SPE 111269 (Ilk) Deconvolution Based on Cum Prod Cont Meas q and Pwf

(SPE 111269 DRAFT DI Version, 071017)

Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the 2007 SPE Eastern Regional Meeting held in Lexington, Kentucky, U.S.A., 1719 October 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.

Abstract

This work presents a new methodology for the analysis of permanent downhole gauge and production data based on the deconvolution of cumulative production and bottomhole pres-sure measurements.

In this work we show that the cumulative production is a con-volution of pressure drop (as if the well were producing at a series of constant flowing bottomhole pressures) and flowrates (as if the well were producing at a constant pressure through-out the production history). We then employ our modified B-spline based deconvolution algorithm which, in this work, is based on constant stepwise pressures. We use the analytical integration of the B-splines, weighted least squares and regu-larization to obtain the constant-pressure rate response func-tion and further to estimate the initial pressure. The Laplace transformation is not required in this work.

We have validated this approach thoroughly using a synthetic case and we have applied this methodology to various field example cases consisting of both high and low frequency pro-duction data. The results indicate that the new method is very efficient in recovering the constant-pressure rate response function in cases where rates are controlled by series of con-stant flowing bottomhole pressures. We also provide a procedure to check the correlation between pressure and rate measurements by the joint use of two deconvolution methods (SPE 95571 and this work).

Introduction

Our primary goal is to provide a robust and error-tolerant deconvolution approach for the diagnosis of "high-frequency" permanent downhole gauge and "long-term" production data. Our approach centers on the use of the cumulative production as the "impulse" in the convolution/deconvolution process.

This approach is in contrast to prior work [Ilk et al (2006)] in which we focused on using the raw rate and pressure history for our "B-spline deconvolution" algorithm.

In the work by Ilk et al (2006) the objective was to provide the constant-rate pressure response function and as noted before, our method focuses on providing the constant-pressure rate response function. As stated, obtaining the constant-pressure rate response function is not the ultimate objective but the most important step in our procedure for the diagnosis of the production data. In fact, in this paper we do not propose to analyze the deconvolved rate response function exclusively obviously these results can be analyzed, but our contention is that the results alone will not be very meaningful unless this procedure (deconvolution) is used in conjunction with another method (i.e. forward modeling).

We believe that no matter how robust a particular deconvolu-tion algorithm may be, the severely ill-conditioned nature of the problem prevents the generation of a high resolution deconvolved response function. At this point regularization (or some other optimization constraint method) becomes the critical element for a given deconvolution algorithm where regularization is used to overcome errors associated with the data. An "ideal" regularization method will yield a "well-behaved" solution however; regularization may yield smoothness, but it may (and probably will) yield biased deconvolved responses. Bias in the raw data functions can yield artifacts in the deconvolved response which may be misinterpreted as reservoir features.

Our primary hypothesis is that the only way to verify if deconvolution is successful or not is to regenerate the pressures and rates using convolution based on the "solution" of the deconvolution problem i.e., the constant-rate pres-sure response function or the constant-pressure rate response function, as appropriate. In such a scheme the measured pressure would be convolved with the constant-pressure (deconvolution) solution to "regenerate" the rate profile. A similar approach would be used for the measured rates to generate the variable-rate pressure profile.

What does this prove? In a simplistic fashion we are using deconvolution as a "filter," then use the filtered results to "regenerate" the unfiltered results this process serves to validate the data, to assess quality and establish how well the measured rates and pressures are correlated. This is not an analysis process. It is a consistency check, and if the

SPE 111269

A Deconvolution Method Based on Cumulative Production for Continuously Measured Flowrate and Pressure Data D. Ilk, SPE, P.P. Valk, SPE, and T.A. Blasingame, SPE, Texas A&M U.

2 D. Ilk, P.P. Valk, and T.A. Blasingame SPE 111269

measured and "regenerated" data do not agree, then the data are not correlated, and any deconvolution or convolution would be invalid. Some might argue that such an approach is prone to artifacts, but as our examples will show, this is not the case.

Used properly, this process should prevent over-smoothing in the deconvolution as an over-smooth deconvolution solution would yield an oversmoothed convolution when we "regenerate" the data functions. The approach has been successful in our experience so far, but we have also observed some cases where the deconvolution methodology could not yield smooth results with optimum regularization. In those cases reproduced data honor the base data to a certain extent but the deconvolved responses are not smooth enough to be analyzed and increasing the effect of regularization in the solution results in over-smoothing.

At this point we do not choose to make changes in the deconvolution methodology (in particular changes related with implementation of the regularization procedure used in metho-dology). We believe that the issues are data related (e.g. synchronization of pressure/rate data, transient break point identification [Nomura (2006)], inconsistencies in pressure/ rate data, etc.) and further modification of the methodology will not significantly improve the results without considering the factors mentioned before. This can be another path for further study but in this work we shift our attention to identify the correlation between pressure and rate data using decon-volution. If the data are correlated then it is possible to perform any type of analysis on the data set.

We note that the significance of deconvolution in well test and production analysis is not underestimated but we put forward another use of deconvolution we believe that the identifica-tion of the correlation between pressure and rate data has significant importance in well test and production data analysis in terms of diagnostics. In particular, this is an important issue in "high-frequency" data sets where data measurements are taken by permanent downhole gauges.

In the light of these remarks, we propose to use variable rate and variable pressure deconvolution approaches together and regenerate the pressure and rate responses and finally check if the data are correlated or not. Ideally, when two approaches are used together, regenerated pressures and rates have to be identical with the base data. Any deviations or mismatches are attributed to data inconsistencies. We believe that tying both the constant rate and constant pressure deconvolution approaches together will provide a "double deconvolution" process, which while independent, should yield a clear correlation of the production rates and pressures if these data are truly correlated. Again, our goal is to provide a simul-taneous data diagnostic and a data quality check.

