This work presents a new methodology for the analysis of permanent downhole gauge and production data based on the deconvolution of cumulative production and bottomhole pressure measurements.
In this work we show that the cumulative production is a convolution of pressure drop (as if the well were producing at a series of constant flowing bottomhole pressures) and flowrates (as if the well were producing at a constant pressure through-out the production history). We then employ our modified B-spline based deconvolution algorithm which, in this work, is based on constant stepwise pressures. We use analytical integration of the B-splines, weighted least squares and regularization to obtain the constant-pressure rate response function and to estimate the initial pressure. The Laplace transformation is not required in this work.
We have validated this approach thoroughly using a synthetic case. We have applied this methodology to various field example cases consisting of both high and low frequency production data. The results indicate that the new method is very efficient in recovering the constant-pressure rate response function in cases where rates are controlled by series of constant flowing bottomhole pressures. We also provide a procedure to check the correlation between pressure and rate measurements by the joint use of two deconvolution methods (SPE 95571 and this work).
Our goal is to provide a robust and error-tolerant deconvolution approach for the diagnosis of production data. Our approach centers on the use of the cumulative production as the "impulse" in the convolution/deconvolution process. This approach is in contrast to prior work [Ilk et al (2006)] in which we focused on using the raw rate and pressure history for our "B-spline deconvolution." In the work by Ilk et al (2006) the objective was to provide the constant rate pressure response function — and as noted, our objective in this work is to provide the constant pressure rate response function. We believe that tying both the constant rate and constant pressure deconvolution approaches together will provide a "double de-convolution" process, which while independent, should yield a clear correlation of the production rates and pressures if these data are truly correlated. Again, our goal is to provide a simultaneous data diagnostic and a data quality check.
To begin this process, we must recall the diffusivity equation, for convenience we cite Van Everdingen and Hurst [Van Everdingen and Hurst (1949)] as this was one of the earliest and most comprehensive treatments on the solution of the diffusivity equation. From Van Everdingen and Hurst, in modern nomenclature, we have:
Equation (Darcy units) (1)
In well test analysis the solution of the diffusivity equation (Eq. 1) is derived using the constant rate inner boundary condition (i.e., the constant-rate pressure response solution). For the variable-rate case, then the pressure response is obtained using the superposition principle (i.e., Duhamel's principle), which is given in the convolution integral form as:
The goal of the variable-rate deconvolution (solving for the constant-rate pressure response, which is the inverse problem) is to estimate the constant flowrate reservoir response using the measured pressure and flowrate data. For most deconvolution algorithms which reside in the petroleum engineering literature [Ilk et al (2006), Ilk (2005), Kuchuk et al (2005), Levitan (2005), Levitan et al (2006), Unneland et al (1998), von Schroeter et al (2002, 2004)], the algorithms are based on the use of constant stepwise flowrates.