SPE Society of PetroIeun Engineers of "IME

SPE 11964

Well Test Interpretation in Heterogeneous Reservoirs With Skin and Afterflow: Some New Theoretical Solutions and General Field Experience by Helmut Niko. Shell EP Laboratories

Member SPE-AIME

Copyright 1983 Society of Petroleum Engineers of AIME

This paper was presented at the 58th Annual Technical Conference and Exhibition held in San Francisco, CA, October 5-8, 1983. The material is sublect 10 correction by the author. Permission to copy is restricted to an abstract of not more than 300 words. Write SPE, 6200 North Central Expressway, Drawer 64706, Dallas, Texas 75206 USA. Telex 730989 SPEDAl.

ABSTRACT

Some new analytical solutions for dimensionless pressure vs. dimensionless time are presented. It is deomonstrated how these new solutions, together with existing ones, can be used for unravelling reservoir heterogeneity patterns, quantitative inter­pretation of selected field tests, and well test design. The field tests cover both naturally frac­tured carbonates and layered sandstones.

Two examples of well test interpretation in naturally fractured formations reinforce the conclu­sions of other authors, namely that well test inter­pretation based only on 2 parallel semi-log straight lines is questionable in cases with afterflow.

The new analytical solutions refer to stratified and heterogeneous systems with inter layer cross­flow. In both cases after-flow and skin factor are included. It is shown how the type curve can be used in certain long pressure build-ups to gain information on the average reservoir pressure in a stratified system. In the pressure behaviour of a heterogeneous system with inter layer cross­flow, there is often quite a long transition zone bemore the system behaves like a single reservoir. This requires careful planning of test duration. In particular, the early part of the transition zone. looked at in isolation, may give a false impres­sion of pressure stabilisation, which could lead to the interpretation of small drainage areas or a gross over-estimation of the system permeability.

INTRODUCTION

Over the past few years, there has been an increasing trend in industry to develop tighter and more heterogeneous reservoirs. Simultaneously, detailed studies of old fields often demonstrated that reservoirs are not as homogeneous as they were originally thought to be. In both cases, the petroleu~ engineer should be in a position to characterise and define the nature of these heterogeneities, in order to select the best production policy for the field, be it primary or enhanced.

References and illustrations at end of paper.

There has also been a significant upsurge of industry interest in well testing recently. The reasons are two-fold. First, the advent of fast computers together with the development of analytical and numerical (=simulator based) flow models has greatly enhanced our theoretical base of well test interpretation. Secondly, the introduc­tion of precision pressure gauges, continuous downhole pressure recording etc. has vastly improved the quality of pressure data available for analysis.

The main types of heterogeneities that may be encountered in reservoirs are summarised in ig. 1, comprising (i) natural fractures, (ii) stratified formations with sealing shale layers, (iii) heterogeneous formations with interlayer cross-flow. Combinations of these three main types of heterogeneities are also possible. The factors which usually complicate well test interpre­tation are near well-bore effects such as skin and bore hole compressibility, and influences of reservoir boundaries such as faults or gas caps/ aquifers.

Although much has been published on well testing in recent years, gaps still exist. For instance, there were significant contributions to the well testing theory in heterogeneous formations during 1982, which did not, however, include the effect of well bore storage. There is thus the need for a study which tries to close some of these gaps. The other, perhaps even more important, objective of the present study is to demonstrate the use of well testing theories for interpretation in realistic field cases and for well test design. In this context, field examples are presented for naturally fractured carbonates. stratified formations and heterogeneous reservoirs with interlaye cross-flow.

All equations were originally solved with the help of a micro computer (HP-85). In order to use these analytical models efficiently for well test design and interpretation, running on faster computers is obviously required.

WELL TEST INTERPRETATION IN HETEROGENEOUS RESERVOIRS WITH SKIN AND AFTERFLOW: 2 SOME NEW ANALYTICAL SOLUTIONS AND GENERAL FIELD EXPERIENCE SPE 11964

NATURALLY FRACTURED FORMATIONS

Fractured formations contain many of today's known oil fields. Such formations have a so-called dual porosity system: a matrix porosity and a frac­ture porosity. Fractures are either solution cracks or tectonic fractures. The objective of well testing in fractured formations is to obtain: (i) the perme­ability of the fracture system, (ii) the skin factor, (iii) average reservoir pressure, (iv) the secondary porosity parameter w, and (v) the matrix-fracture cross-flow coefficient A. Both w 'and A are dimension­less numbers.

THEORY

Well testing theory for naturally fractured formations has undergone significant development in recent years. References 1 to 4 in historical sequence may be quoted amongst many others. Theories presently available for predicting dimensionless pressure vs. dimensionless time include after-flow, skin, transient and semi steady state flow between matrix and fractures, plus a wide range of boundary conditions.

It is generally accepted nowadays that formations with a dual porosity system should be analysed using both semi-log and log-log plots. A log-log plot of P

D vs tn/CD for a flowing well, such as shown

in F~g. 2 tor 4 different skin factors, clearly illustrates the different flow regimes during a well test: (i) well bore storage effects in the early part, when the curve is near the 45° line; (ii) depletion of the fracture system (between 45° line and line labelled 2'), (iii) the transition region, characterised by a nearly horizontal line, and (iv) depletion of the entire system (to the right of the line labelled 2). It has been stated in the past, Ref. 1, that naturally fractured systems usually result in 2 parallel semi-log straight lines on a Horner plot. This is, however, only true in special circumstances. The extent of the transition region (iii) shown in Fig. 2, together with the line labelled 1, showing the beginning of the semi­log straight line for single porosity systems, deter­mines whether or not 2 parallel Horner straight lines will occur. For the conditions prevailing in Fig. 2, this is not the case, as the transition region (iii) starts before a semi-log straight line would have been reached in phase (ii). For 2 parallel semi-log straight lines to occur, the matrix-fracture cross-flow coefficient A generally has to be less than a certain value. This has the effect of moving the transition zone to the upper right hand side of the plot, i.e. the semi-log straight line in flow region (ii) is reached before the start of the transition flow region. The start of the semi­log straight line in flow region (iv) coincides with the end of the transition region, refer to Fig. 2.

