Laboratory experiments of well testing for fracture

Petroleum 3 (2017) 301e308

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Laboratory experiments of well testing for fracture-cave carbonate gas reservoirs Yu Xiong a, Wanli Xiong a, *, Mingjin Cai b, Chengxi Hou a, Chong Wang a a b

The State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation Engineering, Southwest Petroleum University, Chengdu 610500, China Exploration and Development Research Institute, PetroChina Tarim Oilfield, Korla 841008, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 May 2016 Accepted 18 September 2016

It is well known that the flowing of oil and gas in fracture and cave does not obey Darcy law, which makes it unable to interpret parameters correctly when doing well testing for those kinds of formation for having no flowing test used to correct corresponding flowing equations. Based on similarity criterion, a physical experimental method for gas flowing from cave to wellbore through fracture has been built up. The characteristics of fluid flowing in fracture and cave can be seen clearly according to logelog curves with the measured data, which was obtained from the experimental model test and dealt with Savitzky-Golay filtering method. In addition, a new mathematical model reflecting those transient-flow behaviors as well as its solution has been presented in this paper. Logelog curves obtained from our new model could reflect the characteristics of flowing in fracture and cave. The results showed that test experiments can reflect the influence of large-scaled cave and fracture on the flowing characteristics and the new model can be applied to explain parameters of fracture and cave for similar cases. Copyright © 2016, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Fractured-cave reservoir Well testing Mathematical model Similarity criterion

1. Introduction It was complex for well testing of carbonate reservoirs, which had generated great scientific interest and significant challenges [1,2]. Vugular porosity can be subdivided into two kinds, namely, connected and disconnected types [3]. The effect of vuggs on permeability was related to their connectivity [4]. It had been observed in the literature that vugular zones could strongly influence the production performance [5,6]. Well testing interpretation model could not explain those kinds of fractured-cave reservoirs effectively [7,8]. For the fractured-vuggy reservoir, the double [9,10] or triple-continum [11,12] model was widely used in well testing interpretation. But the syntagmatic relationship for large scale of fracture and cave or the fluid exchange

* Corresponding author. E-mail address: [email protected] (W. Xiong). Peer review under responsibility of Southwest Petroleum University.

Production and Hosting by Elsevier on behalf of KeAi

mechanism between them was still not been observed. Most of the existing theories were based on the linearly flow theory and there was no more sophisticated theory to explain the sophisticated flow in fractured-vuggy reservoir. It is in need of experiment to study the flow characteristics of fractured-vuggy reservoir to guide the production of field. Lots of studies had been done about the flow in fracturedvuggy reservoir. In 2008, Peng et al. [13] regarded the cavity as a equipotential body neglecting the flow from the reservoir to the large vug and had created a model of drilling directed into the cavity. In 2009, Cheng et al. [14] proposed a model of single well drilling in isolated cavity, in which the pressure of outer boundary was steady and the cave was regard as the expand of the wellbore. In addition, the flow mechanism between the matrix and the cave was also studied in their paper. In 2010, Zhang et al. [15] established a model of well drilled in big vug and had gained the analytic solution in Laplace space. In addition, the main factors affecting the bottom hole dynamic pressure were also analyzed in their paper. In 2011, Xiong et al. [16] regarded the cave as the expansion of wellbore and had analyzed the flow mechanism between bedrock and isolated cave. In 2014, Lin et al. [17] put forward a suitable model for fractured-vuggy reservoir and the well testing model was classified into four types, namely,

