then new concentration distribution and new level of insolubles in sump are determined basing on the balance of mass and volume for brine, water and insolubles.
So, resuming :
- The model use time t and three spatial variables: radius r, azimuth Φ and depth h (vertical). The depth z is the most important of them. The physical quantities in the model as concentration, brine flow, leaching properties of rock salt are depth-dependent. Leaching properties of rock salt layers may be also variable with the azimuth Φ . Radius r is only used together with depth h , to describe the cavern contour, which is also azimuth-dependent. cavern contours, sump level,
- Finite difference approximation is applied for time t and depth h .
- Cavern depth will be approximated using homogeneous step Δx constant in time, with permanent nodes, i.e. having constant position in time. The computational blocks (cavern slices) are connected with these nodes. Such physical quantities as concentration and brine flow will be determined in them.
- Shape of the cavern is modeled as the sequence of contours in 8/16 azimuthal sectors, 45°/22.5° each, with edges at the cavern axis.
- Cavern contour in each azimuthal sector, can be presented as radius r being a function of depth h. It is assumed, that depth sequence of the cavern contour points is monotonous, possibility of non-monotonic contour (so called "fingers" or "pockets") is excluded.
- Cavern contour is approximated independently of the cavern approximation. For contour approximation, an open polygon will be used with time dependent vertices. These vertices are floating and changeable, i.e. their position (depth and radius) varies in time and their number is not constant. During simulation some vertices may disappear and new ones may be created. The vertices describing cavern contour can be spaced very irregularly appropriately to needs. In some slice few vertices may appear while in some sequence of adjacent slices no one.
- Time will be approximated with an automatically chosen variable step, limited from above by a given maximum value.
- Time and depth approximation will be coupled together by means of so-called explicit scheme of first order, i.e. values in nodal points in a given time, may be functions of their distribution one time step back only, they will not depend each other as well as they have not to depend on their distributions two time step back nor previous.
- Time and depth coupling will be of first range, i.e. values in a given nodal point may be functions of values in the same point and neighboring nodal points (one time step back). They have not to depend on values in the points positioned farther. For practical purpose it will be Lelevier scheme.
- Explicit computational scheme proposed for the model, limits time step to avoid approximation instabilities. That is why a fine depth step, smaller surface of cavern slice and bigger flow rate result in a finer step of time approximation. At sufficiently large cavern dimension the time step becomes homogeneous and equal to its maximum value.
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