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. 2022 Aug 18;7(34):30313–30320. doi: 10.1021/acsomega.2c03573

Pressure Transient Analysis for Hydraulically Fractured Wells with Changing Conductivity in Stratified Reservoirs: Case Study in Xinjiang Oilfield

Jiaqi Li †, Chengwei Zhang ‡, Yun Xia †, Feiwen Wang †, Dengke Shi ‡, Shiqing Cheng ‡,*
PMCID: PMC9434779  PMID: 36061705

Abstract

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The oil development has been oriented toward deep-layer reservoirs and the commingling production and the separate-layer fracturing are important development methods. Currently, limited attention is given to the pressure transient analysis (PTA) of the fractured wells located in a stratified reservoir. Moreover, the proppant is very difficult to move inside the hydraulic fracture in the deep-layer reservoir, leading to the uneven fracture conductivity along the hydraulic fracture and increasing the complexity of PTA. To fill this gap, this work presented a fully analytical well test model for hydraulically fractured wells with changing fracture conductivity in stratified reservoirs, which is convenient to be used for interpreting the recorded pressure data from the oilfield due to its analytical nature. The establishment of this model is based on the trilinear flow model, Duhamel theorem, and pressure superposition principle. A systematic verification is conducted to ensure the validity of the proposed model. Furthermore, we offer a sensitivity analysis to investigate the effect of crucial parameters on pressure and pressure derivative, including the fracture extension, fracture conductivity, transmissibility factor, and storativity factor. Finally, a field case of a four-layer fractured well from Xinjiang Oilfield in Junggar Basin is interpreted to demonstrate the practicability of the presented model.

1. Introduction

China has directed the petroleum exploration to deep-layer unconventional oil and gas reservoirs to meet the increasing demand for energy.1−9 In recent years, several largely deep-layer oil and gas reservoirs have been discovered in Western China, among which Xinjiang Oilfield Company achieved exploration breakthrough in the Nanyuan, Mahu, and Shawan districts of Junggar Basin and commercially developed deep-layer tight reservoirs.10−15

The deep-layer tight reservoirs contain multiple oil layers. Due to the huge investment costs needed for deep-layer drilling, the oilfield adopts commingling production for stratified reservoirs to reduce the number of drilling wells. Moreover, considering that the permeability of deep-layer reservoirs is extremely low, hydraulic fracturing stimulation is applied to each layer of stratified reservoirs to increase the production rate of oil well (separate-layer fracturing). Several papers have presented the pressure transient analysis (PTA) models of un-fractured multilayer systems where different situations are considered.16−19 However, limited attention was given to the fractured wells located in a stratified reservoir. Bennett et al.20 presented the analytical solutions for the response at a well intercepting a layered reservoir without interlayer communication. Osman21 developed a model of a well located in a multilayered infinite acting reservoir and crossed by a finite conductivity vertical fracture. Gao et al.22 presented the derivative responses of commingled systems with mixed inner and outer boundary conditions (BCs), including un-fractured and infinite-conductivity fractured wells. Osman and Abou-Kassem23 evaluated the effect of BCs on pressure behavior of finite-conductivity fractures in bounded stratified reservoirs. Manrique and Poe24 presented a unique methodology designed for the evaluation and optimization of multi-fractured wells in layered reservoirs using rate-transient analyses and historical production data. Ali et al.25 applied an analytical elliptical flow solution to analyze production data from hydraulically fractured vertical tight gas wells producing from multilayer reservoirs.

However, the issue of PTA for hydraulically fractured wells in deep-layer stratified reservoirs is more complicated. Proppant movement inside hydraulic fracture is always restricted during the process of hydraulic fracturing, causing its accumulation near the wellbore segment and poorly propped at the tip of the hydraulic fracture.26 This problem is even worse in deep-layer reservoirs. Once the hydraulic fracturing completed and the oil wells are put into production, the width of poorly propped fracture segments would be rapidly decreased, causing that the conductivity at the fracture tip decreases sharply. Soliman27 analyzed the effect of changing conductivity on production based on the simplified bilinear flow assumptions for single-layer reservoir. Lolon et al.28 clearly pointed out that the effect of poorly propped segments should be considerable for making better interpretations. Gonzalez-Chavez and Cinco-Ley29 investigated the behavior of a well with variable finite-conductivity and skin fracture for a vertical fracture in an infinite slab reservoir. Wanjing and Changfu30 presented a semi-analytical model for a well with variable finite-conductivity fracture and non-Darcy effect in an infinite reservoir.

