When a porous solid is penetrated by a reactive fluid which changes the pore geometry, the macroscopic properties of that porous material may be greatly changed. A model is proposed in which the matrix is visualized as being a number of short cylindrical pores dispersed randomly throughout the solid. The change in the distribution of these cylindrical pores is then represented by an integro-differential equation which is solved for two special cases. The case considered here is that of a surface reaction which dissolves the solid thus continuously enlarging the pores. The rate of reaction is calculated theoretically using a laminar flow diffusion model and this growth rate expression is then taken as the basis for numerical calculations relating to the action of dilute hydrochloric acid on limestone. Abstract: A perturbation series solution is derived for isothermal bubble dissolution and bubble growth from an initially finite size. The accuracy and range of validity of the new results are investigated by comparison with finite-difference solutions of the equations governing bubble growth or dissolution. In addition, previous numerical solutions of the problem are compared to the finite-difference results of this study.Heat transfer from a cylinder in an air-water spray flow stream, Mednick, R. Lawrence, and C. Phillip Colver, AlChE Journal, 15, No. 3, p. 357 (May, 1969).

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

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Long-term conductivity testing at realistic environmental conditions has greatly improved the measurement of proppant pack permeability. However, the use of low flow rates to insure Darcy flow in such measurements has masked the total effect of failed proppant fines on proppant pack permeability. As flow rates increase, corresponding with those commonly found in the field, fines are mobilized and migrate into new positions that reduce the permeability of the proppant pack beyond that normally observed in conductivity measurements. This effect has generally been overlooked in proppant pack design.This paper examines the extent of conductivity reduction caused by migrating proppant fines and the effect of proppant type on the extent of that reduction. The role of fines migration on the conductivity of proppant packs containing two different types of proppants, where the more capable proppant is used near the wellbore, is also evaluated.Representative commercially available proppants, including sand, resin-coated sand, and low density ceramics are included in the study.

In stimulating sandstone formations, if the acid treatment is followed with diesel oil containing a mutual solvent, ethylene glycol monobutyl ether, the water wetness of both fines and formation will be assured. Field results show that EGMBE effects five to six times as much increase in oil productivity as conventional hydrofluoric-hydrochloric acid treatments. productivity as conventional hydrofluoric-hydrochloric acid treatments. Introduction Often the cause of low productivity in sandstone formations is reduced formation permeability near the wellbore. This condition, called "formation damage", has been related to a variety of completion and drilling practices. For example, the perforating operation may practices. For example, the perforating operation may reduce permeability around the perforation by matrix crushing and compaction caused by the shaped charge or by gun debris. The loss of completion fluids, filtrates from drilling mud, or drilling mud particles may cause clay swelling, particle plugging by dispersed formation fines, particle invasion, or adverse changes in fluid saturation and formation wettability. Highly alkaline filtrates occurring with the cementing operation may reduce effective permeability near the wellbore. Finally, reduced permeability may be caused by certain processes that occur during production; an example is the precipitation of scales, asphaltenes, or other insolubles as formation pressures and temperatures are lowered. Many of the adverse permeability effects relate directly to the clay fraction of the matrix. Extensive research has shown that certain effects such as clay swelling, clay dispersion, and particle plugging can be reduced by solutions of inorganic salts such as potassium chloride, calcium chloride, and certain complex potassium chloride, calcium chloride, and certain complex transition metal salts. A more direct elimination of the problem involves solubilizing the clay fraction with problem involves solubilizing the clay fraction with mineral acid solutions of hydrogen fluoride. Mechanics of Acid Attack The value of hydrofluoric acid for removing clays and increasing permeability in sandstone formations has been known for a long time. Unfortunately the potential of this treatment has not been realized in practice because of the nature of hydrogen fluoride attack upon sandstone The mechanics of this reaction was first described by Smith and Hendrickson, who showed that the initial contact of a sandstone core with hydrofluoric acid causes a reduction in permeability. The extent of this reduction was found to be related to the HF concentration of the acid, the flow gradient imposed during acid treatment, and the mineralogical composition of the sandstone core. These results are typified by Fig. 1. The reduction in permeability is caused by the partial disintegration of the sandstone matrix by hydrogen fluoride, which dissolves matrix cementing material and loosens fine particles. These in turn flow downstream and plug pore channels. This phenomenon is not limited to HF attack; it has also been found that permeability reduction can be induced by acids containing no hydrogen fluoride, provided the matrix cementing material is acid soluble provided the matrix cementing material is acid soluble and can be contacted by the acid. Sandstone formations containing calcite often exhibit reduced permeability in the first stages of acid contact even with dilute hydrochloric acid. In core experiments, the initial reduction in permeability resulting from acid attack is overcome when permeability resulting from acid attack is overcome when enough acid is injected to dissolve plugging materials or to enlarge alternate flow channels. p. 571