Diagnosis and Characterization of Crossflow behind the Casing ...

ABSTRACTINTRODUCTION

The cement completions in the vast majority of wells providegood zonal isolation between the adjacent reservoir and non-reservoir layers. Effective zonal isolation is a prerequisite foroptimum production of hydrocarbons from the targeted reser-voirs. Similarly, when transient pressure tests are conducted,zonal isolation helps ensure a proper evaluation of the targetedreservoirs. But on some occasions the cement loses its integrity,and the zonal isolation is compromised. This can be the conse-quence of various operational failures, in situ environmentalconditions or just partially successful cementing jobs. Some-times a massive stimulation job may open up micro-channelsor fractures adjacent to the cement sheath, which can persist ina carbonate environment. Compromised zonal isolation or thepresence of flow channels creates an opportunity for an adja-cent reservoir layer to be in hydraulic communication with thetested reservoir layer. This complicates evaluation of the testedreservoir layer, due to the interference or contribution of theadjacent reservoir layer in relation to the production of thetested reservoir layer. Compromised zonal isolation or flowchannels allow the fluid from the adjacent layer to flow behindthe casing, Fig. 1, when the intent is to have only the testedlayer contribute to production.

In a reservoir system with two neighboring layers, the testedand the adjacent layers that are separated by impermeablestrata, is considered. Fluid may still migrate from the adjacentlayer to the tested layer if the zonal isolation behind the casingis compromised or if flow channels exist in the vicinity. Amethod is presented to diagnose the fluid contribution to thetested layer from the adjacent layer, and to quantify the tran-sient rate of crossflow by utilizing the transient pressure data.

During transient tests on the tested layer, the crossflow fromthe adjacent layer has to be accounted for to ensure reasonablecharacterization of the tested layer. A new analytical solutionfor a two-layer system with crossflow behind the casing hasbeen employed to understand the effects of crossflow on thetransient behavior in the tested layer. Matching of type curveswith measured pressures helps to diagnose the presence ofcrossflow behind the casing and to estimate parameters. Thecrossflow rate as a function of time is also estimated by quan-tifying the hydraulic conductivity of the compromised zonalisolation.

This study shows that the magnitude of the hydraulic con-ductivity of the section of compromised zonal isolation or flowchannels has a strong effect on the rate of crossflow with theflow capacities of individual layers. There is an upper limit ofconductivity beyond which the crossflow rate is limited by thelayer flow capacities. For a given two-layer system, the cross-flow rate increases with time for a constant rate drawdown inthe tested layer due to the increasing pressure differentialacross the compromised zonal isolation. The test duration isanother important aspect. More time lets the crossflow ratereach a critical value, one able to be detected in the log-logplot of the pressure derivative. This means that a long tran-sient test is likely to be substantially affected by the crossflowbehind the casing.

Field examples with buildup test data are presented to illus-trate the methodology. These show that ignoring the flow behind the casing may lead to an overestimation of the flowcapacity of the tested reservoir layer. Matching of the transienttest data also provides estimates of the conductivity of the seg-ment of compromised zonal isolation, which leads to comput-ing the transient crossflow rates.

Diagnosis and Characterization of Crossflowbehind the Casing from Transient Pressure Tests

Authors: Dr. N.M. Anisur Rahman, Saud A. Bin Akresh and Faisal M. Al-Thawad

Fig. 1. A schematic illustrating a probable scenario of flow behind casing.

WINTER 2015 SAUDI ARAMCO JOURNAL OF TECHNOLOGY

This section of compromised zonal isolation essentially createsa conductive medium, which allows fluid to transmit from theadjacent layer to the tested layer due to the pressure differentialat their respective sandface locations. The hydraulic conductivityof the section of compromised zonal isolation between the twolayers is characterized as:

(1)

where Fc is millidarcy feet (md-ft), k0 is the average permeabil-ity in md of the compromised cement sheath or channels, A0 isthe average annular area in ft2 of the cement, and L is the dis-tance in ft between the layers.