To begin this process, we modify our B-spline based deconvolution algorithm [Ilk et al (2006)] to yield constant-pressure rate response functions. For this case we must recall the diffusivity equation, for convenience we cite van Ever-dingen and Hurst [van Everdingen and Hurst (1949)] as this was one of the earliest and most comprehensive treatments on the solution of the diffusivity equation. From van Everdingen and Hurst, in modern nomenclature, we have:

tp

ck

rpr

rr t =

1 (in Darcy (or SI) units).................. (1)

Mathematically, a particular solution to Eq. 1 (i.e. second or-der partial differential equation) is obtained by employing an initial condition and two boundary conditions the outer boundary condition defines the reservoir boundary as the inner boundary condition describes the operating conditions. Well test analysis solutions are generally constructed upon the solu-tion of Eq. 1 using constant rate inner boundary condition. When the inner boundary condition is not constant (i.e. rates are changing with time) then Duhamel principle (superposition principle) applies to yield the flowing well pressures which are affected by the variable flow rates. The simplest example of a variable rate inner boundary condition is a production sequence which consists of a constant rate drawdown followed by a shut-in. Eliminating the variable rate effects on the pres-sure and transforming the entire production sequence into a constant rate equivalent is essentially a variable-rate decon-volution problem and recently robust algorithms have been proposed in the literature to solve this problem [von Schroeter et al (2002, 2004), Levitan (2005), Levitan et al (2006), Ilk et al (2006), Ilk (2005)]. In addition, an investigation on these algorithms is also available in literature [Cinar et al (2006)].

Contrary to the well testing, long term production of a well is controlled by a constant bottomhole flowing pressures and for those cases Eq. 1 is solved by applying constant pressure inner boundary condition to obtain the rate function. In a similar way, the rate function which is affected by the variable pressure changes is acquired by the Duhamel's principle as well and solving for the constant pressure rate function translates into variable-pressure deconvolution. Few attempts have focused on variable-pressure deconvolution in the litera-ture so far. Unneland et al (1998) presents the application of "rate convolution" for reservoir description and well perfor-mance monitoring. On the other hand, Kuchuk et al (2005) provides a modification of the von Schroeter algorithm [von Schroeter et al (2002, 2004)] for obtaining the constant-pres-sure rate function which they call this procedure as the rate/-pressure deconvolution in their work. Kuchuk et al apply this methodology for monitoring the productivity index, produc-tion forecast and parameter estimation.

Our approach is by some means similar to Kuchuk et al methodology. But, our main objective is to use the variable-pressure deconvolution algorithm along with the variable-rate deconvolution algorithm for the diagnostics of the production data. In addition, we prefer to analyze the deconvolved rate response function in "decline-rate" form [Fetkovich (1980)] and its auxiliary functions [Doublet et al (1994)] with the use of "material balance time" [Blasingame and Lee (1988), Doublet et al (1994)]. We note that this procedure is the regu-lar practice in the analysis of the production data [Palacio and Blasingame (1993), Agarwal et al (1999)] either by type curves or modeling (since few solutions of Eq. 1 using constant-pressure inner boundary condition have been deve-loped to date compared to the solutions using constant-rate inner boundary condition).

In short, in this work we provide a deconvolution method to obtain the constant-pressure rate function using the cumulative

SPE 111269 A Deconvolution Method Based on Cumulative Production for Continuously Measured Flowrate and Pressure Data 3

production and flowing bottomhole pressure as the input data for deconvolution. We also regenerate the variable-pressure rates using the constant-pressure rate function (obtained from deconvolution) by superposition and compare the regenerated rates with the base rate data. We tie this procedure with the regenerated variable-rate pressures using the constant-pressure rate function (obtained from variable-rate deconvolution algo-rithm) to check the data consistency. Further, we analyze the deconvolved rate response function using the decline rate and its auxiliary functions by matching with a type curve and/or with a model. In the next section we describe the develop-ment of the modified B-spline based deconvolution metho-dology.

Development of the Modified Deconvolution Method Based on Cumulative Production and Pressures

As mentioned before in well test analysis the solution of the diffusivity equation (Eq. 1) is derived using the constant rate inner boundary condition (i.e., the constant-rate pressure res-ponse solution). For the variable-rate case, then the pressure response is obtained using the superposition principle (i.e., Duhamel's principle), which is given in the convolution integral form as:

dptqt

ptp 'uiwf )()(0

)( = .................................... (2)

The goal of the variable-rate deconvolution (solving for the constant-rate pressure response, which is essentially the in-verse problem) is to estimate the constant flowrate reservoir response using the measured pressure and flowrate data. For most deconvolution algorithms which reside in the recent petroleum engineering literature [Kuchuk et al (2005), Levitan (2005), Levitan et al (2006), von Schroeter et al (2002, 2004)], the algorithms are based on the use of constant step-wise flowrates (the exception is the Ilk et al (2006) algorithm where the rates are represented continuously (in functional form) and Laplace transform is involved).

The rates are convolved (leading to Laplace transformation if rates are represented with functionals) or superposed (discreti-zation in real time domain if rates are represented using con-stant stepwise functions) with the derivative of the unknown, constant-rate pressure response function. Where the constant-rate pressure response function is represented (approximated) using an interpolating function. As the final step, the resulting system of equations is solved for the coefficients of the interpolating function in order to obtain the constant-rate pressure response function.

Given the discussion above, it is reasonable to assume that for the analysis of production data, the B-spline based deconvolution algorithm should work as efficiently as they work in the case of well test analysis (pressure transient test data). As long-term production data sequences are often app-roximated by series of constant bottomhole pressures, we be-lieve that the generalized deconvolution approach should work well for the "step-pressure" case and consequently deconvo-lution approach should be modified to yield the constant-pres-sure rate function instead of constant-rate pressure function.

In such cases, since the rates can not be controlled by a single constant bottomhole pressure during the entire production history, the rate data are distorted by the variable pressures which results in the variable-pressure rate response function. The solution of the diffusivity equation using the constant pressure inner boundary condition (Eq. 1) and using the Duhamel's principle leads us the following convolution integral describing the variable-pressure rate response func-tion:

dqtpt

tq 'uwf )()(0

)( = .......................................... (3)

In Eq. 3, we note that )(tq'u represents the derivative of the constant-pressure rate solution. Eq. 3 can be derived from Eq. 2 and the details are given in the seminal work of van Everdingen and Hurst [van Everdingen and Hurst (1949)]. The work by Kuchuk et al [Kuchuk et al (2005)] presents a deconvolution methodology based on the solution of Eq. 3 as indicated before.

