Pressure buildup test analysis in wells with sustained casing pressure

Accepted Manuscript Pressure buildup test analysis in wells with sustained casing pressure R. Xu, A.K. Wojtanowicz PII:

S1875-5100(16)30933-7

DOI:

10.1016/j.jngse.2016.12.033

Reference:

JNGSE 2007

To appear in:

Journal of Natural Gas Science and Engineering

Received Date: 7 August 2016 Revised Date:

24 November 2016

Accepted Date: 20 December 2016

Please cite this article as: Xu, R., Wojtanowicz, A.K., Pressure buildup test analysis in wells with sustained casing pressure, Journal of Natural Gas Science & Engineering (2017), doi: 10.1016/ j.jngse.2016.12.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Pressure Buildup Test Analysis in Wells with Sustained Casing Pressure

AC C

EP

TE D

M AN U

SC

RI PT

R. Xu, SPE, Schlumberger, [email protected]; A. K. Wojtanowicz, SPE, Louisiana State University, [email protected] total, 22 wells (85%) have SCP problems. And by casing type, Abstract almost half of intermediate casings have SCP problem, When the well’s casing head pressure cannot be permanently followed by surface casings, production casings and conductor bled off with a needle valve, the casing is said to exhibit casings (Fig. 1). The statistical analysis shows the trend Sustained Casing Pressure (SCP). Rebuilding of surface similar to that reported in literature [1] (Fig. 2), with exception pressure (pressure buildup – after closing the valve) results of the intermediate casing strings. From further analysis by from migration of gas in the leaking cement sheath of the casing size, large casings are likely to have SCP problem (Fig. well’s annulus. Problem of sustained casing pressure (SCP) 3). has been widespread in the Gulf of Mexico and also reported The SCP problem in terms of casing pressure values can be in Canada, Norway, and other places. Regulations require 24also sized. Among casings affected by SCP, about 50 percent hr testing of SCP wells comprising pressure bleed-down of the production casings and 35 percent of the intermediate followed with pressure buildup. In these tests, the rate of casings have SCP values less than 1000 psi. For the other pressure buildup is indicative of the size of cement leak – casing strings, about 90 to 100 percent of the strings have SCP prompt buildup implies larger leak on a relative scale with no values less than 500 psi (Fig. 4). quantification. Presented here is the first mathematical model for quantitative analysis of pressure buildup in SCP wells. The Reported Field Patterns of SCP Buildup model simplifies transient gas flow in cement and ignores Typical SCP Buildup Pattern migration time of gas in the annular fluid column above the We have analyzed historical data of casing pressure change cement top. The simplification allows finding the size of from 38 casing string affected by SCP. Fig. 5 shows a typical cement leak and the depth and value of gas pressure source pattern of SCP buildup. (Of the 38 casings with SCP, 82 formation. percent displayed typical pattern of buildup.) By this pattern Also presented is validation of the model with actual field data the casing pressure increases monotonically after the bleed-off from testing SCP wells. Matching the model to field data gives at steadily-reduced rate until stabilizing at a certain constant acceptable estimates of the gas-source formation depth and value. In some cases, a small buildup rate may not change pressure, cement leak size, and expected maximum casing during the test as shown in Fig. 6. Such a low-rate incomplete pressure value. The results also reveal a correlation between buildup would be difficult to analyze. In this study, only the pressure buildup-stabilization pattern and well parameters typical patterns of SCP buildup are quantitatively analyzed. - cement leak size controls the pressure buildup rate, while the gas-source formation pressure controls the stabilized pressure Abnormal Behavior value. Quantitative analysis of SCP buildup with the new Fig. 7 is an abnormal case. The well was shut in at about 500 model could be extremely useful as it provides values of three days. The casing pressure record is scatted significantly. It is parameters (cement leak size, gas source formation depth and very difficult to identify any obvious trends from the data. pressure) that are critically important for designing remedial Moreover, comparison of all casing pressures change vs. time treatment. in this well (Fig. 8) indicates a correlation, i.e. pressure communication between different casings. This could be the Introduction main reason for the erratic pressure data in Fig. 7. Problem of SCP has been common in the Gulf of Mexico (GOM). Over 11,000 casing strings in over 8000 wells have Comparsion of Bleed-off and Buildup Patterns been reported with SCP [1]. The regulations promulgated by Unlike pressure buildup, the bleed-off is very short. Usually, US Minerals Management Service, MMS, (presently, Bureau only two or three data points were recorded from the bleed-off of Safety and Environmental Enforcement, BSEE) 30 CFR (Fig. 9). It is difficult to analyze the bleed-off from such a few 250.517 require remedial operation on a well if any of its data. On the other hand, in a short time, the bleed-off behavior casing string has significant SCP problem. could be only affected by the compressibility of gas MMS has also developed guidelines to tolerate small values of accumulated at the wellhead and that of mud below. Testing of SCP – a departure from 30 CFR 250.517. However, wells with bleed-off can only provide partial information on well approved departure must be frequently tested so that severity parameters. Therefore, this study concentrated on analyzing of SCP could be monitored and controlled. Presently, testing SCP buildup after the bleed-off. of SCP is mostly qualitative and limited to arbitrary criteria Mechanism of SCP Return after the Bleed-off for casing pressure buildup. Such information is insufficient The most significant cause of SCP in the outer casing strings, for operators to quantitatively analyze SCP problem and outside of the production casing, is gas migration outside prevent potential risks. Thus there is a need for improved wells due to poor cementing. In the database of MMS, 50% of analysis that could provide information on the parameters the casing strings exhibiting SCP are outer strings outside the causing gas migration and SCP. production casing (Fig. 2). Field Data Analysis. We have analyzed casing pressure data Gas migration is classified into two distinct groups – “early” from an offshore oilfield in the GOM [2] and compared the and “late”. The former can be defined as those that are related results with MMS data from the whole GOM [1]. Of 26 wells to aspects of the actual cementing operation i. e. slurry