The rate of flow from the adjacent layer will depend on thehydraulic conductivity, Fc, of the cement behind the casing —or the hydraulic conductivity of the existing micro-channels —and the pressure differential between the sandface locations ofthe two layers. The magnitude of Fc can therefore be consid-ered the degree of hydraulic connectivity or conductivity be-tween the layers under consideration. Figure 2 shows that therate of flow behind the casing that originates in the adjacentlayer is additive to the contribution of the tested layer in deter-mining the total production rate during a transient test. If theconductivity is too low (zero or a negligible value), the adja-cent layer may not be able to interfere with the tested layer atall. If the conductivity is too high, both layers will behave likea commingled system, even if the well is completed in thetested layer only. One of the objectives in this study is to em-phasize the estimation of a key parameter, Fc, through tran-sient modeling of the test data.

The literature1, 2 provides petrophysical methods for assess-ing the integrity of cement. These methods do not include anyquantitative evaluations of how the fluid would flow acrossthe section of compromised zonal isolation under dynamicconditions due to the presence of pressure differentials in areservoir system. Kremenetskiy et al. (2008)3 has made somelimited efforts in developing type curves for transient pressure

behavior under the conditions of crossflow behind the casingusing a commercially available numerical package. This work,however, has not considered estimating the rate of crossflowbehind the casing with time.

The purpose of this study is to utilize a new solution for an-alyzing the transient pressure behavior of a two-layer systemsubject to flow behind the casing. Type curves from this ana-lytical solution can help diagnose and characterize the flow be-hind the casing by quantifying the conductivity of the sectionof compromised zonal isolation, in addition to estimating therates of crossflow from the adjacent layer as a function oftime. Also, this solution helps build models for use with actualdata from transient tests. We also intend to show that it is im-portant to account for the crossflow behind the casing, whenthis phenomenon exists, while analyzing data from transientpressure tests to ensure accurate reservoir characterization.

MATHEMATICAL MODEL

In the two-layer reservoir system, previously shown in Fig. 1,addressed by our new analytical solution, each homogeneouslayer of the system has identical fluid properties, but can havedifferent and distinct rock properties. Because the tested andadjacent layers are hydraulically connected by the conductivecement sheath or the created channels in the vicinity aroundthe casing of Fc, the magnitude of this conductivity controlsthe rate of crossflow with time. Because the layers are sepa-rated by impermeable strata, any exchange of fluid betweenthe layers can only occur through the conductive cementsheath or through channels with a non-zero, positive magni-tude of Fc. In this study, Fc is a key parameter to be estimatedthrough modeling of the transient test data.

Wellbore storage and skin effects in both layers have beenincorporated into the solution. Initially, before any productionstarts from the subject well, both layers are in hydrodynamicequilibrium, i.e., no crossflow behind the casing has been go-ing on. We are neglecting any effects of compressibility in theflow through the conductive cement sheath or channels to re-move from the analysis any transient pressures in the flow con-duits between the tested and the adjacent layers, i.e., transientpressures are experienced only within the tested and the adja-cent layers. All the cited pressures in this study are corrected toa datum depth.

Governing equations of the reservoir system are presentedin Appendix A for a constant rate of production from thetested layer, while Appendix B outlines the procedures for de-veloping an analytical solution in the Laplace domain, inver-sion into the time domain, and utilization of the constantproduction rate solution in a variable production rate situa-tion. The final solutions to this transient pressure problem inpressure and crossflow rate, in the Laplace domain, have beenpresented by Rahman (2014)4.

Fig. 2. Accounting for flow behind casing in the total production rate during atransient test.

SAUDI ARAMCO JOURNAL OF TECHNOLOGY WINTER 2015

shown for when the tested layer has about the same flow ca-pacity as the adjacent layer (k1h1≃ k2h2). Properties listed inTable 2 have been used in generating type curves for Case IIfrom the analytical solution.

