Pressure and reciprocal rate transient analysis for finite-conductivity vertical fractured hydrocarbon wells under trilinear flow

Hydraulic well fracturing has been an important practice for well stimulation along the history of the hydrocarbon industry. Therefore, accurate and practical assessment of the fracturing job should be of importance. Interpretation of well pressure and rate test data in vertical finite-conductivity fractured hydrocarbon wells by the TDS methodology and conventional analysis are presented in this paper. The accuracy and practicality of the developed equations are successfully tested with synthetical examples and a field case.


Introduction
Hydraulic fracturing is a very common way to stimulate hydrocarbon wells. Also, well test analysis is the most used practice to evaluate the results of a fracturing job. Well test analysis is mostly based upon analytical solutions which allow us to understand the formation and well flow dynamics. For well pressure tests, analytical models can be grouped into two types according to their solution: semianalytic models and asymptotic analytical models. A semianalytic model was first developed by Gringarten, Ramey and Raghavan (1974) for hydraulically fractured wells with fractures of infinite conductivity, and later, this semi-analytic model was adapted for hydraulically fractured wells with finite-conductivity fractures by Cinco-Ley, Samaniego and Dominguez (1978). They developed a mathematical model to study transient behavior in wells with vertical finite-conductivity fractures for dimensional times greater than 0.001, which may be correlated with the dimensional finite conductivity of fractures. Their provided a type-curve matching interpretation technique. Subsequently, Cinco-Ley and Samaniego (1981) developed an asymptotic analytical solution for the interpretation of transient pressure data for wells intercepted by a vertical fracture. With this, they demonstrated that for dimensional fracture conductivity less than 300, the fracture conductivity cannot be considered as infinite. They also provided type curves for the determination of the reservoir and fracture parameters.
Later, Cinco-Ley (1982) developed a method for fractured wells considering different flow geometries. This work recognized that wells intercepted by a fracture had different flow behaviors than non-fractured wells, and application of traditional pressure analysis methods based upon radial flow theory, could yield erroneous results. His method was based on the bilinear flow rate, which is present when the linear flow of the fracture and the linear flow within the formation occur simultaneously.
Several other models were also introduced. For instance, Cossio, Moridis, and Blasingame (2013) presented a semianalytic solution for a finite-conductivity fractured using fractal geometry theory. Wong, Harrington and Cinco-Ley (1984) presented a mathematical solution for pressure behavior under pseudolinear conditions which has useful applications when the fracture has a skin factor that allows having a linear flow from the fracture tip to the well.
As a final flow model mention, there are more, Lee and Brockenbrough (1986) introduced a new analytical solution, based on a trilinear flow model as shown in Fig. 1. This sketch shows how three linear flows make up the trilinear flow model. Flow in region 1 renders the fracture with a dominant flow in the x direction. If the fracture is considered as a bending line, the flow in region 2 will be controlled by the y direction. With similar reasoning, the flow in region 3 would be approximate in the x direction for a very short period. These reasoning of flow behavior is reasonable for fluid flow before the pseudoradial flow regime shows up. In addition to the formulation of their trilinear flow model, they also conducted a study comparing this trilinear flow solution with that proposed by Cinco-Ley et al. (1981). They found that the trilinear model fits perfectly with the model proposed by Cinco-Ley et al. (1981) up to a value of t D = 1.0. After this value, the deviation becomes more appreciable.
The authors are focused on well test interpretation by the TDS Technique, Tiab (1995), which is a novel, accurate and revolutionary methodology based upon characteristic points and lines found on the pressure and pressure derivative versus time log-log plot -also to the reciprocal rate and reciprocal rate derivative versus time log-log plot. Details of this technique is widely exposed on the textbooks of Escobar (2015Escobar ( , 2019 and a very comprehensive state-of-the-art is presented by Escobar, Jongkittnarukorn and Hernandez (2018). Classical articles on the TDS Technique for hydraulic fracture wells are presented by Tiab (1994) and Tiab et al (1999). Also, Escobar, Gonzalez and Hernandez (2016) extended the TDS Technique for well test interpretation under pseudolinear conditions using the already mentioned model by Wong et al (1984).
In this work, TDS Technique and straight-line conventional analysis are extended for the practical test data interpretation of finite-conductivity vertical fractured wells in hydrocarbons reservoirs using the trilinear flow model of Lee and Brockenbrough (1986). Both pressure transient and rate transient analysis are provided for the TDS Technique and conventional analysis, as well. The proposed equations were successfully tested with synthetic and actual field data. Lee and Brockenbrough (1986) provided a Laplace space pressure solution of a vertical fractured well where the flow is dominated by trilinear flow regime, where;

