Note

This notebook is located in the ./examples directory of the gwtransport repository.

Macrodispersion and Microdispersion in Solute Transport#

This notebook explains the distinction between macrodispersion (spreading from aquifer-scale heterogeneity, captured by the APVD), microdispersion (spreading from pore-scale velocity variations, characterized by \(\alpha_L\)), and molecular diffusion (\(D_m\)), and provides formulas to convert between them. See dispersion scales and advection-dominated assumption for background.

Physical Background: Dispersion as Scale-Dependent Heterogeneity#

All spreading in groundwater transport arises from velocity heterogeneity. What we call “dispersion” vs “heterogeneity” is simply a matter of observation scale:

Scale	Mechanism	Dispersion type	gwtransport representation
Molecular	Brownian motion	Molecular diffusion	\(D_m\) [m²/day] (molecular diffusivity)
Pore (~mm-cm)	Velocity variations in pores	Microdispersion	\(\alpha_L\) [m] (longitudinal dispersivity)
Aquifer (~m-km)	Different streamline paths	Macrodispersion	\(\sigma_V\) [m³] (std of pore volume distribution)

Key insight: Macrodispersion and microdispersion are the same phenomenon (velocity heterogeneity) at different resolutions. The boundaries are not sharp—this is why measured \(\alpha_L\) increases with experiment scale. Molecular diffusion is a separate process (Brownian motion).

Physical nature:

Microdispersion is an aquifer property — determined by pore structure, independent of hydrological boundary conditions
Macrodispersion depends on both aquifer properties and hydrological boundary conditions — when streamlines change due to changed boundary conditions, macrodispersion changes
Molecular diffusion depends on the compound and temperature, not on the aquifer structure

Prerequisite: The aquifer pore volume distribution (APVD) is only well-defined and time-invariant under the steady-streamlines assumption. If the flow field changes in a way that alters streamline geometry (not just velocity), the APVD itself changes.

The dispersion coefficient and flow velocity#

The longitudinal dispersion coefficient is:

\[D_L = D_m + \alpha_L \cdot v\]

\(D_L\) increases with flow velocity because faster flow drives more vigorous pore-scale mixing. However, the actual spreading of a breakthrough curve at a fixed travel distance \(L\) does not increase:

Spreading measure	Molecular diffusion (\(D_m\))	Microdispersion (\(\alpha_L\))
Spatial \(\sigma_x^2\)	\(2 D_m L / v\) — decreases with flow	\(2 \alpha_L L\) — constant
Temporal \(\sigma_t^2\)	\(2 D_m L / v^3\) — decreases strongly	\(2 \alpha_L L / v^2\) — decreases
Volume \(\sigma_V^2\)	\(\propto 1/Q\) — decreases	constant

The microdispersion part gives the same spatial spreading regardless of velocity — the plume traverses the same pore structure. The molecular diffusion part gives less spreading at higher flow because there is less time for diffusion.

When to use each approach#

APVD source	Recommendation
Calibration from measurements	Use APVD only (macrodispersion, microdispersion, and average molecular diffusion already absorbed in \(\sigma_V\)). When calibrating with the diffusion module, the three are taken into account separately.
Streamline analysis (gamma fit)	Can combine \(\alpha_L\) and \(D_m\) as equivalent std (variance addition)
Streamline analysis (discrete volumes)	Must use the `diffusion` module to add microdispersion and molecular diffusion — cannot modify discrete volumes by adding/subtracting variance

Spreading Components#

Breakthrough curve spreading arises from three processes. To compare them, we express each as an equivalent standard deviation in volume units [m³]:

Component	Symbol	Physical cause	Dispersion type	Formula in volume units [m³]
Aquifer heterogeneity	\(\sigma_{V,apv}\)	Different streamline lengths	Macrodispersion	\(\sigma_{V,apv}\) (given)
Molecular diffusion	\(\sigma_{V,diff}\)	Brownian motion	Molecular diffusion	\(\frac{\bar{V}}{L} \sqrt{2 D_m \tau}\)
Mechanical dispersion	\(\sigma_{V,disp}\)	Pore-scale velocity variations	Microdispersion	\(\bar{V} \sqrt{\frac{2 \alpha_L}{L}}\)

where:

