## An Analytical Model for Capturing the Decline of Fracture Conductivity in the Tuscaloosa Marine Shale Trend from Production Data

## Abstract:

Fracture conductivity decline is a concern in the Tuscaloosa Marine Shale (TMS) wells due to the high content of clay in the shale. An analytical well productivity model was developed in this study considering the pressure-dependent conductivity of hydraulic fractures. The log-log diagnostic approach was used to identify the boundary-dominated flow regime rather than the linear flow regime. Case studies of seven TMS wells indicated that the proposed model allows approximation of the field data with good accuracy.

#### Authos:

#### Xu Yang , Boyun Guo, and Xiaohui Zhang

Department of Petroleum Engineering, University of Louisiana at Lafayette, Lafayette, LA 70504, USA.

Received: 26 February 2019 / Revised: 9 May 2019 / Accepted: 17 May 2019 / Published: 21 May 2019

Production data analyses with the model revealed that the pressure-dependent fracture conductivity in the TMS in the Mississippi section declines following a logarithmic mode, with dimensionless coefficient χ varying between 0.116 and 0.130. The pressure-dependent decline of fracture conductivity in the transient flow period is more significant than that in the boundary-dominated flow period.

## Introduction

The Tuscaloosa Marine Shale (TMS) across Louisiana and Mississippi has been an attractive unconventional shale oil reservoir since 2012 [1]. The TMS is a sedimentary formation that consists of organic-rich fine-grained sediments deposited during the Upper Cretaceous [2]. The TMS is one part of the Tuscaloosa group consisting of Upper Tuscaloosa, Middle Marine Shale, and Lower Tuscaloosa [3]. The thickness of TMS ranges from 500 ft in the southwestern Mississippi to more than 800 ft in southeastern Louisiana, within a depth range of 11,000 to more than 15,000 ft [4]. Middle Tuscaloosa is composed of a dark grey, fissile, and sandy marine shale. Experimental results indicated that the permeability ranges from less than 0.01 to 0.06 md, and porosity ranges from 2.3% to 8.0% [4].

More than 80 multi-fractured horizontal wells were drilled in the TMS between 2012 and 2014 [2]. Several TMS wells were recorded to have appealing initial oil production rates exceeding 1000 stb/d. Although such significant high production rates were observed, drilling activities stopped in 2014 due to the high cost of drilling and low price of oil. Previous studies estimated an unproven recoverable oil of seven billion barrels [4], but the real production potential from TMS is still poorly understood. It is of significance to assess the TMS well performance through production decline analysis and identify key factors controlling the productivity of TMS wells.

Mathematical modeling plays an important role in analyzing well behavior and identifying factors that affected well performance in the past. The available models for predicting shale oil well long-term productivity include (1) analytical transient flow models [5,6,7], (2) numerical computer models [8,9,10], and (3) empirical production decline models [11,12,13,14,15]. The analytical transient flow models are utilized in pressure transient test analysis rather than in production data analysis. Although numerical computer models are flexible in handling systems with non-symmetrical fractures and multiphase flow, applications of these models are limited owing to the low efficiency of their numerical nature, especially the numerical treatment of local grid refinement near the fractures.

Uncertainty in locations of natural fractures is another concern regarding the accuracy of the computer simulation result. The empirical production decline models were not derived based on engineering principles. They are mainly utilized for evaluating field development projects on the basis curve fitting to the production history data [16]. Besides, empirical production decline models are mostly applicable to boundary-dominated flow conditions that are often the case for conventional reservoirs. Wells in unconventional reservoirs are characterized by long-term transient flow owing to ultra-low reservoir permeability [17,18]. These models are not suitable for evaluating the real potential and identifying factors controlling the potential of TMS wells.

Hydraulic fracture conductivity plays an essential role in well performance [19,20,21,22,23,24]. It can decline significantly during production owing to proppant embedment and blockage of debris [25,26]. Experimental investigations have indicated that hydraulic fracture conductivity declines logarithmically or exponentially with time [27,28]. Sun et al. [28] demonstrated that the decrease in fracture conductivity could reduce the production decline curve value and lead to a significant production drop. Production decline analysis failing to take time-dependent fracture conductivity into account will lead to significant errors in production prediction of multi-fractured wells in unconventional shale reservoirs.

