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# Fluid flow in naturally fractured reservoirs

This article focuses on interpretation of well test data from wells completed in naturally fractured reservoirs. Because of the presence of two distinct types of porous media, the assumption of homogeneous behavior is no longer valid in naturally fractured reservoirs. This article discusses two naturally fractured reservoir models, the physics governing fluid flow in these reservoirs and semilog and type curve analysis techniques for well tests in these reservoirs.

## Naturally fractured reservoir models

Naturally fractured reservoirs are characterized by the presence of two distinct types of porous media: matrix and fracture. Because of the different fluid storage and conductivity characteristics of the matrix and fractures, these reservoirs often are called dual-porosity reservoirs. **Fig. 1** illustrates a naturally fractured reservoir composed of a rock matrix surrounded by an irregular system of vugs and natural fractures. Fortunately, it has been observed that a real, heterogeneous, naturally fractured reservoir has a characteristic behavior that can be interpreted using an equivalent, homogeneous dual-porosity model such as that shown in the idealized sketch.

Several models have been proposed to represent the pressure behavior in a naturally fractured reservoir. These models differ conceptually only in the assumptions made to describe fluid flow in the matrix. Most dual-porosity models assume that production from the naturally fractured system comes from the matrix, to the fracture, and then to the wellbore (i.e., that the matrix does not produce directly into the wellbore). Furthermore, the models assume that the matrix has low permeability but large storage capacity relative to the natural fracture system, while the fractures have high permeability but low storage capacity relative to the natural fracture system. Warren and Root^{[1]} introduced two dual-porosity parameters, in addition to the usual single-porosity parameters, which can be used to describe dual-porosity reservoirs.

Interporosity flow is the fluid exchange between the two media (the matrix and fractures) constituting a dual-porosity system. Warren and Root^{[1]} defined the interporosity flow coefficient, *λ*, as

where *k*_{m} is the permeability of the matrix, k_{f} is the permeability of the natural fractures, and α is the parameter characteristic of the system geometry.

The interporosity flow coefficient is a measure of how easily fluid flows from the matrix to the fractures. The parameter α is defined by^{[2]}

where *L* is a characteristic dimension of a matrix block and *j* is the number of normal sets of planes limiting the less-permeable medium (*j* = 1, 2, 3). For example, *j* = 3 in the idealized reservoir cube model in **Fig. 1**. On the other hand, for the multilayered or "slab" model shown in **Fig. 2**, ^{[3]} *j* = 1. For the slab model, letting *L* = *h*_{m} (the thickness of an individual matrix block), *λ* becomes

The storativity ratio, ^{[2]} *ω*, is defined by

where *V* is the ratio of the total volume of one medium to the bulk volume of the total system and *ϕ* is the ratio of the pore volume of one medium to the total volume of that medium. Subscripts *f* and *f* + *m* refer to the fracture and to the total system (fractures plus matrix), respectively. Consequently, the storativity ratio is a measure of the relative fracture storage capacity in the reservoir.

Many models have been developed for naturally fractures reservoirs. Two common models, pseudosteady-state and transient flow, that describe flow in the less-permeable matrix are presented here. Pseudosteady-state flow was assumed by Warren and Root^{[1]} and Barenblatt *et al.*^{[4]}; others, notably deSwaan, ^{[5]} assumed transient flow in the matrix. Intuition suggests that, in a low-permeability matrix, very long times should be required to reach pseudosteady-state and that transient matrix flow should dominate; however, test analysis suggests that pseudosteady-state flow is quite common. A possible explanation of this apparent inconsistency is that matrix flow is almost always transient but can exhibit a behavior much like pseudosteady-state, if there is a significant impediment to flow from the less-permeable medium to the more-permeable one (such as low-permeability solution deposits on the faces of fractures).

## Pseudosteady-state matrix flow model

The pseudosteady-state flow model assumes that, at a given time, the pressure in the matrix is decreasing at the same rate at all points and, thus, flow from the matrix to the fracture is proportional to the difference between matrix pressure and pressure in the adjacent fracture. Specifically, this model, which does not allow unsteady-state pressure gradients within the matrix, assumes that pseudosteady-state flow conditions are present from the beginning of flow.