\(\bar{V}\) [m³] is the mean aquifer pore volume
\(L\) [m] is the streamline length
\(\tau = R \bar{V} / Q\) [day] is the mean solute residence time
\(v = L / \tau\) [m/day] is the pore velocity

Important distinctions:

\(\sigma_{V,apv}\) is determined by aquifer geometry and boundary conditions—it changes when streamlines change
\(\sigma_{V,disp}\) (microdispersion) is a fixed aquifer property determined by pore structure—it does not change with flow or boundary conditions
\(\sigma_{V,diff}\) (molecular diffusion) depends on flow: it decreases with higher \(Q\) (less time for diffusion)

Unit conversions#

The same spreading can be expressed in different units:

Unit	Symbol	Conversion from spatial \(\sigma_x\)
Spatial [m]	\(\sigma_x\)	—
Volume [m³]	\(\sigma_V\)	\(\sigma_V = \sigma_x \cdot \bar{V}/L\)
Time [day]	\(\sigma_t\)	\(\sigma_t = \sigma_x / v = \sigma_x \cdot \tau / L\)

Decision Tree: Choosing the Right Approach#

Q1: How are you planning to obtain your APVD (pore volume distribution)?
│
├─► From calibration on measurements
│   │
│   Q2: Is molecular diffusion important? (sigma_{V,diff}² / sigma_{V,disp}² > 0.05)
│   │   AND does flow vary significantly during the tracer test?
│   │
│   ├─► Yes (molecular diffusion important AND variable flow)
│   │   │
│   │   └─► Calibrate using diffusion module (slow)
│   │       Function: diffusion.infiltration_to_extraction
│   │       Fit parameters: mean_apv, std_apv (true APVD = macrodispersion only)
│   │       The module handles microdispersion, molecular diffusion, and variable
│   │       flow correctly. The three components are taken into account separately.
│   │       For predictions: use same module, or convert to combined std.
│   │
│   └─► No (molecular diffusion negligible OR flow relatively constant)
│       │
│       └─► Calibrate using advection module (fast)
│           Function: advection.gamma_infiltration_to_extraction
│           Fit parameters: mean_apv, std_apv
│           NOTE: The fitted std absorbs ALL mixing (macrodispersion +
│           microdispersion + average molecular diffusion).
│           For predictions under similar flow: use advection with same std.
│           For predictions under different flow: see "Key insight" below.
│
└─► From groundwater flow model / streamline analysis
    │
    Q3: Will you approximate APVD as a gamma distribution?
    │
    ├─► Yes (fit gamma to streamline pore volumes)
    │   │
    │   Q4: Is microdispersion + molecular diffusion significant?
    │   │   (sigma_{V,diff}² + sigma_{V,disp}²) / sigma_{V,apv}² > 0.05?
    │   │
    │   ├─► No (< 5% of variance)
    │   │   └─► Use advection.gamma_* with std = sigma_{V,apv}
    │   │
    │   └─► Yes (> 5% of variance)
    │       │
    │       Q5: Is molecular diffusion significant? sigma_{V,diff}² / sigma_{V,disp}² > 0.05?
    │       │
    │       ├─► No → Use advection.gamma_* with std = sqrt(sigma_{V,apv}² + sigma_{V,disp}²)
    │       │
    │       └─► Yes
    │           │
    │           Q6: Is flow relatively constant?
    │           ├─► Yes → Use advection.gamma_* with std = sqrt(sigma_{V,apv}² + sigma_{V,diff}² + sigma_{V,disp}²)
    │           └─► No → Use diffusion module with gamma-distributed pore volumes
    │
    └─► No (use full discrete APVD from streamlines)
        │
        Q7: Is microdispersion + molecular diffusion significant?
        │
        ├─► No → Use advection.infiltration_to_extraction with discrete pore volumes
        │
        └─► Yes
            │
            └─► Use diffusion.infiltration_to_extraction with discrete pore volumes
                NOTE: Variance addition/subtraction CANNOT be used with discrete
                pore volumes. The volumes are specific geometric measurements from
                the flow model—they cannot be meaningfully modified. Microdispersion
                and molecular diffusion must be applied on top via the diffusion module.

