Inverse problem to pressure convolution, performed as a fully or semi-automated search for initial pressure for every well and Unit-rate Transient Responses (UTR) for wells and cross-well intervals in order to fit the sandface pressure response (usually recalculated from PDG data using wellbore flow model for depth adjustment ) to total sandface flow rate variation history (usually recalculated from daily allocations based on surface well tests). Basic conceptThe basic element of deconvolution is the pressure Unit-rate Transient Responses (UTR) to a unit-rate production. specifies two types of : Drawdown Transient Response () and Cross-well transient response (). The Drawdown Transient Response () is the sandface pressure response of a given well to its sandface unit-rate production in absence of the other wells. It is equivalent to conventional drawdown test with unit-rate production. The Cross-well Transient Response () is the sandface pressure response of a given well to the sandface unit-rate production of the offset well in absence of the other wells. is only working in low-compressibility formations, hence before using one should analyze the data to see if this condition is met for the tested area. can be performed in two options: Radial Deconvolution () and Cross-well Deconvolution (). Radial Deconvolution () corrrelates pressure and rate in selected well (called pressure-tested well) and only account for the rates in surrounding wells (called rate-tested wells) in order to reconstruct:
A group of wells with one selected pressure-tested well has transient responses: 1 diagonal transient response and cross-well transient responses.
Only rates are taken into account for offset wells in RDCV. In case a group of tested wells have mulitple pressue gauge installations one may wish to deconvolve the unit-rate transient responses using all of the pressure data which is called Cross-well deconvolution (). The main advantage of over is the ability to simulate and interpret all PDG simultaneiously, resulting in mopre information and better constrain and stability of deconvolution process. The group of pressure-tested wells has transient responses, because every well has 1 diagonal transient response and cross-well transient responses thus having transient responses for each well. The intervals between two wells with pressure gauge instaltions results in two transient response: first well onto the second well and revers. This may indicate anisotropy of pressure propagation in counter directions and shed the light on the resevroir physics between these wells. Once all possible DTR/CTR are deconvolved one can perform a conventional type-curve analysis for each well, defining the type and distance to the boundary, estimating skin, transmissibility and diffusivity around each well. Unlike routine numericial fitting, where pressure responses to complicated rate history are being fit for wells, one can run XDCV to get responses to very simple rate history (unit rate production) and then fit them all with diffusion models (sequentially or in parallel) by varying the same parameters (current formation pressure around every well Pe, skin-factor S for every well, and usually, transmissibility σ + pressure diffusivity χ around each well). Main benefits of are:
Main disadvantages of are:
MathematicsIn linear formation approхimation the pressure response to the varying rates in the offset wells is subject to convolution equation:
where
with assumption:
Hence, convolution is using initial formation pressure , unit-rate transient responses of wells and cross-well intervals and rate histories to calculate pressure bottom-hole pressure response as function time :
The is a reverse problem to convolution and search for functions and numbers using the historical pressure and rate records and provides the adjustment to the rate histories for the small mistakes :
The solution of deconvolution problem is based on the minimization of the objective function:
where
and objective function components have the following meaning:
In practice the above approach is not stable. One of the efficeint regularizations has been suggested by Shroeter One of the most efficient method in minimizing the above objective function is the hybrid of genetic and quasinewton algorithms in parallel on multicore workstation. The also adjusts the rate histories for each well to achieve the best macth of the bottom hole pressure readings. The weight coefficients and control contributions from corresponding components and should be calibrated to the reference transients (manuualy or automatically). The methodology constitute a big area of practical knowledge and not all the tricks and solutions are currenlty automated and require a practical skill. SampleSample #1 – RDCVНа рис. 2.1.2 представлена карта участка с тремя скважинами. Синтетическая история работы добывающей скважины с простым поведением продуктивности.
Пример #2 – КДКВНа Рис. 2.1.5 представлена карта участка с тремя скважинами.
На Рис. 2.1.15 приведена история дебитов и давлений по всем скважинам.
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