ACCEPTED MANUSCRIPT

characteristics, displacement mechanics, and hydrostatic pressure [3]. The late gas migration has little to do with the cementing operation. It may be caused by mechanical and thermal stresses which compromise the integrity of hydraulic bond or the integrity of the cement material leading to gas leakage [4, 5]. Another possible cause might be the use of cement systems with high water/cement ratio (low density, extended system). These cements can exhibit fairly high innate permeability (0.5 – 5 md), even when set [6]. It is possible, therefore, for gas to flow within the matrix (matrix channeling) of such cements and to eventually reach the surface, resulting in SCP. Although lots of researchers have studied the late gas migration, few related it to SCP testing. Using the gas well testing model, Somei [7] modeled the late gas migration through an annulus cemented to the surface (Fig. 10). He considered the gas migration as a vertical flow through permeable porous media. The assumptions in his model were constant formation pressure, zero gas flow rate at wellhead, and steady-state bleed-off. He analyzed effects of cement porosity, temperature, and gas gravity on SCP and concluded that low porosity, low temperature and low gas gravity would enhance SCP. Unfortunately, he has not verified his solution with experimental or field data. Moreover, the limitation of his model is obvious because most of outer casings are not cemented to the surface. The study is based on our first comprehensive model for quantified analysis of SCP [8]. Furthermore, several important corrections have been addressed in this paper.

known. The conceptual process of gas migration in the cement and mud column is illustrated in Fig. 12. Based on above assumptions, we can compute pressure at the wellhead (casing pressure) pt at n-th time step as,

A Model of SCP Buildup in Annuli with Gas-free Mud We developed a numerical model for SCP buildup in the annulus with a mud column above the cement top (Fig. 11). We assumed the gas migration process comprising large number of short time steps. In each time step, gas flow is steady-state in the cement. Also assumed is that the pressure at the cement top remains constant during each time step thus resulting in the constant flow rate for this step. The gas released at the cement top over one time step migrates though the mud and completely accumulates in the casing gas cap during the next time step. Thus, the assumed gas migration time (controlled by gas rising velocity) is shorter than the time step. This assumption is valid only for high gas rising velocity in the mud. So the time steps are small. Otherwise, the procedure of representing transient gas flow as a series of steady-state flow periods would be invalid. However, for a low-viscosity and short mud column, it will take a very short time for gas to reach the top. So we can set a reasonable time step to make this assumption valid. Formation pressure is assumed constant due to the high formation permeability to gas compared with that of cement. The mud is assumed a slightly compressible. In our original model, the mud density is assumed as constant [8], which is not correct. In fact, it should be the hydrostatic pressure exerted by gas-free mud that is constant. Temperatures at the

The derivation and procedure of calculation are shown in Appendix.