http://dx.doi.org/10.1016/j.petlm.2016.09.004 2405-6561/Copyright © 2016, Southwest Petroleum University. Production and hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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a single large-scale vug, a large-scale vug with a connected largescale fracture, two connected large-scale vugs in series through a large-scale fracture and two connected large-scale vugs in parallel with a large-scale fracture. In 2015, Omotayo et al. [18] presented an analytic technique for interpreting pressure falloff tests of non-Newtonian Power-law fluids in wells that were located near boundaries in dual-porosity reservoirs. In 2016, Yao et al. [19] used a multiscale mixed finite element method for fluid flow in fractured-vuggy media using the discrete fractureevug model. Gao et al. [20] developed an efficient well testing analysis model, which was suitable for naturally fractured reservoirs with a well drilled into a large-scale cavity. However, all of those methods were just of theoretical studies and there was no experimental study about them. Based on the previous researches, this paper has studied the influence of fracture and vug on the well testing curves with laboratory experimentation. Furthermore, the gas well of JY401 in Tarim Oilfield was analyzed with the proposed model in this paper. 2. Typical geology model in tarim oil field Lost circulation, one of the most severe problems in drilling fractured formations, caused the most serious formation damage and remained a crucial issue in drilling engineering. Lost circulation problems associated with drilling in fractured and caved zones were linked with the nature of fractures and caves. Dominated by the large-scale Tazhong No. 1 fault zone, a fault system extends across central uplift of the Tarim basin. Many wells are drilled on the fractures next to a big cavity in Tarim Oil field. As can be seen in Fig. 1, JY401 is a typical fractured-vuggy well and the logelog curve is shown in Fig. 2. The well depth is 7068 m and the finished lawyer is the limestone in Ordovician system. The build-up well testing began at November 3 in 2014 and was finished at November 9 in 2014 lasting for 182.4 h. History plot and semi-log plot were shown in Figs. 3 and 4, respectively. Classical radial uniform compound interpreting model in Saphir does not match actual situation of gas reservoir making the results colliding to actual geology conditions. It could not explain the volume of the vug or the fracture length. A way to

solve this kind of reservoir parameter by experiment has been proposed in this paper. 3. Physical model and the experimental procedure 3.1. Physical model design A large number of geological research results showed that in fractured-vuggy reservoir, karst cave is the main space for storage while the fractures are mainly used as the flow channel. There are many kinds of reservoir space in fractured-vuggy reservoir. The effective reservoir space for storage includes eroded pores, large caves and fractures. The matrix only has little capability of permeability and the reservoir space has obvious multi-scale properties. The characteristic size of the eroded pores is usually measured by millimeter while the characteristic scale of large caves is usually measured by meter. When it comes to the caves, the radius of some caves is as long as hundreds of meters. The forming mechanism of the fractures is very complicated and the width of fracture ranges from several microns to a few centimeters while the length can even reach thousands of meters. In view of the phenomena of drill pipe discharge, slurry leakage during drilling operation and the beads-shaped reflection in geological exploration, this paper simulated the situation that the well is located at the large scale of fracture connected to the big carve with experimental measurement. The experiment device was shown in Fig. 5. The highpressure vessel is used to simulate the cave, the thin tube filled with sand simulates the fracture and the empty pipeline is used to simulate the wellbore. The device is designed to test the influence of characteristics of fracture and cave on the well testing curves. 3.2. The establishment of similarity criterion There are some certain defects in the existing mathematical equations describing the flow mechanism in the carbonate reservoir. Then, it is in need of similarity analysis to ensure the correctness and applicability of the results in the experiment.

Fig. 1. Seismic profile plot of JY401.

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movement similar and dynamic similar. The proposed cluster of similarity criterions reflects not only the conventional reservoir similarity analysis but also reservoir characteristic of large scaled vuggy and fracture in carbonate reservoir. Based on the needs of experimental research, we have regrouped these similarity criterions to get main similarity criterions (shown in Table 1), which can reflect the characteristics of fractured-cave reservoir. Therefore, the ratio of cave's volume to wellbore's volume is used as a condition of geometric similarity. By controlling the flow, the movement similarity is obtained and by controlling the pressure, the dynamic similarity could also be obtained. Each of p is obtained:

Kf Lf m p gL Q ; p ¼ ; p ¼ ; p ¼ ; p ¼ ; p6 ¼ 2 ; rvL 2 rv2 3 L2 4 v2 5 L vL df Vv rw tv p7 ¼ 3 ; p8 ¼ Crv2 ; p9 ¼ ; p10 ¼ ; p11 ¼ ; L L L L Wf mg Qg vg H p12 ¼ ; p13 ¼ ; p14 ¼ ; p15 ¼ ; p16 ¼ ; L L Qo vo mo rg Cg p17 ¼ ; p18 ¼ Co ro p1 ¼

Fig. 2. Logelog plot of JY401.

