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 concept

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

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

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

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

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

It is equivalent to the pressure interference test 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 conditionsMDCV is only working in low-compressibility formations, hence before using MDCV one should analyse the data to see if this condition is met for the tested area.

MDCV can be performed in two options: Radial Deconvolution (

Radial Deconvolution (

• Pressure response of the well to its unit rate production in absence of other wells (also called Diagonal Transient Response or 

  Hint
  0 DTR 1 Diagonal Transient Response 2 DTR

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

  Hint
  0 CTR 1 Cross-well transient response 2 CTR

  )
  
  

A group of 

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 (

The main advantage of 

The group of 

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 

Main benefits of 

• 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 

• 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

with assumption:

• LaTeX Math Inline
  body q^{(-1)}_k = 0

   – for any well 

  LaTeX Math Inline
  body k = 1.. \ N

  
  
  

• LaTeX Math Inline
  body p^u_{nk}(\tau) = 0

   at 

  LaTeX Math Inline
  body \tau < 0

   for any pair of wells 

  LaTeX Math Inline
  body n, k = 1.. \ N

Hence, convolution is using initial formation pressure 

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

The 

\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:

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

where

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:

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 

The weight coefficients 

The 

Sample

Sample #1 –  RDCV

На рис. 2.1.2 представлена карта участка с тремя скважинами.

^{Синтетическая история работы добывающей скважины с простым поведением продуктивности.}

На Рис. 2.1.4 приведена история дебитов и давлений по всем скважинам.

Пример #2 – КДКВ

На Рис. 2.1.5 представлена карта участка с тремя скважинами.

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.