In order to aid in well test analysis, several log-log type curve plots were generated, which illus­trate how individual reservoir and well parameters influence the shape of the type curve. Only selected type curve plots are shown.

Results can be summarised as follows:

(i) A negative skin factor usually causes a larger slope at long time values (Fig. 2). This has nothing to do with a boundary effect.

(ii) For small values of A, the transition zone starts later and lasts longer. It thus moves to the upper right hand side of the plot.

(iii) Transient flow between matrix and fracture, as compared to the normally assumed semi steady state flow, gives the transition zone a slight tilt.

(iv) An increased well bore storage coefficient significantly shortens the transition zone.

(v) A decrease in the secondary porosity parameter w lengthens the transition zone.

(vi) Boundary effects show up clearly on the log­log plot such as shown in Fig. 3. In this case, for a well draining a rectangle with width to length ratio 1:3, the lines become steeper at larger times.

APPLICATIONS

Example No.1

The first well chosen is an oil well completed in a European offshore chalk province. The reservoir is known to have natural fractures which are mostly of a tectonic nature. The well was acidized, cleaned up and shut in for a prolonged period prior to being tested at constant rate. The reservoir fluid is undersaturated.

Recorded pressures are given on page 1 of Appendix A which is followed by the interpretation. A semi-log analysis was attempted first, as the Horner plot on Fig. 5 clearly indicated 2 parallel straight lines. This resulted in a fracture perme­ability of 133 mD, a (total) skin factor of -1.2, a secondary porosity value w of 0.33, and a matrix:6racture cross-flow coefficient A of 2.2 x 10 •

A type curve interpretation was tried next. In this context, a sufficient number of type curves was created, whereby parameters such as A, w, CD' S were varied. The curve that gave a nearly perfect match with the field data, s~6 Fig. 4, corresponds to w = 0.001, A = 5 x 10 ,CD -4800, S = -1.7, refer also to table 1 for a comparison with the Horner method. The conclusion is that, except for the value of w, agreement between the two interpretations is excellent. The cause for the discrepancy in the interpreted value of w soon becomes apparent when we inspect Fig. 2 (the type curve in Fig. 5 has been found by interpolation from Fig. 2). As mentioned in chapter 2.1, the prevailing value of A is obviously not small enough, causing the transition zone to start before the semi-log straight line is reached in the initial period. The fact that the Horner plot shows 2 parallel lines is thus misleading. The second straight line is in the correct position, as can be verified with the type curve. The first one is only apparently parallel and still influenced by well bore storage effects. The true first semi-log straight line is also shown on Fig. 5 and was obtained by calculation, accepting the

SPE 11964 H. NIKO 3

type curve interpreted value of w = 0.001. It should be noted that fracture systems.of a tectonic nature rarely exhibit w's in excess of 1%, even in case of high connate water saturations in the matrix (refer to definition of w). Accepting the value of w derived by semi-log analysis would obviously lead to an inflated estimate of oil in place in the fracture system.

The other important lesson that we learn from this type of test is that flow tests have to be conducted for quite a long time to reach a reliable semi-loS straight line. t'~ should be of the order of 5xlO as compared to 2x~0 in the case of homogen­eous formations. In order to obtain real times, tD has to be divided by the diffusivity constant k/~c~r 2. Typical values of the diffusivity constant for a ~ermeability of about 100mD are 25,000 hr-1

for a gas reservoir. In the~ of example 1, the second Horner straight line does not start until ±24 hrs. .

Example No.2

The well chosen is a gas well completed in a carbonate formation onshore Europe. The reservoir is known to have a dual porosity system. The frac­ture system consists mainly of solution cracks. The well has been on production and was subsequently shut-in for a long period of time at the end of which the reservoir pressure was practically fully built-up. A production test was subsequently conducte( which consisted of a 48 hrs flowing period followed by a 204 hrs shut-in.

Pressures recorded during the flowing and shut­in per-dios, together with a detailed interpretation, are given in Appendix B. In order to provide as wide a data band as possible for analysis, the raw pressure data of both periods were subjected to "desuperposition" as suggested in Ref. 5. Prior to that all pressures were converted to real gas pseudo pressures. The result is a continuous unit inflow function, extending over both flowing and shut-in periods, which can then be plotted on both log-log and semi-log paper. This approach requires that the skin factor remains constant over the entire period; secondly, points affected by well bore storage are only allowed at the beginning of the first flow period. The pressure recordings in the beginning of the shut-in period which were obviously affected by afterflow were thus eliminated. The resulting semi-log and log-log plots are shown as Figs. 6 and 7 respectively. "Pseudo-times" according to Ref. 6 are plotted on the time axes to eliminate distortions due to non-constant ~c product.

The type curve match together with the Horner plot indicates that we are dealing with a dual porosit system characterised by a relatively large matrix­facto!3cross-flow coefficient, Le. ), of the order of 10 • A very short and early transition period is indicated on the Horner plot, whereas it is barely visible on the log-log plot. The consequence is that the first Horner straight line hardly appears at all and that, as in example 1, w cannot be estimate from the semi-log plot.

The second semi=log straight line can be drawn on the Horner plot and its position is also reinforced by the type curve match. Both Horner plot and type curve match show very clearly the boundary effect.

Geological data available indicated that the well was draining a rectangular area with a width to length ratio of 1:3. The well is located close to a corner of the drainage area. The type curve chosen for the match shown in Fig. 7 was interpolated among the ones shown in Fig. 3 which were specially created for this type of boundary configuration.