To the best of our knowledge, the well test model for hydraulically fractured wells with changing conductivity in stratified reservoirs has rarely been reported. For better matching the actual situations encountering in Xinjiang Oilfield and solving the Oilfield’s well test problems, this work considers investigating the pressure behavior of a well located in stratified reservoirs, which is intercepted by the finite-conductivity vertical fracture with changing conductivity. From the points of calculation efficiency and interpretation accuracy of field data, the mathematical model is developed based on the classical trilinear flow model presented by Lee and Brockenbrough.31 Following that, the model is validated to ensure its accuracy for general practice and the sensitivity analysis is conducted on crucial parameters, including the fracture extension, fracture conductivity, transmissibility factor, and storativity factor. Finally, a field case of hydraulically fractured well in the stratified reservoir from Xinjiang Oilfield is interpreted to demonstrate the practicability of the presented model.

2. Methodology

2.1. Physical Model

As shown in Figure 1, the stratified reservoir contains m oil layers, which are intercepted by hydraulic fractures. Every layer is homogeneous and isotropic, containing the single-phase, slightly compressible fluid with viscosity μ. The initial formation pressure is pi and the porosity, permeability, thickness, total compressibility, wellbore storage, skin factor, and production contribution of the layer n are ϕn, kn, hn, ct, Cn, Sn, and qn, respectively. Considering the real situations of Xinjiang Oilfield, every layer is separated by interlayers with no communication. Other basic assumptions for facilitating the model establishment are also given as follows:

  • (a)

    The permeability of deep-layer reservoirs is extremely low. It is difficult to observe the pseudoradial flow regime during the real test duration. To achieve a high calculation efficiency, this work considers applying the trilinear flow model to present the flow behavior of finite-conductivity fractured wells.

  • (b)

    The hydraulic fracture is symmetrical along wellbore with permeability kfn, height hfn, half-length xfn, and width wfn. There is no flow at the fracture tip.

  • (c)

    The fracture conductivity along the hydraulic fracture is usually uneven due to the difficulty of proppant movement in deep-layer reservoirs. For the practical purpose, this work considers using two dimensionless fracture conductivities (i.e., FcD1 and FcD2) to characterize the propped and poorly propped fracture segments, as shown in Figure 1b.

  • (d)

    The fluid obeys Darcy’s flow and the gravity and capillary force are neglected in this work.

Figure 1.

Figure 1

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Schematic of stratified fractured well with changing fracture conductivity. (a) Stratified fractured well characterization and (b) trilinear flow with changing fracture conductivity.

2.2. Mathematical Model

Following Lee and Brockenbrough’s work,31 only a quadrant of the flow domain in layer n will be considered due to the symmetry. We further divided the flow domain into regions I, II, and III. Region I represents the hydraulic fracture. The flow is dominated by the x-direction flow. It is worth to mention that region I contains two linear-flow regions due to the changing fracture conductivity, as shown in Figure 1b. If the fracture is regarded as a line sink, the flow in region II will be approximated by the y-direction flow and supply to region I. Similarly, the flow in region III is approximated by the x-direction flow for a short time period and supply to region II. This visualization of the flow behavior is reasonable for fluid flow before the pseudoradial flow period. Note that in our model, the fracture storage effect and fracture skin effect are not considered.

The diffusivity equation for linear flow is now formulated in terms of dimensionless variable in region I of layer n. Note that we neglect the fluid compressibility inside the fracture because the hydraulic fracture volume is very small.32,33

2.2. 1

The initial conditions (ICs) are

2.2. 2

The inner and outer BCs are

2.2. 3
2.2. 4

The diffusivity equation for linear flow is formulated in terms of dimensionless variable in region II of layer n (0 < xD < αn, 0 < yD < ∞).

