Fluid flow in naturally fractured reservoirs: Difference between revisions

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<ref name="r1">_</ref> 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<ref name="r1">_</ref> defined the interporosity flow coefficient, ''λ'', as

The interporosity flow coefficient is a measure of how easily fluid flows from the matrix to the fractures. The parameter α is defined by<ref name="r2">_</ref>

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''', <ref name="r3">_</ref> ''j'' = 1. For the slab model, letting ''L'' = ''h''_''m'' (the thickness of an individual matrix block), ''λ'' becomes

The storativity ratio, <ref name="r2">_</ref> ''ω'', is defined by

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<ref name="r1">_</ref> and Barenblatt ''et al.''<ref name="r4">_</ref>; others, notably deSwaan, <ref name="r5">_</ref> 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).

The pseudosteady-state matrix flow solution developed by Warren and Root<ref name="r1">_</ref> predicts that, on a semilog graph of test data, two parallel straight lines will develop. '''Fig. 3''' shows this characteristic pressure response.

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. <ref name="r1">_</ref> 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. <ref name="r2">_</ref>

The interporosity flow coefficient, ''λ'', is calculated<ref name="r2">_</ref> for a drawdown test by

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.''<ref name="r6">_</ref> type curves developed for pseudosteady-state matrix flow. Initially, test data follow a curve for some value of ''C''_''D''''e''^2''s'' 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''^2''s'' = 1. The data then deviate from the early fit and follow a transition curve characterized by the parameter ''λe''^-2''s''. In '''Fig. 6''', the data follow the curve for ''λe''^–2''s'' = 3×10^–4. When equilibrium is reached between the matrix and fracture systems, the data then follow another ''C''_''D''''e''^2''s'' curve. In the example, the later data follow the ''C''_''D''''e''^2''s'' = 0.1 curve.

'''Fig. 7''' illustrates the derivative type curves for a formation with pseudosteady-state matrix flow. <ref name="r6">_</ref> 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.

Serra ''et al.''<ref name="r3">_</ref> 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.

Serra ''et al.''<ref name="r3">_</ref> 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.

Bourdet ''et al.''<ref name="r6">_</ref> 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''^2''s'' 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''^2''s'' curve.