A Method for Correcting Dimensionless Fracture Conductivity for Non-Darcy Flow Effects

Abstract

Summary This paper provides a method of extending the concepts of dimensionlessfracture conductivity to the design of fracture conductivity in wells to beproduced at rates that cause non-Darcy flow down the fracture. Such flow ratesare common in gas wells and occasionally occur in oil wells. The correctionrequires knowledge of the expected producing rate and calculation of a Reynoldsnumber characterizing flow in the fracture near the wellbore. Dimensionless fracture conductivity, a concept introduced in its currentform in the late 1970's, has become a key parameter in hydraulic fracturingtreatment design. It is used to determine the adequacy of fracture conductivitytaking into consideration formation permeability, fracture length and width, and proppant permeability. permeability. A dimensionless fracture conductivityof 10 typically is assumed to provide a fracture so permeable that formationpermeability, not fracture permeability, controls the flow resistance. Unlessfracture conductivity is corrected for non-Darcy flow effects before thisguideline is used, however, this assumption may provide, ion the case of gaswells, a serious deficiency in fracture conductivity and a stimulated well thatfalls short of the expected productivity. A simplified method for correctingboth the proppant-pack permeability and dimensionless fracture conductivity fornon-Darcy permeability and dimensionless fracture conductivity for non-Darcyflow effects is discussed, and a design equation to ensure adequateconductivity in the fracture to avoid flow restriction is provided. The methodfor correcting the dimensionless fracture conductivity, although approximates, makes this correction at the point where the fracture interests the wellbore. At that point, the non-Darcy effect is the greatest, and this choice ensuresadequacy of fracture conductivity.. The correction of proppant permeability fornon-Darcy flow effects is exact to the extent that the Forchheimer equationcorrectly describes such flow behavior. A problem in the design of hydraulicfracturing treatments is providing sufficient conductivity in the fracture thatformation, not providing sufficient conductivity in the fracture thatformation, not fracture, permeability controls the overall flow resistance. With the introduction in the late 1970's of well-testing methods that permittedanalysis of wells producing from vertical fractures with finite permittedanalysis of wells producing from vertical fractures with finite fractureconductivity, a method for quantifying the conductivity needed in the fracturebecame available. Cinco-Ley et al., Agawal et al., and others presented suchmethods. To quantify the flow capacity of vertical fractures, Cinco-Ley et al.used the dimensionless fracture flow conductivity, (1) where w-fracture width (assumed to be constant throughout the fracture), permeability of the propped fracture under down-hole stress conditions, lengthof one wing of the fracture, and k=formation permeability. This expression forfracture conducetivity is dimensionless; and dimensionally consistent set ofunits may be used. The particular grouping of variables in Eq.1 is aconsequence of the differential equations required in the solution of the flowproblem posed by Cinco-Ley et al. Hence, the term re is an integral part ofthat grouping and should be retained in the use of with figures in Cinco-Ley etal.'s paper of Fig.1 here. The relationship between and the dimensionlesspressure drop is shown in Fig.1 (adapted from Ref.1). Note that =10 merges with Cfd=100 at a dimensionless time of about 1. Both merge with the curve forinfinite fracture conductivity (not shown in Fig.1) at a dimensionless time ofabout 10. Thus because Cfd=10 ultimately implies infinite fractureconductivity, it has become common practice to design fractures treatments withsufficient conductivity to meet this criterion. Montgomery and Steansondescribed such a design procedure in detail. The major limitation of the Cince-Ley et al. analysis is that it is limited to Darcy flow conditions. Non-Darcy flow, however, is commonly encountered in gas wells because of thelow viscosity of gas (about 1/100 that of oil) and the correspondingly higher Reynolds numbers. Thus, a design procedure is needed where non Darcy flowconditions prevail. Guppy et al. presented a welltesting procedure to handlethe pressure-transient testing of vertically fractured wells of finite fractureconductivity flowing within the non-Darcy flow range, but the parametersinvolved do not easily adapt to the design of hydraulic fracturing treatments. Rather, we have found a different approach more useful, but it requiresrexamination of non-Darcy flow relationships. Non-Darcy Flow Fancher and Lewis extensively investigated flow through porous media, including both consolidated and unconsolidated sands, in the early 1930's. Intheir effort to correlate the experimental data, they used a friction-factorchart analogous to that used in pipe flow and a Reynolds number, defined in asimilar manner: (2) where d= characteristic linear dimension of the flowregime, v= Darcy flow velocity (i.e., the volumetric flow divided by thecross-sectional flow area), density of the flowing fluid, and u= flowing-fluidviscosity. In applying this relationship to the experimental data, Fancher and Lewis chose to define as the average grain size of the porous material and useda sieve analysis on unconsolidated sands to determine this quantity. Although Fancher and Lewis method clearly revealed a departure from Darcy flowconditions at NRe 1 by their definition, their plot required a separate curvefor each porous material. A unique correlation for all the data requires abetter definition of the Reynolds number. Such a definition can be derived formthe non-Darcy flow relationship. The principal relationship for describingnon-Darcy flow was proposed by Forchheimer in 1901. This relationship may beproposed by Forchheimer in 1901. This relationship may be expressed as (3) where pressure drop per unit length along the flow path, permeability of theporous medium, and non-Darcy flow factor. The other dimensions are aspreviously defined. If the second term on the right side is omitted, theequation becomes Darcy's law. Hence, the first term clearly accounts forviscous flow. Similarly, the second term, containing the product of density andvelocity squared, accounts for inertial flow effects. Because the fundamentalbasis of the Reynolds number is that it expresses the ratio of inertial toviscous forces, we may define a Reynolds number by dividing the second term in Eq.3 by the first term: (4) where and the term replaces as the characteristiclinear dimension. SPEPE P. 391

A Criterion for Non-Darcy Flow in Porous Media

“…It is obvious that this revised definition can be applied to all types of porous materials, as long as the permeability and non-Darcy coefficient can be determined experimentally, or empirically when no experimental data is available (Li and Engler, 2001). In fact, several researchers have expressed their preference for using this type of criterion (Geertsma, 1974;Martins et al, 1990;Gidley, 1991). For all these reasons, the Forchheimer number revised in Equation (11) is recommended as the criterion for non-Darcy flow in porous media.…”

Hydraulic fracturing in a sedimentary geothermal reservoir: Results and implications

“…The higher N Re the smaller becomes the remaining transmissibility [23]. The magnitudes of NDF were calculated for a fluid production rate of up to 25 m 3 /h as it was observed during production tests.…”

“…Using the approach of Gidley [23] the dimensionless fracture conductivity calculated can be corrected for non-Darcy flow effects (Eq. (2)).…”

Gas Condensate Flow Around Hydraulically Fractured Wells

“…(c) Flow to the wellbore directly from the matrix is negligible compared to that from the fracture. Actual hydraulic fractures are very complex flow geometries to replicate, and hence, to the best of authors' knowledge, the above assumptions are the basis of most existing simulation studies in this area (McGuire and Sikora 1960;Prats 1961;Cinco-Ley and Samaniego-V 1977;Cinco-Ley et al 1978;Guppy et al 1982;Giddley 1991;Settari et al 2002;Romero et al 2003;Huang and Ayoub 2007).…”

“…The inertial or non-Darcy effect in HFWs has also been studied by some investigators and different correlations for the estimation of fracture skin have been introduced for these flow systems (Guppy et al 1982;Giddley 1991;Settari et al 2002;Huang and Ayoub 2007). In all these studies, the negative impact of inertia in increasing the pressure drop in the fracture, i.e., reducing effective fracture conductivity, has been highlighted.…”