3.3. The experiment parameters

Fig. 3. History plot of JY401 (Pressure, liquid rate).

First, considering the flow mechanism of condensed gas in the carbonate reservoir, the selected characteristics of physical parameters are as follows: the cave characteristic length L, the fracture length Lf, fracture width df, fracture opening Wf, fracture permeability Kf, volumetric flow of each phase Q(Qg, Qf), the density of each phaser(rg, ro), the velocity of each phase v(vg, vo), the viscosity of each phasem(mg, mo), the compression coefficient of each phaseC(Cg, Co), the compressibility of rock Cp, pressure p, the volume of cavity Vv, acceleration of gravity g, the radius of wellbore rw, and the height of wellbore H. According to the p theory, r, v, L are selected as basic physical quantities. According to the dimensional analysis, cluster of similarity criterions have been deduced, which includes geometric similar,

According to the similarity criterion shown in Table 1 and the actual parameters of the reservoir, the physical model parameters have been designed, which were shown in Table 2. According to the similarity criterion, field parameters have been analyzed to deduce model parameters in the experiment. Regardless of the subordinate affecting factors, this paper has taken the primary similarity criterion into consideration to design the experimental parameters. The experimental model and the field prototype conformed to the principle of similarity. 3.4. The experiment condition and experimental procedure The experiment system is mainly composed of four parts, namely, the constant pressure pump, the model system, the measuring system and collecting system (shown in Fig. 5). Model system consists of cavity and fracture. The cavity, fracture and wellbore are made up of vessel, pipeline filled with sand and empty thin pipeline, respectively. The wellbore is connected with the pipeline filled with sand to simulate the well located on the fracture next to the cavity. By changing the length of the thin tube filled with sand and the volume of vessel, it is able to simulate different fracture length and different sizes of the largescaled cave, respectively. Metering system mainly includes high precision pressure sensor and gas flow-meter, which can measure the changes of pressure and flow rate in the process of well testing. 4. The characteristic of well testing curve of the fracturedvuggy reservoir 4.1. The method of date processing

Fig. 4. Semi-log plot of JY401.

Practice has proved that the pressure derivative is more sensitive than pressure itself and the small change in pressure curve is not so obvious, which is often overlooked in the pressure analysis. Pressure derivative can magnify the small change of pressure leading to obvious reflection on pressure derivative curve, which can tell the small discriminations and explain

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Fig. 5. The experimental flow graph of well testing.

Table 1 Main similarity criterions for physical modeling. Number

Similarity criterion

Physical significance

Source

Conditions of satisfaction

1

2H Vv =rw

p7 =p210 p13

2

=L3

Ratio of the volume of cavity to wellbore The production rate

Geometric similarity Kinematic similitude

Qt

3

Dp/rgL

4

m/rvL

Ratio of pressure to gravity Reynolds number

p6p11

parameter estimation from the graphics [22]. The algorithm of S - G filter is as follows: A set of 2m þ 1 consecutive values are used to determine the best mean square fiting through these value of a polynomial of degree n (n less than 2m þ 1). This polynomial is of the form:

fi ¼

bnk ik ¼ bn0 þ bn1 þ bn2 þ /bnn in

The sum of the squarer E is

Dynamic similarity

m X

E¼

½fi  xðiÞ2 ¼

i¼m

them. Therefore, it puts forward higher requirements on the accuracy of the pressure data for the small error on the pressure curve will cause obvious deformation on the pressure derivative curve. However, it is inevitable to eliminate the noise in the pressure measurement process. It is necessary to deal with the errors on the pressure date tested. SavitzkyeGolay smooth filtering method (S-G filter) is an effective way to deal with the noise [21]. It is capable to satisfy the demands of smoothness and approximation. Here is a brief introduction to the S e G filtering. The S e G filter is a special kind of low-pass filter and the obvious use is to smooth the dates with noise, which is used to describe the mistakes that can hardly be observed in extracting information. Data smoothing can eliminate larger error of data points and it can make preliminary and rough

