Interpolation and Integration Functions

Overview

Petroleum engineering frequently requires working with tabular data that must be interpolated, differentiated, or integrated:

PVT tables — fluid properties vs. pressure at discrete points
Relative permeability curves — $k_r$ vs. saturation from laboratory measurements
Production history — rate and cumulative data at irregular time intervals
Decline analysis — integrating rate profiles for EUR calculations
IPR curves — finding intersection of inflow and outflow performance

Petroleum Office provides a comprehensive set of spline-based numerical functions powered by the MathNet.Numerics library.

Interpolation Methods

Four interpolation approaches are available, each suited to different data characteristics:

                   Data Points                    Interpolation Behavior
    
    Linear:        •───────•───────•             Piecewise straight lines
                                                  (preserves monotonicity)
    
    Cubic:         •╮     ╭•╮     ╭•             Smooth curves through points
                    ╰─────╯ ╰─────╯              (natural spline, C² continuous)
    
    Step:          •───────┐                     Constant until next point
                           └───────•             (piecewise constant)
    
    Proximal:      Returns value of              Nearest-neighbor lookup
                   closest data point            (no interpolation)

Method Selection Guide

Data Type	Recommended Method	Reason
PVT properties (smooth)	Cubic spline	Smooth physical behavior
Relative permeability	Linear spline	Preserves monotonicity
Production rates	Step interpolation	Constant between measurements
Rock type classification	Proximal	Categorical data
General tables	Linear spline	Conservative, stable

Linear Spline Functions

Linear splines connect data points with straight line segments. This method is first-order continuous (C⁰) — continuous function, but with slope discontinuities at data points.

Mathematical Formulation

For a data point interval $[x_i, x_{i+1}]$ :

y(t) = y_i + \frac{y_{i+1} - y_i}{x_{i+1} - x_i} (t - x_i)

The derivative (slope) is constant within each interval:

\frac{dy}{dt} = \frac{y_{i+1} - y_i}{x_{i+1} - x_i}

The integral uses the trapezoidal rule:

\int_a^t y \, dx = \sum_{i=0}^{n-1} \frac{(y_i + y_{i+1})}{2} (x_{i+1} - x_i) + \text{partial segment}

Functions

Function	Description
LinearSplineInterpolate	Interpolate $y$ at point $t$
LinearSplineDifferentiate	First derivative $dy/dx$ at point $t$
LinearSplineIntegrate	Definite integral from first $x$ to $t$
LinearSplineIntegrateT1T2	Definite integral from $t_1$ to $t_2$
LinearSplinesIntersection	Find $x$ where two splines intersect

Advantages

Monotonicity preservation — no oscillation between data points
Bounded — interpolated values stay within range of adjacent points
Robust — stable with irregular data spacing

Limitations

Discontinuous derivatives at knots
Less accurate for smooth physical phenomena

Cubic Spline Functions

Cubic splines fit a third-degree polynomial between each pair of data points, with smoothness constraints at the connections (knots).

Mathematical Formulation

For each interval $[x_i, x_{i+1}]$ , a cubic polynomial:

S_i(t) = a_i + b_i(t - x_i) + c_i(t - x_i)^2 + d_i(t - x_i)^3

Natural spline boundary conditions:

S''(x_0) = 0, \quad S''(x_n) = 0

This gives C² continuity — continuous function, first derivative, and second derivative.

Functions

Function	Description
CubicSplineInterpolate	Interpolate $y$ at point $t$
CubicSplineDifferentiate	First derivative $dy/dx$ at point $t$
CubicSplineIntegrate	Definite integral from first $x$ to $t$
CubicSplineIntegrateT1T2	Definite integral from $t_1$ to $t_2$
CubicSplinesIntersection	Find $x$ where two splines intersect

Advantages

Smooth interpolation — no kinks at data points
Accurate derivatives — smooth $dy/dx$ for rate calculations
Natural for physical data — most reservoir properties are smooth

Limitations

Can oscillate (overshoot/undershoot) between widely-spaced points
Not monotonicity-preserving
Requires at least 3 data points

Step Interpolation

Step interpolation returns the previous data value until a new data point is reached. This is piecewise constant interpolation.

Mathematical Formulation

For $x_i \le t < x_{i+1}$ :

y(t) = y_i

Function

Function	Description
StepInterpolate	Return value of preceding data point

Use Cases

Production rates — assumed constant between measurements
Well status — on/off, flowing/shut-in
Rock type — categorical property
Economic parameters — prices held constant until updated

Proximal (Nearest-Neighbor) Interpolation

Proximal interpolation returns the value of the closest data point to the query point, regardless of direction.