In this work, our goal is to use the cumulative production profile instead of the rate profile as depicted in Eq. 3. There-fore, taking the integral of both sides of Eq. 3 with respect to time, we obtain

dqtpt

tN uwfp )()(0

)( = ...................................... (4)

For a quick proof, if pwf(t) is assumed constant, it is factored out of the integral and we obtain the definition of the cumula-tive production function. We state two reasons for using the cumulative production data: First, we favor to solve for the rate function (in Eq. 4) rather than solving for the derivative of the rate function with respect to time (in Eq. 3) because the rate-derivative function will more likely to be unstable. Se-cond, cumulative production data are smoother (and generally more reliable) than rate data and therefore considering the ill-conditioned nature of the problem, using the cumulative pro-duction data should provide advantages on the methodology than using the rate profile.

We now create the B-spline methodology using Eq. 4 by representing the unknown constant-pressure rate function as a linear combination of 2nd-order B-splines this gives:

)(2)( tiBicu

litqu =

= .............................................................. (5)

Substituting Eq. 5 into Eq. 3, we have:

dptiBicu

li

ttN wfp )()-(

20

)( =

= .............................. (6)

If we have m cumulative pressure observations collected in a vector pN

~ that were observed at times (t1, t2, tm) then the problem can be written as an overdetermined system of linear equations.

This overdetermined set of equations is written as (we show here the case where the weights are unity):

XcNp =~ ............................................................................ (7)


Where X is the mxn sensitivity matrix (n is the number of unknown B-spline coefficients and n=u-l+1). Theoretically, the elements of the sensitivity matrix are defined as:

dptBt

X wfjij

j,i )( )(0

2 = ...................................... (8)

Where tj is the time when the rate and pressure measurement is taken. We can take the pressure drop function outside the integral when we use constant stepwise pressure drop func-tions for representing the continuous pressure drop profile as expressed in Eq. 8 (similar to superposition). Performing these operations on Eq. 8, we obtain:

=

=j

kkjijjwfj,i ttBttpX

11

2int,1 )( )( .......................... (9)

In this case, Eq 9 can be calculated by use of closed formulas where the analytic integral of the second-order B-spline is available [Cheney and Kincaid (2003)] and hence the Laplace transformation is not required. In this work, the entire methodology (i.e., weighting and regularization) is same with Ilk et al (2006) [Ilk et al (2006)] except for the calculation of the elements of the sensitivity matrix, X. In this work we use constant stepwise pressure drop functions to obtain Eq. 9 from Eq. 8 and therefore the use of the Laplace transformation is not required. Note that the Laplace transformation is also not needed in the previous methodology (variable rate decon-volution) as well when the rate data are denoted by constant stepwise functions.

Proceeding, we solve Eq. 7 (using weighted least squares) for the coefficients of the B-spline function with proper weighting of the data and appropriate levels of regularization. Once the coefficients are obtained, the constant-pressure rate function is constructed (we denote this process as the variable-pressure rate deconvolution throughout this work). We finally regene-rate the rates affected by the variable-pressure profile by convolution/superposition using the obtained constant-pres-sure rate function. The interested reader is referred to Ilk et al (2006) [Ilk et al (2006)] for the details regarding the "B-spline" deconvolution methodology. It is worth to note that the initial pressure can also be estimated while solving Eq. 7 for the unknown B-spline coefficients.

Validation Constant Pressure Production

In this section, we provide the validation of the methodology by comparing the results from the deconvolution procedure with the exact constant-pressure rate solution using a reservoir model. We have selected a homogeneous reservoir with a cir-cular boundary (the reservoir and fluid parameters used for this model are given in Table 1) where the single-phase flow-ing fluid is oil.

For reference, the constant-pressure rate solution of a homogeneous reservoir with circular boundary is given in Laplace domain by:

srKsIssKsrIs

sKsrIssKsrIssqee

eeD

)( )( )( )( )( )( )( )( )(

1001

1111+= ......... (10)

To generate the variable-pressure rate responses, we have specified three different constant-pressure sequences through-out the production history and we generate the rates by convolving the constant-pressure rate solution (Eq. 10) with the prescribed pressure drops in Laplace domain. Finally we take the inverse Laplace transform to get the rate function in time domain (Fig 1a).

Table 1 Reservoir and fluid properties for the Validation Case

Reservoir Properties: Wellbore radius, rw = 0.3 ft Net pay thickness, h = 50 ft Formation permeability, k = 2 md Formation compressibility, ct = 510-6 1/psi Porosity, = 0.15 (fraction) Outer boundary radius, re = 420 ft Initial reservoir pressure, pi = 5000 psi Wellbore storage coefficient, CD = 0 (dimensionless) Skin factor, sx = 0 (dimensionless)

Fluid Properties: Fluid viscosity, o = 1 cp Formation volume factor, Bo = 1 RB/STB

Production Parameters: Reference flowing pressure, pwf,ref = 2500 psia

Figs. 1a and 1b present the base and input data for deconvolu-tion. Our goal is to create a production sequence which is controlled by constant bottomhole pressures to mimic a typical production sequence from a producing well. Note that the changes in pressures cause transient flow periods at the begin-ning of each pressure variation. Traditional decline curve ana-lysis methods [Arps (1945)] make use of fitting functions (hyperbolic/exponential/harmonic) to the rate decline function and proceed to forecast. In addition exponential decline empi-rically corresponds to the condition where the flowing bottom-hole pressure is constant throughout the production sequence during the boundary dominated flow region. In Figs 1a and 1b exponential rate decline is easy to be identified at each production interval because of the constant bottomhole pres-sures.

Figure 1a Generated flowrate data using the constant-pressure rate solution.


Figure 1b Bottomhole pressure data and computed cumu-lative oil production data (input data for deconvo-lution).

Our motivation in this example is to prove that the modified B-spline based deconvolution algorithm successfully converts the variable-pressure rates into a constant pressure equivalent rate function by matching the deconvolved response with the exact (analytical) solution (i.e. Eq. 10). After getting the de-convolved rate response function it is theoretically possible to employ Arps' analysis [Arps (1945)] using the exponential decline function. But again we mention that the deconvolved response rate function will include the transient flow region and exponential decline function is only valid for the boundary dominated flow region. Fig. 2 presents the deconvolved con-stant-pressure rate response function and its auxiliary func-tions (i.e. rate-integral and rate-integral derivative) in log-log coordinates. Note that the auxiliary functions are computed numerically from the deconvolved rate response function. Since this is an "ideal" synthetic case we see an excellent reso-lution of the transient and boundary dominated flow regimes.

Figure 2 Deconvolved rate data functions (constant pres-sure equivalent case).