RI PT

 n −1 Vt n −1  pt − + cmVmn −1  1 n ptn =  2 2 4 T psc qck ∆t ∑ wh   n −1 Vt n −1  k =1  +   pt − n −1  n −1 c V c m m mVm Twb   

SC

In Eq. (1), the pressure at standard condition Psc, instead of Pc (pressure at cement top), is used to correct the imprudence in our original model [8]. The pressure at the top of cement surface casing pressure by:

Twh ) are different and

p c can be related to the

pcn = ptn −1 + 0.052 ρ m0 L0f

(2)

For pressure < 2000 psia or > 4000 psia, the steady-state flow rate at the top of cement is:

M AN U

TE D

EP

AC C

top of cement and top of mud ( Twb and

     (1)   

q cn =

[

( )]

0.003164 kTsc A 2 p f − p cn p sc TL c µ i Z i

2

(3)

For pressure between 2000 and 4000 psia, the flow rate is:

qcn =

[

( )]

0.003164 kTsc A m ( p f ) − m pcn p scTLc

(4)

Recent applications and modifications of the model Since the original conference paper was published [8], it had inspired some research work based on the mathematical model. In addition to analysis on SCP in producing wells, the model has been applied to determine the CO2 leakage rate along the wellbore during its sequestration, in fear of possible emission to the environment. Huerta et al [9] consider physical analogy between the mechanisms of SCP and well leakage of CO2 sequestration using Class VI wells. They simplified the model for quantifying the potential for CO2 leakage. Matching the model with field data provided information on the depth of leakage and effective cement permeability. The CO2 leak source pressure was determined by adding the hydrostatic pressure of Newtonian fluid to the measured stabilized casing pressure. The matching between field data and calculated data involved an iterative process with the guessed gas source depth and cement permeability. And the matched effective permeability was converted into equivalent geometries of discrete pathway such as a gas channel, micro fracture, or micro annulus. Tao et al [10, 11] followed on the CO2 leakage model based on the SCP buildup model to determine effective permeability of the leakage pathway along the cemented section of a wellbore. They determined the depth of gas leakage source and effective permeability by matching the field pressure

ACCEPTED MANUSCRIPT

with a thin, gas-free liquid would improve the testing by reducing the pressure build up time.

RI PT

Effect of Cement Sheath’s Permeability. The “cement sheath’s permeability to gas” was used to represent the quality of cement. From Fig. 15, we can see that the early stage of SCP buildup is very sensitive to this parameter. Reduction of cement sheath’s permeability from 0.36 md to 0.1 md results in the increased SCP stabilization time from 40 months to 90 months, and the pressure is still increasing. The cement sheath’s permeability is also a hard-to-get data in practice. Using the model, we can obtain the permeability and have an assessment of cement quality.

SC

Effect of Formation Pressure. During the SCP buildup, formation pressure is relatively constant because formation permeability to gas is most likely much higher than that of cement. Formation pressure controls the value of stabilized SCP when pressures are in balance and gas migration stops. From Fig. 16, we can see that the higher formation pressure, the higher the stabilized pressure. Its impact on early stage of buildup is relatively low, comparing with other parameters such as casing gas cap size and cement sheath’s permeability.

M AN U

buildup data with the model. They would change the control variables - leakage depth and effective cement permeability to minimize the sum of the squared differences between observed and calculated data. Similar to Huerta study, the gas source pressure was calculated from the hydrostatic pressure of mud column and stabilized casing pressure. Rocha-Valadez et al. [12] derived an analytical model of casing pressure buildup in SCP testing. They assumed constant-pressure gas inflow to the well and instant migration of gas through the gas-free column of liquid from the top of cement to the gas cap below the casing head. In matching field pressure buildup data the objective function was the meansquared error between observed and calculated data. They removed ambiguity by assuming a known value of gas source formation pressure that reduced number of unknown variables to one parameter - cement permeability. The cement permeability values calculated with their analytical model would closely match the results from the numerical model. Xu et al. [13] improved the SCP buildup model by coupling the transient gas flow in cement with the gas migration in nonNewtonian fluid in mud column. The new model considered the important mechanism – gas migration in mud, which would control pressure bleed-down and early buildup in SCP testing. Therefore, the new model was also used to match SCP bleed-down testing.

TE D

Effect of Well Parameters on SCP Buildup Using the model, we analyzed the effect of parameters controlling SCP buildup. The parameters are mud compressibility, formation pressure, size of the casing gas cap above the mud, and the cement sheath’s permeability to gas. The analysis of the effects is very useful to identify input data needed for meaningful testing and reduce the uncertainty of the matching.

AC C

EP

Effect of Casing Gas Cap. The casing gas cap is the top portion of the casing annulus at the wellhead. Usually, the part is filled with gas or gas-cut mud with high gas concentration. Using the model, we plotted theoretical SCP buildup curves with different size of the casing gas cap. Because of the compressibility difference between gas and mud, the cap functions as a “cushion”. The larger the casing gas cap is, the slower the pressure builds up and longer the time is needed to reach the stable pressure value (Fig. 13). Therefore, if we leave a large casing gas cap after bleed-off, we will need a long testing time. Moreover, larger cap would reduce the hydrostatic pressure of mud. More gas would migrate upward after the bleed-off, causing severe SCP problem. On the other hand, by filling up the column with mud, we could remove the effect of gas cap and make the buildup plot analysis nonambiguous. Effect of Mud Compressibility. Using the model, we also studied the effect of mud compressibility. From Fig. 14, we can see that the more compressible the mud is, the slower the casing pressure buildup becomes. Gas-free mud compressibility is easy to measure. Thus, filling the annulus

