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The Nordgren crack model

Extracted from the article Propagation of vertical hydraulic fracture by P. R. Nordgren (Society of Petroleum Engineers Journal 12, 4, 306-314, 1972)

The underlying mathematical problem is a partial differential equation with moving boundary. Let $\omega(x,t)$ be the width of the crack at position x and time t. Let L(t) be the length of the crack at time t, and $\tau(x)$ be the time required for the crack to reach length x. Then $\omega(x,t)$ satisfies the partial differential equation:

\hfill\displaystyle{{{\partial^2 \omega^4}\o...
 ...&0\hfill\hbox{(no volume or fluid flow at the
tip)} \cr}\right.\end{displaymath}

A standard algorithm uses a discretization in x and t to form finite difference approximations to the partial derivatives in the previous equation. At each time step tj, one must solve a nonlinear system of equations for


The solution provided by the numerical methods dealing with these nonlinear systems is not very satisfactory since in many cases problems arise when the program attempts to take the square root of a big negative number[*].