In Fig. 3 we verify our results by matching with the analytical results obtained by using Eq. 10. We get a nearly identical match with the analytical solution confirming the validity of our concept and methodology. It is noted that in this work we do not present deconvolution results when the input data are corrupted by random (Gaussian) errors. In our previous stu-dies ([Ilk (2005)], [Ilk et al (2006)] and [Cinar et al (2006)])

we have presented a comprehensive study on deconvolution results when the base data are corrupted by errors.

Figure 3 Comparison of the deconvolved data with the exact solution validation of the concept.

Therefore, here we do not choose the show the cases with ran-dom errors since the B-spline based deconvolution metho-dology is essentially the same with the previous work (except for not taking the Laplace transformation of the convolution integral in this work we believe that this will not have a significant effect on the deconvolution results when data are corrupted by random errors but a further investigation may be required).

Fig. 4 presents the deconvolved rate response rate function in semi-log coordinates along with the base rate data and the analytical solution. This plot (Fig. 4) simply illustrates the definition of deconvolution. Taking the reference pressure as the first pressure at the beginning of the production sequence, deconvolution provides the equivalent rate response function as if the well is producing with a constant reference bottom-hole pressure. Furthermore, the effect of the pressure varia-tions and the related transients on the rate response is elimi-nated with deconvolution as can be seen in Fig. 4. Finally, traditional decline curve analysis can be performed by using the exponential decline model (constant bottomhole pressure) in the boundary dominated flow region of the deconvolved rate response function.

Figure 4 Comparison of the variable-pressure rate data with the deconvolved (constant-pressure) rate data.


We present the Cartesian plot of the regenerated rates and the base rate data in Fig. 5 we also note the identical match. The regenerated rates are obtained by convolving (superpos-ing) the deconvolved rate response function with the avail-able pressure history during the production sequence.

Figure 5 Regenerated rate data using deconvolution, iden-tical match with the base rate data.

The purpose of this plot is to check if the base rate data are matched with the regenerated rates according to superposition principle. If data are not matched then deconvolution fails for the particular case. In other words any deviations or mis-matches indicate the inconsistency of the data set. As stated before we will mostly focus on the diagnostics (consistency check) of the production data set. As such, we also plot the regenerated pressures obtained from variable-rate decon-volution to check with the base pressure data as well we also call this process in this work as the "double decon-volution" approach. Essentially, the pressure and rate matches indicate the consistency level of the data set therefore we believe that Fig. 5 has significant value as we will show more field examples in the following section.

Application Field Cases

In this section we present four field example applications of the new variable-pressure deconvolution approach. We will demonstrate the use of our modified deconvolution algorithm on two "long-term" production data field cases, one permanent downhole gauge case and on one field case which is a typical well testing example. We have categorized the field examples below:

East Texas (US) (Tight Gas) Well "regular" (daily) surface rate and pressure measurements.

Canada (Tight Gas) Well "continuous" (very high fre-quency) rate and downhole pressure measurements.

South America Oil Well constant surface rates followed by shut-ins and "continuous" (high frequency) downhole pres-sure measurements.

Wyoming (US) (Tight Gas) Well "regular" (daily) surface rate and pressure measurements.

This exercise has two primary purposes: First, we will show that the modified B-spline based deconvolution algorithm without using the Laplace transformation and with the use of cumulative production and pressure data as the input is a

robust mechanism to analyze production data. Second and most importantly, we will demonstrate that the combined use of variable-rate and variable-pressure deconvolution results in a significant diagnostic tool for checking the consistency of the production data. The latter has particular importance in production monitoring applications.

Example 1: East Texas Gas Well

Our first example presents a hydraulically fractured tight gas well from an East Texas gas field. This case was initially presented and analyzed by Pratikno et al [Pratikno et al (2003)] using decline type curves developed for hydraulically fractured wells. We have also analyzed this well in our previous work by variable-rate deconvolution [Ilk et al (2006)]. We were recently provided with an update data for this well extending the production history to almost 6 years. Fig. 6a presents the production data for this well this well has been on continuous surveillance (daily surface gas rates and surface pressures available) for almost 6 years. We obser-ve a few off-trend data some erratic pressures and scattered rates. Other than that production data are considered to be good quality by initial visual inspection and later we are going to prove our statement using our proposed procedure for data consistency check.

In Fig. 6b we present the input data for deconvolution cumulative production and pressure. There is an overall constant pressure trend throughout the production (except for some abnormal pressures) and we note that cumulative pro-duction follows a very smooth trend which should yield a more stable deconvolution process according to our expect-ations.

Figure 6a East Tx gas well, production history plot. Data quality is good in general.

Before performing deconvolution, we need to specify that the initial pressure for this well was available from the operator and later we confirm this value with the initial pressure estimate obtained by the deconvolution algorithm. We use the entire data sequence for deconvolution but weight the data less which we consider unreliable. However, before proceeding further at this point we have to state that we do not weight the data for the consistency check procedure. We will discuss this distinction later in the text.


Figure 6b East Tx gas well, computed cumulative gas production data (input for deconvolution).

Fig. 7 presents the deconvolved rate response function and its auxiliary functions (i.e. rate integral and rate-integral deriva-tive both of which are computed numerically). In our analysis procedure, Fig. 7 has a general diagnostic purpose in this case we are able to identify the fracture flow and the transition to boundary dominated flow. However, early part of the de-convolved response is not analyzable due to the overall data quality and the other issues (discussed before) that could have caused instabilities in the deconvolution algorithm. For these reasons we do not prefer to conclude our analysis only based on this plot. Instead we will convert this plot into its constant-rate equivalent (using material balance time) in order to match with a model which will be a finite conductivity fracture for this well (the available solutions for finite conductivity frac-tured wells are present in constant rate equivalent format).

Figure 7 Deconvolved rate data functions (constant pressure equivalent), excellent diagnostic cha-racter is exhibited.

Fig. 8 presents the deconvolved rate response functions and the finite conductivity fractured well model. We observe an excellent match of the deconvolved data functions for two log cycles but data are not matched with the model for the first two log cycles. We also note that pseudopressure and pseudo-time transformations (both for deconvolution and for model

match processes) are required for linearizing the problem.

Figure 8 Deconvolved data functions (constant rate equi-valent material balance time) and match with a finite conductivity model.

In short, the match is very satisfactory and the obtained para-meters are very consistent with our previous analyses. But we still need to prove that the regenerated rates are consistent with the base rate data to verify if deconvolution has worked for this case or not.