Both type curve and semi-log interpretation give very consistent results, refer to table 2: fracture permeability between 4.4 and 5.6 mD, skin factor between -3 and -3.6, matrix-fracture cross-flow coefficient between between .001 and .0026; w was only available from the type curve match and is approximately 0.001. Noteworthy is the rather high value of the dimensionless well bore storage constant C = 10,000. This is typical for a gas well prgducing at relatively low pressure, but it could also indicate a larger volume (e.g. fractures) connected to the bore hole.

STRATIFIED FORMATIONS

THEORY

The pressure behaviour of a well completed on two layers separated by a shale has been described in Ref. 7. Reference 7 is a thorough investigation into the medium and long term behaviour of such wells, but omits early time effects which are usually influenced by the skin and well bore storage effects. The latter was included by the present author, Ref. 8, for circular formations. The resulting equation for the dimensionless pressure as a function of dimensionless time is repeated in Appendix C. page 1. The theory has recently also been extended by the author to wells located anywhere within a rectangularly bounded area.

In solving for the dimensionless well bore pressures, rather complicated Laplace transforms such as equation C-l had to be inverted. This was accomplished successfully using the numerical scheme of Ref. 9.

A typical log-log plot of P vs t Ic is shown as Fig. 8. In all subsequ~nt di~cu~sions and figures P

D and tD are defined with average

values of k and ~ (PD = P (av) etc). Figure 8 applies to a pressure bRild-up showing P (super­imposed) for a circular drainage area. Twg sand layers with permeability contrast 10, the low permeability layer being 10 times thicker than the high permeability one, were modelled. A common skin factor has been assumed. Figure 8 illustrates how the system is initially dominated by afterflow, enters a brief infinitely-acting period, and ends up with a "hump". The magnitude of the "hump", according to Ref. 7, depends on the producing time and the permeability contrast. It reflects in approximation the difference between the average system pressure and the one of the most depleted layer. It should be pointed out that the "hump" is virtually absent in wells producing only for short times.

APPLICATIONS

Example No.3

The example refers to an oil well in the

WELL TEST INTERPRETATION IN HETEROGENEOUS RESERVOIRS WITH SKIN AND AFTERFLOW: 4 SOME NEW ANALYTICAL SOLUTIONS AND GENERAL FIELD EXPERIENCE SPE 11964

Western Hemisphere in a roughly circular drainage area. The well is completed in a sandstone environmen The geological model is that of highly permeable river channels embedded in low permeability overbank deposits. The completion log indicates the likeli­hood of a two-layer system. The survey was taken after a cumulative production of some 585,000 stb and lasted for 72 hours •. The reservoir contains light oil. The GOR prior to the survey was somewhat above solution GOR.

A complete interpretation was attempted in this case, refer to Appendix D for details. The type curve match is illustrated in Fig. 9 whereas the Horner plot is shown in Fig. 10. The type curve of Fig. 9 was interpolated among the ones shown in Fig. 8.

Both Horner and type curve interpretations gave very consistent results, i.e. an average system permeability of 15 mD and a skin of +15, refer to table 3. Note that the Horner slope in the infinitely acting period always gives the average system perme­ability. What is really remarkable in this test, is, however, the conclusion regarding the average reservoir pressure. Had the survey only lasted 24 hours as is often the case, a reservoir pressure of near 1000 psia would have been interpreted. The use of the type curve in conjunction with the long pressure build-up indicates, however, that the reservo r pressure is more likely to be near 1600 psia. This result was obtained by converting the dimensionless f,inal pressure rise from a reference point, as indicat d by the type curve, into psi. The reservoir engineer who would work with a reservoir pressure of 1000 psia instead of 1600 psia would thus draw the wrong conclusions in his reservoir analysis work, e.g. he would try to look for smaller drainage areas etc. Obviously, a 72 hour pressure build-up survey cannot always be justified. Alternatives for finding representative reservoir pressures could be carrying out static surveys in inactive wells, or carrying out short pressure build-ups in selective zones.

TWO-LAYER FORMATIONS WITH INTERLAYER CROSS-FLOW

THEORY

One of the more recent publications on the subject is Ref. 10, which was based on numerical simulations. Ref. 10 concluded that the pressure transient behaviour of this type of formation can be subdivided into 3 parts: (i) a period during which the system behaves as if it were stratified, (ii) a transition period during which flow behaviour is dominated by interlayer cross-flow, and (iii) a period during which the reservoir behaves as an equivalent single-layer system.

Since neither Ref. 10 nor some of the earlier publications on the subject includes a complete set of realistic boundary conditions such as after­flow and skin, these have been added to the analytical solution. The analytical solution is based on Ref. 11 which could be extended by adding skin factor and well bore storage. The resulting equation for the dimensionless well bore pressure (in Laplace domain) has been derived in Appendix E for the condi­tions of negligible gravity effects and identical skin factors in both layers.

The mathematical expression contains an infinite series which converges very rapidly, and could be inverted using the Stehfest algorithm (Ref. 9). Some problems were encountered with accuracy. In particular, for problems with a permeability contrast> 100, it could not yet be solved with a computer having a l2-digit accuracy. Some numerical experimentation is obviously still required. The solution given in Appendix E can also easily be adapted for the case of anisotropy which must, however, be common to both layers. All that needs to be done is modifying certain equations of Appendix E by introducing the anisotropy ratio ~/kv'

Results and Discussion

Out of the cases run so far, two are shown on Figs. 11 and 12 as log-log and dimensionless Horner plots respectively. The cases investigated refer to a radial reservoir consisting of 2 layers. The reservoir has on purpose been chosen very large (reD = 20,000), and the permeability contrast between fhe 2 layers is 100. The less permeable layer is 10 times as thick as the highly permeable layer. A pressure bu~ld-up after a production period with tD = 5xlO was simulated, with the dimensionless well bore storage constant CD being 1000.

