Porosity – Permeability Relationships Permeability and porosity trends for various rock types [CoreLab,1983]Porosity – Permeability Relationships Influence of grain size on the relationship between porosity and permeability [Tiab & Donaldson, 1996]Porosity – Permeability Relationships • Darcy’s Law (1856) – empirical observations of flow to obtain permeability • Slichter (1899) – theoretical analysis of fluid flow in packed uniform spheres • Kozeny (1927),Carmen (1939) – capillary tube modelPorosity – Permeability Relationships Capillary Tube Model Define porosity Where r is radius of the capillary tube, nt is number of tubes/ unit area Define permeability Porosity-permeability relationship 2rnt84rtnk82rk Porosity – Permeability Relationships Example For cubic packing shown, find  and k. Number of tubes per unit area: 4 tubes/(4r)2 Porosity Tortuosity Permeability r 42*241 rr12LaL32)1(8*48222rrrkCarmen – Kozeny Equation Where Kz, Kozeny constant-shape factor to account for variability in shape and length Porosity – Permeability Relationships Define specific surface area Spv – specific surface area per unit pore volume Spv = 2/r (for cylindrical pore shape) Sbv- …unit bulk volume Sgv- …unit grain volume pvSgvSpvSbvS1*2pvSzkk2LaL82rk Spv = 2/rCarmen – Kozeny Equation Tortuosity,  ko is a shape factor = 2 for circular = 1.78 for square Porosity – Permeability Relationships 2LaL2pvSzkkCarmen – Kozeny Equation Where Kz, Kozeny constant-shape factor to account for variability in shape and length 82rk Spv = 2/r *ozkk Porosity – Permeability Relationships Example: spherical particles with diameter, dp  217223pdk2pvSzkk??Distribution of Rock Properties Porosity Distribution Expected porosity histogram [Amyx,et at., 1960]Distribution of Rock Properties Porosity Distribution Actual porosity histogram [NBU42W-29, North Burbank Field] 0123456789104 6 8 10 12 14 16 18 20 22 24 26 28Porosity , %Frequency0.00.20.40.60.81.01.2Cumulative FrequencyDistribution of Rock Properties Permeability Distribution Expected Skewed normal and log normal histograms for permeability [Craig,1971]Distribution of Rock Properties Permeability Distribution Actual permeability histogram [NBU42W-29, North Burbank Field] 05101520250.01 0.10 1.00 10.00 100.00 1,000.00frequency Permeability, mdDistribution of Rock Properties Permeability Variation Dykstra-Parsons Coefficient Characterization of reservoir heterogeneity by permeability variation [Willhite, 1986] 50k1.84k50kVDistribution of Rock Properties Permeability Variation Example of log normal permeability distribution [Willhite, 1986]Distribution of Rock Properties Permeability Variation Actual Dykstra-Parsons probability plot [NBU42W-29, North Burbank Field] y = 578.37e-4.647x R² = 0.9917 0.0010.0100.1001.00010.000100.0001000.00010000.0000.0 0.2 0.4 0.6 0.8 1.0k,md probability of samples with permeability > Flow unitsDistribution of Rock Properties Flow capacity vs storage capacity distribution [Craig, 1971] Permeability Variation Lorenz Coefficient ADCAAreaABCAAreakL Distribution of Rock Properties Permeability Variation Lorenz Coefficient 643.0ADCAAreaABCAAreaLkActual Lorenz plot [NBU42W-29, North Burbank Field] y = -3.8012x4 + 10.572x3 - 11.01x2 + 5.2476x - 0.0146 R² = 0.9991 00.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1Fraction of total Flow Capacity Fraction of total Volume Flow Capacity DistributionDistribution of Rock Properties depth arranged un-arranged Schematic of statistical approach of arranging data in comparison to true reservoir data, which is not ordered. Drawback of statistical approaches • Sequential ordering of data • reliance only on permeability variations for estimating flow in layers. Does not account for: – phase mobility, pressure gradient, Swirr and the k/ ratioDistribution of Rock Properties Hydraulic Flow Unit • unique units with similar petrophysical properties that affect flow. – Hydraulic quality of a rock is controlled by pore geometry – It is the distinction of rock units with similar pore attributes, which leads to the separation of units into similar hydraulic units. – not equivalent to a geologic unit. The definition of geologic units or facies are not necessarily the same as the definition of a flow unit. HFU1 HFU2 HFU3 HFU4 Schematic illustrating the concept of flow units.Distribution of Rock Properties • Start with CK equation • Take the log where the Reservoir quality index (RQI) is given by, the Flow Zone Indicator (FZI) is, and the pore-to-grain volume ratio is expressed as Plot of RQI vs r for East Texas Well [Amaefule, et al.,1993] )log()log()log( FZIrRQI }{0314.0}{mdkmRQI zkgvSFZI11rgvSokk11Distribution of Rock Properties HFU [NBU42W-29, North Burbank Field] 0.0100.1001.00010.0000.010 0.100 1.000RQI Porosity Ratio 0.0100.1001.00010.0000.010 0.100 1.000RQI Porosity Ratio 4.0 2.6 1.8 0.5 FZIDistribution of Rock Properties y = 578.37e-4.647x R² = 0.9917 0.0010.0100.1001.00010.000100.0001000.00010000.0000.0 0.2 0.4 0.6 0.8 1.0k,md probability of samples with permeability > Flow units 0123456789104 6 8 10 12 14 16 18 20 22 24 26 28Frequency Porosity, % FZI4FZI3FZI2FZI1k = 6E+066.9644 R2 = 0.9014 1E-031E-021E-011E+001E+011E+021E+031E+040.00 0.10 0.20 0.30 0.40permeability porosity 0.0100.1001.00010.0000.010 0.100 1.000RQI Porosity Ratio 4.0 2.6 1.8 0.5