Abstract

The RORA computer program for estimating recharge is 

based on a condition in which ground water flows perpendicular 

to the nearest stream that receives ground-water discharge. The 

method, therefore, does not explicitly account for the groundwater-flow component that is parallel to the stream. Hypothetical 

finite-difference simulations are used to demonstrate effects of 

complex flow conditions that consist of two components: one that 

is perpendicular to the stream and one that is parallel to the 

stream. Results of the simulations indicate that the RORA program can be used if certain constraints are applied in the estimation of the recession index, an input variable to the program. 

These constraints apply to a mathematical formulation based on 

aquifer properties, recession of ground-water levels, and recession of streamflow.

Introduction

The RORA computer program estimates recharge to the water 

table on the basis of daily streamflow (Rutledge, 1998; 2000). 

The program is intended for analysis of a ground-water-flow 

system driven by diffuse areal recharge to the water table and by 

ground-water discharge to a gaining stream. When the program 

is used, regulation and diversion of streamflow should be negligible.

The formulations that RORA uses are derived from a crosssectional-flow model that calculates ground-water discharge per 

unit of stream length on the basis of designated transmissivity, 

storage coefficient, distance from the stream to the hydrologic 

divide, and increase in head caused by recharge to the groundwater system (Rorabaugh, 1964, p. 433). The program is based 

on simple geometry, as illustrated in figure 1A, which represents 

a segment of a basin. The bold line on the left border represents 

the stream, and the right border represents the hydrologic divide. 

Because the stream extends along the length of the model, and 

because its altitude is uniform, the direction of ground-water flow 

is one-dimensional and perpendicular to the stream. A finite-difference-flow model, MODFLOW-96, then is used to generate the 

hydrographs of ground-water level and ground-water discharge 

to the stream (fig. 1B, details about this simulation are given in 

“Finite-Difference Simulation Design”).

Before applying RORA, the user must designate the recession 

index (K). The input variable K is a measure of the time required 

for ground-water discharge to recede by one log cycle when the 

recession becomes linear (or nearly linear) on the semilog 

hydrograph. In the ideal model (fig. 1), K can be determined 

from the following equation, which is derived from Rorabaugh 

and Simons (1966, p. 12) as: 

(1) K 0.933a

2

S

T = -----------------------

Use of RORA for Complex Ground-Water Flow Conditions

U.S. Geological Survey Water Resources Investigation Report 03-4304

 

Figure 1. (A) Finite-difference-flow model (plan view) and (B) simulated flow to drains and simulated ground-water level. 

(Water levels are shown for the finite-difference cell in the middle of the model.)

A B

WATER LEVEL

FLOW, IN CUBIC FEET PER SECOND,

FLOW

AND WATER LEVEL IN FEET

100

10

1

0.1

0.01

0 100 200 300 400

TIME, IN DAYS

flow direction

By A.T. Rutledge
where a is the distance from the stream to the hydrologic divide 

(same as arrows in fig. 1); S is storage coefficient; and T is transmissivity. Hydrographs of ground-water level and ground-water 

discharge (fig. 1) show linearity of recession on the semilog plot. 

When RORA is used in this ideal simulation, the determination 

of K (and subsequent application of RORA) is straightforward 

because:

1. the value of “a” is not ambiguous;

2. the recession of ground-water level (when expressed as vertical distance above the altitude of a known outflow boundary) is clearly definable and is equal to K;

3. streamflow recession is equal to K, some time after 

recharge;

4. there are no departures from linearity in the semilog plot of 

recession.

When a component of ground-water flow is parallel to the 

stream, in the “downvalley” direction toward the next stream in 

the network, that component will cause departures from the ideal 

conditions described in items 1-4. This report demonstrates the 

use of the RORA program for these complex flow conditions. 

Finite-difference simulations are used to evaluate flow conditions. In one simulation, flow is perpendicular to the stream (fig. 

1). In other simulations, there is an additional flow component 

that is parallel to the stream. The latter simulations include a 

variety of outflow-boundary slopes.