Field Data Analysis with Model Using the model to match field data was encouraging and showed that the model could be used to determine the most uncertain parameters controlling SCP: cement conductivity or permeability, formation pressure, and the depth of gas invasion zone. We selected SCP testing in two wells to validate our mathematical model. Field data and well parameters are listed in Table 1. The matched data are shown in Table 2. Analysis of SCP Pressure Buildup in Well 23 Well 23 is a gas production well. From the pressure record, only intermediate casing (Fig. 17) had SCP problem. And SCP testing shows the incomplete buildup pattern. The casing pressure increased from 200 psi to 1600 psi and continued raising after 8 months. The model indicated that only for the formation pressure of 6600 psi, would the casing pressure reach 1600 psi in 8 months. Also, using the model, we predicted that the casing pressure would stabilize at about 2170 psi in 25 months (Fig. 18). The matched value of cement sheath’s permeability for this well is 0.36 md. This value indicates that the SCP may be caused by using the cement systems with high water/cement ratio [6]. Analysis of SCP Pressure Buildup in Well 24 In well 24, only intermediate casing (Fig. 19) had SCP problem. The casing pressure increase to about 1000 psia in one month. Before the buildup, well 24 was frequently bled off. After each bleed-off, heavier mud was pumped into the annulus. Operators recorded the volume and weight of the bled and pumped muds. During the buildup stage, no mud was bled from or pumped into the annulus.

ACCEPTED MANUSCRIPT

•

RI PT

•

Nomenclature A = area of annulus, L2, sq. ft c m = mud compressibility, Lt2/m, psi-1

AC C

EP

TE D

M AN U

Conclusions • Statistical analysis of casing pressure in a single oilfield shows similar trends to those reported by MMS for the whole GOM. Thus, we conclude that the SCP problem is widespread and independent from conditions of specific oilfield in the GOM. Also, the analysis method validated for one oilfield should work anywhere in the GOM. • SCP buildup pattern is controlled by parameters of cement, mud and gas invasion zone. Using the mathematical model, we theoretically analyzed the effects of those parameters and found out as follows: − Large casing gas cap prolongs the SCP buildup cycle and would complicate buildup analysis by reducing the buildup plot resolution. Operators should keep this cap as small as possible by filling up the well after the bleed-off. − Mud compressibility controls the early stage of SCP buildup. Thin drilling mud having low tendency for gas cutting would considerably improve the analysis of SCP buildup by removing the compressibility effect. − Cement sheath’s permeability parameter represents the cement leak size. It controls early stage of SCP buildup. Thus, SCP buildup rate analysis may become an overall measure of the annular seal performance of the well. − Formation pressure controls the maximum value of stabilized SCP, with high formation pressure resulting in high stabilized SCP value. Potentially, a combined analysis of the stabilized SCP value, mud density, top cement depth and formation pressure gradients may identify the gas invasion zone. In case when maximum value of SCP is not attainable (too high) from the field data, the mathematical model presented here could extrapolate the value. • The field validation of the model, presented here, gives acceptable estimates of the gas-source formation pressure, cement conductivity, and expected maximum casing pressure value. Ambiguity of the analysis can be significantly reduced by reducing the number of unknown parameters to two: cement conductivity and formation pressure. Early stage of SCP buildup is controlled by cement conductivity; while stabilized pressure is determined by formation pressure. Usually the depth of gas invasion zone can be provided by casing-hole logging with reasonable confidence. Thus the formation pressure

•

value can be determined with less uncertainty than cement sheath’s permeability. Then the test analysis would be fairly straightforward. Recent applications of the model proves that the model could be a useful tool for analyzing CO2 leakage along wellbores designated for CO2 sequestration. The model has been simplified by disregarding effects of gas migration in the mud and gas cutting of the mud. The two parameters may have strong effects on the rate of SCP buildup. Future study [13] address SCP buildup analysis including the effect of gas migration in nonNewtonian fluids. Using the gas-free mud model, we assumed that gas doesn’t dissolve in mud. However, the model can be readily expanded to accommodate gas solubility in mud by relating mud density change with pressure.

SC

Because of lacking some input data, we had to make several assumptions using available well information. In spite of the assumptions, we could still get a very good match as shown in Fig. 20. We think that the reason of efficient matching is small number of parameters to be determined from the match. Moreover, the early and late pressure buildups are not controlled by the same parameters. Mud compressibility, casing gas cap size and cement sheath’s permeability control early SCP buildup. The late buildup is controlled by formation pressure and mud density.

