Well Test Analysis for Hydraulically-Fractured Wells

This chapter focuses on the application of Tiab ’ s direct synthesis ( TDS ) technique for practical and accurate interpretation of pressure tests on vertical wells in conventional reservoirs, so bilinear, linear, and elliptical flow regimes can be used for fracture characterization. Most fractured well interpretation tests are conducted using nonlinear regression analysis if the pressure model is available. This method has some drawbacks associated with the nonuniqueness of the solution. Also, the conventional straight-line method requires one plot for each individual flow regime observed in the pressure tests, and the estimated parameters cannot be verified. Tiab ’ s direct synthesis ( TDS ) methodology, which uses specific lines and intersection points found on the pressure and pressure derivative plot, is used in some direct equations which are obtained from the solution of the diffusivity equation for a given flow regime. It has been proven to provide accurate results, and its power allows verification of most results which is not possible from any other technique. The methodology has been successfully explained and tested by its application in two examples, although there exists more than a hundred articles that provide many useful applications.


Introduction
Throughout their history, well test analyses for fractured wells have received many contributions. For practical purposes, let us name the most important ones for this chapter. A good place to start is by mentioning the work in [1], which described the pressure behavior for infinite-conductivity and uniform-flux fractured wells, so people started conducting interpretation tests on such wells by using type-curve matching. Later, [2] introduced the concept of finite-conductivity fractures and established the onset value of dimensionless conductivity as 300. Values lower than that are considered finite-conductivity values, and those above 300 are classified as infinite conductivity. In [2], a fine semi-analytical solution was introduced for describing the well-pressure behavior in hydraulically fractured wells. This solution was then applied in [3] to provide a well interpretation method using type-curve matching. Since then, other mathematical solutions have been presented for finite-conductivity fractures. Among them, the work in [4] using fractal theory is worth mentioning.
The way of conducting well test interpretation was changed by the introduction of Tiab's direct synthesis (TDS) technique by [5]. This revolutionary and modern technique focuses on the different flow regimes seen on the pressure derivative curve. Defined lines are drawn through each individual flow regime, and the intersection points found among them are read and used for reservoir characterization. Additionally, reading arbitrary points on the pressure and pressure derivative of each flow regime also serve for reservoir parameter determination. A great number of applications of the TDS technique are given in [6]. The second work [7], by the same author of [5], presented TDS technique for infinite-conductivity and uniformflux fractures in vertical wells. In [7], the elliptical or biradial flow regime was introduced and characterized. This elliptical flow is also seen in horizontal wells and was characterized in [8][9][10]. Because of the similarity between the mathematical models of hydraulic fractures and horizontal wells, this concept was applied by [11] to determine the average reservoir pressure in formations drained by horizontal wells using the TDS technique. The infinite-conductivity model in [7] also included the late-time pseudosteady-state period as well as some equations involved in the drainage area (conventional analysis for this case was included in [12]). This may be disadvantageous for inexperienced users of TDS technique when interpreting pressure tests without reaching reservoir boundaries because the equations involved the use of the unknown reservoir drainage area, although it can be still applied by using the intersection points. To overcome this drawback, [13] presented a new mathematical model excluding the late-time pseudosteady-state period.
TDS technique for finite-conductivity fractured wells is given in [14], with practical field applications to demonstrate the usefulness of the technique. The fracture parameters can be readily obtained by using an arbitrary point on the flow regimes. TDS technique plays an important role when analyzing short pressure tests because a user can "make up" nonexisting flow regimes since, for instance, the radial flow horizontal line can be obtained from the reservoir permeability even though radial flow regime is absent. [15,16] extended the works of finite-and infinite-conductivity fractures in naturally fractured reservoirs. The equations provided by these works can also be applied to either homogeneous or naturally fractured formations since they involve a dummy variable that takes the value of one for the homogeneous case or the value of the dimensionless storativity coefficient for the case of a naturally fractured formation.
TDS technique has also been extended to several scenarios related to hydraulically fractured wells. For instance, when a finite-conductivity fracture intersects with a fault, the pressure trace changes; then, the equations developed in [17] apply for this case. There are cases where a threshold pressure is required to start the flow. The work in [18] includes this concept in uniform-fractured vertical wells, and the work in [19] includes the concept for horizontal wells. Also, when the fractured face is damaged, a pseudolinear flow regime develops along the fracture. [1] included TDS technique to characterize such systems. [16] presented TDS technique for fractured wells in gas composite reservoirs. TDS technique can also be usefully applied to transient-rate analysis, as seen in [20]. Application of TDS technique to horizontally isolated fractured wells was presented and characterized in [21] and in conventional analysis in [22]. The works in [23,24] use TDS technique for shale reservoirs. Other applications of TDS technique to these systems are given by [25] under transient-rate analysis and [26] for pressure-transient analysis conditions. Other important applications of TDS Technique to fractured wells are given by [29,30].
This chapter is devoted to the application of TDS technique to hydraulically fractured wells in either homogeneous or naturally fractured formations. Without given detailed derivations, the expressions for characterizing the hydraulic fracture parameters are presented along with the way they should be used. Important relationships and practical exercises are included.