Why variance combination only works for gamma (continuous) distributions: The equivalent std approach works by adjusting the std parameter of a gamma distribution. For discrete streamline volumes from a flow model, each volume is a specific geometric measurement. You cannot “add variance” to an array of fixed values in a physically meaningful way. When microdispersion and molecular diffusion matter for discrete APVD, use the diffusion module, which applies them along each streamline individually.

Key insight for APVD calibrated from measurements using the advection module: The fitted \(\sigma_{V,apv}\) already absorbs all mixing that occurred during the tracer test (macrodispersion + microdispersion + average molecular diffusion). If you predict transport under very different flow conditions (especially much slower flow), molecular diffusion may contribute more than during calibration. In that case, consider:

Re-calibrating with diffusion module (which takes the three components into account separately), or
Using the correction: std = sqrt(sigma_{V,apv}² + sigma_{V,diff}(Q_pred)² - sigma_{V,diff}(Q_calib)²)

Cross-compound calibration: When the APVD was calibrated using one compound (e.g., temperature with \(D_m \approx 0.1\) m²/day) and is used to predict another (e.g., a dissolved solute with \(D_m \approx 8.6 \times 10^{-5}\) m²/day), the molecular diffusion contribution baked into the calibrated std is too large for the solute. To correct: subtract the calibration compound’s \(\sigma_{V,diff}\) and add the target compound’s. Similarly, retardation affects \(\sigma_{V,diff}\): a more retarded compound spends more time in the aquifer, giving more diffusive spreading.

Modeling Approaches Summary#

Method	Function	Dispersion included	`std` parameter	Speed	When to use
Gamma APVD only	`advection.gamma_*`	Macrodispersion only	\(\sigma_{V,apv}\)	Fast	Calibration from measurements, or gamma APVD with negligible microdispersion
Gamma + microdispersion	`advection.gamma_*`	Macro + micro	\(\sqrt{\sigma_{V,apv}^2 + \sigma_{V,disp}^2}\)	Fast	Gamma APVD; \(\sigma_{V,diff}^2 \ll \sigma_{V,disp}^2\)
Gamma + micro + diffusion	`advection.gamma_*`	Macro + micro + molecular diffusion	\(\sqrt{\sigma_{V,apv}^2 + \sigma_{V,diff}^2 + \sigma_{V,disp}^2}\)	Fast	Gamma APVD; constant flow
Discrete APVD	`advection.infiltration_to_extraction`	Macrodispersion only	N/A	Fast	Streamline APVD; negligible microdispersion
Discrete APVD + micro + diffusion	`diffusion.infiltration_to_extraction`	Macro + micro + molecular diffusion	N/A	Slow	Discrete streamline APVD with significant microdispersion

Note on the equivalent microdispersion std: The equivalent microdispersion and molecular diffusion std computed in this notebook slightly increases the gamma std parameter. This value can also be used to assess whether the diffusion module is needed at all — if the increase is small relative to \(\sigma_{V,apv}\), the advection module alone is sufficient. When aquifer pore volumes are provided directly to the function and come from a streamline analysis, they represent macrodispersion only; the diffusion module can be used to add microdispersion and molecular diffusion.

