Deconvolution Under Normalized Autocorrelation Constraints
- B. Baygü (Salomon Brothers) | F.J. Kuchuk (Schlumberger) | O. Arikan (Bilkent U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 246 - 253
- 1997. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.6.4 Drillstem/Well Testing
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In this paper we describe a time domain algorithm for determining theinfluence function from the measured input and output signals of the system.The deconvolution, which is a highly unstable inverse problem with measurementerrors, is an important step for obtaining the system's influence function thatprovides insight about flow regimes normally masked by the time-dependent inputsignal. The algorithms presented for deconvolution in the literature aregenerally based on data reduction, with the exception of constraineddeconvolution methods.
We propose a constrained least-squares deconvolution method to reconstructthe influence function from noisy data. The constraints are the lower bounds onthe first few lags of the normalized autocorrelation coefficients of theinfluence function. The lower bounds may represent known or desirablesmoothness properties of the function. By choosing the constraint valueslarger, a smoother deconvolution can be obtained. We also impose an energyconstraint on the derivative of the reconstructed signal for furtherregularization.
Deconvolution is a signal processing method in which the effect of thetime-dependent input signal is extracted from the output signal. In testing, itis defined as determining the influence function or unit response behavior of asystem at the wellbore (the constant rate behavior) from measured downholepressure and flow rate.1 For convenience, this deconvolution iscalled pressure-rate deconvolution. Recently, Goode etal.2 presented a different approach using pressure signals fromthe horizontal and vertical probes of the multiprobe wireline formationtester3 to obtain the ratio impulse function of the system. Thedeconvolution method described by Goode et al.2 is calledpressure-pressure deconvolution in this paper. The pressure-pressuredeconvolution as shown in Appendix A can easily be extended to otherwell-testing problems, such as interference testing between wells. For bothmethods, the deconvolution operation can be defined as obtaining solutions forconvolution type linear Volterra integral equations that can be written as
For convenience, we use p instead of ?p. The kernel Gof the convolution integral is called the impulse response or the ratio of theimpulse responses evaluated at two different locations (see Appendix A).Whether the kernel is an impulse response or a ratio of impulse responses, theimpulse response is a solution of the diffusivity equation for a Neumann-typeinternal boundary condition. As discussed by Kuchuk et al.1the deconvolution method given in this paper can also be applied toDirichlet-type problems. The quantities p and f in Eq. 1 aremeasured as functions of time.
The deconvolved kernel G, and its integral (unit response) andlogarithmic derivative forms ( t×G) are used in system identification,which is the first step of the well test interpretation. Many differentdeconvolution algorithms can be developed using different operationaltechniques. The most appropriate method for a particular testing problemdepends on the behavior and the noise characteristics of the measurements. Inthe absence of noise, numerical algorithms for approximating the solution ofEq. 1 for the kernel G would be relatively trivial. However, asdiscussed in Kuchuk et al.,1 even small amounts of noise inthe flow rate have a detrimental effect on the solution. Consequently, Coatset al. 4 used a linear programming method with a number ofconstraints in order to smooth the solution. An alternative constraineddeconvolution method using linear least-square minimizing was presented byKuchuk et al.1 The latter method gives a smooth solution forthe integral of G, but G itself oscillates mildly. Theoscillatory nature of the solution from the method of Kuchuk etal.1 sometimes hinders the identification step.
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