TRANSIENT PRESSURE BEHAVIOR

In this section, we will discuss the transient pressure behaviorunder the conditions of flow behind the casing. An analyticalsolution is utilized to generate pressure drawdown and pressurederivative at the tested and adjacent layers, and the crossflowrate behind the casing, as a function of time. Figure 3 shows thetypical output of the model, presenting key transient measureswith time. In the reservoir dominated flow, following the well-bore storage and transition period, relative production fromthe tested layer keeps on decreasing with time. This means thatthe rate of crossflow behind the casing keeps on increasingwith time, due to the growing pressure differential between thetested and the adjacent layers, which rises to keep the produc-tion rate constant.

Our experience with the model shows that contrasts in theflow capacities of the two layers play an important role intransient behavior with flow behind the casing. Therefore, wehave grouped the transient behavior under three differentcases, based on relative values of the flow capacities of thetested and the adjacent layers. In all three synthetic cases, theproperties of the tested layer are identical, but the properties ofthe adjacent layer have been varied. As such, the flow capacityof the tested layer, k1h1, remains the same for all three cases,but the flow capacity of the adjacent layer is varied in such away that k2h2 in Case I is greater than k2h2 in Case II, which isequal to k1h1 and greater than k2h2 in Case III. Dimensionaltype curves during a constant rate of production are presentedto illustrate the significant features of transient pressure behav-ior when the crossflow behind the casing is active.

Presentation of Three Cases

In Case I, Figs. 4 and 5, the transient pressure behavior isshown for when the tested layer has significantly less flow ca-pacity than that of the adjacent layer (k1h1 << k2h2). Propertieslisted in Table 1 have been used in generating type curves forCase I from the analytical solution.

In Case II, Figs. 6 and 7, the transient pressure behavior is

Fig. 3. Typical output from the analytical model presented on log-log scales.

Fig. 4. Effects on the transient behavior when the tested layer has lower flowcapacity than the adjacent layer (Case I).

Fig. 5. Effects on the contribution of the tested layer when the tested layer has thelower flow capacity than the adjacent layer (Case I).

Tested Layer Adjacent Layer

kj (md) 440 2,200

zj 0.22 0.18

ctj (1/psi) 3e-6 3e-6

hj (ft) 42 100

rwj (ft) 0.3 0.3

sj 2 0

C (bbl/psi) 2e-3

n (ct) 0.74

qB (bbl/d) 3,755

p0 (psia) 2,780

T

Table 1. Properties for generating type curves in Case I


In Case III, Figs. 8 and 9, the transient pressure behavior isshown for when the tested layer has a flow capacity signifi-cantly greater than that of the adjacent layer (k1h1 >> k2h2).Properties listed in Table 3 have been used in generating typecurves for Case III from the analytical solution.

Discussion on Presented Cases of Transient Behavior

Figures 4, 6 and 8 show the log-log plots of pressure drawdown

Fig. 6. Effects on the transient behavior when the tested layer has the same flowcapacity as the adjacent layer (Case II).

Fig. 7. Effects on the contribution of the tested layer when the tested layer has thesame flow capacity as the adjacent layer (Case II).


kj (md) 440 440

zj 0.22 0.18

ctj (1/psi) 3e-6 3e-6

hj (ft) 42 42

rwj (ft) 0.3 0.3

sj 2 0

C (bbl/psi) 2e-3

n (ct) 0.74

qB (bbl/d) 3,755

p0 (psia) 2,780

T

Table 2. Properties for generating type curves in Case II

Fig. 8. Effects on the transient behavior when the tested layer has more flowcapacity than the adjacent layer (Case III).

Fig. 9. Effects on the contribution of the tested layer when the tested layer has moreflow capacity than the adjacent layer (Case III).