2.2. 5

The ICs are

2.2. 6

The inner and outer BCs are

2.2. 7
2.2. 8

The diffusivity equation for linear flow is formulated in terms of dimensionless variable in region III of layer n (αn < xD < ∞, 0 < yD < ∞).

2.2. 9

with ICs

2.2. 10

and BCs

2.2. 11
2.2. 12

The dimensionless definitions in equation system, eq 1 through eq 12, are listed in Table 1.

Table 1. Dimensionless Variable and Definitiona.

dimensionless variable definition dimensionless variable definition
dimensionless pressure
graphic file with name ao2c03573_m024.jpg
 dimensionless time
graphic file with name ao2c03573_m025.jpg
dimensionless production contribution
graphic file with name ao2c03573_m026.jpg
 dimensionless fracture length
graphic file with name ao2c03573_m027.jpg
dimensionless transmissibility factor
graphic file with name ao2c03573_m028.jpg
 dimensionless storativity factor
graphic file with name ao2c03573_m029.jpg
dimensionless wellbore storage
graphic file with name ao2c03573_m030.jpg
 dimensionless fracture conductivity
graphic file with name ao2c03573_m031.jpg
dimensionless coordinates
graphic file with name ao2c03573_m032.jpg
 dimensionless propped fracture length
graphic file with name ao2c03573_m033.jpg
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a

Note that the definitions Inline graphic are used in Table 1.

We solve the partial-difference equation system from the start of region III. The pressure solution is obtained with the auxiliary ICs and BCs, given by

2.2. 13

Making the derivation of eq 13 and substituting it into eq 5, the pressure solution is obtained with the auxiliary ICs and BCs in region II, given by

2.2. 14

where A is defined by

2.2. 15

Making the derivation of eq 14 and substituting it into eq 1, we can obtain the following equations with the auxiliary ICs and BCs in region I, given by

2.2. 16

where B1 and B2 are defined by

2.2. 17

Equation 16 can be written as a matrix form, given by

2.2. 18

Solving eq 18, we can obtain the values of C1, C2, D1, and D2. Eventually, the pressure function p1Dn1 in the Laplace domain can be written as

2.2. 19

If the wellbore storage and skin effect are considered (here, we consider that every layer has different wellbore storage coefficients and skin factors), we obtain the following equation based on the Duhamel theorem and pressure superposition principle, given by34

2.2. 20

Finally, the bottom-hole pressure is obtained with the flowrate condition Inline graphic in the Laplace domain, given by

2.2. 21

and Stehfest numerical inversion method is used to obtain pwfD in the time domain.35 It deserves to clarify that total wellbore storage should be Inline graphic in this work.

3. Results and Discussion

3.1. Model Validation

When m = 1 (i.e., λ = 1, ω = 1, α = 1) and FcDn1 = FcDn2, the proposed model can be simplified as a trilinear flow model presented by Lee and Brockenbrough.31 We compare the pressure and pressure derivative results calculated by two models under different dimensionless fracture conductivities (FcD = 5, 30, 100, 500). The wellbore storage and skin effect are not considered in this subsection. As shown in Figure 2, the results computed by the proposed model match well with that of the trilinear flow model, indicating the correctness of the proposed model.

Figure 2.

Figure 2

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Comparison of the proposed model and Lee and Brockenbrough’s model under different dimensionless fracture conductivity. (a) Pressure results, (b) pressure derivative results.

3.2. Sensitivity Analysis

Taking two-layer reservoir as illustration, this subsection conducts sensitivity analysis on several crucial parameters that are focused by oil engineers, including the transmissibility factor, storativity factor, fracture extension (rn = xfn/xf1), and fracture conductivity. The wellbore storage and skin effect are not considered in the sensitivity analysis.

