Objectives/Scope: Rate and pressure transient analysis is considered a routine process that has been developed and refined over many years. The underlying assumptions of linearity justify the use of superposition (in time and space), convolution and deconvolution. The reality of nonlinearities are handled on a case by case basis depending on their source (fluid, well or reservoir). Shale gas wells are subject to significant nonlinearity over their producing life. We review some of the fundamental equations that govern pressure and rate transient behavior, introduce several new techniques which are suited to the analysis of data from producing wells and apply them to a synthetic example of a shale gas well. Methods, Procedures, Process: First, we use simple calculus to show how the convolution integral is derived from standard multirate superposition. Then, from the convolution integral, we derive an equation that describes the pressure response due to a stepramp rate (i.e. an instantaneous rate change from initial conditions followed by a linear variation in rate). It results in a combination of the pressure change due to a constant rate and it's integral. Applying superposition to this equation allows any rate variation to be approximated by a sequence of ramps with far fewer points than those required to achieve the same level of accuracy using standard constant step rate superposition. Second, we rewrite multirate superposition functions allowing for stepwise linear variable rate which, when applied to flowing data and used to calculate the pressure derivative, can result in a much smoother response and hence an overall improvement in the analysis of rate and pressure transients recorded from producing wells. Third, we review the use of the Laplace transform and how it can be applied to discrete data with a view to deconvolving rate transient data. Finally, we demonstrate how data detrending can remove the impact of long term nonlinearities and apply the methods mentioned above to a synthetic dataset based on a typical shale gas well production profile. Results, Observations, Conclusions: We illustrate the advantages of the newly introduced superposition functions compared to conventional analysis methods when applied to the pressure transients of wells flowing at variable rate. As an example, we have simulated the production of two shale gas wells over twenty years. Both have the same production profile, but one includes pressure dependent permeability. At various intervals during the life of the well, we introduce a relatively short well test which imposes a small variation in rate but does not include a shutin. We detrend the rate transients and then apply the techniques described above to analyse the resulting data. The interpretation allows us to identify nonlinearities that may be influencing well productivity over time and to obtain a better understanding of the physics of shale gas production. The mathematics documented in the paper provides a useful overview of how convolution, superposition, deconvolution and Laplace transforms provide the means to analyse pressure and rate transients for linear systems. Data detrending removes the impact of long term nonlinearities on shorter transient test periods. Novel/Additive Information: We develop and demonstrate some new and improved techniques for rate and pressure transient analysis, and we illustrate how these can provide insight into the nonlinearities affecting shale gas production.
Superposition, Convolution, Deconvolution, and Laplace: Pressure and Rate Transient Analysis Revisited with Some New Insights / Whittle, Timothy; Viberti, Dario; SALINA BORELLO, Eloisa; Verga, Francesca.  ELETTRONICO.  (2019). (Intervento presentato al convegno SPE Europec featured at 81st EAGE Conference and Exhibition tenutosi a London, England, UK nel 3  6 Jun 2019) [10.2118/195554MS].
Superposition, Convolution, Deconvolution, and Laplace: Pressure and Rate Transient Analysis Revisited with Some New Insights
WHITTLE, TIMOTHY;Dario Viberti;Eloisa Salina Borello;Francesca Verga
2019
Abstract
Objectives/Scope: Rate and pressure transient analysis is considered a routine process that has been developed and refined over many years. The underlying assumptions of linearity justify the use of superposition (in time and space), convolution and deconvolution. The reality of nonlinearities are handled on a case by case basis depending on their source (fluid, well or reservoir). Shale gas wells are subject to significant nonlinearity over their producing life. We review some of the fundamental equations that govern pressure and rate transient behavior, introduce several new techniques which are suited to the analysis of data from producing wells and apply them to a synthetic example of a shale gas well. Methods, Procedures, Process: First, we use simple calculus to show how the convolution integral is derived from standard multirate superposition. Then, from the convolution integral, we derive an equation that describes the pressure response due to a stepramp rate (i.e. an instantaneous rate change from initial conditions followed by a linear variation in rate). It results in a combination of the pressure change due to a constant rate and it's integral. Applying superposition to this equation allows any rate variation to be approximated by a sequence of ramps with far fewer points than those required to achieve the same level of accuracy using standard constant step rate superposition. Second, we rewrite multirate superposition functions allowing for stepwise linear variable rate which, when applied to flowing data and used to calculate the pressure derivative, can result in a much smoother response and hence an overall improvement in the analysis of rate and pressure transients recorded from producing wells. Third, we review the use of the Laplace transform and how it can be applied to discrete data with a view to deconvolving rate transient data. Finally, we demonstrate how data detrending can remove the impact of long term nonlinearities and apply the methods mentioned above to a synthetic dataset based on a typical shale gas well production profile. Results, Observations, Conclusions: We illustrate the advantages of the newly introduced superposition functions compared to conventional analysis methods when applied to the pressure transients of wells flowing at variable rate. As an example, we have simulated the production of two shale gas wells over twenty years. Both have the same production profile, but one includes pressure dependent permeability. At various intervals during the life of the well, we introduce a relatively short well test which imposes a small variation in rate but does not include a shutin. We detrend the rate transients and then apply the techniques described above to analyse the resulting data. The interpretation allows us to identify nonlinearities that may be influencing well productivity over time and to obtain a better understanding of the physics of shale gas production. The mathematics documented in the paper provides a useful overview of how convolution, superposition, deconvolution and Laplace transforms provide the means to analyse pressure and rate transients for linear systems. Data detrending removes the impact of long term nonlinearities on shorter transient test periods. Novel/Additive Information: We develop and demonstrate some new and improved techniques for rate and pressure transient analysis, and we illustrate how these can provide insight into the nonlinearities affecting shale gas production.File  Dimensione  Formato  

SPE195554MS_Whittle.pdf
non disponibili
Descrizione: Articolo principale
Tipologia:
2a Postprint versione editoriale / Version of Record
Licenza:
Non Pubblico  Accesso privato/ristretto
Dimensione
2.04 MB
Formato
Adobe PDF

2.04 MB  Adobe PDF  Visualizza/Apri Richiedi una copia 
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/11583/2742571