Mathematical formulation
(1) The dimensionless quantities are defined below as: The dimensionless parameters for an oil reservoir are given below:

141.2q B
The solution of the flow equation for the constant-pressure case; Then, the dimensionless parameters are given by: The dimensionless quantities for gas reservoirs, constantrate case, are given as follows: 1424.26q sc T Fig. 1 A quadrant of the top view of the fractured well system and trilinear flow approximation. Taken from Lee and Brockenbrough (1986) 1 3 The dimensionless parameters for the constant-pressure case, which are used in transient rate analysis, are provided below:

Wellbore rate and pressure behavior
Both dimensionless pressure and dimensionless pressure derivative versus dimensionless time behavior provided in Fig. 2  the reciprocal rate does not display a unit-slope behavior but a variety of slopes going to values higher than 2. This, therefore, difficult the application of conventional analysis. For a better understanding of the flow dynamics let us refer to Fig. 1 which sketches the trilinear model proposed by Lee and Brockenbrough (1986). They consider three flow zone. The first zone provides the pressure response of the fracture. The second zone is responsible for the simultaneous answer of two linear flows: one occurring inside the fracture and the other one from the formation to the fracture. Finally, region 3 consists of linear flow to the well-fracture system. As reflected in both pressure derivative and reciprocal rate derivative, the first flow regime occurs corresponds to a linear flow from formation to fracture. This means that the flow inside the fracture is imperceptible. It looks that the unit-slope behavior results from the transition between the early linear flow regime to the subsequent bilinear flow regime which takes place at later time. This bilinear flow may be formed by the simultaneous flow regime from the formation to the fracture and from zone 3 to the fracturewell system.

Pressure transient analysis by the TDS technique for oil reservoirs
TDS Technique, Tiab (1995), is a methodology based upon finding unique characteristic and points on the pressure and pressure derivative versus time log-log plot (this is extended to reciprocal rate and reciprocal rate derivative). Escobar (2015Escobar ( , 2019 presented two textbooks which clearly illustrate this technique.
For extension of the TDS Technique is necessary to find unified pressure/rate and pressure/reciprocal rate derivative behaviors so unique equations can be developed. For instance, Fig. 4 presents a unified behavior of the pressure derivative multiplied by the dimensionless fracture conductivity. The following equation is.
then obtained: Suffix L in Eq. (18) stands for "Linear". Plugging Eqs. (6), (7) and (9) into Eq. (18) and solving for the fracture conductivity, it yields: Integration of Eq. (18) leads to find the governing pressure equation: where c is a given constant of no interest for this work. Again, replacing Eqs. (6, 7 and 8) into Eq. (20), another equation is found to find the fracture conductivity.
Another unified behavior is found during the unit-slope behavior as shown in Fig. 5. The following expression governs such behavior: Again, we found by integration that the unified dimensionless pressure equation is governed by: Replacement of the dimensionless quantities given by Eqs. 7,8,9 into Eqs. (22) and (23) allows to find two expressions for the determination of the half-fracture length: As seen in Fig. 2 a unified behavior exists by itself during linear flow regime. The pressure and pressure derivative governing equations during the bilinear flow regime are found to be: Replacing the dimensionless parameters in the above equations, it yields: We may also solve for formation permeability,