Finite-Difference Simulation Design

Each of the three finite-difference simulations (fig. 2) includes 

drain cells that represent streams. In simulation A, drain cells are 

configured so that all flow will be one-dimensional, in the “west” 

direction, perpendicular to the stream (same as fig. 1). The altitude of all drain cells is zero in simulation A. In simulations B 

and C, drain cells are configured so that a flow component will 

be parallel to the stream. This configuration includes an absence 

of drain cells in the northern part of the model and additional 

drain cells along the southern boundary. The altitude of the additional drains along that boundary is zero. The altitude of drains 

along the western boundary increases from zero at the southern 

end to some higher altitude at the northern boundary. Slopes of 

drain lines are uniform and are expressed as SB and SC, in feet per 

100 feet (ft/100 ft). Included among simulations are two variations on the flow condition illustrated in figure 2. In the first 

variation, SB = 0.05 and SC = 0.25 ft/100 ft; and in the second 

variation, SB = 0.4 and SC = 2.0 ft/100 ft. (These slopes correspond to about 3, 13, 21, and 106 feet per mile.) The conductance 

of all drain cells is uniform (0.1 feet squared per second). 

Figure 2. The three finite-difference simulations used to evaluate flow 

conditions (plan view): (A) simulation with flow perpendicular to the 

stream, (B and C) simulations that include a flow component that is parallel to the stream. (Each simulation measures 8,000 ft in the northsouth direction and 2,000 ft in the east-west direction. Aquifer properties are uniform. Bold lines indicate drains. The altitude of all drains is 

zero, except for the segments SB and SC, which are sloping drains. The 

altitude of these segments is zero at the southern boundary.)

The program MODFLOW-96 (Harbaugh and McDonald, 

1996) is used to simulate flow. Each model measures 8,000 ft in 

the north-south direction and 2,000 ft in the east-west direction. 

All finite-difference cells are squares measuring 100 ft on the 

side. In each simulation, storage coefficient and transmissivity 

are uniform. Drain segments are intended to represent streams 

that receive ground-water discharge most of the time, with the 

exception of extreme drought. The initial hydraulic-head distribution in each simulation is calculated on the basis of a preliminary steady-state simulation in which ground-water recharges at 

a constant rate of 10 in/yr (inches per year). Subsequently, all 

simulations include 13 recharge events, each lasting 2 days, and 

each uniformly inducing 1 inch of recharge.

Simulation Results

Simulation results are shown for different transmissivities and 

storage coefficients (figs. 3 and 4). In each group of three simulations, recession characteristics of simulations B and C are similar to recession characteristics of simulation A. Characteristics 

in simulation A must be dominated by the flow component that is 

perpendicular to the length of the stream; therefore, it appears 

that component also is dominating recession characteristics in 

simulations B and C. Consequently, if equation 1 is used to determine the recession index before the RORA program is applied, 

the value of “a” might be measured in the “east-west” direction 

in each simulation. To test this hypothesis, the RORA program 

is used to calculate recharge from each model-generated data set 

of ground-water discharge. Before application of RORA, the 

recession index is calculated from equation 1, designating 

a = 2,000 ft. The following tabulation shows the recession index 

and recharge estimated using RORA:

A B C

SB SC
The RORA program does not detect the last recharge event, 

but this is not ordinarily important because the program is 

intended for analysis of long periods (years) of streamflow 

record. The recharge simulated by MODFLOW, for which 

RORA is giving estimates, is equal to 12 inches.

All simulations described include an initial head distribution 

that was calculated by using a preliminary steady-state simulation, in which recharge is 10 in/yr. To test effects of initial conditions, all simulations were run again, but with a steady-state 

recharge of 20 in/yr (not illustrated). The RORA program was 

used to estimate recharge for each of the six simulated flow 

records. Each result varied from its corresponding result previously presented, by 2 percent or less.

Figure 3

Simulation Recession index

(days per log cycle)

Recharge

(inches)

A 93 11.6

B 93 11.3

C 93 11.2

Figure 4

A 373 11.7

B 373 11.5

C 373 11.5

Figure 3. Ground-water discharge in the three simulations, with hydraulic diffusivity = 40,000 feet squared per day (ft2/d). (Simulation A 

is identical to the simulation in figure 1. Variables are transmissivity (T), Storage coefficient (S), and slopes of drain lines in simulations B 

and C (SB and SC, respectively).