In Fig. 9 the deconvolved rate response function is presented in traditional rate decline format (semi-log plot). As seen in the plot, the variations in rates caused by the pressure changes are completely eliminated by deconvolution. One can also employ traditional decline curve analysis using the constant-pressure rate function to estimate the expected ultimate reco-very (EUR) for this well. But we strongly advise caution on using decline curve analysis with the deconcolved rate res-ponse because deconvolved responses are extremely sensitive to data quality.


We have already shown in Figs 7 and 8 that deconvolved results make sense but we need to verify that these results are consistent with the base data. To achieve this, we regenerate the rate responses by superposing the deconvolved constant-pressure rate response function with the input pressure data. If


the results are acceptable with the base rate data within a cer-tain extent then we conclude that deconvolution is successful. This procedure is what we have been doing so far to verify de-convolution results for a specific case. Now we move for-ward to a new step which is the establishment of correla-tion/consistency check between the base rate and pressure data.

In addition to regeneration of rates by the deconvolved con-stant-pressure rate function, we regenerate the pressures by the deconvolved constant-rate pressure function obtained using our previous method variable-rate deconvolution algorithm. For the first method (variable-rate deconvolution) rates are fixed, in this work (variable-pressure deconvolution) pres-sures are fixed in the algorithm. Therefore, regenerated rates by the pressures and regenerated pressures by the rates should be consistent with the measured rate and pressure data in order to satisfy the superposition (Duhamel's) principle (Eqs. 2 and 3). We note that these two processes are independent from each other but indeed provide an apparent correlation of rate and pressure data. In Fig. 10 we apply this methodology to identify the correlation between rate and pressure data. First, the regenerated rates obtained by superposing the available pressure data and the deconvolved rate response function exhibit excellent agreement with the rate data. Next, rege-nerated pressures are obtained and plotted with the base pres-sure data it is seen that in general there is a good match of the pressures except for some minor deviations. We conclude from this practice that by and large, pressure and rate data are well correlated but there are still some issues (we suspect that in fact there had been some unreported variations in the pressure data during the entire production history). Never-theless, in this case our diagnostic check confirms the data quality and provides the analyst with the information that data are correlated and deconvolution (or any other production data analysis methodology) should safely be used for the analysis of the production data of this well. It is worth to note that during the consistency check process, we minimize the intervention to deconvolution algorithm i.e. weights are not assigned and the value of the regularization parameter is either kept as zero or at minimum to prevent bias/artifacts.

Figure 10 Regenerated rate and pressure data by deconvolution excellent data matches confirm the correlation between rate and pressure data.

Example 2: Canada Gas Well

This case is an example of "high-frequency" near perfect pro-duction data which were collected by a permanent downhole gauge system.

Figure 11a Canada gas well production history plot high frequency, excellent quality data.

Fig. 11a presents the base pressure and rate data for this well. The well is producing from a tight gas reservoir in Canada. A careful visual inspection of rate and pressure data confirms the data quality at first sight. Our purpose is to analyze this data set by variable-pressure deconvolution and confirm the quality of the data set.

In Fig. 11b the input data for deconvolution are presented we note a very smooth cumulative production trend except for the break at the instant of shut-in. The entire data are used in deconvolution unlike variable-rate pressure, in this case, it is not practical to use only the data during the shut-in for deconvolution. In fact, we do not prefer to use only portions of the data for deconvolution instead as mentioned before we weight the data (again we remind that weighting is not performed for data quality check).

Figure 11b Canada gas well, computed cumulative gas pro-duction data (input for deconvolution).

Fig. 12 presents the results of variable-pressure deconvolution for this case. As expected, since the data quality is beyond expectations a clear resolution of all flow regimes are present on the plot. The character of finite-conductivity fracture is


identified during early time and it can also be seen from the deconvolved responses that the well is in transition to full boundary dominated flow (late time data).

Figure 12 Deconvolved rate data functions (constant pres-sure equivalent), outstanding diagnostic charac-ter.

We note that these results can also be obtained using modern production data analysis methods developed for analyzing variable rate and pressure data [Doublet (1994), Palacio (1993), Agarwal (1999)]. The goal of all these well-estab-lished methods is to remove the effect of variable-pressures as same with deconvolution.

In Fig. 13 we present the deconvolved responses and its com-puted auxiliary functions in constant-rate equivalent format (using material balance time). We match these data functions with a finite conductivity fracture model and get excellent agreement. This model is also verified using other production data analysis methods as well.


Using the first pressure point as the reference, we present the constant-pressure rate response function in traditional decline curve format (semi-log plot) in Fig. 14. Since pressure was kept during the early days of production, deconvolved con-stant-pressure rate function is identical with the base flowrate

data.


Deviation of the deconvolved rate response function from the rate data is observed when pressure variations are introduced on the data set. The decline of the deconvolved rate response function is similar to exponential decline. However, a few artifacts are observed between 100 and 200 days of product-ion.

We can estimate the "Expected Ultimate Recovery" by using Arps [Arps (1945)] analysis. In this case our estimate of EUR agrees well with other methods but for noisy data sets this may not always be the case. We encourage the interested reader to use deconvolution for this purpose but recommend extreme caution because of the data quality.

In Fig. 15 we present the results for our data quality check procedure. As we have stated previously in the beginning of this example, data quality is extraordinary for this well. This is also confirmed in Fig. 15 using our "double-deconvolution" procedure for data correlation purpose.

Figure 15 Regenerated rate and pressure data by deconvolution almost identical data matches confirm the quality of the data.

There is an outstanding match of the regenerated rates with the base rate data. It is seen that the pressures are not matched (pressure buildup part) exactly as the rates are matched. It is not possible to identify which data have issues since the two


deconvolution algorithms are independent. However, we can conclude that the matches are excellent confirming the overall data quality. This example shows the significance of data qua-lity on deconvolution actually the importance can easily be generalized for the other methods for analyzing production data. In short the better the data quality is more robust/effi-cient will the analysis be.

Example 3: South America Oil Well

This example can easily be treated by the traditional well test analysis methodologies (and obviously by variable-rate decon-volution method) since the production sequence which inclu-des two drawdowns followed by the shut-ins was designed for the pressure transient analysis of this well. Fig 16a presents the measured pressure and rate data for this case. As for this well it was reported that the data were taken from continuous (almost instantaneous) measurements of surface oil rates and bottomhole pressures.

