Abstract:

A pressure deconvolution technique based on ridge regression with existing convolution features was explored and reinterpreted to frame a new optimization problem structure. We found a way to split the feature matrix into a deconvolution part and a smoothing part that handle each effect independently. The deconvolution part is comprised of a convolution matrix (Toeplitz matrix) for discrete-time convolution while the smoothing part is composed of a basis function of the impulse response. Hence, this formulation preserves linearity (a superposition of pressure solution) and the kernel for nonlinear mapping (e.g., polynomial kernel, radial basis function kernel, etc.) is effective only on a smoothing part (a basis function of impulse response). Importantly, the key idea of existing convolution feature is the modelling of the impulse response with a linear combination of a basis functions.
For single-well pressure behavior, ridge regression often struggles with high-frequency measurements that contain multistage responses. An impulse response might share similar functional characteristic like wellbore storage and pseudosteady state but there is only one linear basis function that is always active. In traditional well test interpretation, the engineer is typically focused on certain parts of the impulse response on a log-scale, to identify reservoir behavior (e.g., early-time for wellbore storage, transient-time for infinite acting radial flow, late-time for boundary effect, etc.). Analogously, a basis function could be formed from multiple pressure responses in each log-interval (inherent from a solution of the diffusion equation) then bridged together with a constraint on function value and its derivative at each interval boundary. This is the basic principle of spline regression that has more flexibility in terms of function expression. In addition, it also subsumes ridge regression with Laplacian regularization.
For multiwell pressure behavior, additional basis functions are included to handle interference responses. Typically, the exponential integral is approximated with a logarithm function for single-well pressure response which is also one of the basis functions in previous convolution features. Nonetheless, interference pressure response is more subtle and its solution could not be easily approximated, which previously resulted in a poor fitting using earlier methods. By inspection of the convergent series of the exponential integral, its higher-order terms are associated with 1/t^m that could be easily included as a basis function to enhance the capability of the model and capture the detail of the interference test that is essential for multiwell problems. A full extension to multiwell problems is also presented with the additional derivative constraint for the impulse response and symmetry constraint. Finally, we can quantify the sensitivity of the deconvolution process using singular value decomposition (SVD) and Monte Carlo simulation (MCS). Using singular value decomposition, we could reveal the orthogonal basis of the feature matrix and its corresponding singular value that helped us identify the effectiveness of the basis functions as well as their susceptibility to noise of certain pattern. Nevertheless, singular value decomposition is only suitable for noisy pressure measurement problems without inequality constraint. Thus, Monte Carlo simulation is required for full quantification of noisy measurements in both pressure and flow rate. We can observe a propagation of error in measurement to a band in pressure prediction and pressure derivative of the impulse response.
This study developed a more robust method for deconvolution of pressure measurements. In addition, the mathematical insights obtained in the study allow for a more general understanding of how the deconvolution problem can be deconstructed into its relevant parts – likely allowing for expanded capability of deconvolution algorithms in industrial practice.

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