Fracture conductivity declines exponentially in the Haynesville Shale, which is very close to the TMS [29]. The decline in TMS is of particular concern because of the high level of clay materials in TMS [2]. Clay minerals are water-sensitive, making the fracture face soft and vulnerable to the embedment of fracture proppant, leading to a fracture conductivity drop due to partial closure of the fracture. This study focuses on capturing the pressure-dependent decline rate of fracture conductivity in TMS wells under the boundary-dominated flow conditions using production data.

An analytical well productivity model was developed in this study considering time-dependent fracture conductivity. Case studies of seven TMS wells indicate that the production rates calculated by the analytical model agree with field data very well. Production data analyses with the model revealed that fracture conductivity in the TMS in the Mississippi section declines following a logarithmic mode.

## Mathematical Model

An analytical model was developed in this work for capturing the decline of pressure-dependent fracture conductivity of multi-fractured shale oil wells under boundary-dominated flow conditions. The assumptions on the analytical model include the following,

- The oil formation is isotropic.
- Boundary-dominated flow has been reached within the fracture drainage area.
- Linear flow prevails from the shale matrix to the fractures.
- Fracture and formation damages are negligible.
- No change in fluid composition during production.
- Hydraulic fractures have the same geometry.
- Reservoir pressure is above the bubble point pressure.

Derivation of the analytical model considering the pressure-dependent decline of fracture conductivity is shown in Appendix A. The analytical model for predicting the productivity of multi-fractured shale oil wells is,

Where

where q_{o} is the oil production rate, n_{f} is the number of hydraulic fractures, k_{m} is the matrix permeability, h is the reservoir thickness, p_{i} is the initial formation pressure, p_{w} is the wellbore pressure, B_{o} is the oil formation volume factor, μ_{o} is the oil viscosity, S_{f} is the hydraulic fracture spacing, x_{f} is the fracture half-length, α_{b} is the B_{iot} coefficient, ν is the Poisson’s ratio, N_{p} is the cumulative oil production, N_{i} is the original oil in place within the well drainage area, c_{t} is the total compressibility, C_{f0} is the initial fracture conductivity, and c_{p} is the compressibility of the proppant pack.

The analytical model for capturing time-dependent fracture conductivity of shale wells during the boundary-dominate flow period is,

where C_{f} is the fracture conductivity.

From Equation (4) we can see that fracture conductivity can be predicted if the cumulative oil production data are known.

## Flow Regime Diagnosis

### Flow Regime Diagnosis Method

As stated earlier, the proposed model is only applicable to the wells that have reached the boundary-dominated flow over the production time. Therefore, preliminary analysis of production data should be performed to make sure that the candidate wells are in the boundary-dominated flow regime.

Four flow regimes may exist in multi-fractured reservoirs: (1) early time reservoir linear flow (transient flow), (2) mid-time boundary-dominated flow, (3) late time reservoir radial flow, and (4) very late time bounded flow. For wells in shale plays the third and fourth flow regimes usually are not seen due to the ultralow permeability of shale matrix. To fully understand the performance of multi-fractured horizontal wells, we identify the first two flow regimes between fractures. Figure 1a presents two fractures with an assumption of a virtual boundary between these two fractures. In the transient flow period, pressure propagates outward from the fracture face without encountering the virtual boundary. In the boundary-dominated flow period, the pressure transient has reached the virtual boundary, and the static pressure is declining at the boundary, as shown in Figure 1b.

*Figure 1.** Schematic chart between two fractures. (a) Virtual boundary between two fractures. (b) Schematic chart of pressure distribution.*

In this section, the log-log diagnostic plots of production rate versus real production time as well as material balance time (MBT) are constructed to identify the flow regime of TMS wells. The procedure for identifying the flow regime is as follows [30].

Construct the production rate versus real production time curve (i.e., PT curve) and production rate versus material balance curve (i.e., MBT curve) on the same log-log plot. Then plot a tangent line L_{MBT} using the end of MBT curve. If the slope k_{MBT} of the tangent line L_{MBT} is close to −1, then we have confidence that the boundary-dominated flow regime has been reached. If the slope k_{MBT} is greater than -1, one can move the line L_{MBT} onto the PT curve and get an approximately tangent point P. We draw another tangent line L_{PT} using data between point P and the end of production data. If the slope k_{PT} of this line is less than or equal to −1, it is believed that the well has reached the boundary-dominated regime, and the real production time corresponding to point P is the estimated switching time from transient flow to the boundary-dominated flow. However, if it is difficult to draw a tangent line L_{PT} with slope k_{PT} less than or equal to −1 owing to few points after point P, then one failed to obtain the conclusion that the well has reached boundary-dominated flow.