This well is a horizontal oil well (permeabilities in this field are ranging between 20-30 md) from South America. In Fig. 16b the input data for deconvolution are presented. We can easily analyze this set by variable-rate deconvolution to remove the effects of the variable rates on the pressure data (as we will show the results of variable-rate deconvolution later as well) but our goal is to perform the variable-pressure deconvolution methodology for this type of data set and test the results of the variable-pressure deconvolution metho-dology.

Figure 16a South America oil well production history plot constant production rates, well test sequence.

In fact, for this example we are particularly interested in the correlation between the pressure and rate data. And as such, using the constant-pressure rate function, which is to be obtained by deconvolution, we will observe that if the base rate data are matched with the regenerated rates or not. In addition, variable-rate deconvolution will be performed to obtain the constant-rate pressure function and then pressures will be regenerated by superposing the constant-rate pressure function with the rate data.

Figure 16b South America oil well, computed cumulative oil production data (input for deconvolution).

In an ideal case, the deconvolved responses obtained by the two deconvolution algorithms; variable-rate and variable-pres-sure, have to be consistent (i.e. the deconvolved responses must be matched with solutions using the same model and its parameters). The "near ideal" previous example already proves this statement (see Fig. 15). In this case effect of the issues related with data (or data correlation) on the two deconvolution algorithms is investigated.

In Fig. 17a the results of variable-pressure deconvolution are presented. As seen the deconvolved constant-pressure rate response and its computed auxiliary functions are not suffi-cient enough for an interpretation. Initially the results indicate that variable-pressure deconvolution is not a good candidate for the analysis of production data for this well.

Figure 17a Deconvolved rate data functions (constant pressure equivalent), diagnostic character is ques-tionable.

On the other hand, when variable-rate deconvolution is per-formed the deconvolved constant-rate pressure function is smoother and analyzable. We should note that in this case for these two deconvolution algorithms, algorithmic parameters (i.e. number of knots, weights, value of the regularization parameter) are same.


Figure 17b Deconvolved rate data functions (constant rate equivalent, variable-rate pressure deconvolution is used), very good diagnostic character is exhibited.

In Fig. 17b the signature for a horizontal well can be recog-nized on the well-testing derivative trend (i.e. early and late time radial flow). Fig. 17b (variable-rate deconvolution) should be satisfactory for the interpretation of this well but one should not forget that Fig. 17a contradicts this claim as we mentioned before that results using the two algorithms have to be consistent.

Figure 18a Deconvolved data functions (constant rate equi-valent material balance time) and match with a horizontal well model.

We continue to analyze the responses (Figs. 17a and 17b) we obtained using the two deconvolution algorithms in Figs. 18a and 18b. We match the deconvolved responses with the best fit horizontal well models on these plots. The best fit models are shown on Figs. 18a and 18b. As it can easily be seen, in Fig. 18a data do not match very well with the model very well. In contrast, we see a very good match in Fig 18b. In these two figures we use horizontal well models but with dif-ferrent model parameters. When the model shown in Fig 18b is imposed on the data in Fig 18a (except for wellbore sto-rage), the match gets worse clearly implying the ill-conditioned nature of deconvolution and instability caused by

inconsistent data.

Figure 18b Deconvolved data functions (constant rate equi-valent, variable-rate pressure deconvolution is used) and match with a horizontal well model.

In Fig. 19 as for each case we show the constant-pressure rate response function in traditional decline curve analysis format (semi-log plot). This rate function represents the case as if the pressure is kept constant throughout the production. However, in this case the rates are being kept constant during drawdown periods. Therefore, this plot is not very significant for this case unless the pressure and rate data are perfect. But we al-ready showed that this is not the case in this example.


In addition, when used alone the constant-pressure rate func-tion is not very meaningful because of the artifacts especially seen at earlier times. We did not even attempt to estimate the EUR in this case.

The other way to identify the data inconsistency in this case is to apply our "double deconvolution" procedure to regenerate the pressure and rate data to match with the base pressure and rate data. Using the same algorithmic parameters we run the two deconvolution algorithms to get the deconvolved constant-pressure/rate response functions and with these functions and base rate and pressure data, we regenerate the pressures and rates, respectively. Fig. 20 presents the matches of the regenerated pressures and rates along with the base pressure and rate data. Pressure match is excellent with the


available constant rates but conversely the rate match is not satisfactory with the given pressures. As the two processes are not tied, it is not surprising to get a very good and bad match altogether. It is not the scope of this work to identify which data or part of the data set is inconsistent but obviously, from Fig. 20 we conclude that there are inconsistencies associated with pressure and rate data present in this case.

Figure 20 Regenerated rate and pressure data by decon-volution pressure match is good but the rate match is questionable indicating issues regarding with pressure and rate data correlation.

Example 4: Wyoming Gas Well

In the last example we are going to perform the variable-pressure deconvolution algorithm on a data set taken from a hydraulically fractured Wyoming gas well. This well is pro-ducing from a tight gas sand with permeabilities less than 0.1 md. Fig 21a presents the measured rate and pressure data for this well. The measurements were taken daily (surface rates and surface pressures then converted into bottomhole) and a pressure buildup test for 5 days was performed at a late stage during the production sequence of this well.

It is worth to note that data quality for this well (i.e. daily rates and pressures) is beyond average for a typical production data set. Therefore, our expectations regarding the analysis of the production data are higher for this well. For further reference, transient flow period can be distinguished by inspecting the bottomhole pressures on Fig. 21a.

Figure 21a Wyoming gas well, production history plot. Data quality is good in general.

Upon investigation of shut-in pressures it is observed that significant wellbore storage effects dominate the pressure transient response. For this reason pressure transient analysis most likely will not be sufficient enough to estimate the fracture parameters and most importantly the contacted-gas-in-place of this well. Deconvolution should be a valuable option here for the analysis of the pro-duction data.

In Fig. 21b we present the input data for the variable-pressure deconvolution algorithm. Cumulative production trend is very smooth except for the shut-in part. On the other hand, the pressure data at earlier times might be affected by well clean-up effects.

We present the variable-pressure deconvolution results on Fig. 22. The deconvolved constant-pressure rate function and its auxiliary functions clearly display high resolution. It could be identified from the results that the well is still in transient flow and transition to boundary dominated flow is being estab-lished. Fracture flow signature is exhibited indicating that this data set can be matched using a finite conductivity fractured well model. But again no conclusion is based upon this diag-nostics and further verification of these results is needed.