kj (md) 440 60

zj 0.22 0.18

ctj (1/psi) 3e-6 3e-6

hj (ft) 42 100

rwj (ft) 0.3 0.3

sj 2 0

C (bbl/psi) 2e-3

n (ct) 0.74

qB (bbl/d) 3,755

p0 (psia) 2,780

T Table 3. Properties for generating type curves in Case III


and pressure derivative for different values of Fc for the respec-tive Cases I, II and III. When Fc = 0, only the tested zone iscontributing to production, and no interference of the adjacentzone is observed. Therefore, the pressure derivative profile atthe late time is parallel to the time axis with an intercept valuereflective of the flow capacity of the tested layer only. When Fc= infinite, the adjacent layer does not experience any addi-tional resistance to its contribution to production along withthe tested layer. Therefore, both the tested and the adjacentlayers are simultaneously contributing to production as a com-mingled system. Here, the pressure derivative profile at the latetime is parallel to the time axis with an intercept value reflec-tive of the sum of the flow capacities of the tested and the ad-jacent layers (k1h1 + k1h2), which is true for all three cases.Systems with Fc = 0 and Fc = infinite offer the two extreme sit-uations with conductivity, while in reality a finite value for theupper boundary is needed to make the system behave as if itpossessed an infinite magnitude of conductivity. Notice thatthe profiles of pressure drawdown and pressure derivative forFc = 2�k2h2 in Fig. 5, and for Fc = 2�k1h1 in Figs. 7 and 9, ap-proximately merge with respective profiles for Fc = infinite.Therefore, the upper boundary of conductivity can be approxi-mated as the higher value of the flow capacities of the two lay-ers multiplied by 2� in a given system.

A similar observation as to the upper boundary of conduc-tivity can also be drawn from Figs. 5, 7 and 9, where the rela-tive contribution of the tested layer is plotted for differentmagnitudes of conductivity in the respective cases, Cases I, IIand III. This means that systems with conductivity of Fc ≥2�k2h2 in Case I and of Fc ≥ 2�k1h1 in Cases II and III will be-have as if the conductivity were infinite. With these extrememagnitudes of conductivity, the rate of crossflow behind thecasing reaches its upper boundary, because the lower flow ca-pacity of the layers in the respective systems offers the most re-sistance at this point. Therefore, the crossflow rate is expectedto be at its maximum possible value under this condition of amaximum value of conductivity. In an actual situation with aleaky cement sheath around the casing or flow channels, thevalue of Fc will be greater than zero but smaller than 2�k2h2in Case I, and 2�k1h1 in Cases II and III. As a result, in a typi-cal situation, the crossflow rate is expected to be lower thanthe respective maximum possible value previously mentioned,because the cement sheath or the flow channels would offerconsiderable resistance to flow. Note that the contribution ofthe tested layer decreases with an increase in the conductivity,which means that the crossflow rate increases with the increasein the conductivity.

As just discussed, 0 < Fc < 2�k2h2 can be considered as representing the probable ranges of conductivity in an actualleaky sheath or flow channels in Cases I and II, while 0 < Fc <2�k1h1 can be a range for Case III. Figures 4, 6 and 8 showthat the pressure derivative values keep on declining with timein the reservoir dominated portion following the wellbore stor-age and transition period. Therefore, it is often easier to diag-

nose flow behind the casing by studying the pressure derivativeprofile of the actual transient pressure data in Case I than it isin Cases II and II due to the higher rate of decline in Case I.Note that the rate of decline of the pressure derivative for agiven value of Fc depends on the range of the intercept valueson the respective derivative scales. Recall the ranges of inter-cepts on the pressure derivative scales due to k2h2 in Cases I, IIand III in the log-log plots. Given the way that all three caseshave been set up, we can appreciate the fact that k2h2 in Case I> k2h2 in Case II > k2h2 in Case III. Analysts of transient dataneed to be careful when flow behind the casing exists. If theydeliberately ignore it and try to analyze the data as a single-layer homogeneous reservoir system, it is likely that a higherflow capacity of the tested layer would be reported than is theactual case. Such a probability has also been recognized byKremenetskiy et al. (2008)3. The maximum error in the over-estimated flow capacity thereby reported would be limited tok2h2 in any of the cases.

The declining trend of the pressure derivative values, shownin Figs. 4, 6 and 8, underlines an important fact about the testduration. If the test duration is too short, not allowing thetested layer the time to develop enough drawdown to kickoff asignificant rate of crossflow from the adjacent layer, analystsmay not be able to recognize the declining trend of pressurederivative, and they will be tempted to analyze the data as asingle-layer homogeneous system. It is recommended that thetest duration be extended as much as possible if flow behindthe casing is suspected to give analysts the chance to recognizesome dip in the pressure derivative.