[1]:

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

from gwtransport import advection, diffusion

np.random.seed(42)
plt.style.use("seaborn-v0_8-whitegrid")

1. System Parameters#

The input parameters are organized in three groups:

Aquifer geometry: streamline length \(L\) and aquifer pore volume distribution (mean \(\bar{V}\), std \(\sigma_{apv}\))
Flow: mean discharge \(Q\)
Transport: retardation factor \(R\), molecular diffusivity \(D_m\), and longitudinal dispersivity \(\alpha_L\)

[2]:

# Aquifer geometry
streamline_length = 100.0  # L [m]
mean_apv = 10000.0  # V_mean [m³] Mean aquifer pore volume
std_apv = 800.0  # sigma_apv [m³] Std from aquifer heterogeneity

# Flow
mean_flow = 120.0  # Q [m³/day]

# Transport
retardation = 2.0  # R [-] Retardation factor
diffusivity_molecular = 1e-4  # D_m [m²/day] Molecular diffusion (~1e-9 m²/s)
dispersivity = 1.0  # alpha_L [m] Longitudinal dispersivity

2. Converting Microdispersion and Molecular Diffusion to Equivalent APVD Spreading#

To compare microdispersion and molecular diffusion with macrodispersion (APVD heterogeneity), we convert spatial spreading to equivalent standard deviation in volume units [m³]. This allows direct variance addition.

Conversion from spatial to volume units#

Spatial spreading is \(\sigma_x\) [m]. To convert to volume units [m³]:

\[\sigma_V = \sigma_x \cdot \frac{\bar{V}}{L} \quad \text{[m³]}\]

This uses the relationship: pore volume \(V\) = cross-sectional area × length, so \(\bar{V}/L\) [m²] converts length to volume.

Formulas for equivalent spreading in volume units#

After substitution (see derivation below), the formulas simplify to expressions using only input parameters:

\[\sigma_{V,diff} = \frac{\bar{V}}{L} \sqrt{2 D_m \tau} = \frac{\bar{V}}{L} \sqrt{\frac{2 D_m R \bar{V}}{Q}} \quad \text{[m³]}\]

\[\sigma_{V,disp} = \frac{\bar{V}}{L} \sqrt{2 \alpha_L L} = \bar{V} \sqrt{\frac{2 \alpha_L}{L}} \quad \text{[m³]}\]

\[\sigma_{V,diff+disp} = \sqrt{\sigma_{V,diff}^2 + \sigma_{V,disp}^2} = \bar{V} \sqrt{\frac{2 D_m R}{L Q} + \frac{2 \alpha_L}{L}} \quad \text{[m³]}\]

Flow-dependence:

\(\sigma_{V,disp}\) (microdispersion) is independent of flow \(Q\): the plume traverses the same pore-scale heterogeneity regardless of velocity — this is a fixed aquifer property
\(\sigma_{V,diff}\) (molecular diffusion) decreases with higher \(Q\) (proportional to \(1/\sqrt{Q}\)): faster flow means less time for molecular diffusion

Derivation

Starting from \(\sigma_x = \sqrt{2 D \tau}\) [m] and converting to volume units via \(\sigma_V = \sigma_x \cdot \bar{V} / L\):

Molecular diffusion (\(D = D_m\), constant):

\[\sigma_{x,diff} = \sqrt{2 D_m \tau} = \sqrt{2 D_m R \bar{V} / Q} \quad \text{[m]}\]

\[\sigma_{V,diff} = \frac{\bar{V}}{L} \sqrt{\frac{2 D_m R \bar{V}}{Q}} \quad \text{[m³]}\]

Microdispersion (\(D_{disp} = \alpha_L v = \alpha_L L / \tau\)):

\[\sigma_{x,disp}^2 = 2 D_{disp} \tau = 2 \alpha_L \frac{L}{\tau} \tau = 2 \alpha_L L \quad \text{[m²]}\]

\[\sigma_{x,disp} = \sqrt{2 \alpha_L L} \quad \text{[m]}\]

\[\sigma_{V,disp} = \frac{\bar{V}}{L} \sqrt{2 \alpha_L L} = \bar{V} \sqrt{\frac{2 \alpha_L}{L}} \quad \text{[m³]}\]

Note: The retardation factor \(R\) cancels out for microdispersion because \(D_{disp} \propto v \propto 1/\tau\) and the product \(D_{disp} \cdot \tau\) is independent of \(\tau\). Physically: a retarded compound moves slower through the same pore structure, experiencing the same heterogeneity over the same distance.