Because it assumes a pressure distribution in the matrix that would be reached only after what could be a considerable flow period, the pseudosteady-state flow model obviously is oversimplified. Again, this model seems to match a surprising number of field tests. One possible reason is that damage to the face of the matrix could cause the flow from matrix to fracture to be controlled by a sort of choke (the thin, low-permeability, damaged zone) and, therefore, is proportional to pressure differences upstream and downstream of the choke. In the next two sections, semilog and type-curve analysis techniques are presented for well tests in naturally fractured reservoirs exhibiting pseudosteady-state flow characteristics.

### Semilog analysis technique

The pseudosteady-state matrix flow solution developed by Warren and Root^{[1]} predicts that, on a semilog graph of test data, two parallel straight lines will develop. **Fig. 3** shows this characteristic pressure response.

The initial straight line reflects flow in the fracture system only. At this time, the formation is behaving like a homogeneous formation with fluid flow originating only from the fracture system with no contribution from the matrix. Consequently, the slope of the initial semilog straight line is proportional to the permeability-thickness product of the natural fracture system, just as it is for any homogeneous system. Following a discrete pressure drop in the fracture system, the fluid in the matrix begins to flow into the fracture, and a rather flat transition region appears.

Finally, the matrix and the fracture each reach an equilibrium condition, and a second straight line appears. At this time, the reservoir again is behaving like a homogeneous system, but now the system consists of both the matrix and the fractures. The slope of the second semilog straight line is proportional to the total permeability-thickness product of the matrix/fracture system. Because the permeability of the fractures is much greater than that of the matrix, the slope of the second line is almost identical to that of the initial line.

Similar shapes are predicted for pressure buildup tests (**Fig. 4**). The lower curve, A, represents the ideal buildup test plot predicted by Warren and Root. ^{[1]} The shape of a semilog plot of test data from a naturally fractured reservoir is almost never the same as that predicted by Warren and Root’s model. Wellbore storage almost always obscures the initial straight line and often obscures part of the transition region between the straight lines. The upper curve, *B*, in **Fig. 4** shows a more common pressure response.

The reservoir permeability-thickness product, *kh* [actually the *kh* of the fractures, or (*kh*)_{f}, because (*kh*) m is usually negligible], can be obtained from the slope, *m*, of the two semilog straight lines. Storativity, *ω*, can be determined from their vertical displacement, *δp*. The interporosity flow coefficient, *λ*, can be obtained from the time of intersection of a horizontal line, drawn through the middle of the transition curve, with either the first or second semilog straight line. ^{[2]}

When semilog analysis is possible (i.e., when the correct semilog straight line can be identified), the following procedure is recommended for semilog analysis of buildup or drawdown test data from wells completed in naturally fractured reservoirs. Although presented in variables for slightly compressible fluids (liquids), the same procedure is applicable to gas well tests when the appropriate variables are used.

- From the slope of the initial straight line (if present) or final straight line (more likely to be present), determine the permeability-thickness product,
*kh*. In either case, the slope,*m*, is related to the total*kh*of the system, which is essentially all in the fractures. The permeability-thickness product is given by - If both initial and final straight lines can be identified (or the position of the initial line can at least be approximated) and the pressure difference,
*δp*, established, then the storativity ratio,*ω*is calculated from

If the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by *t*_{l} and *t*_{2}, respectively, the storativity ratio may also be calculated from

For a buildup test, where the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by [(*t*_{p} + Δ*t*)/Δ*t*]_{1} and [(*t*_{p} + Δ*t*)/Δ*t*]_{2}, respectively, the storativity ratio may be calculated from

The interporosity flow coefficient, *λ*, is calculated^{[2]} for a drawdown test by

or for a buildup test by

where *γ* = 1.781.

The terms (*ϕV*)_{m} and (*c*_{t})_{m} in **Eq. 10** are obtained by conventional methods. A porosity log usually reads only the matrix porosity (not the fracture porosity) and thus gives *ϕ*_{m}, while (*c*_{t})_{m} is the sum of *c*_{o}*S*_{o}, *c*_{g}*S*_{g}, *c*_{w}*S*_{w}, and *c*_{f}. *V*_{m} usually can be assumed to be essentially 1.0. From the definition of *ω* in **Eq. 4**,

The second semilog straight line should be extrapolated to *p*_{1hr}, and the skin factor is

where Δ*p*_{1hr} is equal to (*p*_{i} – *p*_{1 hr}) for a drawdown test or [*p*_{1 hr} - *p*_{wf}(Δ*t*=0)] for a buildup test.