FIELD EXAMPLES

Two field examples, Well-A and Well-B, are presented in thisstudy to illustrate how transient pressure data is utilized in diagnosing and characterizing flow behind the casing.

Well-A Example

Two reservoir layers of Well-A are isolated geologically, asshown in the open hole log presented in Fig. 10. AlthoughWell-A was completed over the tested reservoir layer, the trendof the static pressure data from Well-A has been observed tofollow a trend similar to those collected from the wells com-pleted in the adjacent reservoir layer. This observation hasprompted the notion that the adjacent reservoir layer mighthave been contributing to the production of oil while Well-Awas completed over the tested reservoir layer. The buildup testdata from the tested layer has been analyzed to diagnose andcharacterize the flow behind the casing. The well was not stim-ulated at the time of the test, and the zonal isolation was pre-sumed intact, so no contribution from the adjacent layer wasexpected during production. The flow capacity (k2h2) of theadjacent layer is an order of magnitude higher than that (k1h1)of the tested reservoir layer.


A cluster of interpretation plots with the buildup data fromWell-A is presented in Fig. 11. The data points are plotted inred, and the model lines are drawn in black in the log-log,semi-log and history plots. The analytical solution has been

used to model the test data. The log-log plot of the Bourdetpressure derivative shows the derivative is declining with time,confirming the fact that the adjacent reservoir layer was con-tributing during the test. The model that matched the transienttest data suggests the conductivity, Fc of the communicationchannel between the tested and the adjacent layers is 3,350md-ft. This value of Fc can be treated as the characteristic leak-age parameter in the communication channels, which controlsthe rate of crossflow for the given set of reservoir layers. Therate history plot in Fig. 11 shows the comparative contribu-tions of the two layers. The production rate from the testedlayer is drawn in black, and the crossflow rate from the adja-cent layer flowing behind the casing is drawn in red. Under thecircumstances, both layers were contributing almost equally atlate times in the production period. Interestingly, the crossflowfrom the adjacent layer continued into the tested layer forsome time in the early part of the buildup period due to aresidual pressure differential at the time of shutting in the well.

Well-B Example

Two reservoir layers of Well-B are isolated, as shown in the openhole log presented in Fig. 12. Well-B was completed over the

Fig. 10. Open hole log of two reservoir layers intersected by Well-A.

Fig. 11. Interpretation plots of transient test data from Well-A.


tested reservoir layer at the time of the transient pressure test.Test results of the adjacent layer were also available prior totesting the tested layer. The buildup test data on the testedlayer has been analyzed to diagnose and characterize the flow

behind the casing. The well was not stimulated at the time ofthe test, so no micro-channels were expected to be present con-necting the layers hydraulically. The objective of the test was tovalidate the potential of the tested layer indicated in the mud log.The flow capacity (k2h2) of the adjacent layer is two orders ofmagnitude higher than that (k1h1) of the tested reservoir layer.

A cluster of interpretation plots with the buildup data fromWell-B is presented in Fig. 13. The data points are plotted inred, and the model lines are drawn in black in the log-log,semi-log and history plots. The analytical solution has beenused to model the test data. The log-log plot of the Bourdetpressure derivative shows the derivative value is declining withtime, confirming that the adjacent reservoir layer was con-tributing during the test. The model that matched the transienttest data suggests the conductivity, Fc, of the communicationchannel between the tested and the adjacent layers is 1,100md-ft. This value of Fc can be treated as the characteristic leak-age parameter in the communication channel, which controlsthe rate of crossflow for the given set of reservoir layers. Therate history plot in Fig. 13 shows the relative contribution ofthe two layers. The production rate from the tested layer isdrawn in black, and the crossflow rate from the adjacent layerflowing behind the casing is drawn in red. Under the circum-

Fig. 13. Interpretation plots of transient test data from Well-B.

