Analysis of Pressure Transient Tests in Naturally Fractured Reservoirs

Pressure transient tests in naturally fractured reservoirs often exhibit non-uniform responses. Different techniques can be used to analyze the pressure behavior in dual porosity reservoirs in an attempt to correctly characterize reservoir properties. In this paper, the pressure transient tests in naturally fractured reservoirs were analyzed using conventional semi-log analysis, type curve matching (using commercial software) and Tiab’s direct synthesis (TDS) technique. In addition, the TDS method was applied in case of a naturally fractured formation with a vertical hydraulic fracture. These techniques were applied to a single layer naturally fractured reservoir under pseudosteady state matrix flow. By studying the unique characteristics of the different flow regimes appear on the pressure and pressure derivative curves, various reservoir characteristics can be obtained such as permeability, skin factor, and fracture properties. For naturally fractured reservoirs, a comparison between the results semi-log analysis, software matching, and TDS method is presented. In case of wellbore storage, early time flow regime can be obscured that lead to incomplete semi-log analysis. Furthermore, the type curve matching usually gives a non-uniqueness solution as it needs all the flow regimes to be observed. However, the direct synthesis method used analytical equation to calculate reservoir and well parameters without type curve matching. For naturally fractured reservoirs with a vertical fracture, the pressure behavior of wells crossed by a uniform flux and infinite conductivity fracture is analyzed using TDS technique. The different flow regimes on the pressure derivative curve were used to calculate the fracture half-length in addition to other reservoir properties. The results of different cases showed that TDS technique offers several advantages compared to semi-log analysis and type curve matching. It can be used even if some flow regimes are not observed. Direct synthesis results are accurate compared to the available core data and the software matching results.


Introduction
The analysis of pressure data received during a well test in dual porosity formation has been widely used for reservoir characterization. Conventional semi-log analysis and log-log type curve methods are the early techniques used to analyze pressure transient data. However, both methods need certain criteria to give accurate results, such as; all flow regimes must be identified in the pressure and pressure derivative plot. In case some flow regimes are not identified, type curve matching will give a non-uniqueness solution and is essential trial and error, and semi-log analysis cannot be completed. Tiab [1] used a new method to analyze pressure transient tests, called "Direct Synthesis Technique". This method can calculate different reservoir paramters without type curve matching by using pressure and pressure derivative loglog plots. In 1994, Tiab [2] extended the work to vertically fractured wells in closed system. Engler and Tiab [3] developed direct synthesis method to analyze pressure transient tests in dual porosity formation without using type curve matching. They used analytical and empirical correlations to calculate the naturally fractured reservoir parameters. Jalal [4] discussed the analytical solutions of wells in dual porosity reservoirs with a vertical fracture. The direct synthesis method offers manys advatages in analyzing pressure transient tests.
The objective of this paper is to analyze pressure transient in naturally fractured reservoirs using: conventional semi-log analysis, type curve matching (using commercial software), and Tiab's direct synthesis method to correctly characterize the reservoir properties. These techniques were applied to naturally fractured reservoirs, with and without hydraulic (vertical) fracture.

Properties of Dual Porosity Formation
The Dual porosity reservoir consists of primary and secondary porosity which are the matrix and fractures. Warren and Root [5] defined the fractured reservoirs by two key parameters, ω and λ. These dimensionless paramters are defined as follows: The relative storativity, ( ) The interporosity flow parameter, Where the shape factor α, ft -2 , depends on the matrix block geometry (horizontal slab or spherical matrix block). By assuming that the reservoir is infinite acting and producing a single phase, slightly compressible fluid with pseudosteady state matrix flow, the pressure solution is given by [6]:

Conventional semi-log analysis
Naturally fractured reservoirs give two parallel semi-log straight lines in plot of drawdown and build-up tests as shown in Figure 1.

Permeability thickness product:
The permeability thickness product of the total system (actually of the fractures as the matrix permeability can be neglected) can be calculated from the slope of the initial or final straight line, m.
1. The relative storativity ω can be calculated from the pressure difference, ΔP, between the initial and final straight lines when both of them can be identified.
2. By drawing a horizontal line through the middle of transition period to intersect with both semi-log straight lines, the times of intersection with the first and the second semi-log straight lines are donated by t 1 and t 2 , respectively. The storativity ratio also can be determined as follows [7]: 3. The interporosity flow coefficient, λ, can be calculated by [8]: For drawdown tests:

Direct synthesis technique
Direct synthesis method uses a log-log plot of pressure and pressure derivative data versus time to calculate various reservoirs and well parameters. It uses the pressure derivative technique to identify reservoir heterogeneities. In this method, the values of the slopes, intersection points, and beginning and ending times of various straight lines from the log-log plot can be used in exact analytical equations to calculate different parameters as it is shown in the following procedures [6]: Infinite Acting Reservoir without Wellbore Storage The relative storativity can also be calculated by using the characteristic times as in the following equations: Where t e1 is the end time of the early infinite acting radial flow line, t b2 is the beginning time of the late infinite acting radial flow line and t min is the minimum time.