For molecular diffusion, \(R\) does not cancel: a retarded compound spends more time in the aquifer, giving more time for diffusion to act.

Combined (variances add since processes are independent):

\[\sigma_{V,diff+disp}^2 = \sigma_{V,diff}^2 + \sigma_{V,disp}^2\]

[3]:

# Equivalent spreading in volume units from molecular diffusion
sigma_v_diff = (mean_apv / streamline_length) * np.sqrt(2 * diffusivity_molecular * retardation * mean_apv / mean_flow)

# Equivalent spreading in volume units from mechanical dispersion
# Note: R cancels out because D_disp = alpha_L * v = alpha_L * L / tau, and D_disp * tau = alpha_L * L
sigma_v_disp = mean_apv * np.sqrt(2 * dispersivity / streamline_length)

# Combined spreading (direct formula)
sigma_v_diff_disp = mean_apv * np.sqrt(
    2 * diffusivity_molecular * retardation / (streamline_length * mean_flow) + 2 * dispersivity / streamline_length
)

print("Equivalent spreading contributions in volume units [m³]:")
print(f"  sigma_v_diff (molecular diffusion):   {sigma_v_diff:6.0f} m³")
print(f"  sigma_v_disp (mechanical dispersion): {sigma_v_disp:6.0f} m³")
print(f"  sigma_v_diff_disp (combined):         {sigma_v_diff_disp:6.0f} m³")

Equivalent spreading contributions in volume units [m³]:
  sigma_v_diff (molecular diffusion):       18 m³
  sigma_v_disp (mechanical dispersion):   1414 m³
  sigma_v_diff_disp (combined):           1414 m³

3. Combining Macrodispersion and Microdispersion#

Section 2 computed \(\sigma_{diff,disp}\) [m³]—the spreading from microdispersion and molecular diffusion only.

Now we combine this with the original APVD heterogeneity (macrodispersion) \(\sigma_{apv}\) [m³] to get the total effective standard deviation:

\[\sigma_{total} = \sqrt{\sigma_{apv}^2 + \sigma_{diff}^2 + \sigma_{disp}^2} = \sqrt{\sigma_{apv}^2 + \sigma_{diff,disp}^2} \quad \text{[m³]}\]

This is the std parameter to use in advection.gamma_infiltration_to_extraction when you want to include microdispersion and molecular diffusion effects in the fast advection module.

When is this valid?

Only when APVD was obtained from streamline analysis (explicit geometry), not from calibration on measurements (where all mixing is already included in the calibrated std). See dispersion scales.
Only for gamma-parameterized (continuous) distributions where the std parameter can be adjusted. For discrete streamline volumes from a flow model, variance addition is not meaningful—use the diffusion module instead.

[4]:

# Total std in volume units combining all sources
sigma_v_total = np.sqrt(std_apv**2 + sigma_v_diff_disp**2)

# Variance fractions
var_total = sigma_v_total**2
frac_apv = std_apv**2 / var_total * 100
frac_diff = sigma_v_diff**2 / var_total * 100
frac_disp = sigma_v_disp**2 / var_total * 100

print("Variance contributions to total spreading:")
print(f"  sigma_v_apv  = {std_apv:7.0f} m³  ({frac_apv:5.1f}% of variance) - aquifer heterogeneity")
print(f"  sigma_v_diff = {sigma_v_diff:7.0f} m³  ({frac_diff:5.1f}% of variance) - molecular diffusion")
print(f"  sigma_v_disp = {sigma_v_disp:7.0f} m³  ({frac_disp:5.1f}% of variance) - mechanical dispersion")
print("  " + "-" * 60)
print(f"  sigma_v_total= {sigma_v_total:7.0f} m³  (100.0% of variance)")

