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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).


 Contents


Basic concept


The basic element of deconvolution is the pressure  Unit-rate Transient Response (UTR) which is a sandface pressure response to the total sandface unit-rate production.

Multiwell deconvolution (MDCV) specifies two types of UTRDrawdown Transient Response (DTR) and Cross-well Transient Response (CTR).

The Drawdown Transient Response (DTR) is the sandface pressure response of a given well to its total sandface unit-rate production in absence of the other wells.

It is equivalent to conventional Drawdown Test with sandface unit-rate production.


The Cross-well Transient Response (CTR) is the sandface pressure response of a given well to the total 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.


The pressure convolution principle itself has some limitations and may not be adequate for some practical cases.

For example, changing reservoir conditions, high compressibility – everything which breaks linearity of diffusion equations.

There are some workarounds on these cases but the best practice is to check the validity of pressure convolution (and therefore the applicability of MDCV) on the simple synthetic 2-well Dynamic Flow Model (DFM) with the typical for the given case  reservoir-fluid-production conditions.


MDCV can be performed in two options: Radial Deconvolution ( RDCV ) and Cross-well Deconvolution ( XDCV ).


Radial Deconvolution ( RDCV ) correlates 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  DTR )
  • Pressure response of the well to offset well unit rate production in absence of other wells  (also called Cross-well Transient Response or  CTR )

A group of N wells with one selected pressure-tested well has N  transient responses: 1 diagonal transient response and  N-1 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 ( XDCV ).


The main advantage of  XDCV over  RDCV  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 N pressure-tested wells has N^2  transient responses, because every well has 1 diagonal transient response and N-1 cross-well transient responses thus having N 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 N pressure responses to complicated rate history are being fit for N wells, one can run XDCV  to get N^2 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 4N 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  MDCV  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  MDCV  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:


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




1

p_n(t)

pressure at n-th well at arbitrary moment of time t

2

p_{i,n}

initial pressure at n-the well

3

q^{(\alpha)}_n

rate value of \alpha-th transient at n-th well

4

p^u_{nk} (t)

pressure transient response in n-th wel to unit-rate production from k-th well

5

t_{\alpha k}

starting point of the \alpha-th transient in k-th well

6

N

number of wells in the test
7

N_k

number of transients in k-th well

with assumption:

  • q^{(-1)}_k = 0 – for any well k = 1.. \ N

  • p^u_{nk}(\tau) = 0 at \tau < 0 for any pair of wells n, k = 1.. \ N


Hence, convolution is using initial formation pressure p_{i, n}, unit-rate transient responses of  wells and cross-well intervals p^u_{nk} (t) and rate histories \{ q_k (t) \}_{k = 1 .. N} to calculate pressure bottom-hole pressure response as function time p_n(t):

(1) \big\{ p_{i, n}, \{ p^u_{nk} (t), q_k (t) \}_{k = 1 .. N} \big\} \rightarrow p_n(t)


The  MDCV is a reverse problem to convolution and search for N^2 functions p^u_{nk} (t)  and N numbers p_{i, n}  using the historical pressure and rate records \{ p_k(t), \ \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \}_{k = 1 .. N} and provides the adjustment to the rate histories for the small mistakes \{ q_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q_k \}_{\alpha = 1.. N_k}:

(2) \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:

(3) E(\{ p_{i,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) \rightarrow {\rm min}

where

(4) 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:



\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

is responsible for minimizing discrepancy between model and historical pressure data

w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big)

is responsible for minimizing the curvature of the transient response (which reflects the diffusion character of the pressure response to well flow)

w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2

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  MDCV also adjusts the rate histories for each well \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q^{(\alpha)}_k \}_{\alpha = 1.. N_k} to achieve the best macth of the bottom hole pressure readings.


The weight coefficients w_c and  w_q  control contributions from corresponding components and should be calibrated to the reference transients (manuualy or automatically).


The  MDCV 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 –  RDCV

На 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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