D1 = outer diameter of annulus, L, in D2 = inner diameter of annulus, L, in k = cement sheath’s permeability to gas, (Eq. A-4), L2, md

Lc = distance from the leak point to the top of cement, L, ft Lt = length of gas chamber, L, ft L f = length of mud column, L, ft

pb = base pressure for real gas pseudo-pressure, m/Lt2, psia p c = pressure at the top of the cement, m/Lt2, psia

p f = gas-source formation pressure, m/Lt2, psia

psc = pressure at standard condition, m/Lt2, psia pt = pressure on surface, m/Lt2, psia qc = flow rate at the top of the cement, SCF/D T = reservoir condition temperature, K Twb = average wellbore temperature, K Twh = wellhead temperature, K Vm = volume of mud column, L3, cu ft Vt = volume of gas chamber, L3, cu ft Z = gas-law deviation factor µ g = gas viscosity, m/Lt, cp

ρm

= density of mud in wellbore, m/L3, ppg

∆ t = time step, t, day References 1. Bourgoyne, A. T. Jr., Scott, S. L., and Manowski, W.: “A Review of Sustained Casing Pressure Occurring on the OCS”, Report submitted to U. S. Department of Interior

ACCEPTED MANUSCRIPT

7.

8.

9.

10.

11.

12.

13.

14.

RI PT

Mathematical Model of SCP Buildup in Cemented Annulus with Mud Column

Appendix –

Derivations A well schematic for this model is shown in Fig. 12. The boundary conditions can be easily defined as: (A - 1) p ( 0, t ) = P f

SC

6.

Jour. of Petroleum Technology, August 1968, 877-887; Trans. AIME, 243 15. Craft, B. C., Hawkins M. and Terry, R. E. “Applied Petroleum Reservoir Engineering (2nd Edition)”, January 28, 1991; page 221 16. Al-Hussainy, R., and Ramey, H.J., Jr.: "Application of Real Gas Flow Theory to Well Testing and Deliverability Forecasting," Jour. Petroleum Technology, May 1966, 637

q ( L, t ) = 0

(A - 2)

And the initial condition in mud column is:

p( z, t ) = 14.7 + 0.052ρm0 ( z − Lc ) Lc ≤ z < L

(A - 3) For gas steady state flow in cement, the Darcy’s law can be written as:

M AN U

5.

TE D

4.

EP

3.

AC C

2.

Minerals Management Services, Washington, D.C., 2000, Contract Number 14-35-001-30749. Wojtanowicz, A. K., Somei, N and Xu, R.:“ Diagnosis and Remediation of Sustained Casing Pressure in Wells”, LSU report submitted to MMS, June 15, 2000 Rae, P., Wilkins, D and Free, D.: “A New Approach to the Prediction of Gas Flow After Cementing”, SPE/IADC 18622, 1989 SPE/IADC Drilling Conference, New Orleans, Louisiana, February 28 – March 3, 1989 Jackson, P.B. and C.E. Murphey: "Effect of Casing Pressure on Gas Flow through a Sheath of Set Cement", 1993 SPE/IADC Drilling Conference, Amsterdam, February 23-25, 1993. Goodwin, K.J. and R.J. Crook: "Cement Sheath Stress Failure", SPE 20453, SPE Fall Meeting, New Orleans, 1990. Sykes, R. L. and Logan, J. L. : “New Technology in Gas Migration Control”, SPE 16653, SPE Annual Technical Conference and Exhibition, Dallas, 27-30 September 1987 Somei, N.: “Mechanisms of Gas Migration after Cement Placement and Control of Sustained Casing Pressure”, M. S. Thesis, Louisiana State University, Baton Rouge (May 1999) Xu, R. and Wojtanowicz, A. K.: “Diagnosis of Sustained Casing Pressure from Bleed-off/Buildup Testing”, SPE 67194, SPE Production and Operations Symposium held in Oklahoma City, Oklahoma, 24–27 March 2001. Huerta, N.J, Checkai, D., Bryant, S.L.: “Utilizing Sustained Casing Pressure Analog to Provide Parameters to Study CO2 Leakage Rates Along a Wellbore”, SPE 126700, 2009 SPE International Conference on CO2 Capture, Storage, and Utilization, San Diego, California, USA, 2-4 November 2009. Tao, Q., Checkai, D.A., Huerta, N.J. and Bryant, S.L.: “Model to Predict CO2 Leakage Rates Along a Wellbore”, SPE 135483, SPE Annual Technical Conference and Exhibition. Florence, Italy. 20-22 September 2010. Tao, Q., Checkai, D.A., and Bryant, S.L.: “Permeability Estimation for Potential CO2 Leakage Paths in Wells Using a Sustained-Casing-Pressure Model”, SPE 139576, SPE International Conference on CO2 Capture, Storage & Utilization. New Orleans, Louisiana, USA, 10–12 November 2010 Rocha-Valadez, T., Hasan, A. R., Mannan, M. S., and Kabir, C. S., “Assessing Wellbore Integrity in SustainedCasing-Pressure Annulus”, SPE-169814-PA, March 2014 SPE Drilling & Completion; Vol. 29; Issue 1 Xu, R. and Wojtanowicz, A. K.: “Diagnostic Testing of Wells with Sustained Casing Pressure – An Analytical Approach”, The Petroleum Society’s Canadian International Petroleum Conference 2003, Calgary, Alberta, Canada, June 10 – 12, 2003. Wattenbarger, R. A and Ramey, H. J. Jr., “Gas Well Testing with Turbulence. Damage and Wellbore Storage”,