Figure 4. Ground-water discharge in the three simulations, with hydraulic diffusivity = 10,000 feet squared per day (ft2/d). (Variables 

are transmissivity (T), Storage coefficient (S), and slopes of drain lines in simulations B and C (SB and SC, respectively).)

0.01

0.1

1

10

0 100 200 300 400

TIME, IN DAYS

FLOW, IN CUBIC FEET PER SECOND

Simulation C

Simulation B

Simulation A

T = 2,000 ft /d

S = 0.05

S = 0.05 ft/100 ft

S = 0.25 ft/100ft

2

B

C

0.1

1

10

0 100 200 300 400

TIME, IN DAYS

FLOW, IN CUBIC FEET PER SECOND

Simulation A, B, C

T = 100 ft /d

S = 0.01

S = 0.40 ft/100ft

S = 2.00 ft/100ft

2

B

C
Model-generated hydraulic head for a simulation that includes 

complex flow will indicate considerable variation in the direction 

of flow, depending on the time elapsed since the last recharge 

event (fig. 5). In some hydrologic studies, the length of a groundwater-flow path might be used to designate “a” in equation 1. In 

this simulation, the length of a flow path might vary from 2,000 

to 8,000 ft, depending on the time that is represented by the 

water-table map, and the location of the specific flow path. The 

RORA program was used to estimate recharge for all simulations 

illustrated in figures 3 and 4, substituting 4,000 and 8,000 ft for 

the variable “a” in equation 1. When these longer flow paths are 

considered to represent “a,” the estimates of recharge calculated 

by RORA demonstrated greater error than the errors when “a” 

was designated as 2,000 ft. Errors were particularly large when 

8,000 ft was substituted. In this case, the estimate of recharge 

calculated by RORA exceeds the MODFLOW-designated 

recharge by a factor from 1.7 to 2.9. As indicated above, it 

appears that if the recession index is calculated from equation 1, 

then “a” should be measured in the direction that is perpendicular 

to the stream, or transverse from the stream to the hydrologic 

divide, not from flow-path analysis.

Figure 5. Ground-water levels in the finite-difference simulation illustrated in figure 2B, at time = 105, 150, and 190 days (plan view). (Numbered contours represent equal ground-water level, in feet. The 

hydrograph of ground-water discharge is figure 3B.)

For the ideal condition in which all ground-water flows perpendicular to the stream, the recession of the ground-water level 

will equal the recession index. Importantly, the variable considered should be defined as the altitude of the water level in the well 

minus the altitude of the outflow boundary. Because the altitude 

of the outflow boundary in figure 1 is uniformly equal to zero, 

subtraction is not necessary. Simulations of complex flow conditions, however, include variation in altitude of the outflow 

boundary. It could be inferred from such simulations (fig. 5) that 

the orientation and direction of flow paths will change over time. 

This change, in turn, would lead to various interpretations of the 

location of the outflow boundary. Recession rates, expressed in 

days per log cycle, vary depending on selection of the location of 

the outflow boundary for a simulation in which the altitude of this 

boundary changes along its length (fig. 6). It appears that if 

ground-water-level recession is used to ascertain the recession 

index, results can vary appreciably.

Figure 6. (A) A location at which ground-water level is considered and 

five drain locations (plan view), and (B) ground-water-level recession 

expressed as the difference between water level at the point of interest 

and the altitude of the drain. (The five different drain locations are indicated with numbers on frame A, and the recession curves are labeled 

with the corresponding number on frame B. The simulation shown is 

the same as the simulation shown in figures 2B and 3B.)

The most practical method for determining K is measuring 

recession rates directly from the semilog flow hydrograph. Rutledge (2000, p. 14-18) includes discussion of this method and 

potential problems. The method, which was used in a regional 

application of RORA (Rutledge and Mesko, 1996), consists of 

locating a part of the flow hydrograph that shows continuous 

recession, finding a segment of recession that begins a sufficient 

amount of time after recharge so that it will exhibit linearity on 

the semilog plot, then determining the slope in days per log cycle. 

To test this method for the hydrographs in figure 3, the flow from 

day 120 to 130 is considered to represent a linear-recession segment. As an approximation of the graphical method of calculating the recession index, the difference between the logarithm of 

flow on these 2 days is calculated, then this rate of change is converted to days per log cycle. A method using daily flows at 150 

and 160 days was used for the hydrographs in figure 4. The following tabulation shows the recession index, thus determined, 

and the recharge calculated using RORA.

