The first rigorous coupling of fluid flow and the elas- tic response of the formation was reported by
Khristianovich and Zheltov (1955). They used a two- dimensional (2D) formulation based on a complex variable analysis. Their formulation was equivalent to the length becoming the characteristic, or smaller, dimension and provides the initial “K” for the KGD width model discussed later and in Chapter 6. In addition to being the first paper to provide the cou- pling of fluid flow and rock interaction that is the embodiment of the hydraulic fracturing process, the paper also identified the role for a fluid lag region at the fracture tip. This low-pressure region, beyond the reach of fracturing fluid and filling with pore fluid, has a large, negative net pressure and acts as a clamp at the fracture tip. The fluid lag’s clamping effect provides the natural means to lower the potentially
large tip-region stresses to a level that can be accom- modated by the in-situ condition. The presence of the lag region has been demonstrated by field experi- ments at a depth of 1400 ft at the U.S. Department of Energy (DOE) Nevada Test Site (Warpinski, 1985).
Appendix Fig. 4 compares the Khristianovich and Zheltov analytical results for width and pressure to the corresponding parameters from the Warpinski field results. For the analytical results, decreasing values of the complex variable angle ϑ0toward the right side of the figure correspond to relatively smaller lag regions and larger differences between the minimum stress and pressure in the lag region (i.e., generally deeper formations). The width pro- files clearly show the clamping action at the tip, and the field data appear to be represented by a ϑ0valve of about π/8 for the analytical cases. Also notewor- thy of the experimental results is that tests 4 through 7 with water and test 9 with gel show similar behav- ior when test 4, which had a relatively low injection rate, is ignored. Tests 10 and 11 were with a gelled fluid and clearly show progressively different behav- ior from the preceding tests because of the altered tip behavior resulting from prior gel injections and the residual gel filter cakes that fill the fracture aperture after closure. The cakes have the consistency of sili- con rubber and functionally provide an analogous sealing affect for subsequent tests.
The practical importance of the lag region cannot be overemphasized. The extent of the region, which is extremely small in comparison with commercial- scale fractures, adjusts to the degree required to essentially eliminate the role of the rock’s fracture resistance or toughness (e.g., see SCR Geomechanics Group, 1993) and to isolate the fluid path from all but the primary opening within the multitude of cracks (process zone) forming ahead of the fracture (see Chapters 3 and 6). The field data show the width at the fluid front is well established (i.e., gen- erally greater than 5% of the maximum width at the wellbore) and that fluid enters only a well-established channel behind the complexity of the process zone. These aspects of the lag region provide great simpli- fication and increased predictablility for applying commercial-scale hydraulic fracturing processes.
A paper by Howard and Fast (1957), and particu- larly the accompanying appendix by R. D. Carter, provides the current framework for fluid loss. The paper identifies the three factors controlling fluid loss: filter-cake accumulation, filtrate resistance into
the reservoir and displacement of the reservoir fluid (see Fig. 5-17 and Chapters 6 and 8). All three fac- tors are governed by the relation 1/√t (where t is
time) for porous flow in one dimension. The coeffi- cient for this relation was termed the fluid-loss coef- ficient CL. The authors also provided the means to
determine the coefficient for all three factors using analytical expressions for the filtrate and reservoir contributions and to conduct and analyze the filter- cake experiment, which is now an American Petro- leum Institute (API) Recommended Practice.