The dimensionless Horner plot, Fig. 12, in conjunction with the type curve plot Fig. 11, clearly illustrates the various periods. In the first period, the system behaves like a stratified system; the early part is dominated by afterflow, and the semi-log straight line is not yet reached when the pressure behaviour passes into the transient period. The transient period is characterised by an early flat part followed by a somewhat steep line that merges into the third period during which the system behaves like a single-layer system. The semi-log straight line with slope 1.15 in the dimensionless Horner plot is clearly visible. It later bends over due to the boundary effect.

It follows thus that stratified systems and systems with inter layer cross-flow are very similar on semi-log and log-log plots up to and including the initial flat portion. In stratified systems, total kh should be found on the semi-log plot prior to the flat part. The rise after the flat part can be used to estimate the average reservoir pressure of the entire system, by comparing a log-log field plot with an appropriate type curve. In heterogeneous systems with inter layer cross­flow, total kh and average reservoir pressure can be found from the semi-log plot after the flat part.

Figure 11 also indicates that the semi-log straight line starts quite late for the case of the 2 layer system w~th cross-flow: at tB/CD = 500 or tD = 5 x 10 with CD equal to 10 O. This is nearly a factor 10 (!) longer than in the homogen­eous single layer case (also indicated on Fig. 11). The transition time extends thus over one complete log cycle on the dimensionless time axis.

If we are primarily interested in k and S, it may become attractive to install a bottom hole shut-off tool, in order to minimise well bore storage effects and therefore avoid excessively

SPE 11964 H. NIKO S

long testing periods.

APPLICATIONS

Example No.4

The well chosen here is used only for illustra­tion. It is an onshore oil well in the Far East, completed in a sandstone environment. The formation is known to be heterogeneous; however, only a porosity profile, derived from logs, was available. The semi-log plot shown on Fig. 13 corresponds to a shut-in of 24 hours after only a short initial pro­duction period. The plot indicates an initial period in which the system exhibits well bore storage effects and possibly a transition period extending over roughly 1 log cycle. The early flat part in the transition period is clearly visible. This means that in the case of an initial test, the true straight line would appear quite late and would be only a very short line at the end of the Horner plot. It also means that the only reliable way to estimate kh in this type of test is to minimise well bore storage effects such that the semi-log straight line is reached before the onset of the transition zone. A bottom hole shut-off tool may thus be nec­essary. If a well test ends in the early flat part of the transition period, it may, moreover, lead to the interpretation of a boundary effect and/or over-estimation of the kh.

CONCLUSIONS

Theory, reinforced with realistic field examples, has led to the following conclusions in this study:

1. When carrying out well test analyses in hetero­geneous formations, both log-log and semi-log plots of field data should first be compared to theoretical solutions of PD vs tD for recognition of the heterogeneity patterns one may expect from the geological environment.

2. There are many cases in naturally fractured for­mations where 2 parallel semi-log straight lines appear of which only the second is correct. The secondary porosity cannot be estimated from the Horner plot in this case. In many cases, well tests may have to be conducted for twice as long as in homogeneous sandstgne formations to be 3ble to estimate kh (tD -SxlO as compared to 2xlO ).

3. Well tests in stratified formations, such as 2 layers separated by shales, usually fail to indi­cate the true average reservoir pressure of the system. A type curve derived for this situation can be of help to estimate the average reservoir pressure for cases of long pressure build-ups extending up to 72 hours.

4. Well tests in heterogeneous systems with inter­layer cross-flow have to be carried out for a considerable length of t~me to reach the semi-log straight line: tD -SxlO for a typical case with a permeability contrast of 100, nearly a factor 10 longer than in the case of a single layer. If a well test ends earlier (in the transition ,I

zone), it may give the false impression of stabili sation and thus point to a small drainage area. At the same time, the system permeability may be grossly over-estimated. A serious discrepancy between the interpreted system permeability and

that from another source (e.g. cores) may be taken as a warning.

GENERAL LIST OF SYMBOLS AND UNITS

A

B

CA

c

C

h

k

m

Drainage area

Formation volume factor

Shape factor

Compressibility

Well bore storage coefficient

formation thickness

permeability

semi-log slope

Units

-1 psia

bbl/psi

ft

mD

psia~cycle or psia /cP/cycle

m(p) real gas pseudo pressure, psia2/cP

p

q, Q

r

S

S wc s

t

llt

T

Z

0

w

A

)J

Y

Ei

~ pdp o llZ

pressure

flow rate

radial distance

Skin factor

Connate water saturation

Laplace time parameter

Producing time

Shut-in time

Absolute temperature

Gas deviation factor

Porosity

Secondary Porosity

Matrix-fracture cross flow coefficient

Viscosity

Mathematical constant = 1.781

psia

bid or Mscf/d

ft

l/Time

hrs

hrs

cP

Modified Bessel Functions

Exponential integral

Constants (to be used when field units are employed)

fl 0.894 F4 1422

f2 0.000z64 FS 162.6

f3 0.00708 F7 1637

Subscri]2ts

i initial - refers to radial discontinuity in multi­

layer system layer no. in multi-layer system

- referring to individual image well

WELL TEST INTERPRETATION IN HETEROGENEOUS RESERVOIRS WITH SKIN AND AFTERFLOW: 6 SOME NEW ANALYTICAL SOLUTIONS AND GENERAL FIELD EXPERIENCE

j referring to radial region (1 = inner, 2 = outer)

w

1

2

well

refers to layer no. in layered formation or matrix in dual porosity system (also m)

refers to layer no. in layered formation or fractures in dual porosity system (also f~

av refers to average in layered formation

e external

D dimensionless

o oil

g gas

Superscript

Laplace transformed variable

Definitions (fractured formations)

(5) 2 I (5) I' Skin ratio

Reference skin of layer 1

r,.,/ (tphc) 1 . dimensionless well bore storage

Co (av.)

r/rw' dimensionless distance

re/rw' dimensionless outer radius

'0 fZ (iJ~¢) t ~ dimensionless time r ..