Mathematical Formulation

y(t) = y_k \quad \text{where } k = \arg\min_i |x_i - t|

Function

Function	Description
ProximalInterpolate	Return value of nearest data point

Use Cases

Classification lookups — map to closest defined category
Sparse data — when intermediate interpolation isn't meaningful
Quality control — identify closest measured value

Spline Intersection

Finding where two curves intersect is essential for:

Nodal analysis — IPR vs. VLP (Inflow vs. Outflow performance)
Economic crossovers — cost vs. revenue curves
Saturation equilibria — phase behavior intersections

Mathematical Approach

For two splines $S_1(x)$ and $S_2(x)$ , find $x^*$ such that:

S_1(x^*) - S_2(x^*) = 0

The intersection functions use Brent's root-finding method within the overlapping $x$ -range of both datasets.

Functions

Function	Description
LinearSplinesIntersection	Intersection of two linear splines
CubicSplinesIntersection	Intersection of two cubic splines

Note: Returns the first intersection found within the common $x$ -domain. If curves intersect multiple times, only one solution is returned.

Functions Covered

See each function page for detailed parameter definitions, Excel syntax, and usage examples.

Linear Spline Family

Function	Operation	Output
LinearSplineInterpolate	Interpolation	$y(t)$
LinearSplineDifferentiate	Differentiation	$dy/dx$ at $t$
LinearSplineIntegrate	Integration $[x_0, t]$	$\int_{x_0}^t y\,dx$
LinearSplineIntegrateT1T2	Integration $[t_1, t_2]$	$\int_{t_1}^{t_2} y\,dx$
LinearSplinesIntersection	Intersection	$x^*$ where $y_1 = y_2$

Cubic Spline Family

Function	Operation	Output
CubicSplineInterpolate	Interpolation	$y(t)$
CubicSplineDifferentiate	Differentiation	$dy/dx$ at $t$
CubicSplineIntegrate	Integration $[x_0, t]$	$\int_{x_0}^t y\,dx$
CubicSplineIntegrateT1T2	Integration $[t_1, t_2]$	$\int_{t_1}^{t_2} y\,dx$
CubicSplinesIntersection	Intersection	$x^*$ where $y_1 = y_2$

Other Methods

Function	Method	Output
StepInterpolate	Step (piecewise constant)	$y(t)$
ProximalInterpolate	Nearest neighbor	$y(t)$

Petroleum Engineering Applications

Cumulative Production from Rate Data

Using integration to calculate cumulative oil production $N_p$ :

N_p(t) = \int_0^t q_o(t') \, dt'

Use LinearSplineIntegrate or CubicSplineIntegrate on $(t, q_o)$ data.

Rate from Cumulative Data

Using differentiation to calculate instantaneous rate:

q_o(t) = \frac{dN_p}{dt}

Use LinearSplineDifferentiate or CubicSplineDifferentiate on $(t, N_p)$ data.

PVT Property Lookup

Interpolate formation volume factor at an arbitrary pressure:

B_o(P) = \text{CubicSplineInterpolate}(P_{\text{table}}, B_{o,\text{table}}, P)

Nodal Analysis Operating Point

Find the flowing bottomhole pressure where IPR and VLP curves intersect:

P_{wf}^* = \text{LinearSplinesIntersection}(q_{\text{IPR}}, P_{\text{IPR}}, q_{\text{VLP}}, P_{\text{VLP}})

Exponential Integral (Ei) — Special function for PTA
Decline Curve Models — Rate-time analysis using these integration tools
PVT Properties Overview — Interpolating fluid property tables

References

MathNet.Numerics Documentation. "Interpolation." https://numerics.mathdotnet.com/Interpolation
Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes: The Art of Scientific Computing, 3rd Edition. Cambridge University Press. Chapter 3: Interpolation and Extrapolation.
de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag.

Interpolation and Integration Functions

Overview

Interpolation Methods

Method Selection Guide

Linear Spline Functions

Mathematical Formulation

Functions

Advantages

Limitations

Cubic Spline Functions

Mathematical Formulation

Functions

Advantages

Limitations

Step Interpolation

Mathematical Formulation

Function

Use Cases

Proximal (Nearest-Neighbor) Interpolation

Mathematical Formulation

Function

Use Cases

Spline Intersection

Mathematical Approach

Functions

Functions Covered

Linear Spline Family

Cubic Spline Family

Other Methods

Petroleum Engineering Applications

Cumulative Production from Rate Data

Rate from Cumulative Data

PVT Property Lookup

Nodal Analysis Operating Point

Related Documentation

References

Related Excel Functions