- The second semilog straight line should be extrapolated to
*p** (**Fig. 5**). From*p**, can be found using conventional methods (such as the Matthew-Brons-Hazebroek*p** method).

### Type curve analysis technique

Particularly because of wellbore-storage distortion, type curves are quite useful for identifying and analyzing dual-porosity systems. **Fig. 6** shows an example of the Bourdet *et al.*^{[6]} type curves developed for pseudosteady-state matrix flow. Initially, test data follow a curve for some value of *C*_{D}*e*^{2s} where *C*_{D} is the dimensionless wellbore storage coefficient. In **Fig. 6**, the earliest data for the well follow the curve for *C*_{D}*e*^{2s} = 1. The data then deviate from the early fit and follow a transition curve characterized by the parameter *λe*^{-2s}. In **Fig. 6**, the data follow the curve for *λe*^{–2s} = 3×10^{–4}. When equilibrium is reached between the matrix and fracture systems, the data then follow another *C*_{D}*e*^{2s} curve. In the example, the later data follow the *C*_{D}*e*^{2s} = 0.1 curve.

At earliest times, the reservoir is behaving like a homogeneous reservoir with all fluid originating from the fracture system. During intermediate times, there is a transition region as the matrix begins to produce into the fractures. At later times, the system again is behaving like a homogeneous system with both matrix and fractures contributing to fluid production.

**Fig. 7** illustrates the derivative type curves for a formation with pseudosteady-state matrix flow. ^{[6]} The most notable feature, characteristic of naturally fractured reservoirs, is the dip below the homogeneous reservoir curve. The curves dipping downward are characterized by a parameter *λC*_{D}/*ω* (1 − *ω*), while the curves returning to the homogeneous reservoir curves are characterized by the parameter *λC*_{D}/*ω* (1 − *ω*). Test data that follow this pattern on the derivative type curve can reasonably be interpreted as identifying a dual-porosity reservoir with pseudosteady-state matrix flow (a theory that needs to be confirmed with geological information and reservoir performance). Pressure and pressure derivative type curves can be used together for analysis of a dual-porosity reservoir. The pressure derivative data are especially useful for identifying the dual-porosity behavior. Manual type-curve analysis for well in naturally fractured reservoirs is tedious, and the interpretation involved is difficult. Most current analysis uses commercial software.

## Transient matrix flow model

The more probable flow regime in the matrix is unsteady-state or transient flow; that is, flow in which an increasing pressure drawdown starts at the matrix/fracture interface and moves further into the matrix with increasing time. Only at late times should pseudosteady-state flow be achieved, although a matrix with a thin, low-permeability damaged zone at the fracture face may behave as predicted by the pseudosteady-state matrix flow model even though the flow in the matrix is actually unsteady-state.

A semilog graph of test data for a formation with transient matrix flow has a characteristic shape different from that for pseudosteady-state flow in the matrix. Three distinct flow regimes have been identified that are characteristic of dual-porosity reservoir behavior with transient matrix flow. **Fig. 8** illustrates these flow regimes on a semilog graph as regimes 1, 2, and 3.

Flow regime 1 occurs at early times during which all production comes from the fractures. Flow regime 2 occurs when production from the matrix into the fracture begins and continues until the matrix-to-fracture transfer reaches equilibrium. This equilibrium point marks the beginning of flow regime 3, during which total system flow, from matrix to fracture to wellbore, is dominant. The same three flow regimes appear when there is pseudosteady-state matrix flow. The duration and shape of the transition flow regimes, however, is considerably different for the two matrix flow models.

Serra *et al.*^{[3]} observed that pressures from each of these flow regimes will plot as straight lines on conventional semilog graphs. Flow regimes 1 and 3, which correspond to the classical early- and late-time semilog straight-line periods, respectively, have the same slope. Flow regime 2 is an intermediate transitional period between the first and third flow regimes. The semilog straight line of flow regime 2 has a slope of approximately one-half that of flow regimes 1 and 3. If all or any two of these regimes can be identified, then a complete analysis is possible using semilog methods alone. Certain nonideal conditions, however, may make this analysis difficult to apply.

Flow regime 1 often is distorted or obscured by wellbore storage, which often makes this flow regime difficult to identify. Flow regime 2, the transition, also may be obscured by wellbore storage. Flow regime 3 sometimes requires a long flow period followed by a long shut-in time to be observed, especially in formations with low permeability. Furthermore, boundary effects may appear before flow regime 3 is fully developed.