TDS basis
The pioneer publication on the TDS technique, [5], explains in detail the derivation of the equations. The Laplace space solution of the arithmetic pressure derivative for a homogeneous and infinite reservoir with skin and wellbore storage is also presented in [5] and given by However, we know that the pressure derivative is a horizontal line during radial flow regime. The dimensionless pressure derivative during radial line is easier represented by Then, to obtain practical equations, dimensionless parameters must be used. The dimensionless time, based upon half-fracture length and reservoir drainage area, is given below: and The dimensionless pressure and pressure derivative parameters for oil reservoirs are given by and Finally, the dimensionless fracture conductivity introduced in [3] is defined as It is observed from Eq. (5) that the two key parameters of a hydraulic fracture are the half-fracture length, x f , and the fracture conductivity, k f w f . The total length of the fracture is given by 2 x f .
The easiest application of TDS technique is given by replacing the dimensionless pressure derivative defined by Eqs. (6) and (2), to provide an expression to readily determine formation permeability: where (t*ΔP 0 ) R is the pressure derivative value during radial flow regime. The equations for the TDS technique are derived in the same manner Eq. (8) was obtained.

Biradial flow regime
Biradial or elliptical flow normally results in a hydraulically fractured well when areal anisotropy is present. This is recognized on the pressure derivative versus time log-log plot by a straight line with a slope of 0.36. In hydraulic fractures, the flow from the formation to the fracture is described by parallel flow lines resulting in a linear flow geometry better known as linear flow regime and characterized by a slope of 1/2 on the pressure derivative versus time log-log plot.
Both linear flow and biradial/elliptical flow regimes are seen on the plot of dimensionless pressure and pressure derivative versus dimensionless time based on halffracture length for a naturally fractured formation. New expressions for the elliptical flow regime introduced in [13] excluding reservoir drainage area are given by. and being ξ a dummy variable that defines either a homogeneous or naturally fractured formation. When ξ = 1, a homogeneous reservoir is considered. For the case of naturally fractured formations, ξ = ω, the dimensionless storativity coefficient.
Once dimensionless parameters given by Eqs. (3), (5), and (6) are replaced into Eqs. (9) and (10), respectively, and solve for the half-fracture length, which yields and TDS technique is based on drawing a straight line throughout a given flow regime; then, the user is expected to read the pressure, ΔP BR , and pressure derivative, (t*ΔP') BR , at a given time, t BR . A better way to reduce noise effects consists of extrapolating the mentioned straight line (biradial for this case) to the time of 1 h and read the pressure derivative value, (t*ΔP') BR1 , at 1 h. For this case, the pressure and pressure derivative set in Eqs. (11) and (12) is changed to ΔP BR1 and (t*ΔP') BR1 , respectively.
When bilinear flow is unseen, fracture conductivity can be found with an expression presented in [27] k f w f ¼ 3:31739k e s r w À 1:92173 [5] also provided an equation for the determination of the skin factor using an arbitrary point read during radial flow regime: The pseudosteady-state regime governing the pressure derivative equation is given by [7] used the point of intersection, t RPi , of Eqs. (2) and (15) to derive an equation for the estimation of the drainage area: The derivation of Eq. (16) follows a similar idea as that presented later in Section 4 for the use of the points of intersection.

Bilinear and linear flow regimes
Bilinear flow regime takes place when two linear flows, normal one flowing into the other, take place simultaneously. This situation occurs in low conductivity fractures where linear flow along the fracture and linear flow from the formation to the fracture are observed. Bilinear flow is recognized in the pressure derivative curve by a slope of 0.25. However, this is not shown in Figure 1 since bilinear flow is absent. The governing expressions for early bilinear and linear flow regimes for vertical fractures in naturally fractured systems were, respectively, presented in [16] and Linear flow regime can be used to find the half-fracture length, and bilinear flow regime allows finding the fracture conductivity. Once the dimensionless quantities of Eqs. (1) and (3)-(5) are replaced in Eqs. (16)- (19), the fracture conductivity is solved for then Once the fracture conductivity is found, Eq. (7) applies to find the dimensionless fracture conductivity if reservoir permeability and the half-fracture length are known. When bilinear flow is absent, the fracture conductivity may be found from Eq. (13), or the dimensionless fracture conductivity can be read from Figure 2: 1.E-02 1.E-01 1.E+00 1.E+01  Figure 1.