qpscTZ k dp = −0.001127 µ dz 5.615 pTsc A

(A - 4)

And separating variables results in: z

P

qp scT p dz = − ∫ dp ∫ 0.006328 kTsc A 0 µZ Pf

(A - 5)

Wattenbarger and Ramey [14] observed that the gas deviation factor-viscosity product is the function of pressure. Below 2000 pisa, the product, µZ, is nearly constant. And the product µizi at initial pressure can be used. Then the integration of Eq. A-5 can be written as:

qp scT ( µZ ) dz = − ∫ pdp = 12 ( p 2f − p 2 ) ∫ 0 .006328 pkTsc A 0 Pf z

P

(A - 6)

The initial conditions in cement column can be:

p 2 ( z , t ) = p 2f −

q 0 p sc Tz µ i Z i 0 < z < Lc (A - 7) 0.003164 kTsc A

And the gas rate at the top of cement can be written as:

q0 =

[

( )]

0.003164 kTsc A 2 p f − pc0 pscTLc µi Z i

2

(A - 8)

Above 4000 psia, the product ( ⁄ ) is constant. Then the integration of Eq. A-5 can be rewritten as: P

qp scT ( µz p ) dz = − ∫ dp = ( p f − p ) 0.006328 kTsc A ∫0 Pf z

The gas rate at the top of cement can be rewritten as:

(A - 9)

ACCEPTED MANUSCRIPT

( )]

n

(A - 10)

The term (µZ ⁄ p) should be evaluated at the average pressure between Pf and Pc0 [15]. Then Eq. A-7 can be rewritten as:

q0 =

[

( )]

0.003164 kTsc A 2 p f − pc0 pscTLc ( µZ ) avg

2

(A - 11)

It is also assumed that the changes in gas viscosity and Zfactor are small [12] and the product (µZ)avg can be represented by the values at the initial condition, (µiZi). Then, the Eq. A-11 can be the same as Eq. A-8. Between 2000 and 4000 psi, the real-gas pseudo-pressure m(p) should be used [16],

p m( p) = ∫ dp µZ pb

( )]

pcn = ptn −1 + 0.052 ρ mn −1Lnf−1

TE D

(A - 14)

Because the gas-free mud column has constant mass, the hydrostatic pressure term in Eq. A-14 is constant and can be replaced with the value at initial condition:

pcn = ptn −1 + 0.052ρm0 L0f

(A - 15)

EP

This also means that the pressure increase of Pc is caused by the pressure increase at gas chamber only. And, for pressure < 2000 or > 4000 psia, the flow rate is,

( )] 2

AC C

0.003164 kTsc A 2 p f − p cn p sc TL c µ i Z i

(A - 16)

For pressure between 2000 and 4000 psia, the flow rate is:

( )]

0.003164 kTsc A qcn = m ( p f ) − m pcn p scTLc

(A - 17)

The volume of gas is,

(A - 18)

In terms of number of moles, the volume of gas is,

∆n n =

(A - 20)

Zi R 'Twb

Considering the expansion of gas and the compressibility of fluid, we can write the following equations:

(

)

p tn V t n −1 + ∆ Vt n = n tn ZR 'Twh

(A - 21)

∆ V mn = c mV mn −1 ( p tn − p tn −1 )

(A - 22)

∆ V mn = ∆ V t n

(A - 23)

(p )

(A - 13)

At the top of cement, at n-th time step, the pressure is,

∆V gn = q cn ∆t

q ∆t

psc ∆Vgn Z i R 'Twb

Or, cumulative gas for all time steps is,

 V n −1  −  ptn −1 − t n −1  ptn − cmVm  

Twh ∑ p sc qck ∆t k =1

cmVmn −1Twb

(A - 19)

