Multiwell deconvolution deconvolution is procedure of fitting the sandface pressure response in one well to sandface flow rate variation history in the same well or offset well using the linear convolution relashionship.
Multiwell deconvolution ( Hint |
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0 | MDCV |
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1 | Multiwell deconvolution |
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| ) is a fully or semi-automated procedure of fitting the sandface pressure response (usually recalculated from PDG data using wellbore flow model for depth adjustment ) to sandface flow rate variation history (usually recalculated from daily allocations based on surface well tests) using the Constant Transient Response Model.
Basic concept
Multiwell deconvolution ( Hint |
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1 | Multiwell deconvolution |
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| ) is a fully or semi-automated procedure of fitting the sandface pressure response (usually recalculated from Permanent Downhole Gauge( Hint |
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0 | PDG |
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1 | Permanent Downhole Gauge |
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| ) data records) to sandface flow rate variation history (usually recalculated from daily allocations based on surface well tests) using the Constant Transient Response Model.
which reconstructs initial pressure for every well and unit-rate transient responses for wells and cross-well intervals.
The basic element of deconvolution is the pressure Transient Response ( Hint |
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0 | TR |
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1 | Transient Response |
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2 | DTR |
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| ) to a unit-rate flow. Hint |
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1 | Multiwell deconvolution |
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| specifies two types of Hint |
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0 | TR |
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1 | Transient Response |
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2 | DTR |
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| : Drawdown Transient Response ( Hint |
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0 | DTR |
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1 | Drawdown Transient Response |
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2 | DTR |
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| ) and Cross-well transient response ( Hint |
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0 | CTR |
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1 | Cross-well transient response |
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2 | CTR |
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| ).
The Drawdown Transient Response ( Hint |
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0 | DTR |
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1 | Drawdown Transient Response |
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2 | DTR |
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| ) 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 ( Hint |
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0 | CTR |
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1 | Cross-well transient response |
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2 | CTR |
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| ) 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.
It is equivalent to the pressure interference test with the unit-rate production in disturbing well.
Hint |
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| is only working in low-compressibility formations, hence before using Hint |
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| one should analyze the data to see if this condition is met for the tested area.
Hint |
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1 | Multiwell deconvolution |
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| can be performed in two options: Radial Deconvolution ( Hint |
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0 | RDCV |
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1 | Radial Deconvolution |
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| ) and Cross-well Deconvolution ( Hint |
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0 | XDCV |
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1 | Cross-well Deconvolution |
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| ).
Radial Deconvolution ( Hint |
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0 | RDCV |
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1 | Radial Deconvolution |
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| ) 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:- Pressure response of the well to its unit rate production in absence of other wells (also called Diagonal Transient Response or
Hint |
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0 | DTR |
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1 | Diagonal Transient Response |
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2 | DTR |
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| ) - Pressure response of the well to offset well unit rate production in absence of other wells (also called Cross-well Transient Response or
Hint |
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0 | CTR |
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1 | Cross-well transient response |
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2 | CTR |
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| )
A group of wells with one selected pressure-tested well has transient responses: 1 diagonal transient response and cross-well transient responses. The main difference between RDCV and single-well deconvolution (SDCV ) is that it takes into account offset wells impact on tested well pressure.
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 ( Hint |
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0 | XDCV |
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1 | Cross-well Deconvolution |
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| ).