### Semilog analysis techniques

Serra *et al.*^{[3]} presented a semilog method for analyzing well test data in dual-porosity reservoirs exhibiting transient matrix flow (**Fig. 8**). They found that the existence of the transition region, flow regime 2, and either flow regime 1 or flow regime 3 is sufficient to obtain a complete analysis of drawdown or buildup test data. Further, they assumed unsteady-state flow in the matrix, no wellbore storage, and rectangular matrix-block geometry, as Fig. 2 shows. The rectangular matrix-block geometry is adequate, although different assumed geometries can lead to slightly different interpretation results.

The major weakness of the Serra *et al.* method is that it assumes no wellbore storage. In many cases, flow regimes 1 and 2 are partially or even totally obscured by wellbore storage, making analysis by the Serra *et al.* method impossible or difficult. Despite this limitation, the Serra *et al.* method has great practical value when used in conjunction with type-curve methods. These calculations of the Serra *et al.* method apply to both buildup and drawdown test data and are applicable for well test analysis of slightly compressible liquids and gas well tests.

### Type curve analysis technique

Bourdet *et al.*^{[6]} presented type curves for analyzing well tests in dual-porosity reservoirs including the effects of wellbore storage and unsteady-state flow in the matrix. The type curves are useful supplements to the Serra *et al.* semilog analysis. **Fig. 9** gives an example of the pressure and pressure derivative type curves for transient matrix flow. Early (fracture-dominated) data are fit by a *C*_{D}*e*^{2s} value indicative of homogeneous behavior. Data in the transition region are fit by curves characterized by a parameter *β*′. Finally, data in the homogeneous-acting, fracture-plus-matrix flow regime are fit by another *C*_{D}*e*^{2s} curve.

On the derivative type curve, early data also are fit by a derivative curve reflecting homogeneous behavior. **Fig. 10** shows an actual example. If wellbore-storage distortion ceases before the transition region begins (which did not happen in the example but is possible in other cases), the derivative data will be horizontal and should be aligned with the (*t*_{D}/*C*_{D})*p*_{D}′ = 0.5 curve. However, if the transition region is present (recall that its semilog slope is half that of the middle-time straight line), the derivative curve will flatten and should be aligned with the (*t*_{D}/*C*_{D})*p*_{D}′ = 0.25 curve as shown in this example. The homogeneous (fracture-plus-matrix) data should, after wellbore distortion has ceased and before boundary effects have appeared, be horizontal on the derivative type curve and should be aligned with the (*t*_{D}/*C*_{D})*p*_{D}′ = 0.5 curve as this example shows.

Manual type-curve matching is tedious and difficult, especially with the interpolation involved. Analysis ordinarily uses commercially available software to analyze these kinds of tests after the reservoir model has been identified.

## Nomenclature

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}^{1.6}Warren, J.E. and Root, P.J. 1963. The Behavior of Naturally Fractured Reservoirs. SPE J. 3 (3): 245–255. SPE-426-PA. http://dx.doi.org/10.2118/426-PA. - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Gringarten, A.C. 1984. Interpretation of Tests in Fissured and Multilayered Reservoirs With Double-Porosity Behavior: Theory and Practice. J Pet Technol 36 (4): 549-564. SPE-10044-PA. http://dx.doi.org/10.2118/10044-PA. - ↑
^{3.0}^{3.1}^{3.2}^{3.3}^{3.4}Serra, K., Reynolds, A.C., and Raghavan, R. 1983. New Pressure Transient Analysis Methods for Naturally Fractured Reservoirs(includes associated papers 12940 and 13014 ). J Pet Technol 35 (12): 2271-2283. SPE-10780-PA. http://dx.doi.org/10.2118/10780-PA. - ↑ Barenblatt, G.E., Zheltov, I.P., and Kochina, I.N. 1960. Basic Concepts in the Theory of Homogeneous Liquids in Fissured Rocks. J. Appl. Math. Mech. 24: 1286-1303.
- ↑ de Swaan O., A. 1976. Analytical Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing. SPE J. 16 (3): 117–122. SPE-5346-PA. http://dx.doi.org/10.2118/5346-PA.
- ↑
^{6.0}^{6.1}^{6.2}^{6.3}^{6.4}^{6.5}^{6.6}Bourdet, D. et al. 1984. New Type Curves Aid Analysis of Fissured Zone Well Tests. World Oil (April).

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