Pi - p ~ . dimensionless pressure

SPE 11964

f (~) Pi - Pwf ) u ~ dimensionless pressure, at well

f 3k2h(Pi-P"f) . Pwo qoUoBo

f2t to 7

1.10 (:pc) f+mr....,-

CD flC

(.pc) f+mhr.., 2

Horner Slope:

f 5 qo U J B 0

k2 h

Similar definitions for: gas wells

for gas wells

"Pseudo time" (for gas wells, Ref. 6) d,

(;':c)loI'ell

Secondary porosity:

; . l

Definitions (stratified or layered formations)

(\.l~¢) 1 (1J~¢I) 2' ratio of diffusiv1ties

(~)2 (~)1' ratio of transmissibilities

Pwo (av.)

~ote: 1 ref<!["s to layer 1. 2 to layer 2.

SI Metric Conversion Factors

1 bar ,. lOS Pa · l4.5 psi

cP :< l. 0 E-J '" Pa. s

Et l x 2.831685 E-2 • m' bbl x l.589873 E-l • m' ft x 3.048 E-l • m mO.f' x 3.008142 E2 ,. I.HIll.m

pSi x 6.894757 EO ,. kPa

oR x 5/9 • K

SPE 11964 H. NIKO 7

ACKNOWLEDGEMENT

The author is very grateful to Shell Internation­ale Petroleum Maatschappij for granting permission to publish this paper.

REFERENCES

1. Warren, J.E., Root, P.J.: "The Behaviour of Naturally Fractured Reservoirs" AIME Transaction Volume 1963.

2. Mavor, M.J., Cinco, H.: "Transient Pressure Behaviour of Naturally Fractured Reservoirs" Paper SPE 7977 presented at the 1979 California Regional Meeting of SPE.

3. Bourdet, D., Gringarten, A.C.: "Determination of Fissure Volume and Block Size in Fractured Reservoirs by Type Curve Analysis" Paper SPE 9293 presented at the 55th Annual Fall Technical Conference of SPE, 1980.

4. Gringarten, A.C., Burgess, T.M., Viturat, D., Pellisier, J., Aubry, M.: "Evaluating Fissured Formation Geometry from Well Test Data: a Field Example" Paper SPE 10182 presented at the 56th Annual Fall Technical Conference of SPE; 1981.

5. Bostic, J.N., Agarwal, R.G., Carter, R.D.: "Combined Analysis of Post Fracturing Performance and Pressure Build-up Data for Evaluating an MHF Gas Well" Paper SPE 8280 presented at the 54th Annual Fall Technical Conference of SPE, 1979.

6. Agarwal, R.G.: "Real Gas Pseudo Time: A New Function for Pressure Build-up Analysis of MHF Gas Wells." Paper SPE 8279 presented at the 54th Annual Fall Technical Conference of SPE, 1979.

7. Raghavan, R., Topaloglu, H.N., Cobb, W.M., Ramey, H.J.J.: "Well Test Analysis for Wells Producing from two Commingled Zones of Unequal Thickness" AIME Transaction Volume, 1974.

8. Niko, H.: "Interpretation von Ungewtihnlichen Bohrlochmessungen" Erd1.il Erdgas Zeitschrift 11/81.

9.

10.

11.

Stehfest, H.: "Algorithm 368 - Numerical Inver­sion of Laplace Transforms" Communications of the ACM (1970), Vol. 13, No. 1,47.

Prijambodo, R., Raghavan, R., Reynolds, A.C.: "Well Test Analysis for Producing Layered Reservoirs With Cross-Flow" Paper SPE 10262 presented at the 56th Annual Fall Technical Conference of SPE, 1981.

Jacquard, P.: "Etude Mathematique du Drainage d' m: Reservoir Heterogene" Revue de l'Institute Francais du Petrole, XV, No. 10, 1962.

APPENDIX A

DRAWDOWN ANALYSIS IN A CHALK WITH A DUAL POROSITY SYSTEM (OIL WELL)

The recorded pressure data for a well completed in this chalk are as follows:

Time, hrs. Pressure, psia

0.066 3351 0.266 3054 0.533 2991 0.799 2959 1.133 2935 1.466 29£1 1.866 2910 2.266 2896 2.733 2889 3.199 2880 3.6 2873 4 2870 4.47 2870 5.33 2868 5.73 2868 6.47 2869 7.13 2869 7.8 2867 8.47 2866 9.07 2865 9.73 2861

10.47 2862 11. 93 2860 13 2859 14.7 2854 16.27 2845 19.5 2814 21.4 2828 22.8 2825 26.1 2821 36 2809 45.7 2797

Additional data required for the analysis are:

P. 1

3492 psia

qo 4200 stb/d

B 1.25 0

110 1.25 cP

h 70 ft

c .000014 .-1 PS1

r w .292 ft

cp .2

The well was producing somewhat above initial solution gas/oil ratio, and had an acid treatment prior to the prolonged production test.

INTERPRETATION

1. HORNER SOLUTION

"" 115 psi/cycle

plhr. 2985 psi.