The main advantage of Hint |
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| over Hint |
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| 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 Hint |
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| are:
- Reconstruction of formation pressure history
- Rate corrections for random mistake
- The ability to get transient responses without initial knowledge of reservoir geometry
Main disadvantages of Hint |
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| are:
- Uncertainty in DTR/CTR, in case of uneventfull production history or synchronized flow variation of two (or more) wells
- Error increasing with the number of wells in the test
Mathematics
In linear formation approхimation the pressure response to the varying rates in the offset wells is subject to convolution equation:
LaTeX Math Block |
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| p_n(t) = p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - q^{(\alpha-1)}_k \big) \ p^u_{nk}(t - t_{\alpha k}) |
where
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1 | | pressure at -th well at arbitrary moment of time | 2 | | initial pressure at -the well | 3 | | rate value of -th transient at -th well | 4 | | pressure transient response in -th wel to unit-rate production from -th well | 5 | | starting point of the -th transient in -th well | 6 | | number of wells in the test | 7 | | number of transients in -th well |
with assumption: - – for any well
- at for any pair of wells
Hence, convolution is using initial formation pressure , unit-rate transient responses of wells and cross-well intervals and rate histories LaTeX Math Inline |
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body | \{ q_k (t) \}_{k = 1 .. N} |
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| to calculate pressure bottom-hole pressure response as function time : LaTeX Math Block |
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| \big\{ p_{i, n}, \{ p^u_{nk} (t), q_k (t) \}_{k = 1 .. N} \big\} \rightarrow p_n(t) |
The Hint |
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| is a reverse problem to convolution and search for functions and numbers using the historical pressure and rate records LaTeX Math Inline |
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body | \{ p_k(t), \ \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \}_{k = 1 .. N} |
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| and provides the adjustment to the rate histories for the small mistakes LaTeX Math Inline |
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body | \{ q_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q_k \}_{\alpha = 1.. N_k} |
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| : LaTeX Math Block |
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| \big\{ p_k(t), q_k (t) \big\} _{k = 1 .. N} \rightarrow \big\{ p_{i, n}, \{ p^u_{nk} (t), \tilde q_k (t) \}_{k = 1 .. N} \big\} |
The solution of deconvolution problem is based on the minimization of the objective function: LaTeX Math Block |
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| E(\{ p_{i,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) \rightarrow {\rm min} |
where LaTeX Math Block |
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| E(\{ p_{i,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) = \sum_{n=1}^N \Big(p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} (q^{(\alpha)}_k - q^{(\alpha-1)}_k ) \ p^u_{nk}(t - t_{\alpha k})- p_n(t) \Big)^2
+ w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) +
w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2 |
and objective function components have the following meaning:
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LaTeX Math Inline |
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body | \sum_{n=1}^N \Big(p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} (q^{(\alpha)}_k - q^{(\alpha-1)}_k ) \ p^u_{nk}(t - t_{\alpha k})- p_n(t) \Big)^2 |
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| is responsible for minimizing discrepancy between model and historical pressure data | LaTeX Math Inline |
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body | w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) |
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| is responsible for minimizing the curvature of the transient response (which reflects the diffusion character of the pressure response to well flow) | LaTeX Math Inline |
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body | w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2 |
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| is responsible for minimizing discrepancy between model and historical rate data (since historical rate records are not accurate at the time scale of pressure sampling) |
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 Hint |
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| also adjusts the rate histories for each well LaTeX Math Inline |
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body | \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q^{(\alpha)}_k \}_{\alpha = 1.. N_k} |
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| 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 Hint |
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| methodology constitute a big area of practical knowledge and not all the tricks and solutions are currenlty automated and require a practical skill.
Sample
Sample #1 – RDCVНа рис. 2.1.2 представлена карта участка с тремя скважинами.
Синтетическая история работы добывающей скважины с простым поведением продуктивности. | Рис. 2.1.2. Скв. Р1. Мультискважинная деконволюция | | Рис. 2.1.3. Скв. Р1.Сравнение мультискважинной деконволюции с односкважинной деконволюцией |
На Рис. 2.1.4 приведена история дебитов и давлений по всем скважинам.
| Рис. 2.1.4. P1. Сравнение полученной истории дебитов и давления с исходными |
Пример #2 – КДКВ
На Рис. 2.1.5 представлена карта участка с тремя скважинами.
| Рис. 2.1.5. Синтетическая история работы добывающей скважины с простым поведением продуктивности |
| Рис. 2.1.6. Скв. Р1. Сравнение мультискважинной деконволюции и односкважинной деконволюции | | Рис. 2.1.7. Влияние скважины P2 на скважину P1 | | Рис. 2.1.8. Влияние скважины W3 на скважину P1 | | Рис. 2.1.9. Скв. Р2. Сравнение мультискважинной деконволюции и односкважинной деконволюции | | Рис. 2.1.10. Влияние скважины 1 на скважину 2 | | Рис. 2.1.11. Влияние скважины W3 на скважину P2 | | Рис. 2.1.12. Сравнение мультискважинной деконволюции и односкважинной деконволюции | | Рис. 2.1.13. Влияние скважины P1 на скважину W3 | | Рис. 2.1.14. Скв. W3 Влияние скважины P2 на скважину W3 |
На Рис. 2.1.15 приведена история дебитов и давлений по всем скважинам. | Рис. 2.1.15. P1. Сравнение полученной истории дебитов и давления с исходными | | Рис. 2.1.16. P2. Сравнение полученной истории дебитов и давления с исходными | | Рис. 2.1.17. W3. Сравнение полученной истории дебитов и давления с исходными |
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