Basic concept

The basic element of deconvolution is the pressure Transient Response (

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

The Drawdown Transient Response (

It is equivalent to conventional drawdown test Drawdown Test with sandface unit-rate production as if the well is not interfering with surrounding wells.

The Cross-well Transient Response (

is producing/injecting.

Although the UTR may last the infinite time but in reservoir engineering practise the UTR is usually assumed to be captured when it develops a boundary-dominated Late Time Response (LTR). It should be noted that sometimes the duration of production history is too short to sense the geometrical boundary and UTR will not capture it in full or not  see a boundary at all.  

The true UTRs are also difficult to acquire in practise as most wells are noticeably interferring during at long-term scales. The exception is the remote well or the well draining an isolated compartment. 

This defines the application of MDCV which pretends to decipher the UTR from BHP and Production/Injection History.

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

Radial Deconvolution Deconvolution ( Hint0RDCV1Radial Deconvolution) corrrelates ) 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 Self-Response or Drawdown Transient Response or  Hint0DTR1Diagonal Transient Response2DTR)
  
  
• Pressure response of the well to offset well unit rate production in absence of other wells  (also called  Cross-well Transient Response or  Hint0CTR1Cross-well transient response2CTR)
  
  

A group of 

The main difference between 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 installations 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  Hint0MDCV1Multiwell deconvolution 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  Hint0MDCV1Multiwell deconvolution are:

• Uncertainty in DTR/CTR, in case of uneventfull production history or synchronized correlated flow variation of two (or more) wells
  
  
• Error Uncertainty in DTR/CTR is 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:

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:

is responsible for minimizing discrepancy between model and historical pressure data

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

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 

The weight coefficients 

The 

Sample

Sample #1 –  RDCV

На рис. 2.1.2 представлена карта участка с тремя скважинами.

Image Removed

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

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. 2.1.7. Влияние скважины P2 на скважину P1

Image Removed

Image Removed

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

See also

...

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Production Analysis (PA) / Pressure Deconvolution
[ MDCV @model ]
[ RDCV ][ RDCV @model ][ RDCV @sample ]
[ XDCV ][ XDCV @model ][ XDCV @sample ] 
[ Multiwell Retrospective Testing (MRT) ]