The interporosity flow parameter:
The interporosity flow parameter can be also obtained from the characteristic times as following:  Where S T is the product of the average bulk porosity (from cores or logs) and the average compressibility. λ can be also calculated from the minimum coordinates: In case ω less than 0.05, late transition period unit slope straight line is well observed. The interporosity flow parameter can be calculated from: Where t us,i the intersection of the transition period unit slope line with the infinite acting radial flow line.

Skin factor:
The skin factor can be calculated from the early or late time radial flow pressure and pressure derivative data by using the following equations: Where r 1 is any point on the early horizontal radial flow line and r 2 is any point on the late horizontal radial flow line.

Infinite Acting Reservoir with Wellbore Storage
Wellbore storage effects can obscure early flow regimes on log-log plot of pressure and pressure derivative versus time. It is represented by early time unit slope straight line on the log-log plot. This unit slope period is followed by a peak on the pressure derivative curve as shown in Figure 2. The effect of wellbore storage can affect the minimum coordinates of the pressure derivative curve and cause the appearance of "pseudo-minimum" coordinates. Therefore, the effect of wellbore storage should be investigated prior to the analysis to know whether the observed minimum is the real minimum or the pseudo-minimum. For (t dw ) min /(t dw ) x ≥ 10, the wellbore storage doesn't affect the minimum coordinates. [(t dw ) x is the dimensionless time of the peak point, [6].
In case the minimum coordinates are not affected by wellbore storage, calculate the reservoir parameters using the following procedure [6]: 1-Determine the fracture permeability using the late time radial flow line.

Semi-log analysis
Horner plot is shown in Figure 3. This figure depicts the early points that are affected by wellbore storage, however, the first straight line can be observed clearly. The figure shows two parallel straight lines that proves the dual porosity behavior. Therefore, the conventional semi-log analysis can be used to estimate reservoir parameters.
The fracture permeability can be calculated from the slope of the second straight line (m) to give: The storativity ratio (ω) can be calculated from the vertical distance between the two straight lines (Δp) and the slope (m): 2-Calculate the wellbore storage coefficient from the early time unit slope using the following equations: Where t, Δp are any point on the unit slope line. (Δp=p i -p wf for drawdown and Δp=p ws -p wf (Δt=0) for buildup tests) The wellbore storage coefficient can also be calculated from the intersection time of the early time unit slope with the infinite acting radial flow line (t i ).
3-Determine the ω and λ as outlined before.
4-Determine skin factor from the late time radial flow pressure and pressure derivative ratio.
If the minimum coordinates are influenced by wellbore storage, the interporosity flow parameter and the relative storativity can be calculated using the following equations: Determine λ from the peak to minimum time ratio or from the peak to radial pressure derivative ratio: Calcualte ω from the peak to beginning of second radial flow line time ratio:

Case 1
This case presents an oil field in Iran. A build-up test is conducted on a well from naturally fractured reservoir. The average core permeability received from the Iranian oil company ranges from 4 to 6 md. The well was flowing for 72 hours with q=2300 STB/day before shut-in for a build-up test. The build-up data are given in Table 1. The following reservoir and well data are also known:

Direct Synthesis Technique
The log-log plot of pressure and pressure derivative shown in Figure 4. It is clear that there is a wellbore storage with an early time unit slope and the early radial flow period is well defined. However, the late radial flow period not last for long time. The data exhibit a unique behavior which is indicative of a naturally fractured reservoir. (t*ΔPʹ w ) r2 =99.7 psia t r2 =32 hr, ΔP r2 =614 psia (t*ΔPʹ w ) min =27.56 psia, t min =8.8 hr t us =0.054 hr, ΔP us= 49.6 psia t x= 0.42 hr, t b2 =28 hr (t*ΔPʹ w ) US =49.6 psia, The effect of the WBS on the minimum derivative coordinates can be defined by calculating the ratio (t dw ) min / (t dw ) x t min /t x= 8.8/0.42=20.95 (>10).
Therefore the minimum derivative coordinates are the real minimum and not affected by wellbore storage.  Comparison of the results of conventional semi-log analysis, direct synthesis technique, and type curve matching is shown in Table 2. The results of the semi-log analysis are only matching with the direct synthesis and software results in permeability. However, the storage coefficient and the interporosity flow parameter are inaccurate. On the other side, the direct synthesis technique and the software results show an excellent match in all reservoir parameters.