=0

The root of this equation is the casing pressure, step:

M AN U

[

[

k =1

k =1

n 2 t

0.003164 kTsc A m ( p f ) − m pc0 pscTLc

[

k

k sc c

n

(A - 12)

Where, Pb is an arbitrary base pressure. In practice, pseudo-pressure m(p) can be converted to real pressures using a tabulated correlation: p vs. m(p), and vice versa. Replacing the integral term in Eq. A-5 with m(p) and solving results in:

q cn =

n = ∑ ∆n = n t

∑p

Using Equations (A-19), (A-20), (A-21), and (A-22), we have:

p

q0 =

n

RI PT

[

0.006328kTsc A p f − pc0 pscTLc ( µZ p )

SC

q0 =

 n−1 Vt n−1  pt − + cmVmn−1  1 n ptn =  2 2 4Twh ∑ psc qck ∆t n −1   n−1 Vt  k =1 p −  + n −1  n −1   t c V c m m mVm Twb  

(A - 24)

pt at n-th time        

(A - 25)

Calculation Algorithm The flowchart is shown in Fig. 21. At the beginning of each time step, a new value of

p c is computed from Eq. (A-14)

using the value of pt at previous time step. The p c value remains constant during each time step thus resulting in the constant flow rate (Eq. A-16 or A-17) for this step (Steadystate flow assumption). The gas released at the cement top over one time step migrates though the mud and completely accumulates in the casing gas cap during the next time step. Thus, the gas migration time (controlled by gas rising velocity) is shorter than the time step. Using the real gas law, we can get the surface pressure pt at the end of this time step (Eq. A-25). Therefore, in a step-wise manner, we can compute casing pressure as a function of time. SI Metric Conversion Factors cp × 1.0* E-03 = Pa ⋅ s ft × 3.048* E-01 = m ft2 × 9.290 304* E-02 = m2 ft3 × 2.831 685 E-02 = m3 in. × 2.54* E+00 = cm lbf × 2.448 222 E+00 = N md × 9.869 223 E-04 = µm2

ACCEPTED MANUSCRIPT psi × 6.894 757 E+00 = kPa

AC C

EP

TE D

M AN U

SC

RI PT

*Conversion factor is exact.

ACCEPTED MANUSCRIPT

K K K in in ft ft ft cp

Psc cm ρm Z

psia psi-1 ppg

µg

Well 23 575 630 520 9.95 7 1821 8273 27 0.02

Well 24 552 584 520 9.95 7.625 3217 6433 0 0.015

14.7 4.0e-6 10 0.86

14.7 1.5e-6 16 0.92

SC

Twb T Twh D1 D2 Lc Initial Lf Initial Lt

RI PT

Table 1 - Well parameters

Table 2 - Matched results Well 23 0.36 6600

Well 24 1.5 6362

M AN U

md psi

TE D

k Pf

60% (11,498 CASINGS WITH SCP IN 8122 WELLS ARE INCLUDED IN THIS GRAPH) 50% 40%

AC C

% ALL CASINGS WITH SCP

EP

Fig. 1 - SCP Occurrence by Casing Type

30% 20% 10% 0%

PROD

INTER

SURF

COND

STRUCT

Fig. 2 - SCP in combined database by casing string (Ref. 1)

Fig. 3 - SCP Occurrence by Casing Size

RI PT

ACCEPTED MANUSCRIPT

< 500 psi

1.00

< 500 psi

SC

0.90

0.80

0.70

M AN U

0.60 < 1000 psi 0.50

< 1000 psi

0.40

0.30

0.20

0.10

0.00 Production Casing

Intermediate Casing

Surface Casing

Conductor Casing

TE D

Fig. 4 - Cumulative frequency of different SCP magnitude (Ref. 2)

1000

EP

900

700

AC C

Pressure (psi)

800

600

500

400 0

5

10

15

20

25

Time (days)

Fig. 5 - Typical pattern of SCP buildup – Well 24

30

35

ACCEPTED MANUSCRIPT

1200

1000

RI PT

Pressure (psi)

800

600

200

0 5

10

15

M AN U

0

SC

400

20

25

Time (days)

Fig. 6 - SCP buildup without stabilized pressure - Well 25

4000

3500

TE D

2500

2000

1500

EP

Pressure (psi)

3000

1000

AC C

500

0

0

500

1000

1500

2000

Time (days)

Fig. 7 – Abnormal pattern of SCP behavior – Well 10

2500

3000

ACCEPTED MANUSCRIPT

4500

Intermediate Casing

4000

Production Casing Surface Casing

3500

Shut-in Tubinghead Pressure Flowing Tubinghead Pressure

RI PT

Pressure (psi)

