### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 5 (2016)

# Application of direct and inverse Laplace transforms for geothermal heat recovery tasks

### UDK: 517.2

### Pages: 1 - 12

### DOI: 10.31029/demr.5.1

Difficulties of numerical implementation of processing Laplace image are marked and the sources of their origin are pointed out. As a first example the change of the boundary conditions in the similarity problem of extracting dry heat from rocks is given. It is shown that if the temperature on bound gradually increases from initial value to the same constant value exponentially, then the pole appears on negative real axis of the transform parameter, and the integrals on negative axis become improper and the calculation errors increase. The solution of new problem of growth of temperature of thermal water, which gushes at constant velocity in time, is proposed. This is contact problem, which includes both the temperature of rock with the geothermal gradient and the temperature of water in a fracture. It is modeled by Loverier scheme, solved in non-stationary formulation with using Laplas transform and reduced to universal form in dimensionless variables. The features, which are generated with geothermal gradient and water temperature front, are considered.

**Keywords: **
rock, fracture, liquid, temperature, flow.