Pressure transient analysis by the conventional method for oil reservoirs
Once the dimensionless parameters are replaced in Eq. 20, the following is obtained: Which can be expressed as: Unified pressure derivative times dimensionless conductivity versus dimensionless time during the unit-slope flow behavior The above implies that a Car tesian plot of yields a straight line which slope, m L , provides the fracture conductivity. Such slope, m L is given by: From which; The unit-slope pressure equation obtained from Eq. 23 allows finding: Which can be expressed as: A Cartesian plot of ΔP vs. t yields a straight line which slope, m US , provides the half-fracture length. This slope is given by: The bilinear pressure behavior obtained from Eq. 27 allows to develop: Which can be expressed as: W h i ch i m p l i e s t h a t a C a r t e s i a n p l o t o f ΔP = P ws − P wf vs. t p + Δt 0.25 − Δt 0.25 (for drawdown) or ΔP = P ws − P wf vs. t p + Δt 0.25 − Δt 0.25 (for buildup) provides a straight line which slope, m BL , provides the half-fracture length:

Pressure transient analysis by the TDS technique for gas reservoirs
Equations 18 and 20, 22 and 23 and 26 and 27 apply to gas reservoirs. Dimensionless pressure and dimensionless pressure derivative are changed by dimensionless pseudopressure and dimensionless pseudopressure derivative. Then, after replacing the dimensionless time expression given by Eq. 13 and the dimensionless pseudopressure and pseudopressure derivative given by Eqs. 14 and 15 in the abovementioned equations, it yields: The points of intersection between the different pressure derivative equations are very important for the verification of the reservoir parameters. These developed expressions are valid for both oil and gas cases, with the exception that for gas case the product of compressibility and viscosity is given at initial conditions. Thereby, the intersection points between Eqs. 18 and 22, 18 and 27, 22 and 27 and 11 and 27 provides the following expressions: If radial flow is seen, Tiab (1995) found that during the radial flow regime the pressure derivative is governed by the following simple equation: Then, the intersection between Eqs. 18, 22 and 27 with Eq. 53 yields the following:

Transient rate analysis by the TDS technique for oil reservoirs
As shown in Fig. 6, the unified reciprocal rate derivative behavior obeys the following equation: Replacing Eqs. 7 and 12 allows to solve for an expression to find fracture conductivity, Integration of Eq. 56 leads to find the governing expression for the reciprocal rate, After replacing Eqs. 7 and 11 and solving for the fracture conductivity from the resulting replacement leads to, Fig. 6 Unified reciprocal rate derivative versus dimensionless time divided by the dimensionless fracture conductivity squared observed during early linear flow regime Also, another unified reciprocal rate derivative behavior is given in Figs. 7 and 8 from which can be written: Which also provides an expression for the estimation of the half-fracture length once Eqs. 7 and 12 are replaced in the above equation. Therefore, It is observed during the unit-slope behavior that the reciprocal rate does not follow a unit slope, then the Equation for this was found to be, It should be used only if the log-log slope of the reciprocal rate is about 2. From where an expression to find the half-fracture length is found once the dimensionless quantities have been replaced in the above equation, A unified behavior is also found during the bilinear flow regime in Fig. 3. The governing expression for this flow regime is given by: An expression to find the half-fractured length is obtained once the dimensionless time, Eq. 7, and dimensionless reciprocal rate derivative, Eq. 12, are plugged in Eq. 64, therefore: Or, Integrating Eq. 64 results in, Again, the constant, c, has no effect in this work. Replacing Eqs. 7 and 11 in the above expression leads to find either: Or,

Rate transient analysis by the conventional method for oil reservoirs
Equation 58 can be rewritten as: Which can be expressed as: A Cartesian plot of 1/q vs. t 0.5 yields a straight line which slope, m L , provides the fracture conductivity, Therefore, Once the dimensionless parameters have been replaced into Eq. 62, this be written as:  Intersection points also play an important role in rate transient analysis when the TDS Technique is applied. The following expressions resulted from the reciprocal rate derivative intersection points between the different flow regimes: Equations 87-92 apply to either oil or gas wells.