3000

2500

2000

1500

SC

1000

500

0 500

1000

1500

2000

2500

3000

M AN U

0

Time (days)

Fig. 8 - Pressure testing in all strings - Well 10 1600

1400

TE D

Pressure (psi)

1200

1000

EP

800

AC C

600

400

0.0

5.0

10.0

15.0

20.0

25.0

Time (Hours)

Fig. 9 – 10 3/4" casing pressure in Well 18

30.0

35.0

ACCEPTED MANUSCRIPT

Cement Column

Gas Zone

RI PT

Well Head

M AN U

SC

Fig. 10 - Gas migration though an annulus cemented to surface (Ref. 7)

Gas Bubble

Mud

TE D

Cement

Gas Formation

AC C

EP

Fig. 11 - Gas migration though an annulus with a column of mud above the cement top (Ref. 8)

Fig. 12 – Schematic and nomenclature of gas migration through a mud column above the cement top

ACCEPTED MANUSCRIPT

2500

RI PT

Pressure (psia)

2000

1500

1000

Gas Cap Volume = 0 cu ft Gas Cap Volume = 7 cu ft

Gas Cap Volume = 35 cu ft

SC

Field Data

0 0

10

20

30

M AN U

500

40

50

60

70

80

90

100

Time (month)

Fig. 13 – Large size of casing gas cap slows down the rate of SCP buildup

TE D

2500

1000

EP

1500

AC C

Pressure (psia)

2000

Cm = 2.00E-06 1/psia Cm = 4.00E-06 1/psia Cm = 6.00E-06 1/psia Field Data

500

0

0

10

20

30

40

50

60

70

80

Time (month)

Fig. 14 - High mud compressibility prolongs SCP buildup

90

100

ACCEPTED MANUSCRIPT

2500

RI PT

Pressure (psia)

2000

1500

1000

Cement Permeability = 1 md

Cement Permeability = 0.36 md Cement Permeability = 0.1 md

SC

Field Data

0 0

10

20

30

M AN U

500

40

50

60

70

80

90

100

Time (month)

Fig. 15 - Low cement sheath’s permeability results in long SCP buildup

3000

TE D

2500

1500

AC C

1000

EP

Pressure (psia)

2000

Formation Pressure = 7015 psia Formation Pressure = 6615 psia Formation Pressure = 6015 psia Field Data

500

0

0

10

20

30

40

50

60

70

80

90

Time (month)

Fig. 16 - High formation pressure increases the stabilized SCP

100

ACCEPTED MANUSCRIPT

Drive Casing 26”

738’

Conductor Casing Surface Casing 16” 65# H-40 STC

4310’

Intermediate Casing 10 3/4” 45.5# K-55 STC

SC

11196’

Production Casing 7” 29# 55# N-80 LTC

RI PT

1332’

M AN U

Fig. 17 - Schematic of Well 23 2500

TE D

1500

1000

500

0

AC C

0

EP

Pressure (psi)

2000

5

10

Field data Theory data

15

20

25

30

Time (month)

Fig. 18 – Matching and extrapolating SCP buildup in Well 23 with data in Table 1

ACCEPTED MANUSCRIPT

Drive Pipe 26” Conductor Casing 20” 94# H-40

582’ 1061’

Surface Casing 16” 75# K-55

6433’

Intermediate Casing 10 3/4” 45.5# L-80

SC

Production Casing 7 5/8” 33# N-80

RI PT

4776’

9804’

M AN U

Fig. 19 - Schematic of Well 24

1200

1000

TE D

600

400

200

0

5

AC C

0

EP

Pressure (psi)

800

10

15

Field Data Theoretical Data

20

25

30

Time (days)

Fig. 20 - Matching SCP buildup in Well 24 with data in Table 1

35

ACCEPTED MANUSCRIPT

Start Time n = 0

calculate flow rate qc

n=n+1

t
M AN U

Y

SC

calculate SCP - pt

If

N

Stop

EP

TE D

Fig. 21 - Flow chart of mathematical model

AC C

RI PT

calculate pressure pc

ACCEPTED MANUSCRIPT

RI PT SC

•

M AN U

•

TE D

•

EP

•

The first mathematical model for quantitative analysis of pressure buildup in Sustained Casing Pressure wells is corrected and presented. The analysis method validated for one oilfield should work anywhere with sustained casing pressure problem. Using the mathematical model, impact of parameters in cement, mud and gas invasion zone on SCP buildup pattern were theoretically analyzed. Early stage of SCP buildup is controlled by cement sheath’s permeability; while stabilized pressure is determined by formation pressure. Recent applications of the model proves that the model could be a useful tool for analyzing CO2 leakage along wellbores designated for CO2 sequestration

AC C

•