Fig. 12. Open hole log of two reservoir layers intersected by Well-B.


stances, the adjacent layer was contributing almost all of thetotal production during the entire production period, illus-trated where the red line approximately follows the black line.In fact, the tested layer could not compete with the highly pro-lific adjacent layer, even with the latter’s flow passing throughthe cement sheath, and contributed only a negligible amountfor its tightness. Oil fingerprint analysis and other pertinentdata have been utilized to confirm this finding.

The model presented in Fig. 13 has also confirmed that partof the adjacent layer, not the entire interval, contributed to thecrossflow behind the casing to the tested layer. This finding isconsistent with the cement bond log of Well-B.

CONCLUSIONS

1. A lack of cement integrity or zonal isolation behind the casing or the existence of flow channels in the vicinity, both facilitating crossflow from the adjacent reservoir layer to the tested reservoir layer, can be diagnosed under certain conditions by observing the data from transient tests. A finite, non-zero value to the fluid conductivity of the zonal isolation, or the flow channels, can indicate the loss of integrity in the zonal isolation or the existence of a flow path in the vicinity from the reservoir engineering point of view.

2. The crossflow of fluid from the adjacent reservoir layer to the tested reservoir layer has to be accounted for and quantified to ensure a proper characterization of the testedlayer. Under a constant rate drawdown condition, the rate of crossflow increases with time due to the increasing pressure differential between the tested and the adjacent reservoir layer.

3. If a transient test is too short to create a substantial pres-sure differential between the tested and the adjacent reser-voir layers, the pressure derivative profile in the log-log plot may not have the resolution to diagnose a small rate of crossflow.

4. Any failure to detect the crossflow or the flow behind the casing may lead to a substantial overestimation for the flow capacity of the tested reservoir layer.

ACKNOWLEDGMENTS

The authors would like to thank the management of SaudiAramco for their support and permission to publish this article.Valuable assistance from M.D. Al-Odah, H.A. Nooruddin andS.B. Ahmad is gratefully acknowledged. Review comments fromH.S. Gill were particularly helpful for organizing the article.

This article was presented at the SPE Annual TechnicalConference and Exhibition, Houston, Texas, September 28-30,2015.

NOMENCLATURE

B formation volume factor (bbl/stb)ctj total system compressibility of the jth layer (1/psi)C wellbore storage constant (bbl/psi)Fc hydraulic conductivity of the channel behind the casing

(md-ft), defined in Eqn. 1j layer index value, j = for the tested layer and j = 2 for

the adjacent layerkj permeability of the jth layer (md)hj pay thickness of the jth layer (ft)p0 initial reservoir pressure (psia)pj(r, t) pressure at r in jth layer at time, t (psia)pwf wellbore flowing pressure (psi)q1 rate of production from the tested layer (stb/d)q2 rate of production from the adjacent layer (stb/d)r radial distance from the center of the well (ft)rwj wellbore radius in the jth layer (ft)rwej equivalent wellbore radius in the jth layer (ft), defined

in Eqn. A-8sj skin factor in the jth layert elapsed time (hours)

Greek Symbols

f j porosity of the jth layer (fraction)ηj hydraulic diffusivity in the jth layer, defined in Eqn. A-2

(md-psia/cP)m fluid viscosity (cP)

Subscript

j jth reservoir layer (1 refers to the tested layer, and 2 refers to the adjacent layer)

REFERENCES

1. Hayden, R., Russell, C., Vereide, A., Babasick. P.,Shaposhnikov, P. and May, D.: “Case Studies in Evaluationof Cement with Wireline Logs in a Deep WaterEnvironment,” SPWLA paper presented at the SPWLA52nd Annual Logging Symposium, Colorado Springs,Colorado, May 14-18, 2011.

2. Cai, J., Gao, Y., Zhang, M., Cui, L. and Guo, H.:“Solution to Cement Integrity Evaluation in LongExtended Reach Wells: New Record in the South ChinaSea,” OTC paper 25027, presented at the OffshoreTechnology Conference-Asia, Kuala Lumpur, Malaysia,March 25-28, 2014.