115 • 162.6 x 4200 x 1.25 x 1.25

k2h

k2h • 9280 )!d fe

(:~.-.:.-:::-~ )

133 + 3.23) • -1.2 1. 25x. 000014x. 2x.292

2

WELL TEST INTERPRETATION IN HETEROGENEOUS RESERVOIRS WITH SKIN AND AFTERFLOW: 8 SOME NEW ANALYTICAL SOLUTIONS AND GENERAL FIELD EXPERIENCE SPE 11964

Secondary porosity parameter w

Let 6p be the vertical distance between the 2 parallel liWes, then w can be calculated from:

10 • 10

55 ill

• 0.33

This is clearly too high. The most likely reason is that the early part of the survey is still influenced by afterflow. The true 6P can be derived by a comparison with the type ctirve match which is dis­cussed later:

Wtrue .. 0.001 .. 10-

~P • 345 psi ,._~,s.i'~!._ .... _ •• ___ _

OP wt!'ue

-11-5-

Matrix-fracture cross flow parameter A

Let 6P A be the pressure drop from P., down to the horizontal line in the pressure draw~down test, then A follows from:

2.3 6P, 0.56 (I-w) exp (- -m-)

0.56 x (1-.001) x exp (_ 2.3 x (;~;2-2870»

U_~.lQ::

2. TYPE CURVE MATCH

Curve parameters giving best match:

w 0.001

0.000005

l34

Permeability calculation:

Pressure match (P 2D)M = 1 at (Pi

I • 141.2 x 4200 x 1.25 x 1.25'

(k2 • 120 mD)

Well bore storage parameter:

Time Match

~ 2 (-) • 1 at (t)M • 4.5 x 10 hrs CD

M

or

.0002~:2~ ~424 x .045 _ I .'. C - .089 bbl/psi

.894 x 0.089 • 4786 . 2x. 000014x70x. 292 2

APPENDIX B

PRESSURE BUILD-UP IN A CHALK WITH A DUAL POROSITY SYSTEM ( GAS WELL)

According to the geological interpretation, the well appears to drain a rectangular area of 6800 ft length, 2150 ft width (-340 acres). The well is located 6000 ft from the left boundary of the field and 400 ft from the upper boundary.

Prior to the current production test it was produced at 8815 Mscf/d for 5393 hours and closed in for 192 hours. At the end of the long shut­in the rate of pressure build-up was negligible.

The current production test consisted of a 48 houLflor rate at 5430 Mscf/d, followed by a pressure bUild-up which lasted for 204 hours.

Pressure observations are tabulated below:

Time from start production test in hrs

o 1 2 4 8

12 16 20 28 36 48

49 50 52 54 56 60 64 68 76 84 92

100 108 124 140 156 l7~ 188 204 220 228 244 252

Recorded pressure in psia

1249.5 1143.6 1126.17 1107.65 1099.12 1091. 77 1085.45 1081. 92 1071.48 1063.29 1056.64

End Flow Period

1192.61 1199.37 1207.75 1212.46 1217.16 1223.92 1228.48 1232.01 1237.01 1241. 71 1245.83 1249.06 1252.44 1257.43 1263.17 1268.61 1273.61 1278.61 lL83.l6 1287.87 1290.22 1294.92 1296.98

SPE 11964 H. NIKO 9

The following correlations were found to apply: . 2

(1) "pseudo-pressure" in p~~a vs. pressure in psia

m(p) = -37.167 x 106 + .0864 x 106p + .0000427

x lu6p2

(2) gas deviation factor Z vs. pressure in psia:

A = 1.008 - .0001134p + .000000023p2

(3) gas viscosity in cP vs. pressure in psia

~ = .0112 + .000027p + .000000000463p2 g

valid for 600 psia < p < 4100 psia

Additional data required are:

c (at current pressure) -1

.00068 psi

r w

h

(effective) .045

.2 ft

104 ft

1. HORNER INTERPRETATION

2 "Horner slope" m ,. 2190 ~:!~/~cp / cycle

Skin Factor:

The following arbitrary point is chosen on the "Horner" Straight line:

"pseudo-time" 440.000 hr •. c~Sia ~

pSia2

/cp "Bostic's unit function!! : 4500 ~scf/d

S. 1.151 ~~i~~ - log (440.000 x ~.4) + 3.23 .045 x .2

Secondary porosity parameter w

If ~m(PCFN ) is the vertical distance between w

the 2 parallel lines, then w can be calculated from:

Om(p CFN )

(--m-~-} (i~~o) 10 • 10- • 0.5911

This value is clearly too high, which means that the Horner interpretation of w is masked due to the well bore storage effect.

Matrix-fracture cross flow parameter A

With m(PCFN

) being the value of Bostic's unit A

function read off at the plateau, A follows from:

2.3 m(PCFN,}

• 0.56 (1-w) exp (_ A )

• 0.56 x (1-.001) x exp (-2'\~9~100)

• ~.g.~.!2:~

2. TYPE CURVE MATCH

Pressure match: () ( • 1500

pSi.2 !cp m2D X • 1 at m P ern) M ~scf/d

k2h T4ITi' x m(PCFN):-! 1422 x 613 x 1500 • 1

" k2h • 581 cll Et

:~ ... :.-:~:~.~

From choice of type curve:

25

w .001

.001

Well bore storage

Time Match

.. 10 at ("Pseudo time") II '" 130,000 hrs. psia

or .000295 x 581 C x .00068 x 130.000 • 10

The large value of C is rather typical of a gas well producing at relatively low pressure.

.894 C .894 x 1. 52

?Cihr,} •. 045 x .00068 x [04 x .2 2 C • D • [0.640

APPENDIX C

SOLUTIONS FOR THE DIMENSIONLESS PRESSURE IN A BOUNDED CIRCULAR SYSTEM CONSISTING OF 2 LAYERS AND A SHALE BREAK (Ref. 8) (in Laplace domain).

p • wD

+

WELL TEST INTERPRETATION IN HETEROGENEOUS RESERVOIRS WITH SKIN AND AFTERFLOW: 10 SOME NEW ANALYTICAL SOLUTIONS AND GENERAL FIELD EXPERIENCE SPE 11964

(C-l)

APPENDIX D

PRESSURE BUILD-UP IN A STRATIFIED FORMATION

The following data correspond to a PBU taken in a well draining two layers separated by a shale. The drainage area is approximately circular.

