The problem

First of all, the existing deconvolution technology used by Paradigm finds the pressure response by minimising a cost function with a piecewise linear approximation of the pressure response. It often provides an unreasonable and nonphysical result when the noise in the pressure and rate are significantly high. Secondly, in a real-world situation, the total production is measured with much higher confidence compared with individual pressure and rate. Honouring this constraint will make the resulting deconvolution more realistic. The first figure shows pressure (red) and volume rate (black) history. The second figure (below) shows the non-physical pressure impulse (blue) and the impulse derivative (red).

“This internship provided me an excellent opportunity to discover a real-world inverse problem and to work in a great company with wonderful atmosphere. I have experienced the job which could be my career.” said intern Lian Duan, D.Phil. student, OCCAM, University of Oxford).

The approach

Using a piecewise linear approximation of the impulse derivative and the maximum a posterior method under a Bayesian framework, the deconvolution problem is transferred into a nonlinear total least squares (TLS) problem. A cost function that considers pressure misfits, rate misfits, total production misfit and a smoothness penalty is derived analytically. For most synthetic and well testing problems of short duration, excellent results can be achieved by solving the TLS problem directly. However, for real-world problems with long test durations, the numerical algorithms suffer from slow convergence due to the large dimensionality.

Using the special features of the cost function in the TLS problem, an iterative method is formulated using a Coordinate Descent Method. During each iteration, an approximation to the cost function is minimised to initialize the next stage. Apart from the efficient computation time, the ill-conditioned Jacobi matrix, caused by the derivative of the total production misfit can be avoided as well. This is the key to both the performance of the minimisation and a solution of the deconvolution problem.

Preliminary numerical results with synthetic and real-world examples suggest that the method is capable of producing smooth, interpretable and improved reservoir response estimates. The third figure shows the improved pressure impulse (blue) and impulse derivative (red). The computational time of the iterative method is significantly reduced compared to the direct method.

References

[1] L. Duan, R. Stadie, and C. Farmer: Iterative Deconvolution Method for Well Test Data with Cumulative Production Constraint. In Preparation.
[2] T. V. Schroeter, F. Hollaender and A. Gringarten: Deconvolution of Well Test Data as a Nonlinear Total Least Square Problem, SPE 71574, 2001
[3] T. Whittle and A. Gringarten: The Determination of Minimum Tested Volume from the Deconvolution of Well Test Pressure Transients, SPE 116575, 2008
[4] T. Whittle, H. Jiang, S. Young and A. Gringarten: Well Production Forecasting by Extrapolation of the Deconvolution of Well Test Pressure Transients, SPE 1222299, 2009