3.2.1. Transmissibility Factor

Three cases are designed to investigate the effect of the transmissibility factor on pressure and pressure derivative. The results are presented in Figure 3. Although the transmissibility factor has an evident effect on production contribution of single layer as has been recognized by many literature,18,23 it did not show an evident effect on pressure behavior. Nevertheless, with the difference of λ1 and λ2 increasing, one can find that the pressure and pressure derivative increase slightly. We can further conclude from the results of Figure 3 that the stronger heterogeneity due to a change in transmissibility leads to more increase of the pressure and pressure derivative.

Figure 3.

Figure 3

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Effect of the transmissibility factor on pressure and derivative.

3.2.2. Storativity Factor

The effect of the storativity factor on pressure and pressure derivative is also investigated. Three cases are designed in this subsection. The results are presented in Figure 4. With the difference of ω1 and ω2 increasing, one can find that the pressure and pressure derivative increase slightly. We can further conclude from the results of Figure 4 that the stronger heterogeneity due to a change in storativity leads to more increase of the pressure and pressure derivative.

Figure 4.

Figure 4

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Effect of the storativity factor on pressure and derivative.

3.2.3. Fracture Extension

We designed three different cases (r2 = 1, r2 = 10, r2 = 30) to investigate the effect of the fracture extension on pressure and derivative, as shown in Figure 5. The fracture extension affects the lasting time of bilinear and linear flow. The bilinear and linear flow would last longer with a larger fracture extension.

Figure 5.

Figure 5

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Effect of the fracture extension on pressure and derivative.

3.2.4. Poorly Propped Fracture Conductivity

We designed three different cases (FcD2 = 160, FcD2 = 60, FcD2 = 10) to investigate the effect of the poorly propped fracture conductivity on pressure and derivative, as shown in Figure 6. The poorly propped fracture conductivity affects the lasting time of bilinear flow. The bilinear flow would last longer with a lower poorly propped fracture conductivity. Moreover, the pressure and derivative increase with the decrease of the poorly propped fracture conductivity, while this effect becomes nil after linear flow.

Figure 6.

Figure 6

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Effect of the poorly propped fracture conductivity on pressure and derivative.

4. Case Study

In recent years, Xinjiang Oilfield companies have advanced the exploration to deep-layer unconventional reservoirs. Several reservoirs with complex geological structures, and complex fluids have been discovered in Junggar Basin, which will increase the oil and gas production rate in the future. We choose the Mahu-district reservoir as the study area to verify the applicability of the developed model. The Mahu-district reservoir is a deep-layer reservoir with buried depth more than 4 km. The permeability of the rock matrix is very low, and hence, a large number of oil wells is stimulated by separate-layer fracturing. In this subsection, we will use the proposed model to interpret the recorded buildup pressure data of well P1.

As shown in Figure 7, well P1 intercepted a four oil layers. Four oil layers are all fractured after two rounds of hydraulic fracturing operations. The first fracturing operation is conducted on two oil layers ranging from 4056 to 4059 and 4066.5 to 4068.5 m. The second fracturing operation is conducted on remaining two oil layers ranging from 4082 to 4085 and 4087 to 4090 m. The production rate of well P1 is 17 S m3/d. The other basic parameters of reservoir and oil well are given in Table 2. We apply the proposed model to interpret the pressure data of well P1. It can be seen clearly from Figure 8 that the theoretical curves match the recorded transient pressure and derivative data. The interpretation results are given in Table 3. The average fracture half-length is 68.1 m, indicating that the effect of hydraulic fracturing stimulation is good. The average reservoir permeability is ∼0.35 mD.

Figure 7.

Figure 7

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Wellbore structure description of well P1.

Table 2. Reservoir and Well P1 Parameters.

parameters unit layer 1 layer 2 layer 3 layer 4
oil viscosity cP 1.2 1.2 1.2 1.2
reservoir thickness m 3 2 3 3
oil compressibility MPa–1 0.001 0.001 0.001 0.001
reservoir porosity % 7.2 7.2 7.2 7.2
oil volume factor   1.128 1.128 1.128 1.128
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Figure 8.