(1)

k¼0

p2/p4 p1

n X

m X

"

i¼m

n X

#2 bnk ik  xðiÞ

(2)

k¼0

The least squarer criterion requires that the derivative of each factor is 0, i.e.

vE ¼ 0; r ¼ 0; 1; 2; /:n vbnr

(3)

and with respect to bnr, we have

vE v ¼ vbnr vbnr

(

m X

¼2

m X

n X

i¼m

k¼0

n X

k

"

i¼m

"

#2 bnk ik  xðiÞ #)

bnk i  xðiÞ

ir ¼ 0; r ¼ 0; 1; 2; /:n

(4)

k¼0

or Table 2 Physical quantities describing flow and well testing in fractured-vuggy media. Name of parameter

The actual value of reservoir

The value of physical model

Fracture width Fracture length Fracture permeability The volume of the cavity Viscosity of condensed gas Height of the well bore Diameter of the wellbore Reservoir pressure Reservoir temperature Production of condensed gas

0.1e1.0 mm 5e500 m 0.25e155D 0.15e15  105 m3 0.03 mPa s 6000 m 0.2 m 71.94 Mpa 131.79  C 1e5  104 m3/d

0.02e0.5 mm 0.762e12.52 m 0.04 mDe24 mD 500e20000 ml 0.03 mPa s 2m 0.4e0.6 cm 34.8e43.55 MPa 131.79  C 164-823 mL/min

n X

bnk

k¼0

m X

ikþr ¼

i¼m

m X

xðiÞir

(5)

r¼m

where r is the index representing the equation number which runs from 0 to n (there are n þ 1 equations). and

Fr ¼ Skþr

9 > > > n = X r¼m bnk Skþr 0Fr ¼ m X > > k¼0 > ¼ xðiÞir ; m X

xðiÞir

r¼m

(6)

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In practical application, it usually does not need to acquire the coefficient bn0, bn1 … bnn out entirely, but we can get them by equation (8).

fi ji¼0 ¼ bn0 ¼ an0  dfi  ¼ bn1 ¼ an1 di i¼0  d2 fi  ¼ 2bn2 ¼ an2 di2 i¼0

(8)

By removing the noise of pressure data tested in the lab with the S e G filter, it can effectively remove influence of the noise as can be seen in Fig. 6. 4.2. The characteristic of well testing curve

Fig. 6. Comparison of the dates before and after filing.

If the fitting count (m), the degree of the polynomial(n), and data for fitting x(-m) …, x(0)…, x(m) are confirmed, we can get Fr, P substitute Skþr into Fr ¼ nk¼0 bnk Skþr , the coefficient bn0, bn1 … bnn are obtained. So we can get the polynomial of fi. Note that Srþk ¼ 0 for odd values of r þ k.

Skþr ¼

m X

ikþr ¼ 0

(7)

i¼m

Sine Skþr exists for even values of r þ k only. Equations can be separated into two sets, one for even values and one for odd values. Only when both of n and s are even values or odd values can we get. bnk ¼ bðnþ1Þs

In the condition of the well drilled on the fracture next to the cavity, we have changed the length of the fracture and the size of the large-scaled cave to study the influence the size of largescaled cave (shown in Fig. 7) and the crack length (shown in Fig. 8) on the well testing curve, respectively. As can be seen in Fig. 7, if the other conditions are the same, the bigger the volume of the cavity is, the earlier the characteristic of large-scaled cave reflects on the differential of pressure curve and the lower of the valley for the big cave, which can provide more energy for the wellbore than the small one, so the higher of the dimensionless pressure differential curve restore. The drop in the end on the dimensionless pressure differential curve reflects the boundary. As can be seen in Fig. 8, if the other conditions are the same, the longer the fracture is, the later the characteristic of largescaled cave reflects on the differential of pressure curve and the lower of the valley for the long fracture consuming more energy than the short one and the pressure-wave needs more

Fig. 7. (a) Pressure history. (b) Flow rate history. (c) Sensitivity of cave volume displayed on pressure buildup curve.

