Pressure data for field example (taken from ).
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.
- bilinear flow
- linear flow
- elliptical flow
- half-length fracture
- fracture conductivity
- hydraulic fracturing
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 , 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,  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 , a fine semi-analytical solution was introduced for describing the well-pressure behavior in hydraulically fractured wells. This solution was then applied in  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  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 . 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
This chapter is devoted to the application of
The pioneer publication on the
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:
The dimensionless pressure and pressure derivative parameters for oil reservoirs are given by
Finally, the dimensionless fracture conductivity introduced in  is defined as
It is observed from Eq. (5) that the two key parameters of a hydraulic fracture are the half-fracture length,
3. 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 half-fracture length for a naturally fractured formation. New expressions for the elliptical flow regime introduced in  excluding reservoir drainage area are given by.
TDS technique is based on drawing a straight line throughout a given flow regime; then, the user is expected to read the pressure, Δ
When bilinear flow is unseen, fracture conductivity can be found with an expression presented in 
 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
The derivation of Eq. (16) follows a similar idea as that presented later in Section 4 for the use of the points of intersection.
4. 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 
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:
5. 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,
Solving for the half-fracture from Eq. (27), we readily obtain
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.
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:
6. 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 
Additionally, fracture conductivity can be read from the plot given in Figure 2.
7.1 Field example
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
18.104.22.168 Step 1: Obtain the characteristic points
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:
22.214.171.124 Step 2: Estimate permeability and skin factor 126.96.36.199 Step 3: Estimate fracture conductivity
which indicates that the calculation of the fracture conductivity is accurate. Notice that instead of estimating
188.8.131.52 Step 4: Half-fractured length and dimensionless fracture conductivity estimation
Solve for half-fracture length from Eq. (13) and find this:
Find the dimensionless fracture conductivity using Eq. (5):
The above value confirms that the fracture has finite conductivity.
7.2 Synthetic example
 presented a synthetic example of a pressure test run in a bounded homogeneous reservoir with the information given below:
Estimate the half-fracture length by the
184.108.40.206 Step 1: Obtain the characteristic points.
A pressure and pressure derivative versus time log–log plot is presented in Figure 4, from which the following characteristic points are read:
8. Comments on the results
The two given examples show three aspects of the
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
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.
It has been shown that
The author wishes to express his gratitude to Universidad Surcolombiana for providing him the time to write this chapter.
|A||Draining area (ft2)|
|B||Oil volume factor (rb/STB)|
|CfD||Dimensionless fracture conductivity|
|h||Formation thickness (ft)|
|k||Formation permeability (md)|
|kfwf||Fracture conductivity (md-ft)|
|Pwf||Well-flowing pressure (psi)|
|q||Oil flow rate (STB/D)|
|qg||Gas flow rate (MSCF/D)|
|rw||Wellbore radius (ft)|
|xf||Half-fracture length (ft)|
|t||Test time (h)|
|tp||Production time (h)|
|t*∆P′||Pressure derivative (psi)|
|tD*PD’||Dimensionless pressure derivative|
|λ||Interporosity flow parameter|
|ξ||Variable to identify homogeneous (ξ = 1) or heterogeneous (ξ = ω) reservoirs|
|ω||Dimensionless storativity coefficient|
|BL1||Bilinear at 1 h|
|BR1||Birradial at 1 h|
|DA||Dimensionless based on area|
|Dxf||Dimensionless based on half-fractured length|
|DLBRi||Dual linear-birradial intersection|
|RPi||Intersect of radial-pseudosteady-state lines|