Basic reservoir data are:

Well bore radius, ft Net sand thickness, ft Initial oil saturation, % Porosity, % Flowing pressure, psia Production rate q , stb/d Cumulative oil pr8duction, stb Oil formation volume factor, b/stb Oil viscosity, cP Total compressibility*) 10-6 psi- l

0.29 48 65 15 468 105 585,000 1.3 1.1 200

The well appears to drain a circular area with r D ~ 3000. It produced some free gas at the time of shut-in.

Type Curve Analysis

Seems to match the curve a S = +15

Pressure match:

PD

= 1 at liP 30 psi

i.e.

0.1, f3 10,

JOkh 141.2 x lOS x 1.3 x 1.1 • kh • 707 mD ft

,(~.:.!~.Z_~l

Horner slope m 162.6 x 105 x 1.3 x 1.1 707

Horner straight line reached at ±_~_~E~.*)

Horner Analysis

PI hr (on straight line): 1030 psia

*)The slope at this point in the Horner plot is, however, already affected by cross-flow between the layers (see also type curves).

*) Estimate, including pore compressibility plus contribution of gas and water.

P*: LOO psia

S 1.151 (1030 - 468 1 14.7 2 35 - 05 0.15 x 1.1 x 200 x 10 vx.2~

+ 3.23) = ±!~.:.~

Estimate of Average Pressure P from Type Curve Match

(between "start Horner" at lItD/CD'" 600, and nearly

00 CI, lItD/CD '" 104)

This corresponds to 20 x 30 = 600 psia.

Adding 600 psia to Pws at "start Horner", i.e. 1070 psia, gives

Since p* from the Horner graph is 1270 psia, the Mathews-Brons-Ha~e~roek correction is negative in this case (p* - P = -400 psia) , due to the hetero­geneity of the system.

APPENDIX E

DERIVATION OF THE FLOWING PRESSURE FOR BOUNDED CIRCULAR SYSTEMS CONSISTING OF 2 COMMUNICATING LAYERS (Ref 11) (all in Laplace domain).

Differential equation:

api z-a a PI • 3r (r ar-) +

~ ~ PI (layer I) (E-I)

a apz +

aZpz s

3r (r ar-) ~ is Pz (layer Z) (E-Z)

Boundary conditions:

(E-J), (,-4)

(,-5)

(£-6)

(E-7) , (i-a)

SPE 11964

• -"- (- .1 + Csp (s» . 21Tt"w s V

-(rw' s) . P.,.(s) PI z, + r " 'PI (r .., ;)3 1,1'

z, s)

r

Pz ('tw' z, 0) . P..,(s)

ap z + rw$ a ('tw' z, s)

r

Notes:

(1) Consistent units are assumed.

(2) Boundary conditions (E-10, E-ll) assumes a common skin factor over both layers.

General solutions satisfying (3), (4):

PI · (AI Ko (ru) + BI I (ru) ) cos (wI r) 0

I

hI + h2 - z P2 · (A2 Ko (ru) + B2 10 (ru» cos(w

Z h2 )

- 1-2--s-with WI .

hl\" -~' w2 . h~ 2 s

" -~ (£-14) ,

(5) and (6) yield:

or

Taking (E-7) and (E-8) into account, the general solutions for the pressure become:

~ cos(~ WI n)

PI . l: ¢ (s) n I

" n'O n r cos(w

1 n)

hI + hZ - Z w

2 n)

~ CDS(

h2 PI · r ¢ (s) n

n·O r. ur

cos(wZ n)

H. NIKO

(£-9)

(£-10)

(E-II)

(£-1 Z)

(£-13)

(£-15)

(£-16)

(£-17)

(£-20)

(£-11)

wt. w2", \In are solutions to (15). (16). (17) whereas urn "'s defined as:

I (rvn) o

(E-22)

The coefficient ~ (s) together with the transfQrm of the well pressure pn can be found from (19) to­gether with boundary c~nditions (9), (10) after some algebraic manipulations (see Ref. 12).

Defining

n Arw - -II (re v

n) .K l (rw v

n) + Kl (r

e \p) . It (r

v vn

) vn

The transform of the well bore pressure becomes

p,,(s) •

-2L 21ft" S ..,

11

(E-23)

(£-24)

Definitions

K . k (both layers) "c.

X . kZh l

Yz

Note:

n is not an exponent, it refers to the n'th root of equation E-16.

~

SUDlDary of Interpretation: 011 Well in Chalk with Fracture System.

Horner Type Curve

k. mD 133 120 S -1.2 -1.8

"' 0.33 10-6 0.001 -6

), 2.2 x 5 x 10

Co - 4.786

Summary of Interpretation: Gas Well in Chalk with Fracture System.

Horner Type Curve

k. mD 4.4 5.6 S -4.3 -3

"' 0.59 10-3 0.001 -3

), 2.6 x I x 10 CD - 10,700

Summary of Interpretation: 011 Well in Stratified Formation.

Horner Type Curve

k. mD 14 14.5

~ +14.5 +15 P. psi. Interpret. not 1,670

possible

MODEL NO.

t NATURAL FRACTURES

2. STRATIFIED FORMATION

BOUNDARY CONDo AT WELL

WS,S

WS,S

TYPE OF FORMATION

FRACTURE

(~"-'"~' \ NOTES.