Figure 8

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Application of the proposed solution on well test interpretation of Xinjiang Oilfield.

Table 3. Interpretations Results of Well P1.

parameters unit layer 1 layer 2 layer 3 layer 4
wellbore storage coefficient m3/MPa 0.53
skin factor   0 0 0 0
reservoir permeability 10–3 μm2 0.41 0.37 0.31 0.25
fracture half-length m 79.6 69.3 63.6 59.9
fracture conductivity (D)   113.6 97.8 101.6 88.6
poorly propped half-length m 9.4 13.2 15.0 13.6
poorly propped conductivity (D)   62.9 58.3 52.6 46.7
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5. Conclusions

The deep-layer reservoirs generally adopt the separate-layer fracturing and commingling production. However, the proppant is usually difficult to move inside the hydraulic fracture in the deep-layer reservoir during the hydraulic fracturing operation, leading to the decrease of fracture conductivity at the fracture extreme. For giving the suitable PTA method to solve the well test problem encountered in Xinjiang Deep-Layer Oilfield, this work presented an analytical well test model of hydraulically fractured wells with changing conductivity in stratified reservoirs. The contributions and conclusions of this work can be summarized as follows.

  • An analytical well test model for hydraulically fractured wells with changing conductivity in stratified reservoirs has been presented. The model is fully analytical and hence it is convenient to be applied to solve the well test problem of deep-layer reservoirs.

  • The solution is simple and reliable for short-time analysis. When the test pressure data exhibit the radial flow regime, the proposed model is not applicable.

  • The heterogeneity among layers will lead to an increase of the pressure drop and pressure derivative. The stronger heterogeneity can lead to more increase of the pressure drop and pressure derivative.

  • The poorly propped fracture conductivity affects the lasting time of bilinear flow. The bilinear flow would last longer with a lower poorly propped fracture conductivity. Moreover, the pressure and derivative increase with the decrease of the poorly propped fracture conductivity, while this effect becomes nil after linear flow.

  • A case study of hydraulically fractured well in stratified reservoir is conducted in the Xinjiang Oilfield of Junggar Basin, which indicates that the proposed model is feasible to analyze the formation parameters of stratified reservoirs.

Acknowledgments

The authors would like to acknowledge the financial support from the National Natural Science Fund of China (no. 11872073).

Glossary

Nomenclature

wfn

hydraulic fracture width of layer n, m

kfn

hydraulic fracture permeability of layer n, 10–3 μm2

xfn

hydraulic fracture half-length of layer n, m

hfn

hydraulic fracture height of layer n, m

hn

reservoir thickness of layer n, m

xpfn

propped fracture length, m

x

distance along fracture, m

y

distance perpendicular to fracture, m

kn

reservoir permeability of layer n, 10–3 μm2

ϕn

reservoir system porosity of layer n, %

ct

reservoir compressibility, MPa–1

Cn

wellbore storage coefficient of layer n, m3/MPa

μ

oil viscosity, mPa·s

pi

initial reservoir pressure, MPa

pn

reservoir pressure of layer n, MPa

m

oil layer number

qn

flow rate of layer n, m3/d

Q

total flow rate of single well, m3/d

t

time, h

CDn

dimensionless wellbore storage coefficient of layer n

Sn

skin factor of layer n

pDn

dimensionless pressure of layer n

p1Dn1

dimensionless propped fracture pressure of layer n

p1Dn2

dimensionless poorly propped fracture pressure of layer n

tD

dimensionless time

FcDn1

dimensionless propped fracture conductivity of layer n

FcDn2

dimensionless poorly propped fracture conductivity of layer n

qDn

dimensionless flow rate contribution of layer n

xD

dimensionless distance along the fracture

yD

dimensionless distance perpendicular to fracture

λn

dimensionless transmissibility factor

ωn

dimensionless storativity factor

αn

dimensionless fracture length

RDn

dimensionless propped fracture length

rn

fracture extension of layer n

Glossary

Subscripts

D

dimensionless

f

fracture system

i

initial condition

s

Laplace space

The authors declare no competing financial interest.

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