Figure 2.
Effect of skin factor on fracture conductivity (taken from [28]). and

Points of intersection
If bilinear flow also takes place, then the point of intersection between the pressure derivatives of the bilinear and biradial flow lines, t BLBRi , given by Eqs. (10) and (18), respectively, allows the development of an equation to find the halffracture as follows: Simplifying, Replacing the dimensionless quantities, Eqs. (3) and (7) By the same token, the intercept of Eq. (20) with Eq. (18), t LBRi , provides another expression to find the half-fracture length: Bilinear flow regime is absent in the plot of Figure 1. Linear, biradial, and radial flow regimes along with the late pseudosteady-state period are seen. The interception points formed by the possible combinations of such periods can be represented schematically in this plot.
Another way to find the half-fracture length comes from the intersection of Eqs. (2) and (10), t RBRi , and Eq. (10) with Eq. (15), so that The intercept point resulting between linear flow and bilinear flow lines given by the governing pressure derivative solutions, Eqs. (18) and (19), can be used to find either half-fracture length or permeability: t BLRi is the intersection of the bilinear pressure derivative line given by Eq. (18) with the radial flow regime line (Eq. (2)). This intersection point serves as the estimation of either permeability or fracture conductivity:

Other estimations
The expressions for determination of the naturally fractured reservoir parameters cannot be included in this chapter for space reasons. However, they can be found in [15,16], which also used intersection points and maximum and minimum data read from the pressure and pressure derivative curve.
Radial flow regime may be absent in short tests run in fractured wells with the sole purpose of determining fractured parameters. For these cases, the skin factor can be estimated from any of the two empirical correlations presented by [27] s ¼ ln r w 1:92173 and s ¼ ln r w x f þ 1:65 À 0:32u þ 0:11u 2 1 þ 0:18u þ 0:064u 2 þ 0:005u 3 , where u ¼ ln C fD : Additionally, fracture conductivity can be read from the plot given in Figure 2.
7. Examples 7.1 Field example [14] presented a field example of a fractured well test. Pressure and pressure derivative data are given in Table 1 and Figure 3. Other relevant data are provided below: Using a commercial well test software, the following parameters were estimated by nonlinear regression analysis: The objective is to compute the hydraulic fracture parameters using the TDS technique and compare results obtained from the regression analysis. Once the pressure and pressure derivative versus time log-log plot is built and reported in Figure 3, the characteristic points are read from such plot as follows:  Table 1. Pressure data for field example (taken from [14]).

Figure 3.
Pressure and pressure derivative against time log-log plot for field example (taken from [14]).
Step 2: Estimate permeability and skin factor Permeability and skin factor are found in Eqs. (8) and (14) to be 0.76 md and À4.68, respectively.

Step 3: Estimate fracture conductivity
Fracture conductivity is estimated using Eqs. (21) and (22): which indicates that the calculation of the fracture conductivity is accurate. Notice that instead of estimating t BLRi the fracture conductivity can be found instead to obtain another value of fracture conductivity; then, Eq. (33) can also be expressed as Find the dimensionless fracture conductivity using Eq. (5): The above value confirms that the fracture has finite conductivity.

Synthetic example
[13] presented a synthetic example of a pressure test run in a bounded homogeneous reservoir with the information given below: A pressure and pressure derivative versus time log-log plot is presented in Figure 4, from which the following characteristic points are read: Step 2: Half-fractured length estimation.
The half-fracture length is estimated with Eq. (12) and confirmed with Eq. (31), as follows: Pressure and pressure derivative vs. time for synthetic example (taken from [13]).

Comments on the results
The two given examples show three aspects of the TDS technique: (1) practical use, (2) accuracy, and (3) self-confirmation.
As shown in the exercises, the process includes defining flow regimes, drawing a few lines, and finally computing the necessary parameters. Contrary to the conventional straight-line method, which requires a plot for each flow regime, TDS technique uses only the pressure and pressure derivative versus time log-log plot. Computations are straight forward. Table 2 summarizes the main parameters obtained in the two worked examples. The results show a good agreement between the calculated results by TDS technique and the results obtained from commercial software packages, for the field case. The results of the half-fracture length for the synthetic case using TDS technique are even better compared to the input value use to simulate the test. This demonstrates that TDS technique is an accurate methodology which has been also presented in many publications, not only in the list reference but also in others not mentioned here.
The last aspect dealt with is self-confirmation. In the field example, three values of half-fracture length and three values of fractured conductivity were found, and for the synthetic example, two values of half-fracture length were estimated from different equations. All the estimations match with the reference values.

Conclusion
It has been shown that TDS technique is a powerful, practical, and accurate tool for well test interpretation because manipulations are easy to do and parameters can be confirmed from different sources from the same pressure test. Compared to reference values, the worked examples provided accurate results of both halffracture length and hydraulic fracture conductivity. Besides being accurate, TDS technique has the great advantage of being able to estimate a given parameter, such as half-fracture length or fracture conductivity, from more than one source or equation. This provides a means of verifying that the estimated parameter is in a good range.

Acknowledgements
The author wishes to express his gratitude to Universidad Surcolombiana for providing him the time to write this chapter.