### TMS Well Description

The study area in Mississippi is presented in Figure 2. The shapes for Louisiana and Mississippi counties were downloaded from the website (https://www2.census.gov). The production data as of May 2018 were gathered from 55 TMS wells in Mississippi from the Alfred C. Moore, Pearl River, and Henry Fields. These data were downloaded from the Mississippi Automated Resource Information System (MARIS) website (www.maris.state.ms.us).

**Table 1. **TMS wells used in this study.

Quality control of the production data was performed to validate the compliance of the field data with the assumptions used in the analysis. Those wells that had abrupt changes in monthly oil production rate during the production history were removed. This helps minimize the influence of changes in the production operations that would have a great effect on the production history. Following this, seven TMS wells that had reached the boundary-dominated flow regime were analyzed. Other wells could not be analyzed because of absence of a definite trend in the reported production data, most likely dictated by the used production strategy for which no details are available.

* *

*Figure 2.** Map of the study area in Mississippi.*

It should be stated that other wells may also have reached the boundary-dominated flow regime. However, we did not analyze these wells due to lack of completion data and also abrupt changes in production data of these wells over time. Seven TMS wells used in this study are shown as blue dots in Figure 2. The wells’ labels and their locations are listed in Table 1. Proppant has a major impact on fracture conductivity. Unfortunately, the type of proppant used in these seven TMS wells could not be identified due to the lack of completion data. The average fracture spacing can be estimated by using the stages.

### Flow Regimes of TMS Wells

Figure 3 shows the log-log diagnostic plot of the seven TMS wells. Well 1 has a horizontal length of 9102 ft and an effective lateral length of 8442 ft with 29 stages, and each stage has four clusters. We drew a tangent line of MBT curve and found that the production data at the end of material balance time lies on the tangent line L_{MBT} with a slope of −0.779. As k_{MBT} is less than −1, the tangent line, L_{MBT}, was moved onto the PT curve, and it showed tangency to this curve. If we draw a tangent line L_{PT} using production data vs. real production time. The slope kPT is found to be −1.085, indicating a typical behavior of boundary-dominated flow over 16 months of production.

Well 2 was drilled and completed with an effective lateral length of 6757 ft and a 22 stage hydraulic fracturing operation. We can see from Figure 3b that the slopes of MBT curve (k_{MBT}) and PT curve (k_{PT}) are −0.688 and −1.002, respectively, indicating that well 2 should be in the boundary-dominated flow regime with an estimated switching time of 9.5 months.

Well 3 was drilled and completed with a 4,508 ft lateral and a ten stage hydraulic fracturing operation. From this plot, we can see that k_{MBT} and k_{PT} are −0.845 and −1.295, respectively. Therefore, we have confidence that well 3 switched from the transient flow to the boundary-dominated flow over 27 months of production. Figure 3c also shows that well 3 experienced a long-term transient flow for over two years.

Well 4 has a true vertical depth of 11,489 ft and an effective lateral length of 6451 ft with 26 stages. We can see from Figure 3d that well 4 has reached the boundary-dominated flow regime as the slope k_{PT} is less than −1. The estimated switching time from the transient flow to the boundary-dominated flow is 12 months.

*Figure 3.** Log-log diagnostic plot of TMS wells. (a) well 1. (b) well 2. (c) well 3. (d) well 4. (e) well 5. (f) well 6. (g) well 7.*

Well 5 has a true vertical depth of 12,245 ft and an effective lateral length of 5601 with 24 stages. It can be seen from Figure 3e that the diagnostic plot that the slope k_{MBT} and k_{PT} are −0.733 and −1.123, indicating a boundary-dominated flow regime.

Well 6 has a true vertical depth of 12,016 ft and an effective lateral length of 6,681 ft with 25 stages. From the diagnostic plot Figure 3f, we can see that k_{MBT} and k_{PT} are −0.787 and −1.173, indicating it reached the boundary-dominated flow regime in over 13 months of production.

The true vertical depth and effective lateral length of well 7 are 11,841 and 5681 ft, respectively. Slope k_{MBT} and k_{PT} are found to be −0.714 and −0.988, respectively. As k_{PT} is approximately close to −1, well 7 should be in the boundary-dominated flow regime in over 13 months from its first production month.