Transient rate analysis by the conventional method for gas reservoirs
Equation 58 can be rewritten as: Which can be expressed as Eq. 71 . A Cartesian plot of 1/q vs. t 0.5 yields a straight line which slope, m L , provides the fracture conductivity, Once the dimensionless parameters have been replaced into Eq. 62, this be written as: Or, A Cartesian plot of 1/q vs. t 2 yields a straight line which slope, m US , provides the half fracture length, Equation 67 is rewritten as: Which can be written as Eq. 78. A Cartesian plot of 1/q vs. t 0.25 yields a straight line which slope, m BL , provides the half fracture length, From Eq. 53, Tiab (1995), found an expression to find reservoir permeability, For gas wells, According to Tiab (1995) the mechanical skin factor can be found from: Or for gas wells, the apparent skin factor is given by Escobar,  Escobar et al. (2016) presented a couple of expressions to find the mechanical skin factor when radial flow regime is absent. Also, a correlation to find half-fractured length as a function of fracture conductivity of vice versa.

Example 1: Synthetic example
The second column of Table 1 presents the input data used to generate the simulated drawdown reported in Fig. 9.

TDS Technique
From Fig. 9 the following information was read: The linear flow regime was used to calculate the fracture conductivity by means of Eq. 19 and 21 using pressure derivative and pressure data, respectively: The half-fracture length is then found from the unit-slope trend on the pressure derivative curve by using Eq. 24 and 25, Use of Eq. 49 leads to verify the half-fracture length:

Conventional analysis
Figures 10 and 11 present Cartesian plots from which the slope of the linear flow regime, m L , and the unit-slope behavior, m US , were found to be:

Example 2: Synthetic example
The third column of Table 1 were used to generate the reciprocal rate and reciprocal rate derivative data presented in Fig. 12. Solution to this problem has two limitations which can be drawn from Fig. 12. During the unit-slope behavior the slope of the reciprocal rate is different than 2.0, therefore it is not recommended to use conventional analysis, Eq. 67 nor TDS Technique with the reciprocal rate value Eq. 63. Conventional analysis will only be applied to the linear flow regime.

TDS Technique
The following information was read from Fig. 12.
From the linear flow regime fracture conductivity was estimated using Eq. 57 and 59 to be 1646.7 md-ft and 1657.92 md-ft respectively, as indicated below: The reciprocal rate derivative during the unit-slope behavior was used to find the half-fracture length with Eq. 61 to be 81.9 ft.
Also, Eq. 87 allows us to verify the half-fracture length:

Conventional analysis
As commented previously, conventional analysis was applied only to the linear flow regime. A Cartesian plot of the reciprocal rate versus the square root of time is shown in Fig. 13 which slope m L = 0.0009336 hr 0.5 /STB. With this the fracture conductivity is computed with Eq. 73 to be 1661.76 md-ft.

Example 3: Field example
The fourth column of Table 1 presents information of a massive hydraulically fractured gas well reported by Agarwal, Carter, and Pollock (1979) which time and reciprocal rate are given in Table 2. The reciprocal rate derivative was estimated in this work and it is also reported in Table 2.

TDS technique
The reciprocal rate and reciprocal rate derivative versus time log-log plot is provided in Fig. 14

Conventional analysis
The Cartesian plot of the reciprocal rate versus the fourth root of time are given versus time is given in Fig. 15. The slope of this plot: The half-fracture length is found from Eq. 100 to be,  Table 1 was obtained by using type-curve matching which is not the best well test interpretation method. It is customary to obtain fracture conductivity from the classical bilinear flow regime and half-fracture length from the classical linear flow regime. For our case, the trilinear model, the situation is inverted.

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Funding
The authors thank Universidad Surcolombiana for funding this research.
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