A. OPTION FOR RADIAL DISCONTINUITIES INCLUDED

B. UP TO 10 LAYERS POSSIBLE

EXTERNAL BOUNDARY CONDo

A. - RESERVOIR B. AS (Al. BUT WITH FAULT C. RECTANGULAR BOUNDED,

ARBITRARY WELL POSITION

A. - RESERVOIR B. BOUNDED CIRCULAR

RESERVOIR C. RECTANGULAR BOUNDED,

ARBITRARY WELL POSITION

3. TWO - LAYER WS, S '"1

" I

A. RESERVOIR SYSTEM ALLOWING CROSS-FLOW

c B. BOUNDED CIRCULAR - OR - RESERVOIR

NOTE, WS" WELLBORE STORAGE

S" SKIN ~" INFINITE

Fig. 1-Summary of well lest situations investigated

TYPE CURVES FOR NAT. FRAC. RES. BASE CASE

EFFECT OF SKIN FACTOR

/ CD

SEMI-LOG LINE REACHED: CD HOMOG. RES. <i> NAT. FRAC. RES.

Ii' - <i> TRANSITION ZONt:

LOG (To/Col

\

\ '.

,/

OMEGA ".001 LAMBDA" .OOOOO~ CD" 1000

Fig_ 3-Type-curve match in chalk-drawdown test. ,

S·+~

$.-2

$'-!>

103

Pwf in psia

10'

TYPE CURVES FOR NAT. FRAC. RES WITH BOUNDARY EFFECT .

RECTANG. DRAINAGE AREA: 340 ACRES

OMEGA ".001 LAMBDA".OOI CD" 10,000 LENGTH: WIDTH" 3:1

Fig. 2-Type curves for naturally fractured reservoirs Part 1

CHALK WITH DUAL POROSITY SYSTEM TYPE CURVE MATCH

Fig_ 4-8emilog analySIS In chalk-drawdown test

4

3500 Pwf

3400 in psio

l 3300

3200

3100

2900

2800

2700

2600

2500

11

10

"0 "-t;

9 ., " "-"-u

B "-N 0 'in .. a. 7

Q z 0 f=

6

u z ::> "-~

Z ::>

'" 4

U f= '" 3 0 m

CHALK WITH DUAL POROSITY SYSTEM HORNER ANALYSIS

o

o

o

~--::'-INABSENcE~'- _.­--_ WELL BORE STORAGE

10-2

------------

in Hrs

Fig. 5-Type curves for naturally fractured reservoirs. Part 2

CHALK WITH DUAL POROSITY SYSTEM HORNER ANALYSIS

105 106

PSEUDO TIME. Hrs psia / cP

Fig. 6-Type·curve match In carbonate formation-gas well.

105~~---------------------------------------------------------,

"0 "-'t3 ., ::E

10' "-"-~ 10·

N 0 Ul 'in X a. ct z a 0

t3 "-z ::> "-t::

0 z ::> 103 Ul

'"' f= Ul 0 m

2

TYPE CURVE MATCH GAS WELL IN CHALK WITH DUAL POROSITY SYSTEM

Fig. 7-Semilog analysIs In carbonate formation-gas well.

TYPE CURVES FOR TWO-LAYER SYSTEM WITHOUT CROSS FLOW IN RESERVOIR

LOG (tl TO/CO)

H = START HORNER FOR HOMOG. RES.

TpO • 5.101/ Co = 10' Reo' 3000 AlIa • .1 / Beta = 10

3

Fig. 8-Type curves for tWO-layer system without crossflow.

4

4 r---+---------------------------------------------------------------.

(/) 3 a. .!: !l.

~

0

9

2

2

ci .S Q> a. :::l (/)

0 !l.

01

9

o

WEll TEST INTERPRETATION IN A lAYERED FORMATION WITHOUT INTERlAYER CROSS FLOW

0

TYPE CURVE MATCH

2

lOG (t:,. TO/CD)

1L-__ ~~~_LLLLU ____ L-_L_L~~~ __ ~ __ ~~LLLLU_ __ ~ __ L_~~LLU

pws. In PSIO

-2 -1 0 2

1500

1400

1300 o

1200

HOO

WOO

900

800

700

600

500 3

lOG (t. t in Hrs)

Fig. 9-Type-curve match for two-layer system.

WEll TEST INTERPRETATION IN A lAYERED FORMATION WITHOUT INTERlAYER CROSS FLOW

o

o

---------~--Q_-

4

SEMI-lOG ANALYSIS

---1'L-..o.._a.-o..a.a.ooo

5

lOG t+ t.t t.t

o

o

Fig. 10-Semllog analYSIS for two-layer system.

o

o

6

2r---------------------------------------------------~----------------------------------_,

ci. E .c; Q) a. :::J (/)

Cl a..

<.!)

9 0

-1

o

Po

-5

-10

o

BOUNDARY EFFECT

0

45-

CD ® / ®

/ / I S= +5 /

/ I

/ I 5=0 / /

/ I / / / / I

/'

To = 5 x lOB CD = 1000 kdk1 = 0.01 hdh1 = 10 re/rw = 20,000 ( Large)

START OF SEMI-LOG LINE

CD HOMOGENEOUS RESERVOIR

® HETEROGENEOUS SYSTEM ~-® TRANSITION ZONE

LOG (ATD/Col 2 3

Fig. 11-Type curves for two-layer system with crossflow.

ooACTING, AS UNIFORM RES.

2 LOG t +l:. t

l:.t

3

TO = 5xl0B

Co = 1000 k2/k1 = 0.Q1 h2/hl = 10 re Irw = 20,000 (Large) 5=0

/ TRANSITION

4

/ WELL BORE

STORAGE

Fig. 12-Dimensionless Horner plot for two-layer system with crossflow

4

5

2400r----------------------------------------------------------------------------,

2350

pws • psig

2300

2250

2

24

FIELD EXAMPLE IN WHICH INTERLAYER CROSS-FLOW IS SUSPECTED

o o

o o

3 4 5 6 7 89101 2

6 2

o

o

3

o

4 5 6 7 89102

t+A t At ---

-l::. t, Hrs

Fig. 13-Example of Horner plot in a heterogeneous formation.