105 days 150 days 190 days

10

8

6

4

2

6

4

2

4

2

0.1

1

10

100 120 140 160 180

TIME, IN DAYS

1

2

3

4

5

WATER LEVEL IN FEET

1

2

3

4

5

O

A

Location

at which

water

level is

considered

B
The estimates for the recession index (K) from the hydrograph 

in figure 3 almost are equal to the result of equation 1 (the equation gives 93 days/log cycle). The estimates for the recession 

index from the hydrograph in figure 4, however, vary considerably from the result of the equation (the equation gives 373 days/

log cycle). Nonetheless, the estimates of recharge from this 

method are similar to estimates of recharge presented earlier.

Given the quantitative similarity between hydrographs 

affected by complex flow conditions and hydrographs not 

affected by them, recession characteristics under extreme lowflow conditions should not be used when K is determined. This 

would apply to departures from the linear model of recession, 

such as exhibited by curves B and C in figure 3, in the time 

between 170 and 190 days.

As discussed, the ideal condition (fig. 1) exhibits a perfect loglinear relation in the recession of the streamflow hydrograph, 

after a period since the last recharge event. Complex flow causes 

slight nonlinearity (figs. 3 and 4). The program user should evaluate variation in recession characteristics for the basin of interest. 

In many cases, the recharge estimate may not be extremely sensitive to variation in K.

Conclusions

The RORA computer program for estimating recharge is 

based on a condition in which ground water flows perpendicular 

to the nearest stream that receives ground-water discharge. The 

program does not explicitly account for the component of flow 

that is parallel to the stream. The program can be used for these 

flow conditions, if certain constraints are applied in the estimation of the recession index (K), an input variable to the program. 

These constraints are as follows:

1. If the equation 0.933(a2)S/T is used to determine K, the 

value of “a” should be measured as the transverse distance 

from the stream to the hydrologic divide. Estimation of “a” 

based on flow-path lengths may introduce errors.

2. If K is determined from the recession of ground-water levels, results can vary depending on the user’s selection of the 

outflow boundary location. The boundary location can be 

open to interpretation because there may be multiple flow 

paths.

3. If K is determined from streamflow data, extreme low-flow 

conditions should not be used.

The most practical method for estimating K is from analysis of 

the streamflow data set. Program users should be cautious when 

attempting to evaluate or replace this estimate on the basis of 

flow paths or ground-water levels.

References Cited

Harbaugh, A.W., and McDonald, M.G., 1996, User’s documentation for MODFLOW-96, an update to the U.S. Geological 

Survey modular finite-difference ground-water flow model: 

U.S. Geological Survey Open-File Report 96-485, 56 p.

Rorabaugh, M.I., 1964, Estimating changes in bank storage and 

ground-water contribution to streamflow: International Association of Scientific Hydrology, Publication 63, p. 432-441.

Rorabaugh, M.I., and Simons, W.D., 1966, Exploration of methods relating ground water to surface water, Columbia River 

Basin–second phase: U.S. Geological Survey Open-File 

Report 66-117, 62 p.

Rutledge, A.T., 1998, Computer programs for describing the 

recession of ground-water discharge and for estimating mean 

ground-water recharge and discharge from streamflow data–

update: U.S. Geological Survey Water-Resources Investigations Report 98-4148, 43 p.

Rutledge, A.T., 2000, Considerations for use of the RORA program to estimate ground-water recharge from streamflow 

records: U.S. Geological Survey Open-File Report 00-156, 44 

p.

Rutledge, A.T., and Mesko, T.O., 1996, Estimated hydrologic 

characteristics of shallow aquifer systems in the Valley and 

Ridge, the Blue Ridge, and the Piedmont Physiographic Provinces based on analysis of streamflow recession and base flow: 

U.S. Geological Survey Professional Paper 1422-B, 58 p.

Figure 3

Simulation Recession index

(days per log cycle)

Recharge

(inches)

A 89 11.5

B 94 11.3

C 94 11.2

Figure 4

A 314 11.4

B 319 11.3

C 319 11.3