Linear vs Radial Darcy Flow: Choosing the Geometry

Use a linear Darcy model for flow through an idealized region with constant cross-sectional area. Use a radial model when fluid converges toward a well through cylindrical surfaces whose area changes with radius. Both apply Darcy's relationship, but integrating the pressure gradient over different geometry leads to different equations and input requirements.

A core-like linear example and a steady-state oil-well example are shown separately below. Their rates are not a comparison of model accuracy: their dimensions, drawdowns and volume bases differ. The explorer lets you select a geometry and change permeability within that one case, making a proportional sensitivity visible without pretending the two examples represent the same physical system.

For a horizontal, uniform, single-phase linear region, permeability k and area A promote flow, while viscosity μ and flow length L provide resistance. The pressure difference drives the rate. On the field-unit basis used by this calculator, k is in mD, A in ft², ΔP in psi, μ in cP and L in ft. The result is flowing-volume bbl/day.

Area is the cross section perpendicular to flow, not the total exposed surface area of a rock sample. Use a length in the flow direction and pressures referring to the two corresponding endpoints. The displayed equation does not divide by an oil formation volume factor. If a surface oil rate is required, the reservoir-to-stock-tank conversion must be handled explicitly on a compatible fluid basis rather than silently renaming the output STB/day.

For radial flow through a reservoir layer of thickness h, the area perpendicular to flow is 2πrh. Farther from the well, a cylindrical flow surface has greater circumference. Integrating the changing area between wellbore radius rw and external radius re produces ln(re/rw). Replacing that term with a single straight-line distance would discard the geometry that the radial calculation represents.

The oil-well form used here also includes Bo and a supplied skin term s. Bo converts the liquid volume basis so the result is STB/day; skin represents an additional modeled near-well pressure-drop contribution in the denominator. The example is steady state with an external boundary pressure pe. An average reservoir pressure in a different boundary regime is not automatically interchangeable with that boundary condition.

The linear case uses k = 100 mD, A = 500 ft², ΔP = 100 psi, μ = 2 cP and L = 1,000 ft. Substitution gives 2.8175 bbl/day. The radial case uses k = 100 mD, h = 50 ft, pe = 3,000 psi, pwf = 2,500 psi, μo = 2 cP, Bo = 1.2 bbl/STB, re = 1,000 ft, rw = 0.328 ft and s = 2. It gives about 735.8446 STB/day.

The different rates follow different problems. They do not demonstrate that radial flow is inherently better, that one calculator is more accurate, or that one reservoir would deliver both values. To compare physical models, first define common boundary conditions and compatible volume bases. Here the useful shared observation is narrower: doubling k while holding either case's other inputs fixed doubles that case's computed rate.

These examples assume a supported Darcy-flow regime, one fluid phase and fixed properties over the modeled region. They do not solve gas compressibility, changing saturation, transient well testing or a complete production system. A positive numerical rate alone cannot establish that these assumptions describe a particular sample or well. Start with the physical problem and then select the corresponding analytical relationship.

For the radial equation, check re greater than rw and a positive total resistance term. The logarithm is natural, and its argument must use radii in the same units so the ratio is dimensionless. A diameter substituted for only one radius changes the ratio. Check pressure direction as well: the production examples require external pressure above flowing pressure. Use a different model when the actual boundary regime or fluid behavior differs.