print("\n" + "=" * 62)
if frac_diff + frac_disp < 5:
    print("RECOMMENDATION: Use APVD only (sigma_v_apv)")
    print("  -> advection.gamma_* with std=sigma_v_apv")
elif frac_diff < 1:
    print("RECOMMENDATION: Use APVD + dispersion")
    print("  -> advection.gamma_* with std=sqrt(sigma_v_apv^2 + sigma_v_disp^2)")
elif frac_diff + frac_disp < 25:
    print("RECOMMENDATION: Use APVD + diff + disp (constant flow assumed)")
    print("  -> advection.gamma_* with std=sigma_v_total")
else:
    print("RECOMMENDATION: Use full diffusion module for accuracy")
    print("  -> diffusion.infiltration_to_extraction")
print("=" * 62)

Variance contributions to total spreading:
  sigma_v_apv  =     800 m³  ( 24.2% of variance) - aquifer heterogeneity
  sigma_v_diff =      18 m³  (  0.0% of variance) - molecular diffusion
  sigma_v_disp =    1414 m³  ( 75.8% of variance) - mechanical dispersion
  ------------------------------------------------------------
  sigma_v_total=    1625 m³  (100.0% of variance)

==============================================================
RECOMMENDATION: Use APVD + dispersion
  -> advection.gamma_* with std=sqrt(sigma_v_apv^2 + sigma_v_disp^2)
==============================================================

Including Microdispersion via Variance Addition (Fast Approach)#

When microdispersion and molecular diffusion effects are significant but you want the speed of the advection module, use the combined standard deviation. This is valid because macrodispersion and microdispersion represent the same physical phenomenon (velocity heterogeneity) at different observation scales—their variances can be added. Molecular diffusion variance can also be added as it is an independent process. See dispersion scales.

# Option 1: Macrodispersion + microdispersion only (when molecular diffusion is negligible)
std_apv_disp = np.sqrt(sigma_apv**2 + sigma_disp**2)

# Option 2: Macrodispersion + microdispersion + molecular diffusion (when flow is relatively constant)
std_total = np.sqrt(sigma_apv**2 + sigma_diff**2 + sigma_disp**2)

# Use in advection module:
cout = advection.gamma_infiltration_to_extraction(
    cin=cin, flow=flow, tedges=tedges, cout_tedges=cout_tedges,
    mean=mean_apv,
    std=std_total,  # or std_apv_disp
    n_bins=nbins,
    retardation_factor=retardation,
)

When is this approach valid?

APVD was obtained from streamline analysis (explicit geometry), not calibration from measurements
APVD is parameterized as a gamma distribution (continuous). For discrete streamline volumes, variance cannot be added — use the diffusion module instead
Flow is relatively constant (for \(\sigma_{diff}\) calculation which depends on \(\tau\))
You want fast computation with reasonable accuracy

When to use full diffusion module instead:

APVD consists of discrete pore volumes from a flow model (variance combination is not possible)
Flow varies significantly over time
High precision is required
You need to model different dispersivities for different pore volumes
See advection-dominated assumption for background

4. Numerical Validation#

[5]:

n_days = 350
_tedges = pd.date_range("2019-12-31", periods=n_days + 1, freq="D")

_flow = np.full(n_days, mean_flow)
_cin = np.zeros(n_days)
_cin[50] = 100.0

cin_itedges = np.flatnonzero(np.diff(_cin, prepend=1.0, append=1.0))
flow_itedges = np.flatnonzero(np.diff(_flow, prepend=1.0, append=1.0))
itedges = np.unique(np.concatenate([cin_itedges, flow_itedges]))
tedges = _tedges[itedges]

cin = _cin[itedges[:-1]]
flow = _flow[itedges[:-1]]

cout_tedges = _tedges.copy()

nbins = 5000
streamline_lengths = np.full(nbins, streamline_length)

[10]:

# 1. APVD only (no diffusion/dispersion) - baseline showing aquifer heterogeneity
cout_apvd = advection.gamma_infiltration_to_extraction(
    cin=cin,
    flow=flow,
    tedges=tedges,
    cout_tedges=cout_tedges,
    mean=mean_apv,
    std=std_apv,
    n_bins=nbins,
    retardation_factor=retardation,
)

# 2. Full diffusion solution with a SINGLE pore volume (no APVD heterogeneity)
# This is the reference for validating the equivalent std approximation
cout_diffusion_single = diffusion.infiltration_to_extraction(
    cin=cin,
    flow=flow,
    tedges=tedges,
    cout_tedges=cout_tedges,
    aquifer_pore_volumes=np.array([mean_apv]),
    streamline_length=np.array([streamline_length]),
    molecular_diffusivity=diffusivity_molecular,
    longitudinal_dispersivity=dispersivity,
    retardation_factor=retardation,
    asymptotic_cutoff_sigma=4.0,
)

# 3. APVD approximation using sigma_v_diff_disp (diffusion + dispersion spreading only)
# This should match cout_diffusion_single since both have no APVD heterogeneity
cout_apvd_approx = advection.gamma_infiltration_to_extraction(
    cin=cin,
    flow=flow,
    tedges=tedges,
    cout_tedges=cout_tedges,
    mean=mean_apv,
    std=sigma_v_diff_disp,  # Equivalent std from diffusion + dispersion
    n_bins=nbins,
    retardation_factor=retardation,
)

# 4. Full diffusion solution with MULTIPLE pore volumes (full APVD + diffusion/dispersion)
# This is the slow but physically rigorous approach
from gwtransport import gamma

gamma_bins = gamma.bins(mean=mean_apv, std=std_apv, n_bins=nbins)
pore_volumes = gamma_bins["expected_values"]
cout_diffusion_multi = diffusion.infiltration_to_extraction(
    cin=cin,
    flow=flow,
    tedges=tedges,
    cout_tedges=cout_tedges,
    aquifer_pore_volumes=pore_volumes,
    streamline_length=streamline_lengths,
    molecular_diffusivity=diffusivity_molecular,
    longitudinal_dispersivity=dispersivity,
    retardation_factor=retardation,
    asymptotic_cutoff_sigma=4.0,
    suppress_dispersion_warning=True,  # We know what we're doing here
)

# 5. APVD approximation with combined std (APVD + diffusion + dispersion)
# This is the fast approximation - should match cout_diffusion_multi
cout_apvd_total = advection.gamma_infiltration_to_extraction(
    cin=cin,
    flow=flow,
    tedges=tedges,
    cout_tedges=cout_tedges,
    mean=mean_apv,
    std=sigma_v_total,  # Combined std: sqrt(sigma_apv^2 + sigma_diff^2 + sigma_disp^2)
    n_bins=nbins,
    retardation_factor=retardation,
)

[7]:

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Plot around breakthrough
plot_start, plot_end = 100, 349
t_plot = cout_tedges[plot_start:plot_end]

# Left panel: Single pore volume validation
ax1.plot(
    t_plot[:-1],
    cout_apvd[plot_start : plot_end - 1],
    label=f"APVD only (std={std_apv:.0f} m³)",
    linewidth=2,
    alpha=0.6,
    color="gray",
)
ax1.plot(
    t_plot[:-1],
    cout_diffusion_single[plot_start : plot_end - 1],
    label="Diffusion (single pore volume)",
    linewidth=2.5,
    linestyle="-",
    color="black",
)
ax1.plot(
    t_plot[:-1],
    cout_apvd_approx[plot_start : plot_end - 1],
    label=f"Approx: std={sigma_v_diff_disp:.0f} m³",
    linewidth=2,
    linestyle="--",
    color="C0",
)
ax1.set_xlabel("Date")
ax1.set_ylabel("Concentration")
ax1.set_title("Single Pore Volume: Diffusion vs Equivalent Std")
ax1.legend()
ax1.grid(True, alpha=0.3)