Figure 21b Wyoming gas well, computed cumulative gas production data (input for deconvolution).

In Fig. 23 we analyze the deconvolved responses in constant-rate equivalent format. At first glance we note the differences on the earlier part of the auxiliary functions (i.e. rate-integral and rate-integral derivative). This is probably due to the change from constant-pressure to constant-rate equivalent functions affecting the numerical integration and the different-tiation scheme for obtaining the auxiliary functions with respect to material balance time. In addition, we note that we use pseudopressure and pseudotime transformation for the analysis in Fig 23.

Deconvolved response functions are then matched with an appropriate fractured well model. We have used a finite con-ductivity well model and obtain a good match of the data functions with the model. The discrepancies at the earlier times are most possibly a result of the erratic behavior of the data (perhaps caused by well cleanup effects or issues related with the conversion of the pressures from the surface to bot-tomhole conditions).


Figure 22 Deconvolved rate data functions (constant pressure equivalent), very good diagnostic cha-racter is observed.

Nevertheless, the imposed model according to our diagnosis in Fig 22 can be verified using other production data analysis methods (i.e. we obtain similar results with the imposed model when we employ other methodologies).


We should also note that the data can simply be analyzed by variable-rate deconvolution alone as well in particular, only the pressure buildup portion can be taken as the input during variable-rate deconvolution process. However, we do not pre-fer to do so. We do not perform variable-rate deconvolution in this case but we make sure of the data quality using the procedure we propose in this work.

Fig. 24 presents the deconvolved constant-pressure rate func-tions on semi-log plot compared with the base rate data affected by variable pressures throughout the production se-quence. Variable-pressure deconvolution algorithm effecti-vely changes the base rate data into its constant-pressure equi-valent and we observe the very good resolution of the decon-volved rate response function in Fig. 24. Once again we con-firm the transient flow region and the transition to the boun-

dary dominated flow in this plot. No estimation of the EUR is attempted using decline curve analysis for this well because of the low permeability prohibits the complete establishment of the boundary dominated flow.


Finally, we apply the procedure for pressure and rate data correlation in Fig. 25. Some artifacts/mismatches previously indicated that some parts of the data might be inconsistent (see Fig. 23). When the rates are regenerated by constant-pressure rate function with the available pressures, outstanding match is observed.

Figure 25 Regenerated rate and pressure data by decon-volution data matches are acceptable confirm-ing the correlation between the rate/pressure data.

On the other hand, the regeneration of the pressures by the constant-rate pressure function with the available rates does not yield an excellent match as the regenerated rates yield with the base rate data. This confirms our doubts about the incon-sistency related with parts of the data set. Yet, the inconsis-tency does not prohibit the analysis since the pressure match is acceptable. Although the proposed procedure does not ob-viously let us to identify which data (pressures or rates) are wrong, we can identify that some of the pressures in this data set are not correct by evaluating Figs 21-25 and most impor-tantly by inspecting the match of the pressures in Fig. 25. In this case we believe that the rates are measured accurately and some of the pressure data are affected by well cleanup effects. All in all, we believe that despite of the minor inconsistencies


related with the pressure data, deconvolution works for this case with appropriate weighting and regularization as illus-trated in Fig 22 and confirmed in Fig. 25. Again, we also prove the significance of the data quality check with our proposed procedure in this example.

Summary and Conclusions

Summary: The most important and most substantive contribu-tion of this work is the development of a "variable-pressure" deconvolution algorithm that is based on the cumulative fluid production. This achievement may seem minor or redundant in light of the fact that essentially all other deconvolution methods to date have relied on variable-rate deconvolution, where the output is the "constant rate" pressure profile. In this case, the output is the "constant pressure" rate profile.

This approach permits us to characterize each input data stream (e.g., flowrate or bottomhole pressure) as a convolution of the other data. This capability provides us with very strong (and theoretically consistent) diagnostic method. The pro-posed "double deconvolution" method is less of an analysis tool than it is a diagnostic tool (for model identification) and a "filter" for confirming the consistency of the flowrate and bot-tomhole pressure data.

Conclusions:

1. The "cumulative production" formulation of the proposed rate deconvolution approach is both robust and error tolerant.

2. The proposed "double deconvolution" method (rates from pressures; and pressures from rates) provides a consistency check that has been missing from the literature.

3. The application of this methodology has been successfully demonstrated using typically available field data. This methodology provided insight as the accuracy (and pre-cision) of the measured production data.

Nomenclature

Variables: B = Oil formation volume factor, RB/STB

)(tkiB = k-th degree B-spline starting at bi, dimensionless

CD = Wellbore storage coefficient, dimensionless cf = Formation compressibility, psi-1 cg = Gas compressibility, psi-1 co = Oil compressibility, psi-1 = effective porosity, fraction G = Original gas-in-place, MSCF Gp = Cumulative gas production, MSCF h = Net pay thickness, ft k = Formation permeability, md l = Index of the first knot, dimensionless N = Original oil-in-place, STB Np = Cumulative oil production, STB qg = Gas production rate, MSCFD qnom = Nominal flowrate, STBD or MSCFD qo = Oil production rate, STBD qu = Constant-pressure rate (deconvolution), STB/psia m(p) = Pseudopressure, psi2/cp m(p) = Pseudopressure drop [m(pi)- m(pwf)], psi2/cp pi = Initial reservoir pressure, psia pwf = Flowing bottomhole pressure, psia pwf,ref = Reference flowing bottomhole pressure, psia p = Pressure drop (pi-pwf), psi pu = Constant-rate pressure (deconvolution), psia/STB re = Outer radius, ft

rw = Wellbore radius, ft s = Laplace domain variable sx = Skin factor, dimensionless t = Time, days ta = (gas) Pseudotime, days tmb = [Np/q] (oil) Material balance time, days tmb,gas = [Gp/ qg] (gas) Material balance time, days tmba,gas = (gas) Material balance pseudotime, days u = Index of the last knot, dimensionless z = Gas compressibility factor = Liquid viscosity

Pseudofunctions:

dpz

pp

ppm

base

21)(

= dt

pcp

tct

gggigia )()(

1

0 =

dtpcp

tqt

tqc

tgg

gigigasmba )()(

)(

0)( ,

=

References

Agarwal, R.G., Gardner, D.C., Kleinsteiber, SW., and Fussell, D.D.: "Analyzing Well Production Data Using Combined-Type-Curve and Decline-Curve Analysis Concepts," SPEREE (Oct. 1999) 478-486.

