Experiments and Modeling of High-Velocity Pressure Loss in Sandstone Fractures
- Erik Skjetne (Statoil) | Trygve Kløv (Norwegian U. of Science and Technology) | J.S. Gudmundsson (Norwegian U. of Science and Technology)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2001
- Document Type
- Journal Paper
- 61 - 70
- 2001. Society of Petroleum Engineers
- 1.2.3 Rock properties, 4.3.4 Scale, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 1.6.9 Coring, Fishing, 2.4.3 Sand/Solids Control, 7.5.3 Professional Registration/Cetification, 2.5.2 Fracturing Materials (Fluids, Proppant), 4.2 Pipelines, Flowlines and Risers, 5.6.4 Drillstem/Well Testing, 5.3.1 Flow in Porous Media, 3 Production and Well Operations, 4.1.5 Processing Equipment, 5.1 Reservoir Characterisation, 4.6 Natural Gas, 4.1.2 Separation and Treating
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High-velocity pressure loss in Bentheimer sandstone fractures of varying fracture widths was studied. Direct measurements of the roughness showed that the fractures are self-affine. The new results support a rough fracture high-velocity pressure-loss model. The high-velocity pressure loss was described by a Forchheimer equation with a dominating square term. The square term is a power law in fracture width, and the power is given by a roughness exponent. For low velocities, the pressure loss was not described by the Forchheimer equation. In agreement with theory, a weak-inertia-flow regime exists that separates the Darcy-flow regime from the Forchheimer-flow regime. An expression for the incremental high-velocity skin of a pinched-out hydraulic fracture was derived. Asymptotically for small fracture widths the skin is a power law in fracture width.
High-velocity flow in porous media and fractures results in a rate-dependent skin and has a detrimental effect on well performance. High-velocity flow gives a greater pressure loss than that described with Darcy's law and is often called non-Darcy flow. High-velocity flow is also called inertial flow because it is the inertia of the fluid that activates the nonlinear effects in the microscopic momentum balance of the fluid. Nonlinear effects are present for high velocities in the laminar-flow regime except when the flow paths are rectilinear. Hence, the term turbulent (chaotic) flow would be misleading.
Unpropped fractures with a potential for high-velocity pressure loss include the following.
Hydraulic fractures that are devoid of proppants in a near-well section (pinched-out fractures) because of proppant production (flowback) during cleanup. The pinched-out section has a smaller fracture width than the propped fracture because an in-situ closure stress (least principal stress) acts on the fracture. The pinched-out and proppant-filled sections are separated by a stable arch of proppants.1,2
Thermal fractures in water injectors that are kept open by the injection pressure.
Access to the geometry of such fractures is limited; therefore, models are required.
Traditionally, fracture surfaces were modeled as two parallel planes. Darcy's law describes rectilinear laminar flow in planar fractures with permeability equal to w2/12, where w=fracture width. Experiments show, however, that fracture surfaces are rough, with profiles that are similar to cracks in concrete walls. Fracture surfaces z(x,y) in brittle materials are statistically self-affine (fractal) surfaces. This means that the statistical description of the surface z(x,y) does not change when it is resized according to ?-? z(?x, ?y), where ?>0 is a scale parameter, and ??[0,1]=is the roughness exponent. A variety of brittle materials have ?˜0.8. Sandstone exponents were reported in the range from 0.5 to 0.8.4-6
In this article, we first present our current understanding of high-velocity flow in porous media and fractures. Next, we derive an expression for the rate-dependent skin in a pinched-out fracture that is based on a recent model for high-velocity flow in a self-affine fracture (Appendix A) and on experiments. Then, measurements are presented of high-velocity pressure loss in (unpropped) fractures that are induced in a Bentheimer core. The measurements are analyzed with the recent model.
High-Velocity Flow in Porous Media and Fractures
Steady laminar flow is fully described by the mass and momentum conservation (Navier-Stokes) equations plus the fracture geometry (boundary conditions). The ratio between inertial (linear and centripetal acceleration) and viscous forces is estimated by the Reynolds number NRe
where ?=fluid density, µ=fluid viscosity, l=a characteristic microscopic length scale, and v=a characteristic microscopic velocity. High-velocity (nonlinear) flow in porous media comprises flows with NRe>0.1.
We discuss high-velocity flow in fractures in a context of flow in porous media because a fracture may be viewed as a long tortuous pore. A rough fracture has a relatively simple geometry. Still the flow paths are nonlinear, and we believe that this is the main source of non-Darcy pressure loss. Traditionally, high-velocity flow in porous media is described by the Forchheimer7 equation.
where p=pressure, x=distance in the flow direction, k=permeability, V=superficial velocity (total volume averaged velocity; only equal to interstitial velocity if porosity is 100%), and ß=inertial resistance (also called ß factor). In the literature, the inertial resistance of porous media, including propped hydraulic fractures, typically is correlated to the permeability by a power law.8,9
where a, b, and c are constants, and f=porosity. Often, correlations are made without the porosity dependency, that is with c=0. Because the unit of k is L2 and the unit of ß is 1/L, where L=length, media that are scaled copies of each other have a=0.5. Correlations for natural porous media typically have a˜1, which is far from the dependency of scaled copies. This indicates that, for the natural porous media contributing to the correlations, a permeability reduction is not only the result of decrease in pore size but also because of an increase in complexity of the pore geometry (pore shape).
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