Also of significance was presentation of the Carter area equation, with area defined as the product of the
height and tip-to-tip length. This equation, based on the assumption of a spatial and temporal constant fracture width, provided the first rigorous inclusion of fluid loss into the fracturing problem (see Chapter 6). Equation 6-18, which is solved by Laplace trans- formation, is in terms of exponential and comple- mentary error functions and is not “engineer friendly.” This difficulty was soon overcome by developing a table for the more complicated terms in the equation using a dimensionless variable (see Eq. 6-19) that is proportional to the fluid-loss coefficient (loss vol- ume) divided by the width (stored volume) and hence also related directly to the fluid efficiency
Appendix Figure 4. Comparison of Warpinski (1985) field data (left) and Khristianovich and Zheltov (1955) analysis
(right). woand poare the wellbore values of width and pressure, respectively; x is the distance from the well.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 w/w o 0.6 0.5 0.4 0.3 0.2 0.1 0 w/w o 1.0 0.8 0.6 0.4 0.2 0 p/p o 1.0 0.8 0.6 0.4 0.2 0 p/p o 4 5 6 7 9 10 11 Test 4 5 6 7 9 10 Test Width at fluid arrival 0.25 0.20 0.15 0.10 0.05 0
Normalized distance from tip, (L – x)/L
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Normalized distance from well, x/L
0.5 0.4 0.3 0.2 0.1 0
Normalized distance from tip, (L – x)/L
0 0.2 0.4 0.6 0.8 1.0
Normalized distance from well, x/L ϑ0 = π 4 ϑ0 = π 8 ϑ0 = π 16 ϑ0 = 3π 16 ϑ0 = 3π 8 ϑ0 = π 4 ϑ0 = π 8 ϑ0 = π 16 ϑ0 = 3π 16 ϑ0 = 3π 8
ηillustrated in Appendix Fig. 2. Nomographs for the complete equation were also developed (e.g., figs. 4-17 and 4-18 of the Howard and Fast Mono- graph). Eventually a simple and approximate expres- sion (Harrington et al., 1973) for the Carter equation provided the basis for fracture design into the 1980s. The approximate expression is based on the relation at the top of Appendix Fig. 2. For these applications, the average width was first determined from either the KGD or PKN model, as discussed in the following.
Another 1957 paper was by Godbey and Hodges (1958) and provided the following prophetic phrases: “By obtaining the actual pressure on the formation during a fracture treatment, and if the inherent tec- tonic stresses are known, it should be possible to determine the type of fracture induced. . . . The observation of both the wellhead and bottomhole pressure during fracturing operations is necessary to a complete understanding and possible improvement of this process.” These statements anticipated two of the important enablers for the second generation of fracturing: the use of pressure in an manner analo- gous to well test characterization of a reservoir and employment of a calibration treatment to improve the subsequent proppant treatment (see Chapters 5, 9 and 10).
In 1961 Perkins and Kern published their paper on fracture width models, including the long aspect ratio fracture (length significantly greater than height) and radial model (tip-to-tip length about equal to height) as described in Section 6-2.2. They considered, for the first time, both turbulent fluid flow and non- Newtonian fluids (power law model) and provided validating experiments for radial geometry and the role of rock toughness.
Perkins and Kern also discussed fracture afterflow that affects the final proppant distribution within the fracture. After pumping stops, the stored compres- sion in the rock acts in the same fashion as com- pressible fluids in a wellbore after well shut-in. After fracture shut-in, fluid flow continues toward the tip until either proppant bridges the tip or fluid loss reduces the fracture width and stored compression to the extent that the fracture length begins to recede toward the wellbore (Nolte, 1991). The magnitude of the fracture afterflow is large compared with the wellbore storage case, as discussed later for Appendix Eq. 4.
The one shortcoming acknowledged by Perkins and Kern was not rigorously accounting for the flow rate change in the fracture required by continuity (i.e., material balance). They assumed that the volu- metric flow rate was constant along the fracture’s length, which does not account for the effects of fluid loss and local rates of width change (storage change). This assumption was later addressed by Nordgren (1972), who provided closed-form equa- tions for the bounding cases of negligible fluid loss and negligible fracture storage (i.e., most fluid injected is lost during pumping) for a long-aspect fracture and Newtonian fluid (see Section 6-2.2). The initial letters of the last names of the authors of these two papers form the name of the PKN model.
The remaining paper of historic importance for width modeling is by Geertsma and de Klerk (1969). They used the Carter area equation to include fluid loss within the short-aspect fracture, as previously considered by Harrison et al. (1954) and Khristian- ovich and Zheltov (1955). Their initials coupled with those of the authors of the latter paper form the name of the KGD (or KZGD) width model.