Arps, J.J: "Analysis of Decline Curves," Trans., AIME (1945) 160, 228-247.

Blasingame, T.A., and Lee, W.J.: "The Variable Limits Testing of Gas Wells," paper SPE 17708 presented at the 1988 SPE Gas Technology Symposium, Dallas, TX, 13-15 June.

Cinar, M., Ilk, D., Onur, M., Valk, P.P., and Blasingame, T.A.: "A Comparative Study of Recent Robust Deconvolution Algorithms for Well-Test and Production-Data Analysis," paper SPE 102575 presented at the 2006 SPE Annual Technical Conference and Exhibition, San Antonio, TX, 24-27 October.

Cheney, E.W. and Kincaid, D.R.: Numerical Mathematics and Computing, Pacific Grove, California: Brooks Cole (2003).

Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Oil Well Production Data Using Material Balance Time: Application to Field Cases," paper SPE 28688 presented at the 1994 Petroleum Conference and Exhibition of Mexico held in Veracruz, Mexico, 10-13 October.

Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," JPT (June 1980) 1065-1077.

Ilk, D., Anderson, D.M., Valko, P.P., and Blasingame, T.A.: "Analysis of Gas Well Reservoir Performance Data Using B-Spline Deconvolution," paper SPE 100573 presented at the 2006 SPE Gas Technology Symposium, Calgary, Alberta, Canada, 15-17 May.

Ilk, D., Valko, P.P., and Blasingame, T.A.: "Deconvolution of Variable-Rate Reservoir Performance Data Using B-Splines," SPEREE (October 2006) 582.

Ilk, D.: Deconvolution of Variable Rate Reservoir Performance Data Using B-Splines, MSc. Thesis, Texas A&M University, College Station, Texas, USA (December 2005).

Kuchuk, F.J., Hollaender, F., Gok, I.M., and Onur, M.: "Decline Curves from Deconvolution of Pressure and Flow-Rate Measure-ments for Production Optimization and Prediction," paper SPE 96002 presented at the 2005 SPE Annual Technical Conference and Exhibition, Dallas, TX, 9-12 October.


Levitan, M.M., Crawford, G.E., and Hardwick, A., "Practical Con-siderations for Pressure-Rate Deconvolution of Well-Test Data," SPEJ (March 2006) 35.

Levitan, M.M.: "Practical Application of Pressure/Rate Decon-volution to Analysis of Real Well Tests," SPEREE (April 2005) 113.

Nomura, M.: Processing and Interpretation of Pressure Transient Data from Permanent Downhole Gauges, PhD. Dissertation, Stanford University, Palo Alto, California, USA (September 2006).

Palacio, J.C. and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves Analysis of Gas Well Production Data," paper SPE 25909 presented at the 1993 Joint Rocky Mountain Regional/Low Permeability Reservoirs Symposium, Denver, CO, 26-28 April.

Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "De-cline Curve Analysis Using Type Curves Fractured Wells," paper SPE 84287 presented at the SPE annual Technical Conference and Exhibition, Denver, Colorado, 5-8 October 2003.

Unneland, T., Manin, Y., and Kuchuk, F.J.: "Application of Permanent Pressure and Rate Measurement Systems in Well Monitoring and Reservoir Description: Field Cases," SPEFE (June 1998) 224.

van Everdingen, A.F., and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME (1949) 186, 305.

von Schroeter, T., Hollaender, F., and Gringarten, A.C.: "Decon-volution of Well Test Data as a Nonlinear Total Least Squares Problem," SPEJ (December 2004) 375.

von Schroeter, T., Hollaender, F., and Gringarten, A.C.: "Analysis of Well Test Data From Downhole Permanent Downhole Gauges by Deconvolution," paper SPE 77688 presented at the 2002 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 29 September-2 October.

Appendix A Formulation of the Modified B-Spline Deconvolution Algorithm in Real Time Domain

We start with Duhamel's principle for the rate function which is distorted by the variable pressures throughout the produc-tion sequence.

dtqpt

tq 'uwf )( )(0

)( = ....................................... (A-1)

We will first prove that Eq. A-1 can be written in terms of cumulative production (i.e., cumulative production is a convo-lution of pressure drop and constant-pressure rate function). Integrating both sides of Eq. A-1 with respect to time yields:

dtqpt

tN uwfp )( )(0

)( = .................................. (A-2)

Our next step is to derive the formulation of the deconvolution algorithm without the use of the Laplace transform. We can discretize Eq. A-2 into n segments and assume constant pres-sure steps across these segments cumulative production is then given by:

dqpp

dqpp

dqpptN

u

nt

ntnwfnwf

u

t

twfwf

u

t

wfwfnp

)( )(

)( )(

)( )( )(

11

2

112

1

001

+

++

+=

KKKK

KKKK

...................................................................................... (A-3)

We now approximate the constant-pressure rate function (qu()) as a linear combination of B-splines.

dtBcpptN ji

jt

jt

u

lii

n

jjwfjwfnp )()()( 2

111 =

== .. (A-4)

The analytical integral of a second order B-spline function is available in closed form [Cheney and Kincaid (2003)] and is given by:

)( 3

1 )( )( 33

-

22,int tB

bbdBtBij

ji

t

ii =

== ..................... (A-5)

Substituting Eq. A-5 into Eq. A-4 gives us the cumulative production (at a prescribed time (tn)) in terms of the specified constant pressure steps and the B-spline integral term. This result is given as:

)( )( )( 12int,

11

= = = jniin

j

u

lijwfjwfnp ttBcpptN ... (A-6)

Eq. A-6 is a linear system of equations and can be solved by linear (weighted) least squares (using the singular value decomposition technique) for the unknown vector of B-spline coefficients, c. For reference, the pressure drop function is represented as pwf = pi pwf where pi is the initial pressure and pi can also be obtained as a result of the minimization process (i.e. weighted least squares). Eq. A-6 can be written in vector-matrix form as:

XcNp =~ ........................................................................ (A-7)

The interested may refer to Ilk [Ilk (2005)] and Ilk et al [Ilk et al (2006)] for details on the deconvolution algorithm and the regularization procedure.

Documents

SPE 111269 (Ilk) Deconvolution Based on Cum Prod Cont Meas q and Pwf