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Fig. 8. (a) Pressure history. (b) Flow rate history. (c) Sensitivity of fracture length displayed on pressure buildup curve.

time to spread in the long fracture. The drop in the end on the dimensionless pressure differential curve reflects the boundary.

The cavity is connected with the well by the fracture. At x ¼ x1, the pressure of the cavity is the same as it is in the fracture:

4.3. Mathematical models for vuggy media at multiple scales

  pf 

With the center of the cave as the origin, the coordinate system is established, which is shown Fig. 9. The general form of the one-dimensional seepage equation is shown in equation (9): 2 3:6kf vpf vpf ðx  x  x2 Þ ¼ 2 4f mcf vx vt 1

where: kf ¼ fracture permeability, D; Cf ¼ fracture compressibility at initial conditions, psi; m ¼ fluid physical viscosity, cp; pf ¼ fracture pressure, psi; 4f ¼ fracture porosity; t ¼ time, hour;

(9)

x¼x1

¼ pv

(10)

where: pv ¼ cavity pressure, psi. In the period of dt, the pressure in the cavity drops dpv, the output by gas elastic energy of the cavity is 43 pR3 4v Cvt dpv and v the quantity of flow is 43 pR3 4v Cvt dp . And the output by gas dt elastic energy of the cavity flows into the fracture at the place of x ¼ x1. The flow meets the Darcy's Low at x ¼ x1:

   vpf  m 4 dpv   pR3 4v Cv ¼  86:4kf L2 W 3 vx x¼x1 dt

(11)

where: Cv ¼ cavity compressibility at initial conditions, psi1; R ¼ cavity radial distance, ft; L2 ¼ facture width, ft; W ¼ facture opening, ft; 4v ¼ cavity porosity At x ¼ x2, the pressure at wellbore is the same as the pressure of the fracture at x ¼ x2:

  pf 

x¼x2

¼ pw

(12)

where: pw ¼ pressure at well bottom. At x ¼ x2, the flow meets the Darcy low and the gas flow rate is the same as the output of the well:

 vpf  m  qBg ¼ 86:4 kf L2 W vx x¼x2 Fig. 9. Schematic plot of the model.

(13)

where: q ¼ gas flow rate, STB/D; Bg ¼ gas formation volume factor, RB/STB.

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In addition, the pressure of the cavity, fracture and the downhole is the same at the initial time:

  pfD 

pf jt¼0 ¼ pv jt¼0 ¼ pw jt¼0 ¼ pi

  pfD 

(14)

The dimensionless definition are as follows:

x¼x1D

x¼x2D

307

¼ pvD

(31)

¼ pwD

(32)

The solution in Laplace space is shown in equation (33):

x xD ¼ rw

(15) pwD ¼

a3 a1 þ a3 a2 a4 a4 a2  a1

(33)

pf ;v;wD

 86:4 kf rw  pi  pf ;v;w ¼ qmBg

(16)

uf ;v ¼

4f ;v Cf ;v 4f Cf þ 4v Cv

(17)

a1 ¼

uv R3D pffiffiffiffiffiffiffiffi suf  s L2D WD

(34)

(18)

a2 ¼

uv R3D pffiffiffiffiffiffiffiffi suf þ s L2D WD

(35)

a3 ¼

1 pffiffiffiffiffiffiffiffi s L2D WD suf

(36)