Naturally fractured reservoirs with a vertical fracture
The pressure behavior of a dual porosity formation intersected by uniform flux and infinite conductivity fracture can be investigated using log-log plots of pressure and pressure derivative functions. The direct synthesis technique can be used to calculate reservoir parameters such as skin, wellbore storage, permeability, interporosity flow parameter, relative storativity and half-fracture length without type curve matching. The applied assumptions are: the reservoir is isotropic, horizontal, and has constant thickness and fracture permeability. The fractured well is producing at constant rate with constant viscosity, slightly compressible fluid. In addition, the fracture fully penetrates the vertical extent of the formation and has the same length in both sides of the well. A pseudosteady state interporosity flow between the matrix and the fracture system is also assumed. The equation of pressure derivative during this flow regime is:

Uniform Flux Fracture
By taking logarithm of both sides of the equation gives: Based on Eq. (34) the log-log plot of pressure derivative versus time gives half slope straight line during the linear flow period. The fracture half-length can be calculated by: is the value of pressure derivative at t=1hr on the linear flow line.

2) Pseudoradial flow period:
The infinite acting radial flow period is dominated only for (X e /X f ) > 8, as shown in Figure 5. This flow regime is identified by a horizontal straight line on the pressure derivative plot and can be used to calculate permeability and skin [4].
The pressure derivative equation during this flow regime is: The above equation in dimensional from yields: R stands for radial flow. Solving the above equation for permeability gives: The skin can be determined by: 3) Pseudosteady state flow period: In case of a vertically fractured well inside a closed system, a third straight line of unit slope appears. This line corresponds to the pseudosteady state flow regime is used to calculate the drainage area and shape factor.
The pressure derivative equation describing this flow period is: By taking logarithm of both sides of the above equation, the dimensional form is: By substituting t=1hr, the drainage area can be calculated using the following equation:

4) Transition period:
The transition can occur during the infinite acting radial flow as shown in Figure 5. In this case, the relative storativity, ω, and the interporosity flow parameter, λ, can be estimated by several methods as previously described in the previous section. If the transition takes place during the linear flow period as shown in Figure 6, two parallel straight lines of slope equal 0.5 can be observed. The first line represents the expansion of the fracture network, this flow period is called "fracture storage dominated flow period". While the second line appears during the total system dominated flow period (for this period ω=1). Also, a straight line of unit slope is observed during late transition period. The intersection time of the straight lines of different flow regimes have been used in several equations to calculate reservoir parameters in case one of the flow regimes is missing or for verification purposes. These equations are presented in the following procedure: Step 1 -Plot the pressure difference ΔP and the pressure derivative (t*ΔPʹ w ) versus time on log-log plot and identify different flow regimes.
Step 3 -Calculate ω and λ as outlined before.
Step 4 -If the transition occur during linear flow regime and the two parallel straight lines of slope 0.5 observed, verify ω using the following two equations: where 2L1 stands for the linear flow at the total system dominated regime, and L1 stands for fracture storage dominated flow regime.
where t 2LUSi stands for the intersection point between the late transition period unit slope line and total system dominated flow line, and t LUSi stands for the intersection point between the late transition period unit slope line and the fracture storage dominated flow period.
Step 5 -Read the value of (t*Δpʹ w ) at time 1 hr from the linear flow line (extrapolated if necessary), (t*Δpʹ w ) L1 .
Step 6 -Calculate the fracture half-length, X f , from the linear flow straight line (Eq. 35). If the linear flow not observed (due to wellbore storage or noise), then fracture half-length can be calculated from the half slope pressure Δp w instead of pressure derivative as then draw a straight line of slope 0.5 parallel to the pressure straight line to cross the (t*Δpʹ w ) L1 .
Step 7 -Determine the intersection between the linear and radial flow line t LRi from the log-log plot of the derivative (t*Δpʹ w ) curve.
Step 8 -Calculate the ratio Compare this ratio with the previously calculated values of X f and K f . If the two ratios are nearly equal, then the values are correct. If they are different, shift one or both straight lines then repeat the previous steps until their values approach.
Step 9 -Determine the value of (t*Δpʹ w ) PSS1 from the pseudosteady state line and find the drainage area A from: Step 10 -Read the intersection time of the infinite acting line and the pseudosteady state line (t RPSSi ) from the plot and calculate the drainage area A: Areas from steps 10 and 11 should be equal. If they are not equal, shift the lines left or right and repeat the calculations.
Step 11 -Determine the interporosity flow parameter after stimulation (λ f ) from: Step 12 -Verify (λ f ) using the late transition period unit slope line by the following equations: where t RUSi stands for the intersection time between the late transition period unit slope line and the infinite acting line.