Abstract The producing capacity of a naturally fractured reservoir is governed by interporosity flow since most of the fluid resides in therock matrix. A distributed formulation is introduced to account forvariability in the strength of interporosity flow across the reservoir. The model can account for deterioration or enhancement of reservoir producibility resulting from development of interporosity skin andclosure of fractures or hydraulic fracturing and acidizing. This formulation suggests a reason why certain naturally fractured reservoirs do not experience sudden rate decline as predicted byclassical double porosity models.
Introduction Interporosity flow is a process by which fluid flows from the'tight' rock matrix into the fracture network which is in communication with the wellbore. This process is time dependent because the fluid expulsion capacity of the matrix changes as fluid flows from the portions of the reservoir uncontacted by fractures to matrix that is adjacent to fractures. The strength of interporosity flow can change over time due to reduction of fracture permeability, precipitation of minerals at the matrix-fracture interface, or fluid redistribution via fractures. The strength of interporosity flow can also vary throughout the reservoir due to geologic heterogeneity. Variability in lithology, degree of diagenesis, bed thickness, fracture pervasiveness and other factors resulting nonuniformity of matrix-fracture property contrast. Hence, a naturally fractured reservoir is not depleted uniformly, i.e., it is differentially depleted in that certain portions of the matrix may be depleted while the rest may be partially depleted or even remain virgin. The Warren and Root model and other double porosity models of naturally fractured reservoirs assume that a coefficient of interporosity flow,, expresses matrix-fracture fluid transport capacity. Coefficient is inversely related to the response-time coefficient of the matrix.
= .........................(1)
Parameter is the response time coefficient which is a measure of the time delay associated with onset of matrix flow. Therefore, flow is fracture dominated when (dimensionless) producing time TD << and total matrix-fracture flow prevails when TD » and interporosity flow is significant when T D =. Thus, 'large' (small) indicates a dormant matrix whereas 'small' (large) indicates the matrix responds quickly. The Warren and Root formulation assumes the interporosity flow coefficient is uniform across the reservoir (i.e., single valued)and is independent of time. The consequences of these assumptions areas follows:Because interporosity flow is governed by a single valued coefficient (or), it is not possible to account for variability in the response-time behavior of the matrix across the reservoir, i.e., a spectrum from fast to slow response.
Because the Warren and Root formulation models interporosity flow in a nondistributed (concenterated) form, the transientrate and pressure response exhibit characteristic profiles quite distinct from that of nonfractured homogeneous reservoirs. Fractured reservoirs, however, often exhibit transient response similar to that of nonfractured homogeneous reservoirs.
Figure 1 illustrates a typical rate decline profile for constant pressure production using the Warren and Root formulation. The sudden drop in flow rate following fracture dominated flow is characteristic of the response. Although many wells producing from naturally fractured reservoirs exhibit such behavior, many others do not. Chen et al examined the production history of 1235 wells in the fractured Austin chalk and found 56% of the wells exhibit rapid production decline (Type I), 13.4% exhibit gradual decline (Type II), and 31.6% could not be analyzed. Gradual rate decline is also reported for certain wells producing from the fractured Monterey formation.
Figure 2 illustrates a typical pressure drawdown response based on the Warren and Root model. The time rate of change of pressure (semilog derivative) is illustrated in Figure 3. A rapid change in the rate of pressure drawdown is indicated following fracture dominated flow. Although few examples of pressure response similar to that depicted in Figure 2 have been reported in the literature, data from naturally fractured reservoirs often do not exhibit such pronounced features and might resemble the response of nonfractured homogeneous reservoirs.
Because the nondistributed formulation of Warren and Root yields a shorter interporosity flow period compared to the distributed formulation (discussed below), pace of recovery is overestimated.