3. Kremenetskiy, M.I., Ipatov, A.I. and Kokurina, V.: “WellTest Interpretation in Case of Behind-the-CasingCrossflow,” SPE paper 115323, presented at the SPERussian Oil and Gas Technical Conference and Exhibition,


Moscow, Russia, October 28-30, 2008.

4. Rahman, N.M.A.: “Measuring behind Casing HydraulicConductivity between Reservoir Layers,” U.S. Patent andTrademark Office, Patent Application Number 14/182,430,2014.

5. Stehfest, H.: “Algorithm 368 Numerical Inversion ofLaplace Transforms,” Communications of the ACM, Vol.13, No. 1, October 1970, pp. 47-49.

APPENDIX A – GOVERNING EQUATIONS OF ANALYTI-CAL SOLUTION

The diffusion equation governing the transient flow in the jthhomogeneous layer is given by:

(A-1)

where

(A-2)

Note that subscript j = 1 refers to properties of the testedreservoir layer, and j = 2 refers to properties of the adjacentreservoir layer.

Eqn. A-1 is subject to the following initial condition:

(A-3)

Eqn. A-3 ensures that both layers have been at a hydrody-namically equilibrium condition before any production startsat the tested layer. Note that all the pressures in this model arecorrected to a datum depth.

The inner boundary condition at the sandface due to a time-dependent rate of production can be presented as:

(A-4)

where(A-5)

The production string of the well is directly connected tothe sandface of the tested layer (j = 1) through the completion.The influx or crossflow from the adjacent layer joining thesandface of the tested layer gets produced with the streamfrom the tested layer. Therefore, considering the downholeconditions, the total rate of production through the productionstring (qB) is being replenished by the rate of fluid expansiondue to wellbore storage and the rate of production (contribu-tions) from individual layers. This can be expressed with thematerial balance equation at a given time, t, as:

(A-6)

In this problem, qB is constant (not time-dependent), while qj(t)B — the contribution from the respective layer — is time-de-pendent. Refer to Fig. 2 for the spirit of material balance ex-pressed in Eqn. A-6. Note that j = 1 and j = 2 refer to the twocontributing layers in Eqns. A-2 through A-6. The second term

in Eqn. A-6 is due to the wellbore storage constant, whichdominates at the early time and dissipates with time as theflow from the reservoir layers contribute. The contributionfrom the adjacent layer is related to the sandface pressures ofthe two layers as:

(A-7)

In Eqn. A-7, Fc is the hydraulic conductivity of the cementbetween the tested and the adjacent layers, defined in Eqn. 1.Therefore, the value of Fc controls the influx rate (q2(t)B) fromthe adjacent layer and the difference in the sandface pressuresof the layers. One should appreciate that Eqn. A-7 is a mate-rial balance equation at reservoir conditions, having neglectedany pressure transient effects in the compromised cementsheath or flow channels by assuming the compressibility valuesof rock and fluid to be zero in the flow conduits between thelayers. Note that the sandface pressures in Eqn. A-7 and else-where later are to be evaluated at the respective equivalentwellbore radius due to the presence of the skin factors. There-fore, the skin factor, sj, in each layer is considered through theequivalent wellbore radius with a physical wellbore radius, rwj,as:

(A-8)

This effective wellbore radius can deal with both positiveand negative skin factors. Also, the sandface pressure of thetested layer will be equal to the flowing wellbore pressure,which means that:

(A-9)

The outer boundary condition for each layer can be pre-sented as:

(A-10)

APPENDIX B – DEVELOPMENT OF AN ANALYTICALSOLUTION

Governing equations presented in Appendix A are transformedinto the Laplace domain. Final solutions for the pressure at thewellbore, the rate of crossflow behind the casing and the rateof flow from the tested layer are obtained in the Laplace do-main. The final solutions are available in the work of Rahman(2014)4. Numerical inversion of these solutions in the Laplacedomain into those in the time domain can be performed read-ily with the Stehfest algorithm5.

Although the solutions for pressure and rates have been pre-sented for a constant rate of production from the tested layer,these can be utilized for stepped variations of the productionrates, including any zero rates, by using the principle of super-position.