3:6 kf t  tD ¼  2 m 4f Cf þ 4v Cv rw  p 1 R 3

RD ¼

18 rw

(19)

L2D ¼

L2 rw

(20)

WD ¼

W rw

(21)

Then, the dimensionless model is obtained as the following:

v2 pfD vx2D

¼ uf

vpfD ðx  xD  x2D Þ vtD 1D

(22)

 vpfD  uv R3D dpvD  ¼ vxD x¼x1D L2D WD dtD

(23)

 vpfD  1  ¼ L2D WD vxD x¼x2D

(24)

  pfD    pfD 

x¼x1D

x¼x2D

¼ pvD

(25)

¼ pwD

(26)

pfD jt¼0 ¼ pvD jt¼0 ¼ pwD jt¼0 ¼ 0

where:

a4 ¼ e2

pffiffiffiffiffiffi pffiffiffiffiffiffi suf x1D þ2 suf x2D

(37)

4.4. Typical curve analysis Typical curve of the model above is shown in Fig. 10. The first stage reflects the effect of wellbore storage and both the pressure and pressure derivative's slope is 1. The second stage reflects the pressure propagation in the fracture, which is a stage of fracture linear flow. Both the pressure and pressure derivative's slope is 0.5 at the second stage. The third stage reflects the pressure propagation along the fracture to the cavity, in which the cavity begins to supply gas for the fracture. An important feature observed on this graph is that a concave

(27)

The dimensionless model after Laplace transformation is shown as follows:

d2 pfD vx2D  dpfD   dxD   dpfD   dxD 

¼ suf pfD ðx1D  xD  x2D Þ

x¼x1D

uv R3D ¼s p L2D WD vD

¼ x¼x2D

1 sL2D WD

Fig. 10. Typical curve of the model.

(28)

(29)

(30)

Table 3 Exact values of the parameters solved by the model. Cavity radius (ft)

Fracture length (ft)

Fracture length (ft)

191.9291339

30.83989501

0.754593176

Cavity storativity ratio

Fracture storativity ratio

Fracture permability (mD)

0.98

0.02

186

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appeared in the pressure derivative curve with a step showing in the pressure curve. The last stage reflects the pseudosteady-state flow of the system and the pressure spreads to the boundary soon. Both the pressure and pressure derivative's slope is 1 at the last stage. 5. Application to analysis of field date JY401 is a typical fractured-cave well in north Tarim Oilfield. The well depth is 7068 m and the finished lawyer is the limestone in Ordovician. The build-up well testing began at November 3 in 2014 and finished at November 9 in 2014 lasting for 182.4 h. The logelog well test curve of JY401 is shown in Fig. 2. The geological information showed that the well of JY401 has an obvious feature of cavity in seismic profile and fracture and cavity caved map. The well testing interpretation results (shown in Fig. 2) matches the experimental test shown in Figs. 7 and 8. The example of JY401 proved that the experiment designed can reflect the features of complex fractured-cave reservoir. According to the pressure and production data, the exact values of the reservoir are obtained in Table 3. The material balance method is used to calculate the reserve volume with result of 3.652  108 ft3. The deviation of the cavity radius is 6.8% and the fracture length is 30.8 ft. 6. Conclusions In this study, the effects of different length of the fracture and the size of the large-scaled cave were evaluated on the behavior of well testing curve. The following remarks can be concluded: (1) Based on the example of the condensed gas in Tarim Oilfield, this paper has studied the well testing characteristics of fractured-cave reservoir. According to the similarity criterion, the experiment parameters are designed to reflect the reservoir condition. (2) SavitzkyeGolay filter could remove the noise from date tested in the lab effectively. It provides to be a computational method by using the polynomials convolution algorithm to compute the smoothing coefficients. (3) The experimental results showed that test experiments can reflect the influence of large-scaled cave and fracture characteristics on well testing curve. The mathematical models can solve the parameters of the reservoir effectively. (4) The experiment result matches the actual geological conditions, which proves the correctness of established experiment model and mathematical model.

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