# Right panel: Multiple pore volumes validation
ax2.plot(
    t_plot[:-1],
    cout_apvd[plot_start : plot_end - 1],
    label=f"APVD only (std={std_apv:.0f} m³)",
    linewidth=2,
    alpha=0.6,
    color="gray",
)
ax2.plot(
    t_plot[:-1],
    cout_diffusion_multi[plot_start : plot_end - 1],
    label="Diffusion (multiple pore volumes)",
    linewidth=2.5,
    linestyle="-",
    color="black",
)
ax2.plot(
    t_plot[:-1],
    cout_apvd_total[plot_start : plot_end - 1],
    label=f"Approx: std={sigma_v_total:.0f} m³",
    linewidth=2,
    linestyle="--",
    color="C1",
)
ax2.set_xlabel("Date")
ax2.set_ylabel("Concentration")
ax2.set_title("Multi Pore Volume: APVD + Diffusion vs Combined Std")
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

../_images/examples_05_Diffusion_Dispersion_13_0.png

Summary#

Formulas for equivalent APVD spreading [m³]#

All formulas convert spatial spreading to APVD units using \(\sigma_V = \sigma_x \cdot \bar{V}/L\):

\[\sigma_{diff} = \frac{\bar{V}}{L} \sqrt{\frac{2 D_m R \bar{V}}{Q}} \quad \text{(molecular diffusion — decreases with higher } Q \text{)}\]

\[\sigma_{disp} = \bar{V} \sqrt{\frac{2 \alpha_L}{L}} \quad \text{(microdispersion — independent of } Q \text{; } R \text{ cancels)}\]

\[\sigma_{total} = \sqrt{\sigma_{apv}^2 + \sigma_{diff}^2 + \sigma_{disp}^2} \quad \text{(total spreading)}\]

Key takeaways#

Macrodispersion (APVD) depends on aquifer properties and hydrological boundary conditions; microdispersion (\(\alpha_L\)) is an aquifer property; molecular diffusion (\(D_m\)) depends on the compound
The dispersion coefficient \(D_L = D_m + \alpha_L v\) increases with flow velocity, but the actual breakthrough curve spreading does not increase — it stays constant (microdispersion) or decreases (molecular diffusion)
Variance combination (adding \(\sigma_{disp}\) and \(\sigma_{diff}\) to \(\sigma_{apv}\)) adds microdispersion and molecular diffusion to macrodispersion. This is only valid for gamma-parameterized APVD from streamline analysis. For discrete streamline volumes, use the diffusion module
When APVD is calibrated from measurements, all mixing is already included (macrodispersion + microdispersion + average molecular diffusion) — do not add on top. When calibrating with the diffusion module, the three components are taken into account separately
When calibrating with one compound and predicting for another, account for differing \(D_m\) and \(R\)

References#

Macrodispersion and microdispersion — Background on dispersion types
Dispersion scales — Scale-dependent dispersion
Advection-dominated assumption — When advection-only is appropriate
Steady-streamlines assumption — Prerequisite for APVD to be time-invariant
advection module — Fast advection transport
diffusion module — Full advection-dispersion solutions

Macrodispersion and Microdispersion in Solute Transport

Contents

Macrodispersion and Microdispersion in Solute Transport#

Physical Background: Dispersion as Scale-Dependent Heterogeneity#

The dispersion coefficient and flow velocity#

When to use each approach#

Spreading Components#

Unit conversions#

Decision Tree: Choosing the Right Approach#

Modeling Approaches Summary#

1. System Parameters#

2. Converting Microdispersion and Molecular Diffusion to Equivalent APVD Spreading#

Conversion from spatial to volume units#

Formulas for equivalent spreading in volume units#

3. Combining Macrodispersion and Microdispersion#

Including Microdispersion via Variance Addition (Fast Approach)#

4. Numerical Validation#

Summary#

Formulas for equivalent APVD spreading [m³]#

Key takeaways#

References#