In 1987, Saud received his B.S. degree in PetroleumEngineering from King Saud University, Riyadh, SaudiArabia. He then went to work at King AbdulAziz City forScience and Technology as a researcher at their Petroleumand Petrochemical Research Institute.

Saud is a member of the Society of Petroleum Engineers(SPE). In May 2014, Saud was a co-chair of the SteeringCommittee of the SPE Reservoir Testing Workshop thatwas held in Dubai, UAE.

Faisal M. Al-Thawad is a GeneralSupervisor of the Well Testing Divisionand a Senior Consultant in pressuretransient analysis in Saudi Aramco’sReservoir Description and SimulationDepartment. He has extensiveexperience in operations and

supervision of exploratory tests, and in transient modelingof faults and fractures and multilateral wells.

Faisal has more than 25 years of experience in variouspetroleum disciplines, including drilling, reservoirmanagement and simulation, and he also completed a 6-year Specialist Development Program on “ReservoirTesting/New Technologies” with a focus on fracture identi-fication, characterizing and modeling.

He has authored several technical papers in the field ofwell testing and pressure transient analysis.

Faisal is a member of the Society of Petroleum Engineers(SPE) and has participated in a number of internationalconferences. He was a SPE Distinguished Lecturer duringthe 2013/2014 cycle, delivering the lecture “FracturesNetwork: Reservoir’s Friend or Foe?” Faisal participated asa panelist and was a speaker invited to deliver expertopinions and speeches at many national and internationaltechnical meetings and at universities.

Faisal received his B.S. degree in Petroleum Engineeringfrom King Fahd University of Petroleum and Minerals(KFUPM), Dhahran, Saudi Arabia, and his M.S. degree inPetroleum Engineering from Heriot-Watt University,Edinburgh, U.K. Currently, Faisal is finalizing his Ph.D.degree with a dissertation on Fracture Network Modelingat Heriot-Watt University.

BIOGRAPHIES

Dr. N.M. Anisur Rahman is aPetroleum Engineering Consultant withthe Well Testing Division of theReservoir Description and SimulationDepartment at Saudi Aramco, wherehe designs and interprets transient testson oil production and water injection

wells. His interests include production technology, reservoircharacterization and pressure transient analysis.

Before joining Saudi Aramco, Anisur Rahman workedfor the Bangladesh University of Engineering andTechnology, the University of Alberta, Fekete Associate Inc.and Schlumberger. He developed analytical solutions for anumber of pressure transient models, including methods forshort well tests.

Anisur Rahman has authored or coauthored 26technical papers for publication in refereed journals andconference proceedings, and seven patent documents, ofwhich one patent has already been granted by the U.S.Patent and Trademark Office. Recently, he has beennominated as a PTA/Well Testing Champion under thePetroleum Engineering Technical Excellence (PETE) andSupport Network in Saudi Aramco.

Anisur Rahman received both his B.S. and M.S. degreesin Mechanical Engineering from the Bangladesh Universityof Engineering and Technology, Dhaka, Bangladesh, andhis Ph.D. degree in Petroleum Engineering from theUniversity of Alberta, Edmonton, Alberta, Canada.

He is registered as a Professional Engineer in theProvince of Alberta, Canada.

Saud A. Bin Akresh joined SaudiAramco in 1990 as a PetroleumEngineer in the Reservoir DescriptionDivision of the Reservoir Descriptionand Simulation Department, where heprovided analyses and interpretationsof open hole and cased hole logs. Saud

also had several technical assignments with differentorganizations within Saudi Aramco, including a role as aReservoir Engineer where he was responsible for managingoil production from the Ain Dar area of the giant Ghawaroil field. Saud also looked after well testing operations inexploration and delineation wells. He has served as a peerreview member for numerous integrated reservoir studies inthe company.

Saud is now a Supervisor in the Well Testing Division,where he leads a technical group that annually completesmore than 500 pressure transient analysis reports on oilproduction and water injection wells. He has 25 years ofexperience in petroleum engineering.


Diagnosis and Characterization of Crossflow behind the Casing ...

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